Analysis/synthesis windowing function for modulated lapped transformation

Description 1. Background

Lapped transforms have been developed for several audiocoding applications. These transforms are normally performed on consecutive blocks of a larger dataset (e.g., two consecutive frames) for an audio signal. Subsequent blocks may be overlapped. Accordingly, a last part of one block coincides with a first part of a subsequent block. A windowing function may be used.

Asymmetric modified discrete cosine transform (MDCT) windows, asymmetric modified discrete sine transform (MDST), time to frequency transformations, have been developed which embody modulated lapped transforms.

Asymmetric modified discrete cosine transform (MDCT) windows, asymmetric modified discrete sine transform (MDST) windows and other types of windows have been developed in the past years as they provide improved frequency responses compared to symmetric shapes, esp. for low delay signal applications, such as audio coding applications.

The first generation of window shapes just focused on the design of the frequency response, e.g. the G.7 8 or MPEG-4 AAC-ELD window. Recent developments also take the temporal shape of the window into account which is responsible for the temporal modulation of the quantization error, e.g. the ALDO window as used in the 3GPP EVS codec.

However, the design approach presented there comes along with the problem that the window shapes show a non-continuous differentiation which leads to a suboptimal frequency response. The described invention in this document presents a solution to overcome this problem. 2. Summary of the invention

In accordance to an aspect there is provided an apparatus for encoding an information signal comprising a plurality of frames, the apparatus comprising: a modulated lapped transform tool for transforming a time domain, TD, representation of the information signal, or a processed version thereof, into a frequency domain, FD, representation, the modulated lapped transform tool being configured to perform a modulated lapped transform analysis using an analysis windowing function having a meandering portion crossing a linear function in correspondence of at least four points and a bitstream writer configured to prepare a bitstream based on the FD representation of the information signal or a processed version thereof, wherein the analysis windowing function is defined so as to be asymmetric, wherein the analysis windowing function is defined so as to be, in the meandering portion greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point, wherein the analysis windowing function is defined so that the absolute maximum value is in one of the first and third interval, wherein the linear function is a constant function with constant value 1.

The analysis windowing function may be defined so that the maximum of the analysis windowing function is less than 25% greater than the value of the linear function at the same time instant. The apparatus for encoding an information signal may comprise a plurality of frames, the apparatus may comprise: a modulated lapped transform tool for transforming a time domain, TD, representation of the information signal, or a processed version thereof, into a frequency domain, FD, representation, the modulated lapped transform tool being configured to perform a modulated lapped transform analysis using an analysis windowing function having a meandering portion crossing a linear function in correspondence of at least four points; and a bitstream writer configured to prepare a bitstream based on the FD representation of the information signal or a processed version thereof, wherein the analysis windowing function is defined so as to be asymmetric, wherein the analysis windowing function is defined so that the maximum of the analysis windowing function is less than 25% greater than the value of the linear function at the same time instant, wherein the linear function is a constant function.

The apparatus for encoding an information signal may comprise a plurality of frames, the apparatus may comprise: a modulated lapped transform tool for transforming a time domain, TD, representation of the information signal, or a processed version thereof, into a frequency domain, FD, representation, the modulated lapped transform tool being configured to perform a modulated lapped transform analysis using an analysis windowing function having a meandering portion crossing a linear function in correspondence of at least four points; and a bitstream writer configured to prepare a bitstream based on the FD representation of the information signal or a processed version thereof.

The apparatus may comprise the modulated lapped transform tool is configured to: scale time input buffers and/or cosine or sine values with values of the analysis windowing function.

The apparatus may comprise the modulated lapped transform tool is configured to use input buffers in the form of t(n) = x(Z - Np + n) for n = 0. 2N _F - 1 - Z, and t(2N - Z + n) = 0 for n = 0.. Z - l wherein x(n) is a TD sample of the information signal or a processed version of the information signal at the time instant n, N _F is the number of samples processed in one frame, and Z is the number of leading zeros in modulated lapped transform window.

The apparatus may comprise the modulated lapped transform tool that is configured to perform:

2N _F -1

2 π ( 1 N _F

X(k) ^ w _N (ri) ^■ t(n) cos— n + - +— for k = 0. . N _F - l

N _F

n=0 where XQt) is the modulated lapped transform frequency value at a frequency index k, n is the time instant, w _N (n) is the analysis windowing function, tin) is a time input buffer, and N _F is the number of samples processed in one frame.

The apparatus for decoding an information signal, or a processed version thereof, defined in the frequency domain, FD, the apparatus may comprise: a bitstream reader configured to read a bitstream encoding the information signal; and an inverse modulated lapped transform tool configured to perform an inverse modulated lapped transform synthesis on the information signal, or a processed version thereof, using a synthesis windowing function having a meandering portion crossing a linear function in correspondence of at least four points, wherein the synthesis windowing function is defined so as to be

asymmetric, wherein the synthesis windowing function is defined so as to be, in the meandering portion: greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point, wherein the synthesis windowing function is defined so that the absolute maximum value is in one of the first and third interval, wherein the linear function is a constant function with constant value 1.

The apparatus may comprise the synthesis windowing function that defined so that the maximum of the synthesis windowing function is less than 25% greater than the value of the linear function at the same time instant.

The apparatus for decoding an information signal, or a processed version thereof, defined in the frequency domain, FD, the apparatus may comprise: a bitstream reader configured to read a bitstream encoding the information signal; and an inverse modulated lapped transform tool configured to perform an inverse modulated lapped transform synthesis of the information signal, or a processed version thereof, using a synthesis windowing function having a meandering portion crossing a linear function in correspondence of at least four points, wherein the synthesis windowing function is defined so as to be

asymmetric, wherein the synthesis windowing function is defined so that the maximum of the synthesis windowing function is less than 25% greater than the value of the linear function at the same time instant, wherein the linear function is a constant function.

An apparatus for decoding an information signal, or a processed version thereof, defined in the frequency domain, FD, the apparatus may comprise: a bitstream reader configured to read a bitstream encoding the information signal; and an inverse modulated lapped transform tool configured to perform an inverse modulated lapped transform synthesis on the information signal, or a processed version thereof, using a synthesis windowing function having a meandering portion crossing a linear function in correspondence of at least four points.

The apparatus wherein the inverse modulated lapped transform tool may be configured to: scale values at a time domain aliasing buffer with values of the synthesis windowing function. The apparatus wherein the inverse modulated lapped transform tool may be configured to generate a time domain, TD, signal representation in the form of wherein t(n) is an aliasing buffer, (/c) is the information signal or a processed version thereof, and N _F is the number of samples for a TD frame. The apparatus wherein the inverse modulated lapped transform tool may be configured to: perform a windowing of the time-aliased buffer.

The apparatus wherein the inverse modulated lapped transform tool may be configured to perform a windowing operation by performing: t(n) = w _N (2N - 1 - n) ^■ t(n) for n = 0. 2N _F - 1

The apparatus wherein the inverse modulated lapped transform tool may be configured to perform an overlap-and-add operation.

The apparatus wherein the inverse modulated lapped transform tool may be configured to perform an overlap-and-add operation as: x(n) = mem_ola_add(n) + t(Z + n) for n = 0.. N _F — Z— 1

3c n) = t(Z + n) for n = N _F - Z. . N _F - 1

mem_ola_add(n) = t(N _F + Z + n) for n = 0. . N _F — Z— 1 wherein x(n) is the output value, t(. ) is a windowed time-aliasing buffer, and N _F is the number of samples in one frame.

The apparatus may be so that: the analysis windowing function and/or the synthesis windowing function may be defined so as to be, in the meandering portion: greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point. The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so that the absolute maximum value is in the first or third interval.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so that, in the meandering portion, a relative maximum value is in the first or third interval and a relative minimum value is in the second interval.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so that, in the meandering portion, the value of the meandering window function in correspondence to at least one of the first and third interval is greater than 0.9.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so as to present, in the meandering portion, a value greater than the linear function in an interval comprised of the 30% and 50% of two frames.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so that the maximum of the analysis windowing function and/or the synthesis windowing function is less than 25% greater than the value of the linear function at the same time instant.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so that the maximum of the analysis windowing function and/or the synthesis windowing function is less than 5% greater than the value of the linear function at the same time instant.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so as to present a second numerical differentiation between -3*10 ^{_ 4} and +3*10- ⁴ .

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so as to present a third numerical differentiation between -2*10 ⁵ and +2 ^* 10 ^"5 .

The apparatus may be so that the analysis windowing function and/or synthesis windowing function is defined so as to present a first numerical differentiation between -0.01 and +0.01 .

The apparatus may be so that the analysis windowing function and/or synthesis windowing function is defined so as to present a second numerical differentiation between -10 ^"4 and +10 ^"4 .

The apparatus may be so that the analysis windowing function and/or synthesis windowing function is defined so as to present a second numerical differentiation between -10 ^-5 and +10 ^"5 .

The apparatus may be so that the linear function may be defined so as to be a non-increasing function. The apparatus may be so that the linear function may be defined so as to be a strictly decreasing function.

The apparatus may be so that the linear function may be defined so as to be a non-decreasing function. The apparatus may be so that the linear function may be defined so as to be a strictly increasing function. The apparatus may be so that the linear function may be defined so as to have a value which is constant or varies of maximum +2% or -2%.

The apparatus may be so that the linear function may be defined so as to have increments between -0.05 and -0.20.

The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be defined so as to be asymmetric. The apparatus may be so that the analysis windowing function and the synthesis windowing function may be defined to be time reversed versions of each other.

The apparatus may comprise: a storage space to store the values of the analysis windowing function and/or the synthesis windowing function.

The apparatus wherein the modified lapped transform may be a modified discrete cosine transform, MDCT, or a modified discrete sine transform, MDST, and the inverse modified lapped transform is an inverse modified discrete cosine transform, IMDCT, or inverse modified discrete sine transform, IMDST. The apparatus may be so that the analysis windowing function and/or the synthesis windowing function may be a constant function with constant value 1.

A system which may comprise: an encoder as a decoder as the apparatus.

The system wherein the encoder may comprise a communication unit configured to transmit a bitstream and/or the decoder comprises a communication unit configured to receive a bitstream. According to an aspect there is provided a method which comprises performing a modified cosine transformation, MDCT, analysis for transforming a time domain, TD, representation of an information signal, or a processed version thereof, into a frequency domain, FD, representation, using an analysis windowing function having a meandering portion which passes a linear function in correspondence of at least four points.

According to an aspect there is provided a method which comprises performing a modified cosine transformation, MDCT, analysis for transforming a time domain, TD, representation of an information signal, or a processed version thereof, into a frequency domain, FD, representation, using an analysis windowing function having a meandering portion which passes a linear function in correspondence of at least four points, wherein the analysis windowing function is defined so as to be asymmetric, wherein the analysis windowing function is defined so as to be, in the meandering portion: greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point, wherein the analysis windowing function is defined so that the absolute maximum value is in one of the first and third interval, wherein the linear function is a constant function with constant value 1.

According to an aspect there is provided a method which comprises performing a modified cosine transformation, MDCT, analysis for transforming a time domain, TD, representation of an information signal, or a processed version thereof, into a frequency domain, FD, representation, using an analysis windowing function having a meandering portion which passes a linear function in correspondence of at least four points, wherein the analysis windowing function is defined so as to be asymmetric, wherein the analysis windowing function is defined so that the maximum of the analysis windowing function is less than 25% greater than the value of the linear function at the same time instant, wherein the linear function is a constant function.

According to an aspect there is provided a method which comprises performing a modified cosine transformation, MDCT, synthesis for transforming a frequency domain, FD, representation of an information signal, or a processed version thereof, into a time domain, TD, representation, using a synthesis windowing function having a meandering portion which passes a linear function in

correspondence of at least four points.

According to an aspect there is provided a method which comprises performing a modified cosine transformation, MDCT, synthesis for transforming a frequency domain, FD, representation of an information signal, or a processed version thereof, into a time domain, TD, representation, using a synthesis windowing function having a meandering portion which passes a linear function in

correspondence of at least four points, wherein the synthesis windowing function is defined so as to be, in the meandering portion: greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point, wherein the synthesis windowing function is defined so that the absolute maximum value is in one of the first and third interval, wherein the linear function is a constant function with constant value 1 .

According to an aspect there is provided a method comprising performing a modified cosine transformation, MDCT, synthesis for transforming a frequency domain, FD, representation of an information signal, or a processed version thereof, into a time domain, TD, representation, using a synthesis windowing function having a meandering portion which passes a linear function in

correspondence of at least four points, wherein the synthesis windowing function is defined so as to be

asymmetric, wherein the synthesis windowing function is defined so that the maximum of the synthesis windowing function is less than 25% greater than the value of the linear function at the same time instant, wherein the linear function is a constant function. A non-transitory storage unit storing instructions which, when running on a processor, may cause the processor to execute a method.

There is provided an apparatus for encoding an information signal comprising a plurality of frames, the apparatus comprising: a modified discrete cosine transformation, MDCT, tool for transforming a time domain, TD, representation of the information signal, or a processed version thereof, into a frequency domain, FD, representation, the MDCT tool being configured to perform an MDCT analysis using an analysis windowing function having a meandering portion crossing a linear function in correspondence of at least four points; and a bitstream writer configured to prepare a bitstream based on the FD representation of the information signal or a processed version thereof. In examples, the MDCT tool is configured to scale time input buffers and/or cosine values with values of the analysis windowing function.

In examples, the MDCT tool is configured to use input buffers in the form of t(n) = x(Z - N _F + n) for n = 0. .2N _F - 1 - Z, wherein x(n) is a TD sample of the information signal or a processed version of the information signal at the time instant n, N _F is the number of samples processed in one frame, and Z is the number of leading zeros in MDCT window.

In examples, an initialization may be performed as: t(2N - Z + n) = 0 for n = 0. . Z - l

In examples, the MDCT (or MDST) tool is configured to perform: where X{k) is the MDCT (or MDST) frequency value at a frequency index k, n is the time instant, w _N (n) is the analysis windowing function, tin) is a time input buffer, and N _F is the number of samples processed in one frame.

In examples, there is provided an apparatus which comprises a bitstream reader configured to read a bitstream encoding the information signal; and an inverse modified cosine transformation, IMDCT (or IMDST), tool, configured to perform an IMDCT (or IMDST) synthesis on the information signal, or a processed version thereof, using a synthesis windowing function having a meandering portion crossing a linear function in correspondence of at least four points.

In examples, the IMDCT (or IMDST) tool is configured to scale values at a time domain aliasing buffer with values of the synthesis windowing function. In examples, the IMDCT (or IMDST) tool is configured to generate a time domain, TD, signal representation in the form of wherein t(n) is an aliasing buffer, X(k is the information signal or a processed version thereof, and N _F is the number of samples for a TD frame. In examples, the IMDCT (or IMDST) tool is configured to perform a windowing of the time-aliased buffer.

In examples, the IMDCT (or IMDST) tool is configured to perform a windowing operation by performing: t(n) = w _N 2N - 1 - n) ^■ t(n) for n = 0..2N _F - 1 In examples, the IMDCT (or IMDST) tool is configured to perform an overlap-and- add operation, which may be, for example, as: x(n) = mem_ola_add(n) + t(Z + n) for n = 0.. N _F - Z - 1

(n) = t(Z + n) for n = N _F - Z.. N _F - 1

mem_ola_add(n) = i(N _F + Z + n) for n = 0.. N _F - Z - 1 wherein x(n is the output value, t(. ) is a windowed time-aliasing buffer, and N _F is the number of samples in one frame.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so as to be, in the meandering portion: greater than the linear function in a first interval between a first crossing point and a second crossing point; lower than the linear function in a second interval between the second crossing point and a third crossing point; greater than the linear function in a third interval between the third crossing point and a fourth crossing point. In examples, the analysis windowing function and/or the synthesis windowing function is defined so that the absolute maximum value is in the first or third interval.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so that, in the meandering portion, a relative maximum value is in the first or third interval and a relative minimum value is in the second interval.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so that, in the meandering portion, the value of the meandering window function in correspondence to at least one of the first and third interval is greater than 0.9. In examples, the analysis windowing function and/or the synthesis windowing function is defined so as to present, in the meandering portion, a value greater than the linear function in an interval comprised of the 30% and 50% of two frames.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so that the maximum of the analysis windowing function and/or the synthesis windowing function is less than 25% (in particular 5%) greater than the value of the linear function at the same time instant.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so as to present a second numerical differentiation is between - 3*10 ^"4 and +3*10 ^"4 and/or the third numerical differentiation is between -2 ^* 10 ^"5 and +2 ^* 10- ⁵ .

In examples, the linear function is defined so as to have a value which is constant or varies of maximum +2% or -2%. In examples, the linear function is defined so as to have increments between -0.05 and -0.20.

In examples, the analysis windowing function and/or the synthesis windowing function is defined so as to be asymmetric. In examples, the analysis windowing function and the synthesis windowing function is defined are symmetric with each other.

In examples, a storage space to store the values of the analysis windowing function and/or the synthesis windowing function.

In examples, there is provided a system comprising: an encoder as an apparatus as above and/or below; and a decoder as an apparatus as above and/or below.

In examples, the encoder comprise a communication unit configured to transmit a bitstream and/or the decoder comprises a communication unit configured to receive a bitstream. In examples, there is also provided a method comprising performing an MDCT (MDST), analysis for transforming a time domain, TD, representation of an information signal, or a processed version thereof, into a frequency domain, FD, representation, using an analysis windowing function having a meandering portion which passes a linear function in correspondence of at least four points. In examples, there is also provided a method comprising performing an MDCT (or MDST), synthesis for transforming a frequency domain, FD, representation of an information signal, or a processed version thereof, into a time domain, TD, representation, using a synthesis windowing function having a meandering portion which passes a linear function in correspondence of at least four points. In has been noted that the analysis and/or synthesis windowing functions used for the invention are particularly suitable for performing MDCT (or MDST) synthesis and analysis. - -

Examples of analysis/synthesis windowing functions for modulated lapped transformation and methods and apparatus for using them are disclosed hereinafter.

3. Description of the drawings Fig. 1 and 2 show schemes according to the prior art.

Fig. 3 to 12 show schemes of functions according to examples.

Figs. 13 and 13A show encoders according to examples.

Fig. 14 and 14A shows decoders according to examples.

Figs. 15 and 16 show methods according to examples. Figs. 17 and 18 show apparatus according to examples.

Fig. 19 shows a scheme according to examples.

Fig. 20 shows a scheme according to the prior art.

Figs. 21 to 25 shows schemes according to examples.

Figs. 26 to 31 show schemes of functions according to examples. 4. Examples

An information signal may be described in the time domain, TD, as a succession of samples (e.g., xt>(n) for a block b and index, e.g., instant, n). The TD

representation may be made of a plurality of frames, each associated to a plurality of samples. In the frequency domain, FD, a frame may be represented as a succession of bins (e.g., X (k), each associated to a particular frequency (each frequency being associated to an index k).

It is possible to convert a TD representation into an FD representation using a modulated lapped transform technique (such as a modified discrete cosine transform, MDCT, technique or a modified discrete sine transform, MDST, technique, for example). Such a technique may be implemented, for example, at an encoder side, so as to transform sampled values into frequency values.

It is possible to convert an FD representation into a TD representation using an inverse modulated lapped transform technique (such as an inverse modified discrete cosine transform, IMDCT, technique or an inverse modified discrete sine, IMDST, technique). Such a technique may be implemented, for example, at a decoder side, so as to transform frequency values into TD values (e.g., for performing reproduction).

The table below indicates symbols used in the following sections.

Figs. 13 and 13A show encoder apparatus 130 and 130A, respectively. Each of the encoder apparatus 30 or 130A may comprise a plurality of tools, such as functional blocks capable of performing signal processing techniques. Each of these blocks may be a standalone device or may be a functional block which divides a hardware structure with other tools of the encoder apparatus 130 and 130A.

The encoder apparatus 130 or 130A may comprise a modulated lapped transform tool 131 (such as a low delay modified discrete cosine transform, MDCT, tool, or a low delay modified discrete sine transform, MDST, tool, or another type of modulated lapped transform tool) which may convert an information signal (e.g., an audio signal) from a time domain, TD, representation to a frequency domain, FD, representation. The modulated lapped transform tool 131 (e.g., MDCT or MDST tool) performs modulated lapped transform analysis (e.g., an MDCT analysis, MDST analysis). Other tools may be provided, e.g., downstream to the modulated lapped transform tool 131 or operating in TD. Some of them are here mentioned.

The encoder apparatus 130 may comprise a linear predictive coding, LPC, tool 132 for performing an LPC analysis in the FD. The encoder apparatus 130A may comprise a spectral noise shaping, SNS, tool 132A for performing an LPC analysis in the FD.

SNS may be seen as a generalization of LPC and, therefore, in some examples, the LPC block 132 may be seen as a particular example of the SNS tool 132A (and the encoder apparatus 130 may be seen as a particular example of the encoder apparatus 130A).

Each of the encoder apparatus 130 and 130A may comprise a temporal noise shaping, TNS, tool 133, to control the temporal shape of noise within each window of the information signal (e.g., as output by the modulated lapped transform tool) in the FD. Each of the encoder apparatus 130 and 130A may comprise a spectral quantizer 134 processing signals in the in the FD. The signal as output by the TNS tool 133 may be quantized, e.g., using dead-zone plus uniform thresholds scalar quantization.

Each of the encoder apparatus 130 and 130A may comprise a noise level estimator 136, e.g., downstream to the spectral quantizer 134.

Each of the encoder apparatus 130 and 130A may comprise a coder 135 processing signals in the FD, for example, to perform entropy coding, e.g., to compress a bitstream. The coder 135 may, for example, perform entropy coding.

Each of the encoder apparatus 130 and 130A may comprise a bandwidth detector 137a which may control, for example, a bandwidth controller at the decoder.

Each of the encoder apparatus 130 and 130A may comprise tools which process signals in the time domain, TD. For example, the encoder apparatus 130 or 130A may comprise a re-sampling tool 138a (e.g., a downsampler) and/or a long term postfiltering, LTPF, tool 138b, for controlling an LTPF active in TD at the decoder side.

Each of the encoder apparatus 130 and 130A may comprise a bitstream

multiplexer tool (bitstream writer) 137 to prepare a bitstream with data obtained from TD and/or FD tools placed upstream. The bitstream may comprise a digital representation of an information signal together with control data to be used at the decoder side. The bitstream may be compressed or include portions which are compressed.

Fig. 14 and 14A show decoder apparatus 140 and 140A, respectively. Each of them may decode a digital representation of an information signal, e.g., as encoded by the encoder 130 or 130A, for example.

Each of the decoder apparatus 140 and 140A may comprise an inverse modulated lapped transform tool 147 (e.g., a low delay inverse MDCT tool or a low delay inverse DMST tool) to transform signal representations from FD to TD. The modulated lapped transform tool 147 performs a modulated lapped transform synthesis (e.g., an MDCT synthesis and/or an MDST synthesis, etc.).

Other tools may be provided, e.g., upstream to the inverse modulated lapped transform tool 147 or operating in TD. Some of them are here mentioned.

Each of the decoder apparatus 140 and 140A may comprise a bitstream multiplex tool 141 to obtain a bitstream (e.g., by transmission) from an encoder apparatus (e.g., the apparatus 130 or 130A). For example, an output from the encoder apparatus 130 or 130A may be provided as an input signal to the decoder apparatus 140 or 140A.

Each of the decoder apparatus 140 and 140A may comprise a decoder 142 which may, for example, decompress data in the bitstream. Arithmetic decoding may be performed. A residual decoding may be performed.

Each of the decoder apparatus 140 and 140A may comprise a noise filling tool 143 processing signals in the FD. Each of the decoder apparatus 140 and 140A may comprise a global gain tool 144 processing signals in the FD.

Each of the decoder apparatus 140 and 140A may comprise a TNS decoder tool 145 processing signals in the FD. The decoder apparatus 140 may comprise an MDCT (or MDST) shaping tool 146 (other lapped transformation tools are possible). The MDCT (or MDST) shaping tool 146 may process signals by applying gain factors computed from decoded LP filter coefficients (obtained from an LPC decoding tool 146a, for example) transformed to the FD spectrum (e.g., MDCT or MDST spectrum, etc.). The decoder apparatus 140A may comprise an SNS decoder tool 146A', for example, obtaining LPC coefficients from the SNS tool 132A.

Each of the decoder apparatus 140 and 140A may comprise an LTPF tool 148 for performing a postfilter in the TD.

The decoder apparatus 140A may comprise a decoder bandwidth controller 149 which may obtain bandwidth information from the bandwidth detector 137a, for example.

The encoder apparatus 30 (or 130A) and the decoder apparatus 140 (or 140A) may be composed to each other to form a system.

Basically, the tools downstream to the modulated lapped transform tool 131 in the encoder apparatus 130 or 130A and the tools upstream to the inverse modulated lapped transform tool 147 in the decoder apparatus 140 or 140A may perform signal processing in the FD and are therefore FD tools.

Techniques are here discussed regarding the conversion techniques, e.g., at the tools 131 and 147. In general terms, the MDCT, MDST, etc., are discrete transforms which have the additional property of being lapped transforms. One of these transforms is performed on consecutive blocks of a larger dataset (e.g., two consecutive frames). Subsequent blocks may be overlapped. Accordingly, a last part of one block may coincide with a first part of a subsequent block. A windowing function w _n (n = 0, ..., 2N-1 ) may be used. In particular, the windowing function may be used as a weight. An advantage is that it is possible to avoid or reduce discontinuities at the borders of the blocks. Fig. 15 shows a method 150 for performing a modulated lapped transform analysis, e.g., at the modulated lapped transform tool 131 (which may be a MDCT tool). The input signal Xb(n) of the current frame b may consist of N audio samples, where the newest one is located at x _b (N - 1).

At step S 51 , input buffers may be updated. The time input buffer for the modulated lapped transform t may be updated according to t(n) = x _b _i(Z + n) for n = 0. . N - 1 - Z t(N - Z + n) = x _b (n) for n = 0.. N - 1 t(2N - Z + n) = 0 for n = 0.. Z - l

The latter is an initialization just required for consistency. At step S152, a block of N time samples may be transformed to frequency coefficients X(k) using the following equation: where w _N is an asymmetric modulated lapped transform window (e.g., MDCT window) according to the used frame size.

Fig. 16 shows a method 160 for performing a modulated lapped transform synthesis (e.g., MDCT synthesis), e.g., at the modulated lapped transform tool 147 (which may be an IMDCT tool). Here, a spectrum ¾(k) (i.e., a representation in the FD) may be transformed to TD by the following steps:

At step S161 , a generation of time domain aliasing buffer x _T,b (n) of frame b may be performed. For example:

At step S162, windowing of time domain aliasing buffer may be performed. For example:

= w _N (2N - 1 - n) ^■ x¾n) for n = 0. .2N - 1 At step S163, overlap-add operation to get reconstructed time samples ¾(n) of frame b may be conducted. For example: b(n) = + Z + n) + ¾Z + n) for n = 0.. N - Z - 1

¾(n) = x^T _b (Z + n) for n = N - Z.. N - 1

Notably, the windowing function w _N (n) at the analysis (analysis windowing function) and the windowing function w _N (2N - 1 - n) at the synthesis (synthesis windowing function) may be defined so as to be time reversed with each other. The window coefficients of the analysis and synthesis window are time reversed versions of each other.

It has been noted that a particularly effective windowing function (e.g., synthesis windowing function) may be an analysis windowing function having a meandering portion crossing a linear function in correspondence of four points.

Fig. 4 shows an analysis windowing function 40. The analysis windowing function 40 may be defined between 0 and 2N or between 0 and a value intermediate between the 80% and the 85% or 90% of 2N (e.g., 81.25%). In some examples, the analysis windowing function 40 may be defined in a different (e.g., broader) range.

The analysis windowing function 40 may be defined, at least for one portion, with reference to a linear function 40'. In Fig. 4, the linear function 40' is a constant function whose value is 1. However, the linear function 40' may be in general a linear function, e.g., defined in terms of y = an + b, where n is the index between the 0 and 2N (or its non-zero portion) and a and b are constants. The analysis windowing function 40 may comprise a meandering portion 44. The meandering portion 44 may be so that the analysis windowing function 40 encounters and/or crosses the linear function 40' at four points #1 , #2, #3, #4.

The meandering portion 44 may be so that the analysis windowing function 40 is different from the linear function 40' at the majority of indexes e.g., for more than the 99% of the indexes.

The meandering portion 44 may be so that, at least at one index immediately preceding one of the indexes associated to the points (crossing points #1 , #2, #3, #4, the value of the analysis windowing function 40 is greater than the value of the linear function (e.g., 1 ) at the crossing point (#1 , #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the analysis windowing function 40 is smaller than the value of the linear function at the crossing point.

The meandering portion 44 may be so that, at least at one index immediately preceding one of the indexes associated to the crossing points #1 , #2, #3, #4, the value of the analysis windowing function 40 is smaller than the value of the linear function (e.g., 1 ) at the crossing point (#1 , #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the analysis windowing function 40 is greater than the value of the linear function at the crossing point.

The meandering portion 44 may be such that each of the crossing points #1 , #2, #3, #4 is not immediately consecutive with the previous and the following ones. For example, between two crossing points #1 , #2, #3, and #4, there is at least one index of the analysis windowing function 40 whose value does not coincide with the value of the analysis windowing function 40 at the same index.

The analysis windowing function 40 may be defined so as to be, in

correspondence to the meandering portion 44:

- greater than the linear function 40' in a first interval 41 between a first

crossing point #1 and a second crossing point #2; - lower than the linear function 40' in a second interval 42 between the second crossing point #2 and a third crossing point #3;

- greater than the linear function 40' in a third interval 43 between the third crossing point #3 and a fourth crossing point #4. The first crossing point #1 may precede the second crossing point #2, which may precede the third crossing point #3, which may precede the fourth crossing point #4.

The meandering portion 44 may extend between the first and the last (e.g., fourth) crossing point. The crossing points may be only four, in some examples. The analysis windowing function 40 may also comprise an initial decreasing portion 45, in which a negative minimum may be reached. The negative minimum may be localized between the 6.25% of 2N and the 18.75% of 2N. The analysis windowing function 40 may also comprise a rapidly increasing portion 46, rapidly increasing from the negative minimum towards the meandering portion 44 (e.g., towards the first crossing point #1 ). The analysis windowing function 40 may also comprise a rapidly decreasing portion 47, rapidly decreasing from the meandering portion 44 to 0. The analysis windowing function 40 may also comprise a constantly null portion 48, which may start between the 80% and the 85% or 90% of 2N (size of two frames) and continue to the end of 2N. The absolute maximum value 41 ' is in the first interval 41 (or in the third interval).

The analysis windowing function 40 may, in the meandering portion 44, have a relative maximum value 43' in the third interval 43 (or in the first interval) and a relative minimum value 42' is in the second interval 42.

The analysis windowing function 40 may, in the meandering portion 44, have the values of the meandering window function 40 in correspondence to the second interval 42 greater than 0.9. The second interval 42 may be above 0.9.

The analysis windowing function 40 may present, in the meandering portion 44, a value lower than the linear function 40' in an interval comprised of the 30% and 50% of two frames (2N). In examples, after the 50% of two frames (2N), the windowing function 40 may greater than the linear function 40'. Hence, point #3 may be in correspondence to the last index of the first frame or in correspondence to the first index of the second frame. In the interval 30% to 50% of two frames, the meandering window portion 44 may be below the linear function.

The maximum 41 ' of the meandering window function 44 may be less than 25% (and preferably less than 5% or 3%) greater than the value of the linear function 40' at the same index n.

The analysis windowing function 40 may present a first numerical differentiation between 0.02 and -0.02. The second numerical differentiation may be between -5 ^* 10 ^"4 and +5 ^* 10 ^"4 , and preferably between -3*10 ^"4 and +3*10 ^'4 . The third numerical differentiation may be between -2*10 ^"5 and +2 ^* 10 ^"5 .

The linear function 40' may be a non-increasing function. In their examples, the linear function 40' may be a non-decreasing function. The linear function 40' may have a value which is constant or varies of maximum +2% or -2%.

The analysis windowing function may be defined so as to be asymmetric.

In some examples, the analysis windowing function may be such that some of the interval (subsections) of the meandering portion 44 are in reverse order with respect to the description above. Figs. 26 and 27 show an analysis windowing function 40 (Fig. 27 is an enlarged view of a portion of Fig. 26). The analysis windowing function 240 may be defined between 0 and 2N or between 0 and a value intermediate between the 80% and the 85% or 90% of 2N (e.g., 81.25%). In some examples, the analysis windowing function 40 may be defined in a different (e.g., broader) range. The analysis windowing function 40 may be defined, at least for one portion, with reference to a linear function 240'. In Fig. 27, the linear function 240' is a constant function whose value is 1. However, the linear function 40' may be in general a linear function, e.g., defined in terms ofy ~ an + b, where n is the index between the 0 and 2N (or its non-zero portion) and a and b are constants (e.g., a=0, b=1 ). The analysis windowing function 240 may comprise a meandering portion 244. The meandering portion 244 may be so that the analysis windowing function 240 encounters and/or crosses the linear function 240' at four points #1, #2, #3, #4.

The meandering portion 244 may be so that the analysis windowing function 240 is different from the linear function 240' at the majority of indexes e.g., for more than the 99% of the indexes.

The meandering portion 244 may be so that, at least at one index immediately preceding one of the indexes associated to the points (crossing points) #1, #2, #3, #4, the value of the analysis windowing function 240 is greater than the value of the linear function 240' at the crossing point (#1, #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the analysis windowing function 240 is smaller than the value of the linear function 240' at the crossing point (#1, #2, #3, of #4).

The meandering portion 244 may be so that, at least at one index immediately preceding one of the indexes associated to the crossing points #1, #2, #3, #4, the value of the analysis windowing function 40 is smaller than the value of the linear function 240' at the crossing point (#1, #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the analysis windowing function 40 is greater than the value of the linear function 240' at the crossing point.

The meandering portion 244 may be such that each of the crossing points #1, #2, #3, #4 is not immediately consecutive with the previous and the following ones. For example, between two crossing points #1, #2, #3, and #4, there is at least one index of the analysis windowing function 240 whose value does not coincide with the value of the analysis windowing function 240 at the same index.

The analysis windowing function 240 may be defined so as to be, in

correspondence to the meandering portion 244:

- greater than the linear function 240' in a first interval 241 between a first crossing point #1 and a second crossing point #2; - lower than the linear function 240' in a second interval 242 between the second crossing point #2 and a third crossing point #3;

- greater than the linear function 240' in a third interval 243 between the third crossing point #3 and a fourth crossing point #4. The first crossing point #1 may precede the second crossing point #2, which may precede the third crossing point #3, which may precede the fourth crossing point #4.

The meandering portion 244 may extend between the first and the last (e.g., fourth) crossing point. The crossing points may be only four, in some examples. The analysis windowing function 240 may have all positive values. The minimum may be a 0 value, which may be at the first index of the analysis windowing function 240 and/or at the last indexes of the analysis windowing function 240. The analysis windowing function 240 may comprise a rapidly increasing portion 246, rapidly increasing towards the meandering portion 244 (e.g., the first crossing point #1). The analysis windowing function 240 may also comprise a rapidly decreasing portion 247, rapidly decreasing from the meandering portion 244 to 0. The analysis windowing function 240 may comprise a constantly null portion 248, which may start between the 84% and the 90% of 2N (size of two frames) and continue to the end of 2N. The absolute maximum value 241 ' is in the first interval 41 (or in the third interval).

The analysis windowing function 240 may, in the meandering portion 244, have a relative maximum value 243' in the third interval 243 (or in the first interval) and/or a relative minimum value 242' is in the second interval 242.

The analysis windowing function 240 may, in the meandering portion 244, have the values of the meandering window function 240 in correspondence to the second interval 242 greater than 0.95. The second interval 242 may be above 0.95. The first and/or third interval 241 and/or 243 may be less than 1.05. The analysis windowing function 240 may present, in the meandering portion 244, a value lower than the linear function 240' in an interval comprised of the 30% and 50% of two frames (2N). In examples, after the 50% of two frames (2N), the windowing function 240 may greater than the linear function 240'. Hence, point #3 may be in correspondence to the last index of the first frame or in correspondence to the first index of the second frame. In the interval 30% to 50% of two frames, the meandering window portion 244 may be below the linear function.

The maximum 241 ' of the meandering window function 244 may be less than 25% (and preferably less than 5% or 3%) greater than the value of the linear function 240' at the same index n.

The analysis windowing function 240 may present a first numerical differentiation between -0.01 and +0.01. The second numerical differentiation may be between -10 ^~4 and + 0 ^"4 . The third numerical differentiation may be between -10 ^"5 and +10 ^"

5 The linear function 40' may be a non-increasing function. In their examples, the linear function 40' may be a non-decreasing function. The linear function 40' may have a value which is constant or varies of maximum +2% or -2%.

The analysis windowing function may be defined so as to be asymmetric.

In some examples, the analysis windowing function 240 may be such that some of the interval (subsections) of the meandering portion 244 are in reverse order with respect to the description above.

Figs. 28 and 29 compare the analysis windowing functions 40 and 240.

Fig. 6 shows another example of analysis windowing function 60. Fig. 7 shows an enlarged view of the analysis windowing function 60 in correspondence of a meandering portion 64.

In this case, the analysis windowing function 60 may be defined, at least for one portion, with reference to a linear function 60'. In Fig. 7, the linear function 60' is a linear decreasing function ("diagonal line"). The meandering portion 64 may be so that the analysis windowing function 60 encounters and/or crosses the linear function 60' at four points.

The analysis windowing function 40 may be defined so as to be, in

correspondence to the meandering portion 64: - greater than the linear function 60' in a first interval 61 ' between a first

crossing point #1 ' and a second crossing point #2';

- lower than the linear function 60' in a second interval 62' between the

second crossing #2' point and a third crossing point #3';

- greater than the linear function 60' in a third interval 63' between the third crossing point #3' and a fourth crossing point #4';

- lower than the linear function 60' in a fourth interval 64' between the fourth crossing point #4' and a fifth crossing point #5';

- greater than the linear function 60' in a fifth interval 65' between the fifth crossing point #5' and a sixth crossing point #6'. The absolute maximum value may be in the first interval 61 ' (in other examples, e.g., where the linear function is increasing, may be in the fifth interval).

The analysis windowing function 60 may, in the meandering portion 64, have a relative maximum value in the third interval 63' and/or in the fifth interval 65', a relative minimum value being in the second interval 62' and/or in the fourth interval 64'.

The analysis windowing function 60 may, in the meandering portion 64, have the value of the windowing function in correspondence to at least one of the first and third interval greater than 0.9 and in particular 0.95 (in other examples, e.g., where the linear function is increasing, may be in the third and fifth interval). The analysis windowing function 60 may present, in the meandering portion 64, a value greater than the linear function 60' in an interval comprised of the 30% and 50% of two frames (2N). The maximum of the windowing function 60 may be less than 25% (and preferably less than 5%) greater than the value of the linear function 60 at the same index n.

The linear function 60' may be a non-increasing function, in particular a strictly decreasing function. The analysis windowing function may be defined so as to be asymmetric.

In examples, the analysis windowing function may have the intervals which are reversed with respect to Fig. 6.

In examples, a synthesis windowing function may be construed to be symmetric with respect to the analysis windowing function. For example, the symmetry axis may be in the middle of the 2N range (479 or 480, for example).

An example of synthesis windowing function 90 is provided in Fig. 9 (e.g., the synthesis windowing function associated to the analysis windowing function 40 of Fig. 4). The synthesis windowing function 90 may be defined between 0 and 2N or between a value intermediate between the 10% or 15% and the 20% of 2N and 2N.

The synthesis windowing function 90 may be defined, at least for one portion, with reference to a linear function (which in this case is not shown but is a constant value 1). In Fig. 9, the linear function may be a constant function whose value is 1. However, the linear function may be in general a linear function, e.g., defined in terms of y=an+b.

The synthesis windowing function 90 may comprise a meandering portion 94. The meandering portion 94 may be so that the synthesis windowing function 90 encounters and/or crosses the linear function 90' (constant 1 , only partially shown for the sake of clarity) at four crossing points #1 , #2, #3, #4 (only #2 being illustrated). The synthesis windowing function may cross the value 1 in the middle of the window (point #2), i.e. between sample N-1 and N. The meandering portion 94 may be so that the synthesis windowing function 90 is different from the linear function 90' at the majority of indexes e.g., for more than the 99% of the indexes.

The meandering portion 94 may be so that, at least at one index immediately preceding one of the indexes associated to the points #1 , #2, #3, #4, the value of the synthesis windowing function 90 is greater than the value of the linear function (e.g., 1) at the point (#1 , #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the point), the value of the synthesis windowing function 90 is smaller than the value of the linear function at the crossing point. The meandering portion 94 may be so that, at least at one index immediately preceding one of the indexes associated to the points #1 , #2, #3, #4, the value of the synthesis windowing function 90 is smaller than the value of the linear function at the point (#1 , #2, #3, of #4), and, at ieast at the index

immediately subsequent the same index (associated to the crossing point), the value of the synthesis windowing function 90 is greater than the value of the linear function at the crossing point (#1 , #2, #3, of #4).

The meandering portion 94 may be such that each of the points #1 , #2, #3, #4 is not immediately consecutive with another of the points #1 , #2, #3, #4. For example, between two points of #1 , #2, #3, and #4, there may be at Ieast one index of the synthesis windowing function 90 whose value does not coincide with the value of the synthesis windowing function 90 at the same index.

The synthesis windowing function 90 may be defined so as to be, in

correspondence to the meandering portion 94:

- greater than the linear function 90' in a first interval 91 between a first

crossing point #1 (not shown in Fig. 9, but representable as symmetrical to point #4 in Fig. 4) and a second crossing point #2 (which, in this case, is at the 480 ^th sample, e.g., in the middle of the 2N range);

- lower than the linear function 90' in a second interval 92 between the

second crossing point #2 and a third crossing point #3 (not shown in Fig. 9, but representable as symmetrical to point #3 in Fig. 4); - greater than the linear function 90' in a third interval 93 between the third crossing point #3 and a fourth crossing point #4 (not shown in Fig. 9, but representable as symmetrical to point #1 in Fig. 4).

The synthesis windowing function 90 may also comprise a constantly null portion 98, which may be located between the 10% and the 15% of 2N. The synthesis windowing function 90 may also comprise a rapidly increasing portion 97, rapidly increasing towards the meandering portion 94 from 0. The synthesis windowing function 90 may also comprise a rapidly decreasing portion 96, rapidly decreasing towards a negative minimum from the meandering portion 94. The negative minimum may be localized between the 81 .25% of 2N and the 93.7% of 2N. The synthesis windowing function 90 may also comprise a final increasing portion 95, which reaches 0 from the negative minimum.

The absolute maximum value may be in the third interval 93 (or in the first interval). The synthesis windowing function 90 may, in the meandering portion 94, have a relative maximum value in the first interval 91 (or in the third interval) and a relative minimum value is in the second interval.

The synthesis windowing function 90 may, in the meandering portion 94, have the value of the meandering window function in correspondence to at least one of the first and third interval (91 , 93) greater than 0.9. The third interval may be above 0.9.

The synthesis windowing function 90 may present, in the meandering portion 94, a value greater than the linear function in an interval comprised of the 30% and 50% of two frames (2N). The maximum of the synthesis windowing function 90 may be less than 25% (and preferably less than 5%) greater than the value of the linear function at the same index n.

The synthesis windowing function 90 may present a numerical differentiation between 0.02 and -0.02. The second numerical differentiation may be between - 2

- 35 -

5*10 ^"4 and +5 ^* 10 ^"4 , and preferably between -3*10 ^"4 and +3 ^* 10 ^'4 . The third numerical differentiation may be between -2*10 ^-5 and +2 ^* 10 ^"5 .

The synthesis linear function 90 may be a non-increasing function. In other examples, the synthesis linear function 90 may be a non-decreasing function. The synthesis linear function may 90 have a value which is constant or varies of maximum +2% or -2%.

The synthesis windowing function 90 may be defined so as to be asymmetric.

In some examples, the synthesis windowing function 90 may be a function symmetrical with respect to the shape shown in Fig. 9. E.g., the value at 0 would be the value which is at 960 in Fig. 9; the value at 1 would be the value which is at 959 in Fig. 9, and so on.

An example of synthesis windowing function 290 is provided in Fig. 30 (e.g., the synthesis windowing function associated to the analysis windowing function 240 of Figs. 26 to 29). The synthesis windowing function 290 may be defined between 0 and 2N or between a value intermediate between the 10% or 15% and the 20% of 2N and 2N.

The synthesis windowing function 290 may be defined, at least for one portion, with reference to a linear function 290' (which in this case is not shown but is a constant value 1). In Fig.30, the linear function 290' may be a constant function whose value is 1. However, the linear function may be in general a linear function, e.g., defined in terms ofy=an+b.

The synthesis windowing function 290 may comprise a meandering portion 294. The meandering portion 294 may be so that the synthesis windowing function 290 encounters and/or crosses the linear function 290' (constant 1, only partially shown for the sake of clarity) at four crossing points #1, #2, #3, #4 (only #2 being illustrated). The synthesis windowing function may cross the value 1 in the middle of the window (point #2), i.e. between sample N-1 and N. The meandering portion 294 may be so that the synthesis windowing function 290 is different from the linear function 290' at the majority of indexes e.g., for more than the 99% of the indexes.

The meandering portion 294 may be so that, at least at one index immediately preceding one of the indexes associated to the points #1, #2, #3, #4, the value of the synthesis windowing function 90 is greater than the value of the linear function (e.g., 1) at the point (#1, #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the synthesis windowing function 90 is smaller than the value of the linear function at the crossing point.

The meandering portion 294 may be so that, at least at one index immediately preceding one of the indexes associated to the points #1, #2, #3, #4, the value of the synthesis windowing function 290 is smaller than the value of the linear function at the crossing point (#1, #2, #3, of #4), and, at least at the index immediately subsequent the same index (associated to the crossing point), the value of the synthesis windowing function 290 is greater than the value of the linear function at the crossing point (#1, #2, #3, of #4).

The meandering portion 294 may be such that each of the crossing points #1, #2, #3, #4 is not immediately consecutive with another of the points #1, #2, #3, #4. For example, between two crossing points of #1, #2, #3, and #4, there may be at least one index of the synthesis windowing function 290 whose value does not coincide with the value of the synthesis windowing function 290 at the same index.

The synthesis windowing function 290 may be defined so as to be, in

correspondence to the meandering portion 294: - greater than the linear function 290' (e.g., greater than 1) in a first interval 291 between a first crossing point #1 and a second crossing point #2 (which, in this case, is at the 480 ^th sample, e.g., in the middle of the 2N range); - lower than the linear function 290' in a second interval 292 between the second crossing point #2 and a third crossing point #3;

- greater than the linear function 290' in a third interval 293 between the third crossing point #3 and a fourth crossing point #4. The synthesis windowing function 290 may comprise a constantly null portion 298, which may be located, at one border of the window (2N samples) and may comprise between 10% and 15% of the samples. The synthesis windowing function 290 may also comprise a rapidly increasing portion 297, rapidly increasing towards the meandering portion 294 from 0. The synthesis windowing function 290 may comprise a rapidly decreasing portion 296, rapidly decreasing towards 0 from the meandering portion 294

The absolute maximum value may be in the third interval 293 (or in the first interval).

The synthesis windowing function 290 may, in the meandering portion 294, have a relative maximum value in the first interval 291 (or in the third interval) and a relative minimum value is in the second interval.

The synthesis windowing function 290 may, in the meandering portion 294, have the value of the meandering window function in correspondence to at least one of the first and third interval (291, 293) greater than 0.9 or 0.95. The third interval may be above 0.95.

The synthesis windowing function 290 may present, in the meandering portion 294, a value greater than the linear function in an interval comprised of the 30% and 50% of two frames (2N).

The maximum of the synthesis windowing function 290 may be less than 25% (and preferably less than 5% or 3%) greater than the value of the linear function at the same index n. The synthesis windowing function 290 may present a numerical differentiation between -0.01 and +0.01. The second numerical differentiation may be between - 10 ^"4 and + 0 ^"4 . The third numerical differentiation may be between -10 ^"5 and +10 ^-5 .

The synthesis linear function 90 may be a non-increasing function. In other examples, the synthesis linear function 90 may be a non-decreasing function. The synthesis linear function may 90 have a value which is constant or varies of maximum +2% or -2%.

The synthesis windowing function 90 may be defined so as to be asymmetric.

In some examples, the synthesis windowing function 90 may be a function symmetrical with respect to the shape shown in Fig. 9. E.g., the value at 0 would be the value which is at 960 in Fig. 9; the value at 1 would be the value which is at 959 in Fig. 9, and so on.

5. Discussion

It has been noted that this function has particularly interesting features which are extremely suited for the MDCT (or MDST or other) and IMDCT (or IMDST or other) techniques, in particular for audio coding.

As a metric for the smoothness of MDCT window shape, we use the numerical differentiation like w(t + h) - w(t - h)

dw(t) =— —— for t = h. .2N - 1 - h

2h Another metric in order to assess the quality of MDCT window is temporal shape of the quantization error which is described by

S _q (t) = w ² (t + N) + w ² (t) for t = 0.. N - l

The quantization error may be introduced in the spectral domain as part of an audio coding process. The ALDO window is here discussed. ALDO is a design approach for asymmetric window providing an improved temporal shape. The design process of the ALDO window as used in the EVS codec is described in (3GPP) and [2]. Fig. 1 shows a schematic window shape for a ALDO window scheme.

An ALDO window consists of four sections, a raising function, a section of strict ones, a decay function and zeros. The number of zeros in w also determines the number of ones in W2, with W2=2*w ₄ . How to obtain wi and W3 is described in [1] and [2].

Figure 2 shows an ALDO window designed according to [1] and [2] for parameters Lz=180 (number of leading zeroes in the MDCT window, W ₄ (n) in Fig. 1) and the total window length is 960 samples. As can be seen in the fourth subplot of Fig. 2 ("Time modulation quantization error"), the ALDO shows a good temporal shape of the quantization error which mean the energy sum of two subsequent overlapping synthesis windows is close to one. The optimal result would require only ones which are achieved by symmetric window shapes. The drawbacks of this design approach are mainly visible in the second subplot of Fig. 2 ("Numerical differentiation"), where the numerical differentiation is outlined. It can be seen that the window shape is a not a continuously differentiable mathematical function and is therefore non-optimal regarding the frequency response. Due to static ones in the window shape, smooth transitions coming from the raise function and going to the decay function are not possible.

The present invention breaks up with the design constraint to maintain a section of strict ones in order to optimize the temporal shape. Instead, the ones are replaced by a sequence of values meandering around a linear function (e.g., a constant value, such as "1"). This solution allows the smooth and continuous window shape while keeping the temporal shape close to an optimal state.

Fig. 3 shows an example window designed with a meandering section (portion) according to the invention. As can be seen in the fourth subplot of Fig. 3 ("Time modulation quantization error"), the window's property regarding the temporal shaping of the quantization error is similar to the ALDO window. However, comparing the second subplot of Fig. 3 ("Numerical differentiation") to second subplot of Fig. 2 (ALDO window), it may be understood that the invention permits to generate window shapes which are continuously differentiable. This enables for instance the design of window shapes with higher stop band attenuation, so far not possible with design process used for the ALDO window.

Fig. 4 outlines an example window shape 40 in more detail. In this example, the meandering section is based on crossing, in correspondence of four points, a linear function (in this case, a constant value "1 "). Even a higher number of crossings is possible but four is considered as the minimum number. Such a window can be the result of a mathematical optimization process

combining one error function for the frequency response with and error function representing the deviation from the optimal temporal shaping of the quantization noise.

Some implementation required specific data formats, e.g. fix-point

implementations. For such implementation, the value range of the window coefficients might be scales as values above one require a different scaling. In such cases, the meandering section might not be around one. Only after descaling, the values get into the obvious range.

Fig. 5 shows an example how the MDCT window coefficients 50 are represented inside an fix-point implementation. It is an example representation where some values are scaled by 2 ^X and other by 2 ^{x 1} due to fix point architecture.

Considering a general case, the window does not necessarily meander around 1 but may meander around a virtual diagonal line without crossing the one. Fig. 6 shows three different windows, i.e. - wi : New optimized meandering around 1 (windowing function 70);

- W2: G.718 window (not meandering at all) (windowing function 70');

- W3: New optimized window for special case: Meandering around a diagonal (windowing function 60). The described MDCT window fulfils the perfect reconstruction property. For that, we consider the MDCT to be decomposed as a windowing operation, a time domain alias cancellation (TDAC) step and a discrete cosine transform (DCT) type IV kernel, as for instance described in [3]. Therefore, the windowing and TDAC can be seen as a folding step on analysis and an unfolding step on synthesis side. At the latter one, the unfolded sequences of two blocks are combined in the overlap-and-add operation. See Fig. 8, which refers to a scheme of a block 1 (81 ) and a block 2 (82).

For the overlapping region on the analysis side, the windowing and TDAC of block 1 can be described as

Pi(n)=Wa(N+n)-Wa(2N-1 -n) for n=0... N-1 and block 2

P2(n)=Wa(N+n)+w _a (N-1 -n) for n=0... N-

For the syntheses side, P1 and P2 are folded-out, windowed and overlap-and-add is performed:

0(n)= Ws(N+n)Pi(n)+w _s (n)P2(n) for n=0... N-1

This can be written as

O(n)= Wa(N+n)w _s (N+n)-Wa(2N-1 -n)ws(N+n)+Wa(n)w _s (n)+w _a (N-1 -n)w _s (n) for n=0... N-1 which can be separated into two components:

PR1 : Perfect reconstruction component

Wa(N+n)w _s (N+n)+Wa(n)ws(n)=1

for n=0... N-1 PR2: Alias cancellation component

w _a (N-1 -n)w _s (n)-Wa(2N-1 -n)w _s (N+n)=0

for n=0... N-1

Figs. 9-12 show analysis and synthesis sequences. Reference is now made to Figs. 19 and 20 with reference to the numerical differentiation of the present analysis and synthesis windowing functions.

By considering also the 2 ^nd and 3 ^rd numerical differentiation, it is possible to clearly distinguish the properties of present examples (e.g., Fig. 19 with reference to the example of Fig. 4, and Fig. 31 with reference to the example of Fig. 26) with those of the prior art (ALDO, Fig. 20).

As a metric for the smoothness of modulated lapped transform window shape (e.g., MDCT or MDST window shape), we may use the numerical differentiation like: w(t + h) - w(t - h)

dw{t) =— —^- - for t = h. .2N - l - h In order to weight the importance of each coefficient, i.e. higher amplitudes have a higher weight, the following describes a weighted w(t + h) + w(t - h)

wdw f) = d (t) — —— for t = h. .2N - l - h

This allows neglecting low level window coefficients in the analysis of the continuity of the window shape. In Figs. 19 and 20 and 31 , the 1 ^st numerical differentiation is plotted in the regular and weighted version (dashed). 1902 and 3102 refer to the 1 ^st numerical differentiation for the inventive windowing function; 1904 and 3104 (dashed) refer to the weighted version of the 1 ^st numerical differentiation for the inventive windowing function. 2002 refers to the 1 ^st numerical differentiation for the ALDO windowing function; 2004 (dashed) refers to the weighted version of the 1 ^st numerical differentiation for the ALDO windowing function. For the 2 ^nd and 3 ^rd numerical differentiation (ND), the weighted version of the 1 ^st one is used. 1910 and 31 0 refer to the weighted version of the 2 ^nd numerical differentiation for the inventive windowing function. 2010 refers to the weighted version of the 2 ^nd numerical differentiation for the ALDO windowing function. 1920 refers to the weighted version of the 3 ^rd numerical differentiation for the inventive windowing function. 2020 and 3120 refer to the weighted version of the 3 ^rd numerical differentiation for the ALDO windowing function.

As can be observed, the 2 ^nd ND version shows some clear peaks 2021 for the ALDO window while the inventive windows show only moderate changes. For the 3 ^rd ND, also the amplitude level of both windows clear differentiate and can be expressed.

An analysis of the degree of freedom for window design for ALDO and for the inventive window function is here presented.

The ALDO design approach [2] offers only the parameters C1 , C2 and the number of zeros Lz as degree of freedom. Any focus regarding near field or far field attenuation is not possible by the proposed method as the given degree of freedom in the design process is very limited.

In order to extend the ALDO algorithm and allowing a higher degree of freedom, the design approach in [2] has to be given up and has to be exchanged by a numerical optimization method. In a first step the section of strict ones in the middle of the window is maintained while the free window coefficients (or design parameter) are optimized.

Assuming that analysis and synthesis window are always time reversed versions of each other, the perfect reconstruction constraint can be expressed by Wa(N+n)ws(N+n)+w _a (n)ws(n)=1 for n=0... N-1 w(N+n)w(2N-n)+w(n)w(N+n)=1 for n=0... N-1

This can always be fulfilled by restricting the number of free coefficients to 1 .5N, e.g. by w(2N-n) = (1 - w(n)w(N+n)) / w(N+n) for n=0... N-1

The zero section is located inside w(2N-n). For the constraint of the strict ones in the middle, 2*Lz of the window coefficients are considered as non-free

coefficients. For a transformation length of N=480 and a window size of 2N and Lz=180, this leads to a degree of freedom of 1.5N-2 ^* Lz = 360 free coefficients. This window is referred to as intermediate window.

The new invention increases the degree of freedom to the maximum by

exchanging the portion of strict ones by a meandering section (e.g., 44, 64, 94) around one or around any other linear (e.g., diagonal) line (e.g., 40', 60', 90'). Thus, only half of the middle section needs to be excluded from the free

coefficients. Consequently, the number of available coefficients is increased to 1.5*480-180=540 free coefficients. This allows to maximally optimize the frequency response towards any indented focus.

Figs. 21 and 22 show a comparison of the design approaches. In particular, Fig. 21 shows a comparison window shape ALDO (21 1 ) vs intermediate (212) vs new invention (213). Fig. 22 shows a zoom (enlarged view) of window shapes ALDO (21 1 ), intermediate (212), new invention (213).

The intermediate window and the new invention window are optimized towards the same focus. Fig. 23 shows a plot frequency response with ALDO (231 ), intermediate (232), new invention (233). The intermediate (231 ) and new invention window (233) can focus on different areas, achieving in this case a better far field attenuation. This is required for instance to avoid any aliasing distortions caused by band limited signals or whenever signals are resampled through a filter bank. This focus is impossible for the ALDO window, as there are only two parameters available controlling the shape. Comparing the intermediate and the new invention window, it becomes evident that the new invention achieves much less fluctuations in the response due to the very continuous shape in the time domain. Fig. 24 shows a zoom (enlarged view) of frequency response comparing

intermediate (242) and new invention (243). As can be observed, the new invention window (243) always shows a significant higher attenuation of all side lopes. This confirms that the meandering portion (44, 64, 94) is beneficial in the window design over a portion of strict ones.

With reference of the time modulation of window, it is noted that the main motivation for the ALDO window was to combine an asymmetric shape with a maximum amplitude of 1 , in order to avoid large temporal fluctuations of the quantization error. The quantization error is added in the frequency domain and its temporal shape is controlled by the synthesis window shape.

Actually, here also the meandering sequence even improves this window property. Fig. 25 shows a direct comparison the time modulation of the quantization error caused by an ALDO window (253) and a proposed invention (251 ). As can be seen, the maximum temporal distortion is lower for the new invention and therefore provides less noticeable distortions.

In examples above, the meandering portion may be, for example, in

correspondence (at least partial) with the DCT kernel. In examples, at least one first crossing point (e.g., #1 ) is in the first frame, while at least another crossing point (e.g., #3) is in the subsequent frame. In examples, one crossing point (e.g., #2) at the border between the first and the subsequent frames.

In examples, "meandering" refers to the fact that there are at least four

intersections with a linear function (the meandering portion crosses the linear function in at least two or in some cases in at least four points). For example, a meandering portion may have an increasing part, followed by a decreasing part, followed by an increasing part, and so on, or a decreasing part, followed by an increasing part, followed by a decreasing part, and so on.

6. Other examples

Fig. 17 shows an apparatus 1 10 which may implement the encoder apparatus 130 (or 130A) and/or MDCT tool 131 and/or perform at least some steps of the method 150. The apparatus 1 10 may comprise a processor 1 1 and a non-transitory memory unit 112 with a block 1 14 storing instructions which, when executed by the processor 1 1 1 , may cause the processor 1 1 1 to perform a MDCT synthesis. The non-transitory memory unit 1 12 may also store, block 1 13, an analysis windowing function, e.g., as discussed above. The apparatus 1 10 may comprise an input unit 116, which may obtain an input information signal (e.g., an audio signal). A transitory storage space 1 18 may be reserved for saving data to be updated (e.g., input buffers).

Fig. 18 shows an apparatus 120 which may implement the decoder apparatus 140 (or 140A) and/or the IMDCT 147 and/or perform the method 160. The apparatus

120 may comprise a processor 121 and a non-transitory memory unit 22 with a block 124 storing instructions which, when executed by the processor 121 , may cause the processor 121 to perform, inter alia, an MDCT synthesis process. The apparatus 120 may comprise an input unit 126, which may obtain a decoded representation of an information signal (e.g., an audio signal) in the FD. The processor 121 may therefore perform processes to obtain a TD representation of the information signal. This decoded representation may be provided to external units using an output unit 127. The output unit 127 may comprise, for example, a communication unit to communicate to external devices (e.g., using wireless communication, such as Bluetooth) and/or external storage spaces. The processor

121 may save the decoded representation of the audio signal in a local storage space 128.

In examples, the apparatus 1 10 and 120 may be the same device. In examples, the apparatus 1 10 and 120 exchange information signals and/or control data e.g., wirelessly, e.g., using protocol Bluetooth.

Depending on certain implementation requirements, examples may be

implemented in hardware. The implementation may be performed using a digital storage medium, for example a floppy disk, a Digital Versatile Disc (DVD), a Blu- Ray Disc, a Compact Disc (CD), a Read-only Memory (ROM), a Programmable Read-only Memory (PROM), an Erasable and Programmable Read-only Memory (EPROM), an Electrically Erasable Programmable Read-Only Memory (EEPROM) or a flash memory, having electronically readable control signals stored thereon, which cooperate (or are capable of cooperating) with a programmable computer system such that the respective method is performed. Therefore, the digital storage medium may be computer readable. Generally, examples may be implemented as a computer program product with program instructions, the program instructions being operative for performing one of the methods when the computer program product runs on a computer. The program instructions may for example be stored on a machine readable medium.

Other examples comprise the computer program for performing one of the methods described herein, stored on a machine-readable carrier. In other words, an example of method is, therefore, a computer program having a program instruction for performing one of the methods described herein, when the computer program runs on a computer.

A further example of the methods is, therefore, a data carrier medium (or a digital storage medium, or a computer-readable medium) comprising, recorded thereon, the computer program for performing one of the methods described herein. The data carrier medium, the digital storage medium or the recorded medium are tangible and/or non-transitionary, rather than signals which are intangible and transitory. A further example comprises a processing unit, for example a computer, or a programmable logic device performing one of the methods described herein.

A further example comprises a computer having installed thereon the computer program for performing one of the methods described herein.

A further example comprises an apparatus or a system transferring (for example, electronically or optically) a computer program for performing one of the methods described herein to a receiver. The receiver may, for example, be a computer, a mobile device, a memory device or the like. The apparatus or system may, for example, comprise a file server for transferring the computer program to the receiver. In some examples, a programmable logic device (for example, a field

programmable gate array) may be used to perform some or all of the

functionalities of the methods described herein. In some examples, a field programmable gate array may cooperate with a microprocessor in order to perform one of the methods described herein. Generally, the methods may be performed by any appropriate hardware apparatus.

The above described examples are illustrative for the principles discussed above. It is understood that modifications and variations of the arrangements and the details described herein will be apparent. It is the intent, therefore, to be limited by the scope of the impending patent claims and not by the specific details presented by way of description and explanation of the examples herein.

7. Some examples of windowing functions

Numerical examples are here provided. As no mathematical formula in closed form for obtaining the function has been found, examples are herewith provided. The values hereinbelow may be provided in backword or forward order, in the sense that in some examples the first value is associated to the time instant n=0 and the last value is associated to the last time instant n=2N-1 (forward direction), while in some examples the first value is associated to the time instant n=2N-1 and the last value is associated to the first time instant n=0 (backward direction). In some cases, the analysis windowing function and the synthesis windowing function are taken from the same list, but the analysis windowing function is read in the backward (or forward) direction, while the synthesis windowing function is read in the forward (or backward) direction.

In order to embody at least one of the examples below, it is possible to extract a sub-succession of at least 10 values (e.g., consecutive values), when the 10 values are different from 0 (e.g., all the samples or at least the majority thereof may form the at least 10 values). A tolerance or ± 1 % may be possible; in some cases ± 0.5%, ± 0.05%, in other cases, ± 2%, ± 5%, in other cases 0%.

7.1 Examples associated to Figs. 4 and 9 In the examples, a notation such as -7.078546706512391 e-04f means

-7.078546706512391 *10 ^"4 . "f" refers to the notation in floating point (in some cases, it may be omitted).

A numerical example of windowing function w _B0 , for frame size N=80, is herewith provided by the 160 entries for n = 0...2N-1. As explained above, the last values may be constantly 0.

-7.078546706512391 e-04f, -2.098197727900724e-03f, -4.525198076002370e-03f, -8.233976327300612e-03f, -1.337713096257934e-02f, -1.999721557401502e-02f, -2.800909464274782e-02f, -3.721502082245055e-02f, -4.731768261606175e-02f, -5.794654834034055e-02f, -6.867606753531441 e-02f, -7.904647440788692e-02f, -8.859705468085925e-02f, -9.688303623049199e-02f, -1.034961241263523e-01 f, -1.080766457616878e-01f, -1.103242262600913e-01f, -1 .099809851424550e-01f, -1 .068172142230882e-01f, -1.006190418791648e-01f, -9.1 16452506492527e-02f, -7.820617483254730e-02f, -6.146688124166948e-02f, -4.063362855701623e-02f, -1.536329520788766e-02f, +1.470155068746303e-02f, +4.989736509080558e-02f, +9.050369257152079e-02f, +1.36691 1019414417e-01f, +1.884686389218322e-01f, +2.456456803467095e-01f, +3.077789078889820e-01f, +3.741642373060188e-01f, +4.4381 14799213576e-01f, +5.154735456539700e-01f, +5.876661722564289e-01f, +6.587619767809000e-01f, +7.27057669984 359e-01f, +7.908752989295335e-01f, +8.486643364959733e-01f, +8.99 320235484349e-01f, +9.413348145272842e-01f, +9.747634827941575e-01f, +9.9941 14730415857e-01f, +1 .015760373791603e+00f, +1 .024736164069697e+00f, +1.027634294456205e+00f, +1 .025991493983836e+00f, +1.021427210603284e+00f, +1.015439859549357e+00f, +1.009366925499550e+00f, + .003508162416449e+00f, +9.988898206257559e-01f, +9.953133902427869e-01f, +9.925943919208190e-01f, +9.905771957917731 e-01f, +9.891371616557014e-01f, +9.881790747212391 e-01f, +9.876249269174586e-01f, +9.874056275509585e-01f, +9.874524849192456e-01f, +9.876951 134084213e-01f, +9.880640617030884e-01f, +9.884926873551375e-01f, +9.889230031022089e-01f, +9.893074965384659e-01f, +9.896146331889107e-01f, +9.898319269347060e-01f, +9.899693102025342e-01f, +9.900603352632121e-01f, +9.901575015155720e-01f, +9.903255289051605e-01f, +9.906303787150326e-01f, +9.911298894709990e-01f, +9.918665491 182922e-01f, +9.928619727154252e-01f, +9.941156069136238e-01f, +9.956033775539884e-01f, +9.972793109558521 e-01f, +9.990784840729244e-01f, +1.000922365901945e+00f, +1.00272811 1386909e+00f , +1.004416038098237e+00f, +1.005919224127911 e+OOf

+1.007189345025525e+00f, +1.008200146369426e+00f +1 ,008949493525753e+00f +1.009458241425143e+00f, +1.009768980817384e+00f +1.009940336228694e+00f +1.010039453539107e+00f, +1.010132323996401 e+OOf +1.010272524848519e+00f +1.010494354532353e+00f, +1.010808068774316e+00f +1.011201071127927e+00f +1.01 1641272406023e+00f, +1.012080125934687e+00f +1.012458183122033e+00f + .012706955800289e+00f, +1.012755013843985e+00f +1.012530134411619e+00 +1.01 1962331 100864e+00f, +1.010982135506986e+00 +1.00951243804951 Oe+OOf + 1.007460860286395e+00f, +1.004708677491086e+00 +1.0011114 3242302e+00f +9.965041017623596e-01f, +9.907199995730845e-01f, +9.823765865983288e-01 +9.708821747608998e-01f, +9.546732976073705e-01f, +9.321553861564006e-01f +9.018003682081348e-01f, +8.623984077953557e-01f, +8.132817365236141 e-01f, +7.544551974836834e-01 f, +6.866580716267418e-01f, +6.113488038789190e-01 f +5.306181649316597e-01f, +4.471309850999502e-01f, +3.639114681156236e-01f +2.841647033392408e-01f, +2.110209448747969e-01f, +1.472287968327703e-01f +9.482665349502291 e-02f, +5.482436608328477e-02f, +2.701461405056264e-02f +9.996743588367519e-03f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f

A numerical example of windowing function w ₁₆₀ , for frame size N=160, is herewith provided by the 320 entries for n = 0...2N-1. As explained above, the last values may be constantly 0.

-4.619898752628163e-04f, -9.747166718929050e-04f, -1.664473096973725e-03f

-2.597106916737789e-03f -3.806285163352241 e-03f -5.32460872 716763e-03f -7.175885277771099e-03f -9.382480860899108e-03f, -1.195270300743193e-02f -1 .489528159506296e-02f -1.820666399965468e-02f -2.187570925786862e-02f, -2.588471937157619e-02f -3,020862738245264e-02f -3.481597793538342e-02f -3.967067992672979e-02f -4.472698045914417e-02f , -4.994225863256500e-02f -5.526334794593565e-02f -6.063717235243996e-02f -6.600961519440657e-02f -7.13 966266443390e-02f -7.651 178225890490e-02 -8.152964005319532e-02 -8.631 137544905677e-02f -9.08041 1291245728e-02f -9.495377758870335e-02f -9.870736514214426e-02f -1.020202684361974e-01f -1 .048438825017798e-01 -1.071382314127799e-01f -1.088690135027248e-01f -1.099969655786929e-01f -1.104898474883336e-01 -1.103225838568563e-01f -1.094621746650760e-01f -1 .078834293141886e-01f -1.055612509762041 e-01f -1.024650162703341 e-01f -9.857014566194629e-02f _; -9.384684920715425e-02f -8.826309993000785e-02f -8.178792716809512e-02f -7.43878560021 1463e-02f -6.602189797715241 e-02f -5.665655641 133161 e-02f , -4.624456893420224e-02f -3.474585776145929e-02f, -2.21 1581608120528e-02f -8.310425696208936e-03f, +6.717697635290676e-03f, +2.300642061077823e-02f, +4.060106462625085e-02f, +5.953239090915557e-02f, +7.98335418981651 1 e-02f, +1.015233140203748e-01f, +1 .246171387327525e-01f, + 1.491 152519299797e-01f, +1.750067399059861 e-01f, +2.022699854906251 e-01f, +2.308655379767671e-01f, +2.607365124918583e-01f, +2.918144694729168e-01f, +3.240095704645023e-01f, +3.572175180786021 e-01f, +3.913146885756875e-01f, +4.261571642320424Θ-01 f , +4.615925445090212e-01 f , +4.97447159290 086e-01 f , +5.335326819631583e-01f, +5.696546730080154e-01f, +6.056083823929643e-01f, +6.41 1830842823245e-01f, +6.761653499550255e-01f, +7.103400549562944e-01f, +7.434943718765665e-01f, +7.754281892901473e-01f, +8.059437233154637e-01f, +8.348589373399948e-01f, +8.620108336276733e-01f, +8.872599706865 23e-01f, +9.104863121445679e-01f, +9.315962496426278e-01f, +9.505220861927248e-01f, +9.672366712325431 e-01f, +9.817397501303696e-01f, +9.940557180662704e-01f, +1 .004247514102417e+00f, +1.012407428282884e+00f, +1 .018650990561848e+00f, +1.0231 18841384460e+00f, +1.025972450969440e+00f, +1.02739752393921 Oe+OOf, +1.027585830688143e+00f, +1.026738673647482e+00f, +1.025061777648234e+00f, + 1.022756514615106e+00f, +1.020009 39549275e+00f, +1.016996499560845e+00f, + 1.013915946100629e+00f, +1.01 1044869639164e+00f, +1 .007773858455400e+00f, +1.004848753962734e+00f, +1.002245009135684e+00f, +9.999393169239009e-01f, +9.9790554 5627330e-01f, +9.961203379971326e-01f, +9.945597525471822e-01f, +9.932031606606762e-01f, +9.920297273323891 e-01f, +9.910230654424902e-01f, +9.901668953434221 e-01f, +9.894488374513719e-01f, +9.888556356037892e-01f, +9.883778520531268e-01f, +9.880051626345804e-01f, +9.877295459610343e-01f, +9.875412739766566e-01f, +9.874329809802893e-01f, +9.873949921033299e-01f, +9.874197049003676e-01f, +9.874973205882319e-01f, +9.876201238703241e-01f, +9.877781920433015e-01f, +9.879637979933339e-01f, +9.881678007807095e-01f, +9.883835200189653e-01f, +9.886022219397892e-01f, +9.888182771263505e-01f, +9.890247977602895e-01f, +9.892178658748239e-01f, +9.893923680007577e-01f, +9.895463342815009e-01f, +9.89677201 1542693e-01f, +9.897859 95209235e-01f, +9.898725363809847e-01f, +9.899410789223559e-01f, +9.899945557067980e-01f, +9.900394023736973e-01f, +9.900814722948890e-01f, +9.901293790312005e-01f, +9.901902265696609e-01f +9.902734448815004e-01f, +9.903862280081246e-01f +9.905379830873822e-01f +9.907348826312993e-01f, +9.909842592301273e-01f +9.9129051 18607647e-01f +9.916586940166509e-01f, +9.920906151219310e-01f +9.925887208794144e-01f +9.931516528513824e-01f, +9.937790866568735e-01f, +9.944668184371617e-01f +9.9521 16634297566e-01f, +9.960068616185641 e-01f +9.968461329825753e-01f +9.977203369515556e-01f, +9.986213520769593e-01f +9.995382582242990e-01f +1.000461955079660e+00f, +1.001380551217109e+00f

+1 .002284871786226e+00f, +1.003163845364970e+00f. +1 .004009147462043e+00f

+1 .00481 1375053364e+00f +1 .005563968008037e+00f +1 .006259855360867e+00f +1 .006895570408563e+00f _; + 1.007466616298057e+00f +1.007972441990187e+00f +1 .00841 1468616852e+00f +1.008786009787269e+00f +1.009097763850333e+00f +1 .009351762546296e+00f + .009552401900961 e+OOf +1.009707093778162e+00f +1 .009822090220407e+00f +1.009906958448099e+00f + 1.009969021400474e+00f + 1 .010017890428877e+00f +1.010060809299530e+00f +1.010106564965965e+00f +1 .010161 131093372e+00f + 1.010231078494249e+00f +1.010319484524512e+00f +1 .010430470494512e+00f +1.010564099281 OOOe+OOf +1.010721360243234e+00f + 1 .010899655674578e+00f +1.01 1096993993037e+00f +1.01 1308167670753e+00f +1 .01 1529185153809e+00f +1.01 1753008569803e+00f +1.01 197387651 1603e+00f +1.012182837094955e+00f +1.012373028737774e+00f +1.012535058602453e+00f +1 .012660975529858e+00f +1.012740575296603e+00f +1.012765922449960e+00f +1 .012726958954961 e+OOf +1.0126159041 16265e+00f + 1.012422888521601 e+OOf +1 .01214046021 1 194e+00f +1.01 1758810583150e+00f +1.01 1269960947744e+00f +1 .010663676735228e+00f +1.009930754807923e+00f +1.009058249873833e+00f + 1.008034308295421 e+OOf +1.006843352506855e+00f + 1.005470005637052e+00f +1 .003894772403371 e+OOf +1.002098854400575e+00f +1.000060686758758e+00f +9.977600196406868e-01f, +9.951746430061 121 e-01f, +9.922861082472264e-01f +9.890757868707590e-01f, +9.847362453480265e-01f, +9.798613526271561 e-01f +9.741378617337759e-01f +9.673331975559332e-01f +9.592539757044516e-01f

+9.496984081652284e-01f +9.38463416382671 1 e-01f +9.253567968750328e-01f,

+9.101986790930605e-01f +8.928338316495705e-01f +8.731437835983047e-01f

+8.510420440685049e-01f +8.26483991 1291 133e-01f +7.994681492797084e-01f

+7.700431275216928e-01f +7.383028603058783e-01f +7.043814340356083e-01f

+6.684616478236647e-01f +6.307755329382612e-01 +5.915799587176216e-01f,

+5.51 1703155400274e-01f, +5.098915423728179e-01 +4.6810171 10047964β-01ί,

+4.261772971493010e-01f +3.845172335531009e-01f +3.435228672445613e-01f

+3.036004651973099e-01f +2.651434678028531 e-01f +2.285283969438072e-01

+1 .941021906320984e-01f +1.621735416384830e-01f +1 .330015240938615e-01

+ 1.067840430193724e-01 f , +8.365057236623041 e-02f, +6.3651881 1 1381356e-02

+4.676538412257621 e-02f +3.288072750732215e-02f +2. 83057564646270e-02f

+1 .336381425803019e-02f +6.758124889697787e-03f +0.000000000000000e+00f,

+0.000000000000000e+00f +0.000000000000000e+00f. +0.000000000000000e+00f.

+0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+OOf , +0.000000000000000e+00

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f, +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00 +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f,

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f, +0.000000000000000e+00 +0.000000000000000e+00f,

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f

+0.000000000000000e+00f +0.000000000000000e+00f +0.000000000000000e+00f. +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f

A numerical example of windowing function ₂ 4o, for frame size N=240, is herewith provided (480 samples):

-3.613496418928369e-04f -7.078546706512391 e-04f -1 ,0744436371 0903e-03 -1 .533478537964509e-03f -2.098197727900724e-03 -2.778420871815740e-03f, -3.584129920673041 e-03f -4.525198076002370e-03f -5.609327243712055e-03f -6.843234536105624e-03f -8.233976327300612e-03f -9.785314755557023e-03f -1 .149880303071551 e-02f -1 .337713096257934e-02f -1 .54218167951 1618e-02f -1.762979910961727e-02f -1.999721557401502e-02 -2.252080561390149e-02 -2.519406300389030e-02f -2.800909464274782e-02f -3.095765092956728e-02 -3.402996266948349e-02f -3.721502082245055e-02f _: -4.050053247568393e-02 -4.387219218706189e-02f -4.731768261606175e-02f -5.082325342672667e-02f -5.437166635159518e-02f -5.794654834034055e-02f -6.153426201732499e-02 -6.51 1708163 13709e-02f -6.867606753531441 e-02f -7.219447805250771 e-02f -7.565695975592170e-02f -7.904647440788692e-02f -8.234442557322251 e-02f -8.553324579905185e-02 -8.859705468085925e-02f -9.152091 100798199e-02f -9.428847446755965e-02f -9.688303623049198e-02f -9.929123258537813e-02f -1.015008467688577e-01f -1 .034961241263523e-01f -1 .052637003544443e-01f -1 .067939984687745e-01f _; -1.080766457616878e-01f -1.090997300590506e-01f -1.098524491515805e-01 -1.103242262600913e-01f, -1.105084619148789e-01f -1 .103977408741932e-01f -1.099809851424550e-01f -1.092492774392824e-01f -1.081974227416502e-01f -1.068172142230882e-01f -1 .050995803285455e-01f -1 .0303601 1 1 1 1 1 103e-01f -1 .006190418791648e-01f. -9.78412002341 1771 e-02f -9.469304216883027e-02f, -9.1 16452506492527e-02f, -8.724644532866996e-02f, -8.293043914044632e-02f, -7.8206 7483254730e-02f, -7.306142427456862e-02f, -6.748468182105991 e-02f , -6.146688 24166948e-02f, -5.499497258200362e-02f, -4.805444424454820e-02f, -4.063362855701623e-02f, -3.272045590229335e-02f, -2.430122582451853e-02f, -1.536329520788766e-02f, -5.89 434269890659e-03f, +4.126595858583295e-03f, +1.470155068746303e-02f, +2.584738191459814e-02f, +3.757652772246801 e-02f, +4.989736509080558e-02f, +6.282034030592902e-02f, +7.635397728566121 e-02f, +9.050369257152079e-02f, +1 .0527471 18478660e-01f, +1.206703467513333e-01f, +1.3669 1019414417e-01f, +1.533343890681390e-01f, +1.705954709184399e-01f, + 1.884686389218322e-01f, +2.069449962574092e-01f, +2.260093000067393e-01f, +2.456456803467095e-01f, +2.658346019332584e-01f, +2.865543814049772e-01f, +3.077789078889820e-01f, +3.294769437072290e-01f, +3.51617148 750350e-01f, +3.741642373060188e-01f, +3.97073959121 1551 e-01f, +4.203043046885219e-01f, +4.4381 14799213576e-01f, +4.675442291623012e-01f, +4.914498631045615e-01f, +5.154735456539700e-01f, +5.395557644293222e-01f, +5.636399817032525e-01f, +5.876661722564289e-01f, +6.1 15695310143157e-01f, +6.352890592874099e-01f, +6.587619767809000e-01f, +6.819230974423550e-01f, +7.047092819314779e-01f, +7.270576699841359e-01f, +7.489068963384272e-01f, +7.701990187606995e-01f, +7.908752989295335e-01 f, +8.108788692151807e-01f, +8.301579139160681e-01f, +8.486643364959733e-01f, +8.663548164329093e-01f, +8.831896853053627e-01f, +8.99 320235484349e-01f, +9.141540563656075e-01f, +9.282282546151819e-01f, +9.413348145272842e-01 f, +9.534619388400459e-01f, +9.646048250501910e-01f, +9.747634827941575e-01f, +9.839435385219192e-01f, +9.921529097154242e-01f, +9.9941 147304 5857e-01f, +1 .005746084650236e+00f, +1 .01 183971347815e+00f, +1 .015760373791603e+00f, +1.019515072412387e+00f, +1 .022490937034641e+00f, +1.024736164069697e+00f, +1 ,026304095700693e+00f, +1 .027250978292214e+00f, +1.027634294456205e+00f, +1.02751 1063644843e+00f, +1 .0269427951 15598e+00f, +1.025991493983836e+00f, +1.024716149969084e+00f, +1.023175976163407e+00f, +1.021427210603284e+00f, +1.019521566634239e+00f, +1.0175101 18327508e+00f, +1.015439859549357e+00f, +1.013460916839174e+00f, +1.01 1654901040475e+00f , +1.009366925499550e+00f, +1.007263182132894e+00f, +1 .005313192386866e+00f, +1 .003508162416449e+00f, +1.001840787319378e+00f, +1 .000303927234380e+00f, +9.988898206257559e-01f, +9.975915283480670e-01f, +9.964015284765968e-01f, +9.953133902427869e-01f, +9.943201078053212e-01f,

+9.93415895918601 1e-01f +9.925943919208190e-01f, +9.918510277326026e-01f, +9.91 1797988363887e-01f +9.905771957917731 e-01f, +9.900381047643838e-01f, +9.895594394179152e-01f +9.891371616557014e-01f, +9.887684373604154e-01f, +9.884497924570929e-01f +9.881790747212391 e-01f, +9.879528358230726e-01f, +9.877691368590689e-01f +9.876249269174586e-01f, +9.875179947346887e-01f, +9.874458127312921 e-01 f +9.874056275509585e-01f, +9.873951 1 15886979e-01f, +9.8741 15368168944e-01f +9.874524849192456e-01f, +9.875149888347144e-01f, +9.875968894760857e-01f +9.876951 134084213e-01f, +9.878075819424549e-01f, +9.87931 1998177238e-01f +9.880640617030884e-01f, +9.882032571565917e-01f, +9.883471084085503e-01f +9.884926873551375e-01f, +9.886386592120545e-01f, +9.887825578295630e-01f +9.889230031022089e-01f, +9.890581715933395e-01f, +9.891867674284610e-01f +9.893074965384659e-01f, +9.894196399062921 e-01f, +9.895220757174378e-01f +9.896146331889107e-01f, +9.896970346678272e-01f, +9.897692596535289e-01f +9.898319269347060e-01f, +9.898852572653667e-01f, +9.899307640365727e-01f +9.899693 02025343e-01f, +9.900025692522435e-01f, +9.900321562263099e-01 f +9.900603352632121 e-01f, +9.900889812894406e-01f, +9.901206586012907e-01f +9.901575015155720e-01f, +9.902023946214220e-01f, +9.902575406142213e-01 f +9.903255289051605e-01f, +9.904087914462694e-01 f , +9.905096491583045e-01f +9.906303787150326e-01f, +9.907727108894024e-01f, +9.909387444078919e-01f +9.91 1298894709990e-01f, +9.913476318763218e-01f +9.915928560402563e-01f, +9.918665491 182922e-01f, +9.921691315380984e-01f +9.925010851461232e-01f, +9.928619727154252e-01f, +9.932519181564613e-01f, +9.936700207375173e-01f, +9.941 156069136238e-01f, +9.945873147903244e-01f, +9.950837402063278e-01f, +9.956033775539884e-01f, +9.961439922621 166e-01f +9.96703453392 340e-01f, +9.972793109558521e-01f, +9.978690858367024e-01f +9.984697087896268e-01f, +9.990784840729244e-01f, +9.99691901 1206490e-01f +1 .000308193833526e+00f, +1.000922365901945e+00f, +1.001532636590676e+00f

+1.002135464655177e+00f +1.0027281 1 1386909e+00f + 1.003307449770187e+00f +1.003870934089686e+00f +1.004416038098237e+00f +1.004940548815171 e+OOf +1.005442141810160e+00f +1.00591922412791 1 e+OOf +1 .006370303149314e+00f +1 .006793927824538e+00f + 1.007189345025525e+00f + 1.007555573455895e+00f +1.007892674961336e+00f + 1.008200146369426e+00f + 1 .008478423284851 e+OOf +1.008727884997619e+00f +1.008949493525753e+00f +1 .0091441 12734761 e+00f +1.009313224929575e+00f +1.009458241425143e+00f +1 .00958 280555682e+00f + 1.009684090687164e+00f +1 .009768980817384e+00f + .009838308708799e+00f +1.009894548257807e+00f +1.009940336228694e+00f +1.009977916643680e+00f +1.010010230290263e+00f +1.010039453539107e+00f +1.010068202038694e+00f +1 .010098388689342e+00f +1.010132323996401 e+OOf +1.010171656775640e+00f +1 .010218096148412e+00f +1.010272524848519e+00f +1.010336490294771 e+OOf +1 .010410221483215e+00f +1.010494354532353e+00f +1.010588873699422e+00f +1 .010693501 186928e+00f +1.010808068774316e+00f +1.010931436739342e+00f +1.01 1062876503041 e+OOf +1.01 1201071 127927e+00f +1.01 1344700694417e+00f +1.01 1491904228184e+00f +1.01 1641272406023e+00f +1 .01 1790282474963e+00f +1 .01 1937567254485e+00f +1.012080125934687e+00f +1 .012216235487353e+00f +1 .012342907951334e+00f +1.012458183122033e+00f + 1 .012558879696851 e+OOf +1 .012642857380847e+00f +1.012706955800289e+00f +1 .012748952907404e+00f +1 .012765799894453e+00f + 1.012755013843985e+00f +1.01271379867821 1 e+OOf +1.012639775003457e+00f, +1.01253013441 1619e+00f, +1 .012382309473470e+00f, +1 .012 940681 17524e+00f, +1.01 196233 100864e+00f, +1 .01 1685173724601 e+OOf, +1 .01 1359143572147e+00f, +1.010982135506986e+00f, +1.010550715971368e+00f, +1.010062133151922e+00f, +1.00951243804951 Oe+OOf, +1 .008898689394160e+00f, +1.008215923600973e+00f, +1.007460860286395e+00f, + .006627741823389e+00f, +1 .005712337656749e+00f, +1 .004708677491086e+00f, +1 .00361 1467285588e+00f, +1.002414286392268e+00f, +1.001 1 1 1413242302e+00f, +9.996961651093181 e-01f, +9.981625949525345e-01f, +9.965041017623596e-01f, +9.947148884277037e-01f,

+9.927891912841345e-01f +9.907199995730845e-01f +9.884793707533194e-01f +9.855347660016696e-01f +9.823765865983286e-01f +9.789747333404933e-01f _: +9.75162381 1486372e-01 +9.708821747608998e-01 +9.660805524695870e-01f +9.606976399184645e-01f +9.546732976073706e-0 f , +9.479479345282376e-01f +9.404609052933396e-01f +9.321553861564006e-01f +9.229775478442888e-01f +9.128745354570823e-01 f +9.018003682081348e-01f +8.897163275605041 e-01 +8.765908974996186e-01 +8.623984077953557e-01f +8.471200801854385e-01f +8.307479727020245e-01f +8.132817365236141e-01f, +7.947291447585267e-01f +7.751 10884189 807e-01f +7.544551974836834e-01f +7.327963552921717e-01f +7.101790843209148e-01f +6.8665807 6267418e-01 f +6.622962432368731 e-01 f +6.3716841 19604742e-01f +6.1 13488038789190e-01 +5.849206604934815e-01f +5.579747428663487e-01 f +5.306181649316717e-01f +5.029523957059122e-01f +4.75086882551 1614e-01f +4.471309850999535e-01f +4.192049917945288e-01f +3.914252910998820e-01f +3.6391 1468 156252e-01f, +3.367837772954476e-01f +3.101627843160973e-01f +2.841647033392418e-01f +2.58903371 1808454e-01f +2.344880603710975e-01f +2.1 10209448747974Θ-01 f, +1.885997642296488e-01f +1 .673100807904834e-01f +1.472287968327706e-01f _; +1.284223074167396e-01f +1 .109422548710344e-01f +9.482665349502306e-02f +8.009914366829558e-02f +6.676765847398403e-02f +5.482436608328485e-02 +4.424588851571281 e-02f +3.499361000717621 e-02f, +2.701461405056267e-02f, +2.024370180670145e-02f, +1 .460796755137538e-02f, +9.996743588367531 e-03f, +5.305235098871444e-03f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +O.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0. OOOOOOOOOOOOOOOe+OOf ,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +O.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0. OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +O.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0. OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0. OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +O.000000000000000e+00f,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f, +0.0000000000000Q0e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f,

+0.000000000000000e+00f, +0,000000000000000e+00f, +0.000000000000000e+00f,

+0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f,

+0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f,

+0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f}

A numerical example of windowing function w ₃₂ o, for frame size N=320, is herewith provided (640 samples):

-3.021153494057143e-04f -5.867737487939294e-04f -8.366504004139796e-04f, -1.126635355725494e-03f -1.470492941694331 e-03 -1.873473391018495e-03f, -2.339292362082021 e-03f -2.872008069419264e-03f -3.476256385086407e-03 -4.155963816705528e-03f -4.914563787665504e-03f -5.755172503953251 e-03f -6.680623380533122e-03f -7.693816924650567e-03f -8.796760749750191 e-03f -9.990503073705982e-03f -1.127574117138621 e-02f -1.265334152129685e-02f -1.412438986522702e-02 -1.568889620430290e-02f -1.734512089366117e-02f -1.909097368362797e-02 -2.09254671 1 168754e-02f -2.284684792818856e-02f -2.485207716234951 e-02f -2.693746704328349e-02f -2.909952486193999e-02f -3.133504629493832e-02f -3.363960728361352e-02f -3.600820974457969e-02 -3.843601741746971 e-02f -4.091746034850161e-02f -4.344654894948344e-02f -4.601786724624048e-02f -4.862598509282497e-02f, -5.126474204655663e-02f -5.392644753556616e-02f -5.66038431 1081047e-02f -5.929116747072080e-02f -6.19826820251 1926e-02f -6.467025548071 184e-02f, -6.734542216184526e-02f -7.000099017198280e-02 -7.26305701 1354321 e-02f -7.522784961377151 e-02f -7.778525942347714e-02f -8.029480247839878e-02f -8.274924535373614e-02f -8.514125464087215e-02f -8.746379123238275e-02f -8.971069341834263e-02f -9.187564084638347e-02f _: -9.395176975347193e-02f, -9.593137735886889e-02f -9.780843257659243e-02f -9.957851303827886e-02f, -1.012361165314596e-01f -1.027741036495644e-01f -1.041861222641 1 19e-01f -1.054680247057000e-01f -1.066160875985523e-01f. -1.076255384835563e-01f. -1.084912299471198e-01f -1 .092087422379003e-01f, -1.097736146613313e-01f, -1 .101808861640070e-01f, -1.104271876052675e-01f, -1.105108362290460e-01f, -1.104281465492726e-01f, -1.101739218186236e-01 f, -1.097437360338336e-01f, -1 .091353 2557251 1 e-01f, -1 .083467335729228e-01f, -1 .073739938306107e-01f, -1.062130155324388e-01f, -1 .048606145834788e-01f, -1.033132401525343e-01f, -1.015673163469357e-01f, -9.962005506126154e-02f, -9.746803229469267e-02f, -9.510723623306666e-02f, -9.253303383231506e-02f, -8.974125216128212e-02f, -8.6728776891 19252e-02f, -8.349213839083708e-02f, -8.002639902061687e-02f, -7.632679536516856e-02f, -7.238806162166744e-02f, -6.820576796149519e-02f, -6.37761 1429172260e-02f, -5.909386001558149e-02f, -5.415316322402774e-02f, -4.894812724598650e-02f, -4.3473471 12195197e-02f, -3.772461300253332e-02f, -3.169587609244436e-02f, -2.538179830690266e-02f, -1.877689096555516e-02f, -1 .187461378850388e-02f, -4.669099247423082e-03f, +2.844096748870385e-03f, +1.066976124794342e-02f, +1.881355950582949e-02f, +2.728156010437695e-02f, +3.607810469851272e-02f , +4.520702759803914e-02f, +5.467238802204326e-02f, +6.447866054615346e-02f, +7.462862199422061 e-02f, +8.512490568723846e-02f, +9.596983987496970e-02f, +1 .071650779014335e-01f, +1.1871 15850305241 e-01f, +1.306101067250375e-01f, +1 .428596447589721 e-01 f, +1.554584725339102e-01f, +1 .684041609371527e-01f, +1 .816947894623263e-01f, +1.953273880886783e-01f, +2.092963206850239e-01f, +2.235945635254679e-01f, +2.382160219461597e-01f, +2.531529721334063e-01f, +2.683961570569586e-01f, +2.839361392493072e-01f, +2.99762425517781 1 e-01f, +3.158619077906196e-01f, +3.322210551086769e-01f , +3.488264676990591 e-01 f, +3.656640377499646e-01f, +3.827152968157059e-01f, +3.99961 1859760947e-01f, +4.173843265025887e-01f, +4.349669624916473e-01f, +4.526876397402144e-01f, +4.705242008503956e-01f, +4.884539254831315e-01f, +5.064545550235134e-01f, +5.245006748662190e-01f, +5.425674372882107e-01f, +5.606312044701524e-01f, +5.786672646386708e-01f, +5.966477035050948e-01f, +6.145458904162185e-01f, +6.323361944662236e-01f, +6.49992631921 1774e-01f +6.674874032292857e-01f

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+1 .001894024615659e+00f, +1.006474342231823e+00f, + .010552057109195e+00f, + 1 .014142538208007e+00f, +1.017262593268930e+00f, +1 .019928842669923e+00f, +1.02215986701 1 177e+00f, +1.023976320927187e+00f, +1.025400734608122e+00f, + 1 .026455340400072e+00f, +1.027164510654160e+00f, +1.027552729180790e+00f, +1 .027644462380432e+00f, +1.027463246660797e+00f, +1.027035903410657e+00f, +1.026389068000259e+00f, +1.02554820 799728e+00f, +1.024537134749709e+00f, + 1 .023380803775376e+00f, +1.022103695693341 e+OOf, +1 .020728359657958e+00f, + 1 .019275334687329e+00f, +1.017765178792830e+00f, +1.016217355867531 e+OOf, +1.01466531 1686846e+00f, +1.013249071090664e+00f, +1 .01 1948006992127e+00f, +1.010189090179223e+00f, +1.008557961 167850e+00f, +1 .00701 1287608451 e+OOf, +1 .00554876457591 Oe+OOf, +1.004168417268956e+00f, +1.002867268893035e+00f, +1 .0016417691 15897e+00f, +1.000489068954641 e+OOf, +9.994060799749374e-01f, +9.983898865406841 e-01f, +9.974370849972721 e-01f, +9.96544483691 1705e-01f, +9.957098545943852e-01f, +9.949302413030897e-01f, +9.942024045863540e-01f, +9.935241604969254e-01f, +9.928930430130044e-01f, +9.923068103443909e-01f, +9.917633778190438e-01f, +9.912597642374404e-01f, +9.907954498484041 e-01f, +9.903677893656558e-01f, +9.89975161 1066148e-01f, +9.896160337369861 e-01f, +9.892890160408989e-01f +9.88992851 1 129679e-01f +9.887260333430423e-01f +9.884868721088945e-01f +9.882751039537586e-01f _; +9.880892168751595e-01 +9.8792771 14724612e-01f +9.877898261218510e-01f +9.876743442038471 e-01f +9.875807496078497e-01f +9.875072021876561 e-01f +9.874529447589979e-01f +9.874169741527905e-01f +9.873984685207834e-01f +9.873958301311858e-01f +9.874080027710336e-01f +9.874343401290739e-01f +9.8747362353870 8e-01f, +9.875243137719285e-01 +9.875856201221 135e-01f, +9.876563785063032e-01f, +9.877358921 155149e-01f +9.878225576787804e-01f +9.879150968481590e-01f +9.880132731565830e-01f +9.881 156946084619e-01f +9.88221 1314188272e-01 +9.883289032519310e-01f +9.884378310018685e-01f +9.885476787868710e-01 f _: +9.886568414746639e-01f +9.887645868459630e-01f, +9.888708540445242e-01f +9.889744320992592e-01f +9.890747269455915e-01f +9.891710038703801 e-01f +9.892631024032380e-01f +9.893507219573624e-01f +9.894330645494204e-01f +9.895096919388534e-01f +9.895810813422480e-01f +9.896467469067676e-01f +9.897067365020641 e-01f +9.897606930400666e-01f +9.898094478563998e-01f _; +9.898530133261707e-01f +9.898914705684924e-01f +9.899254194103574e-01f +9.899554202030650e-01f +9.899824494486951 e-01f +9.9000651 16928948e-01 +9.900284805353695e-01f +9.900497484789281 e-01f +9.900709561632662e-01f +9.90092835861 1601e-01f +9.901 163920607219e-01f +9.901427479709606e-01f +9.901734275350572e-01f +9.902087332329851 e-01f +9.902498637985275e-01f +9.902983686695558e-01f +9.903548501470234e-01f +9.904205084933333e-01f, +9.904959297726740e-01f +9.905825150202904e-01f, +9.906812569810133e-01f, +9.907922087340426e-01f +9.909165464981378e-01f +9.910550740962871 e-01f, +9.912084614290896e-01f, +9.913768610980639e-01f +9.915605826937839e-01f, +9.917604214872976e-01f +9.919767175562684e-01 +9.922091 101818779e-01f +9.924579135466506e-01f +9.927231225056266e-01f, +9.930049538427406e-01f +9.933027281437943e-01f +9.936161084869942e-01f +9.939453714404443e-01 f +9.942895145656371e-01f, +9.946481676207727e-01f, +9.950203031067961 e-01f, +9.954058173659507e-01f, +9.958038713694317e-01 f, +9.9621302710171 17e-01 f, +9.966324689957675e-01f, +9.970615306490058e-01f, +9.974990583293081 e-01f, +9.979437430375855e-01f, +9.983940572002874e-01f, +9.9884931 16887893e-01f, +9.993083430214909e-01f, +9.997689221333534e-01f, +1.000231 131275969e+00f, +1 .000692135698996e+00f, +1.001 152013920163e+00f, +1.001608526000461 e+00f,

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+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf,

+0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f

A numerical example of windowing function w ₄₈₀ , for frame size N=480, is herewith provided (960 samples):

-2.353032150516754e-04f, -4.619898752628163e-04f, -6.262931535610879e-04f,

-7.929180432976445e-04f -9.747166718929050e-04f -1.180256894474562e-03f, -1 .409209039594871 e-03f -1.664473096973725e-03f -1 .946591608170231 e-03f -2.257081732588478e-03f -2.597106916737789e-03f -2.967607624839524e-03f -3.370454877988472e-03f -3.806285163352241 e-03f -4.276873767639064e-03f -4.78246990450 813e-03f -5.324608721716763e-03f -5.903403814095400e-03f -6.520419726599805e-03f -7.175885277771099e-03f -7.871422820642307e-03f -8.606586039759667e-03f -9.382480860899108e-03f -1 .019827182163307e-02f -1.105520547739066e-02f -1.195270300743193e-02f -1.289205910303846e-02f -1 .387263484323160e-02f -1.489528159506296e-02f -1.595856621933800e-02f -1 .706288556735433e-02f -1.820666399965468e-02f -1 .939065975232718e-02f -2.061355417582714e-02f -2.187570925786862e-02f -2.31752631526641 1 e-02 -2.451227449041489e-02f -2.588471937157619e-02f -2.729263737090799e-02f -2.873390902713615e-02f -3.020862738245264e-02f -3.171440372994384e-02f -3.325098858986303e-02f -3.481597793538342e-02f -3.640892406933019e-02f -3.802742318209150e-02f -3.967067992672979e-02f -4.133575417353826e-02f -4.302203371734278e-02f -4.472698045914417e-02f -4.645022292934329e-02f -4.818891490266687e-02f -4.994225863256500e-02f -5.170690802826666e-02f -5.348162036097223e-02f -5.526334794593565e-02f , -5.705123152423822e-02f -5.884271749745559e-02f -6.063717235243996e-02 -6.243104027829089e-02f -6.422303545004304e-02f -6.600961519440657e-02 -6.778962269634495e-02f -6.955996868581379e-02f -7.131966266443390e-02f, -7.306581273272733e-02f -7.479758913001458e-02f, -7.651 178225890490e-02f, -7.82071 1420768856e-02f, -7.98801069341 1644e-02f, -8.152964005319532e-02f, -8.315237353264004e-02f, -8.474728946770714e-02f, -8.631 137544905677e-02f, -8.784374452959058e-02f, -8.934164364321417e-02f , -9.08041 1291245728e-02f, -9.222795761428432e-02f, -9.361232867223340e-02f, -9.495377758870335e-02f, -9.625155313139856e-02f, -9.750284620437569e-02f, -9.870736514214426e-02f , -9.98627128827 026e-02f, -1.009680221406219e-01f, -1.020202684361974e-01f, -1.030183804850491 e-01f, -1 .039596356759290e-01f, -1.048438825017798e-01f, -1.056686838192766e-01f, - 1.064342821660323e-01 f , -1.071382314127799e-01f, -1.077799961 21537e-01f, -1 .083570625865931 e-01f, -1.088690135027248e-01f, -1.093135588677235e-01f, -1.096903559498340e-01f, -1.099969655786929e-01f, -1.10233226 219973e-01f, -1 .103972812085189e-01f, -1.104898474883336e-01f, -1.105086416532167e-01f, -1 . 04537426996073e-01f, -1.103225838568563e-01 f, -1 .101 145827722143e-01f, -1 .098276928170364e-01f, -1.09462 746650760e-01f, -1 .090163960055733e-01f, -1.084908852561722e-01f, -1.078834293141886e-01f, -1.071937180231978e-01f, -1 .064196358069465e-01f, -1.055612509762041e-01f, -1.046162812518618e-01f, -1 .035849043557610e-01f, -1.024650162703341 e-01f, -1.012568997532046e-01f, -9.995864571932928e-02f , -9.857014566194627e-02f, -9.70891 1 135857967e-02f, -9.551545820689084e-02f, -9.384684920715425e-02f, -9.208300062891550e-02f, -9.022171021406450e-02f, -8.826309993000785e-02f, -8.620493821803937e-02f, -8.404742152815330e-02f, -8.178792716809512e-02f, -7.942625026703617e-02f, -7.695980775819990e-02f, -7.43878560021 1463e-02f, -7.170797002873608e-02f, -6.891994783815969e-02f, -6.602189797715241e-02f, -6.301349420724424e-02f, -5.989191912667712e-02f, -5.665655641 133161 e-02f, -5.330406164482222e-02f, -4.983427241976235e-02f, -4.624456893420224e-02f, -4.253455686336916e-02f, -3.870195772538443e-02f, -3.474585776145929e-02f, -3.066341518682682e-02f , -2.645425077642105e-02f, -2.21 1581608120528e-02f, -1 .764740541599136e-02f, -1 .304581363895818e-02f, -8.310425696208936e-03f, -3.438268661 133170e-03f, +1 .570315476576933e-03f, +6.717697635290676e-03f, +1.200477020244778e-02f, + 1 .743398319747869e-02f, +2.300642061077823e-02f, +2.872481423270595e-02f, +3.458896350634671 e-02f, +4.060106462625085e-02f, +4.676102915752826e-02f, +5.30713391 1821893e-02f, +5.953239090915557e-02f, +6.614647812869151 e-02f, +7.291293184312803e-02f, +7.983354189816511 e-02f, +8.690807412770696e-02f, +9.4138 3765275064e-02f, +1.015233140203748e-01f, +1.0906515 8336202e-01f, +1 .167626546016197e-01f, +1.246171387327525e-01f, +1 .3262729489381 13e-01f, +1.407938190608664e-01f, +1 .491 152519299797e-01f, +1.575921408388593e-01f, +1 .662224799248571 e-01f, +1.750067399059861 e-01f, +1 .839431938620024e-01f, + 1 .930318183054904e-01f, +2.022699854906251 e-01f, +2.1 16567430906184e-01f, +2.21 1888523410642e-01f, +2.308655379767671 e-01f, +2.406837992341654e-01f, +2.506420640291662e-01f, +2.607365124918583e-01f, +2.709659073501 196e-01f, +2.813259021832532e-01f, +2.918144694729168e-01f, +3.024270279840051 e-01f, +3.131603499997996e-01f, +3.240095704645023e-01f, +3.349719592361666e-01f, +3.460422935204829e-01f, +3.572175180786021 e-01f, +3.684915649120530e-01f, +3.7985951 19591716e-01f, +3.913146885756875e-01f, +4.028532873867052e-01f, +4.144688328137527e-01f, +4.261571642320424e-01f, +4.3791 13897565727e-01f, +4.497256320417501 e-01f, +4.615925445090212e-01f, +4.735067030065239e-01f, +4.854600184866710e-01f, +4.974471592901086e-01f, +5.094597228333853e-01f, +5.214909841729947e-01f, +5.335326819631583e-01f, +5.45578981 1615239e-01f, +5.576217157959890e-01f, +5.696546730080154e-01f, +5.816685576268035e-01f, +5.936560624526468e-01f, +6.056083823929643e-01f, +6.175192060085208e-01f, +6.29379661 1336280e-01f, +6.41 1830842823245e-01f, +6.529203544876097e-01f, +6.645840786371451 e-01f, +6.761653499550255e-01f, +6.876573952173626e-01f, +6.99051 15391 19996e-01f, +7.103400549562944e-01f, +7.215149331458728e-01f, +7.325691772738999e-01f, +7.434943718765665e-01f, +7.542846327442048e-01f, +7.649313654540612e-01f, +7.754281892901473e-01f, +7.857670170752049e-01f, +7.959414651061612e-01 f, +8.059437233154637e-01 f, +8.157687070715176e-01f, +8.254086223972127e-01f, +8.348589373399948e-01f, +8.441 125827416620e-01f, +8.531651 194538425e-01f, +8.620108336276733e-01f, +8.706456337542150e-01f, +8.790631561061 171 e-01f, +8.872599706865123e-01f, +8.952313288619367e-01f, +9.029751680353524Θ-01 f, +9.104863121445679e-01 f, +9.177625550620636e-01f, +9.247997426966093e-01f, +9.315962496426278e-01f, +9.381494858921667e-01f, +9.444588390359354e-01f, +9.505220861927248e-01f, +9.563402921286364e-01f, +9.6191 14522936701 e-01f, +9.672366712325431 e-01f, +9.723156637834687e-01f, +9.771501 187120180e-01f, +9.817397501303696e-01f, +9.860865871353246e-01f, +9.901906380163595e-01f, +9.940557180662704e-01f, +9.976842395284637e-01f, +1.00108096125701 Oe+OOf, +1.004247514102417e+00f, +1.007188578458507e+00f,

+1 .009906654565108e+00f +1.012407428282884e+00f, +1 .014694702432600e+00f, + 1 .016774659209400e+00f + 1.018650990561848e+00f, +1.0203304644631 1 1 e+OOf, +1 .02181732891 1793e+00f +1.0231 18841384460e+00f, +1 .024240262467000e+00f, +1.025189721888128e+00f +1.025972450969440e+00f, +1.026596938589443e+00f, +1 .027069179375841 e+OOf + 1.02739752393921 Oe+OOf, +1 .027587902203109e+00f, +1 .027648951922701e+00f +1.027585830688143e+00f, +1.027408519661012e+00f, +1 ,027122986826984e+00f +1 .026738673647482e+00f, +1.026261663878092e+00f, +1.025701002415063e+00f +1.025061777648234e+00f, +1.024353980976701 e+OOf, +1.023582385618774e+00f +1.022756514615106e+00f, +1 .021880604350422e+00f, +1.020963871317665e+00f +1.020009139549275e+00f, +1.019027285501251 e+OOf, +1.018019442784231 e+OOf +1.016996499560845e+00f, +1.015957433206324e+00f, +1.014923441259795e+00f +1.013915946100629e+00f, +1.013047565149327e+00f, +1.01221613036561 Oe+OOf +1.01 1044869639164e+00f, +1.009914592130044e+00f, +1 ,008824888092573e+00f +1.007773858455400e+00f, +1.006761700412993e+00f, +1 .005786648810854e+00f +1.004848753962734e+00f, +1.003946083413733e+00f, - -

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+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf

+0 OOOOOOOOOOOOOOOe+OOf +0 OOOOOOOOOOOOOOOe+OOf, +0 OOOOOOOOOOOOOOOe+OOf +0.000000000000000e+00f, +0.000000000000000e+00f, +0.000000000000000e+00f

A numerical example of windowing function w ₉₆₀ , for frame size N=960, is herewith provided (1920 samples):

-1 .596869453315999e-04, -3.021 153494057143e-04, -4.142323860121641 e-04, -5.058484031439142e-04, -5.867737487939294e-04, -6.666034929771656e-04,

-7.499813122143762e-04 -8.366504004139794e-04 -9.273096237893370e-04, -1.023773491532728e-03 -1.126635355725494e-03 -1.235247794937702e-03, -1.3496278535106066-03 -1.470492941694331 e-03 -1.598145399394582e-03, -1.732450700747454Θ-03 -1.873473391018495e-03 -2.0214931331 13876e-03, -2.176721793926887e-03 -2.339292362082021 e-03 -2.509298051958086e-03, -2.686801873698675e-03 -2.872008069419264e-03 -3.065254982136409e-03, -3.266686736002951 e-03 -3.476256385086407e-03 -3.694123506258334e-03, -3.920651988943251e-03 -4.155963816705528e-03 -4.399940204176431e-03, -4.652692827862344e-03 -4.914563787665504e-03 -5.185653529531935e-03, -5.46582 495567872e-03 -5.755 72503953251 e-03 -6.054056765330685e-03, -6.362583863449497e-03 -6.680623380533122e-03 -7.008304674718039e-03, -7.346013465075499e-03 -7.693816924650567e-03 -8.051465262788033e-03, -8.419001443263380e-03 -8.796760749750191 e-03 -9.184751507972289e-03, -9.582657230262597e-03 -9.990503073705982e-03 -1.040865597419619e-02, -1.0837161 13754551Θ-02 -1.1275741 17 38621 e-02 -1.172444049589271 e-02, -1.218362542421523e-02 -1.265334152129685e-02 -1.313331245904876e-02, -1.362355747138776e-02 -1 .412438986522702e-02 -1.463576680723352e-02, -1.515729094197161 e-02 -1.568889620430290e-02 -1.623084814367250e-02, -1.678306524600559Θ-02 -1.7345120893661 17e-02. -1.791696516251699Θ-02, -1.849892453107246Θ-02 -1.909097368362797e-02 -1.969273308274951 e-02, -2.030414376031 160e-02 -2.09254671 1 168754e-02 -2.155660557689242e-02, -2.219710359529686e-02 -2.284684792818856e-02 -2.350606779320675e-02, -2.417463749934085e-02 -2.485207716234951 e-02 -2.553826580898371 e-02, -2.623344061522424e-02, -2.693746704328349e-02, ^■ 2.764983997782559e-02, -2.837043858580717e-02, -2.909952486193998e-02, ^■ 2.983695676902939e-02, -3.058218787751 172e-02, -3.133504629493832e-02, ^■ 3.2095736417421 17e-02, -3.286409853195186e-02, -3.363960728361352e-02, ^■ 3.442207835192332e-02, -3.521 66997276568e-02, -3.600820974457969e-02 _: ^■ 3.68 1 193 7395579e-02, -3.762042487257546Θ-02, -3.843601741746971 e-02 •3.925772507976759e-02, -4.008494689009130e-02, -4.091746034850161 e-02 ^■ 4.175542154425013e-02, -4.259863408038441 e-02, -4.344654894948344e-02 ^■ 4.4298992875231 13e-02, -4.515616616948602e-02, -4.601786724624048e-02 ^■ 4.688349274164030e-02, -4.775281564045485e-02, -4.862598509282498e-02 ^■ 4.950274425624717e-02, -5.038242975874127e-02, -5.126474204655663e-02 ^■ 5.214974857230398e-02, -5.303718358615044e-02, -5.392644753556616e-02 ^■ 5.481729406855359e-02, -5.57098305756521 Oe-02, -5.66038431 1081047e-02 ^■ 5.749879655200370e-02, -5.839451321393058Θ-02, -5.9291 16747072080e-02 ^■ 6.018848884279070e-02, -6.108576415855094e-02, -6.19826820251 1926e-02 ^■ 6.287933502598060e-02, -6.377542359879805Θ-02, -6.46702554807 184e-02 ^■ 6.556352962710620e-02, -6.645533804325808e-02, -6.734542216184526e-02 ^■ 6.8233171 3969737e-02, -6.91 1832800881393e-02, -7.000099017198280e-02 ^■ 7.088091415536939e-02, -7.175751581450866e-02, -7.26305701 1354321 e-02 ^■ 7.350020923324756e-02, -7.4366181 16832702Θ-02, -7.522784961377151 e-02 ^■ 7.608493286734087e-02, -7.693750413779520e-02, -7.778525942347714e-02 ^■ 7.862751920094575e-02, -7.946399769633587e-02, -8.029480247839879e-02 ^■ 8.1 1 1965623403654e-02, -8.193789640255647e-02, -8.274924535373614e-02 ^■ 8.355381 170261278e-02, -8.435134905678848e-02, -8.514125464087215e-02 ^■ 8.592328028689659e-02, -8.669753354886602e-02, -8.746379123238275e-02 ^■ 8.822149831804039e-02, -8.897043074644968Θ-02, -8.971069341834263e-02 ^■ 9.044201341 183719e-02, -9.1 16374329727489e-02, -9.187564084638347e-02 ^■ 9.257786640497777e-02, -9.327014538321736e-02 -9.395176975347193e-02 -9.462246851310178e-02,

-9.528240981433814e-02 -9.593137735886889e-02 -9.656876884228789e-02,

-9.719438533739366e-02 -9.780843257659243e-02 -9.841075978146442e-02,

-9.900085534600200e-02 -9.957851303827886e-02 -1.001438325530028Θ-01 ,

-1.006965654660066e-01 -1.012361 165314596e-01 -1.017622262660318e-01 ,

-1.022749718234132e-01 -1.027741036495644Θ-01 -1.032590485916072e-01 ,

-1.037296416381658e-01 -1.041861222641 1 9e-01 -1.046283222746241 e-01 ,

-1.050556674062305Θ-01 -1.054680247057000e-01 -1.058657014776354e-01 ,

-1.06248574752641 1 e-01 -1.066160875985523Θ-01 -1.069680384993124e-01 ,

-1.073045851468520e-01 -1.076255384835563e-01 -1.079303623240916e-01 ,

-1.082188709931267e-01 -1.084912299471 198e-01 -1.087472718159103e-01 ,

-1.089864959432877e-01 -1.092087422379003Θ-01 -1.094141900597718e-01 ,

-1.096026598727515e-01 -1 .097736146613313Θ-01 -1.099268894481712e-01 ,

-1.100626919008929Θ-01 -1.101808861640070e-01 -1.102809952914802e-01 ,

-1.103629709312858e-01 -1 .104271876052675e-01 -1.104735471815890Θ-01 ,

-1.105014789286600e-01 -1.105108362290460e-01 -1.105018987914554e-01 ,

-1.104745246005427e-01 -1.104281465492726Θ-01 -1.10362546401 1331 e-01 ,

-1.102778601442104e-01 -1.101739218186236Θ-01 -1.100502639286607e-01 ,

-1.099067826823344e-01 -1.097437360338336e-01 -1.095610353779043Θ-01 ,

-1.093582462693348e-01 -1.09135312557251 1 e-01 -1.08892553 745051 e-01 ,

-1 .086298673699289e-01 -1.083467335729228Θ-01 -1.080429791821427e-01 ,

-1.077187842899125e-01 , -1 .073739938306107e-01 -1.070081 184450032e-01 ,

-1.066210413471652e-01 -1.062130155324388e-01 -1.057838954855888e-01 ,

-1.053331389663558e-01 -1.048606145834788e-01 -1.043666017588026e-01 ,

-1.038509823636319e-01 -1.033132401525343e-01 -1.027532566706549e-01 ,

-1.021713124102434e-01 _; -1.015673163469357e-01 -1.009408032844788e-01 ,

-1.002916459024019e-01 -9.962005506126154e-02 -9.892587447037898e-02, -9.820858231 161539e-02 -9.746803229469267Θ-02 -9.670445315437526Θ-02, -9.591769653314886e-02 -9.510723623306666e-02 -9.427290387369003e-02, -9.341489003828900e-02 -9.253303383231506e-02 -9.162682426346705e-02, -9.069614104466586e-02 -8.974125216128212e-02 -8.876201954716677e-02, -8.775790063882420e-02 -8.6728776891 19252Θ-02 -8.56749516153351 Oe-02, -8.459627389188201 e-02 -8.349213839083708e-02 -8.236235749389448Θ-02, -8.120716401834208e-02 -8.002639902061687e-02 -7.881950797067065e-02, -7.758626878445929e-02 -7.632679536516856e-02 -7.504088747769501 e-02, -7.372803129789816e-02 -7.238806162166744e-02 -7.1021 1621 1954443e-02, -6.962720872934396e-02 -6.820576796149519e-02 -6.675671622392269e-02, -6.528023606863240e-02 -6.37761 1429172260e-02 -6.22437461 1498080e-02, -6.068292908100376e-02 -5.909386001558149e-02 -5.747627430502456e-02, -5.582945016301614e-02 -5.415316322402774e-02 -5.244768784057990e-02, -5.071287102956145e-02 -4.894812724598650e-02 -4.715324945932696e-02, -4.532841381541464Θ-02 -4.3473471 12195197e-02 -4.158794238398249e-02, -3.967162909873025e-02 -3.772461300253332e-02 -3.574664020854284e-02, -3.373712599806719e-02 -3.169587609244436e-02 -2.962308489442885e-02, -2.751857552683018e-02 -2.538179830690266e-02 -2.321255714428171 e-02, -2.101 100771772769e-02 -1.877689096555516e-02 -1.650953674438219e-02, -1.4208710831 18669Θ-02 -1.187461378850388Θ-02 -9.507008996713286e-03, -7.105225746748432e-03 -4.669099247423082e-03, -2.198970714838444e-03, +3.052506679844099e-04, +2.844096748870384e-03, +5.417800316971459e-03, +8.026287894631955e-03, +1.066976124794342e-02, +1.334869917032908e-02, +1 .606331521084708e-02, +1.881355950582949e-02, +2.159970134663663e-02, +2.442232147839131 e-02, +2.728156010437695e-02, +3.017712138573929e-02, +3.310912443052597e-02, +3.607810469851272e-02, +3.908419391833869e-02, +4.212712008909140e-02, +4.520702759803914e-02, +4.832447852355055e-02, +5.1479661 18159461e-02 +5.467238802204326e-02 _: +5.790284080082958e-02 +6.1 17156717220185e-02 +6.447866054615346e-02 _; +6.782376212130986e-02 +7.120692575100979e-02 +7.462862199422061 e-02 +7.808896236959446e-02 +8.158769578329894e-02 +8.512490568723846e-02 +8.870101069974845e-02 +9.231608110264714e-02 +9.596983987496970e-02 +9.966236893806717e-02 +1.033941063338643e-01 +1.071650779014335e-01 +1.109748994439795e-01 +1.148235927259266e-01 +1.1871 15850305241 e-01 +1.226388834300689e-01 +1.266050731283809e-01 +1.306101067250375e-01 +1.346543064817742e-01 +1.387376249190578e-01 +1.428596447589721 e-01 +1.470203170721216e-01 +1.512199592605080e-01 +1.554584725339102e-01 +1.597353438179875e-01 +1.640504702561671 e-01 +1.684041609371527e-01 +1.727963503982179e-01 + 1.772265953627290e-01 +1.816947894623263e-01 +1.86201 1624788949e-01 +1.907455489457267e-01 + 1.953273880886783e-01 + 1.999464539304762e-01 +2.046028558464008e-01 +2.092963206850239e-01 +2.140261955702662e-01 +2.187922274259156e-01 +2.235945635254679e-01 +2.284329840640989e-01 +2.333068981094213e-01 +2.382160219461597e-01 +2.431603785542250e-01 +2.481396077816722e-01 +2.531529721334063e-01 +2.582001739497298e-01 +2.632813508034234e-01 +2.683961570569586e-01 +2.735437630266799e-01 +2.787237550375912e-01 +2.839361392493072e-01 +2.891805257722012e-01 +2.944561228780392e-01 +2.99762425517781 1 e-01 +3.050992195689690e-01 +3.104659867604553e-01 +3.158619077906196e-01 +3.212865260033979e-01 +3.267397428773339e-01 +3.322210551086769e-01 +3.377295561726898e-01 +3.432647335893443e-01 +3.488264676990591 e-01 +3.544142074632647e-01 +3.600269656734629e-01 +3.656640377499646e-01 +3.713250061671226e-01 +3.770091 190097 18e-01 +3.827152968157059e-01 +3.884429045609680e-01 +3.941917466962361 e-01 +3.99961 1859760947e-01 +4.057501420991821 e-01 +4.1 15579429637846e-01 +4.173843265025887e-01 +4.232286297916428e-01 +4.290897820858789e-01 +4.349669624916473e-01 +4.408596056709005e-01 +4.467668696607166e-01 +4.526876397402144e-01 +4.58621 1456831769e-01 +4.645669547719424e-01 +4.705242008503956e-01 +4.764915897192549e-01 +4.824682887794052e-01 +4.88453925483 315e-01 +4.944476636179056e-01 +5.004481977619742e-01 +5.064545550235134e-01 +5.124661018959751 e-01 +5.184818933365749e-01 +5.245006748662190e-01 +5.305215796893860e-01 +5.365441251369431e-01 +5.425674372882107e-01 +5.48590250685281 1 e-01 +5.5461 16774259739e-01 +5.606312044701524e-01 +5.666478652857275e-01 +5.726602391687222Θ-01 +5.786672646386708e-01 +5.846682453966959e-01 +5.906621680078100e-01 +5.966477035050948e-01 +6.026239249993590e-01 +6.085903032056262e-01 +6.145458904162185e-01 +6.204893201838491e-01 +6.264196163765745e-01 +6.323361944662236e-01 +6.382380808188793e-01 +6.441239094712802e-01 +6.49992631921 1775e-01 +6.558435212204078e-01 +6.616755522321309e-01 +6.674874032292857e-01 +6.732781 108033908e-01 +6.790470657166684e-01 +6.847932667399612e-01 +6.905153235578969e-01 +6.962122668445681e-01 +7.018835463513400e-01 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+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00 _; +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000Θ+00 H-O.OOOOOOOOOOOOOOOe+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000Θ+0Ο H-O.OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00. +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, H-O.OOOOOOOOOOOOOOOe+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00. +0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _: +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00 +0.000000000000000e+00 _; +0.000000000000000Θ+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000Θ+00 +0.000000000000000Θ+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0. OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00 +0.000000000000000Θ+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+OO +0.000000000000000Θ+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00. +0.000000000000000e+00. +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.0000000000000006+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO.

7.2 Examples associated to Figs. 26 to 30

A numerical example of windowing function w ₄₀ , for frame size N=40, is herewith provided by the 80 entries for n = 0...2N-1 . As explained above, the last values may be zeros.

+9.959086585790517e-04, +3.819056787237678e-03, +9.540832613229890e-03,

+1 .921659800166160e-02, +3.382719081038548e-02, +5.424831667522354e-02,

+8.12077766877561 Oe-02, +1 .152171887125930e-01 , +1 .564942331034909e-01 ,

+2.049363422022628e-01 , +2.601 166575816199e-01 , +3.212814164616093e-01 , +3.873472997948746Θ-01 , +4.569497078592333e-01 , +5.285192958868393e-01 , +6.003522489375573e-01 , +6.706896380227332e-01 , +7.378044458510402e-01 , +8.000925313431716e-01 , +8.561409184410547e-01 , +9.048272294524792e-01 , +9.453685031730190e-01 , +9.773507430600533e-01 , +1.000800872826561 e+00, +1 .0161715901 2097e+00, + .024315247630982e+00, +1.026415431432931 e+00, +1 .023858366571912e+00, +1.018135705524407e+00, +1.0 0794822557756e+00, +1 .003406509762925e+00, +9.967831265986109e-01 , +9.920995520917141 e-01 , +9.8922069428 6891 e-0 , +9.879658322200813e-01 , +9.881273531631907e-01 , +9.894805541465801 e-01 , +9.917849916000535e-01 , +9.947847580943504e-01 , +9.9821 19669301 160e-01 , + .001791235858836e+00, +1.005242583245485e+00, +1 .008283053756130e+00, + .01063128 038659e+00, +1.0120 5300253356e+00, +1 .012180753005270e+00, +1.010896765282633e+00, +1.007963362035220e+00, +1 .003227255072391 e+00, +9.966050551498514e-01 , +9.868284225039941 e-01 , +9.731250287581631 e-01 , +9.540636479502398e-0 , +9.283864275822276e-01 , +8.950916858157935e-01 , +8.534769362643825e-01 , +8.032090930429980e-01 , +7.444735201251689e-01 , +6.780787033699449e-01 , +6.053970453856138e-01 , +5.282077505750667e-01 , +4.486552956056635e-01 , +3.691875990296312e-01 , +2.924566408966777e-0 , +2.210718537 10463e-01 , +1.573148583944309e-01 , +1 .030525757797768e-01 , +5.982732244758054e-02, +2.871831923385133e-02, +9.683884928956490e-03, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w ₈₀ , for frame size N=80, is herewith provided by the 160 entries for n = 0...2N-1. As explained above, the last values may be zeros.

+6.143388180964179e-04, +1.489582832987000e-03, +2.884104959764029e-03,

+4.934298832466617e-03, +7.7791304641549 5e-03, +1.154910606525086e-02, +1.637155619860352e-02, +2.2371 16158648752e-02, +2.966159685753317e-02, +3.835663329277230e-02, +4.855610986150206Θ-02, +6.035055738891727Θ-02, +7.382288203064732e-02, +8.90356368721 1 1 19e-02, +1.060356225286319e-01 , +1.248534855777947e-01 , +1 .454931890869180e-01 , +1.679435556337752e-01 , +1.92172862263441 1 e-01 , +2.181238261985594e-01 , +2.457259744642953e-01 , +2.748839432649996e-01 , +3.054824712370942e-01 , +3.373873799614014e-01 , +3.704415932452488Θ-01 , +4.044749630814483e-01 , +4.393004362003260Θ-01 , +4.747225454237193e-01 , +5.105341492548225Θ-01 , +5.465201916422433e-01 , +5.824658100332457Θ-01 , +6.181452662624718e-01 , +6.53341 1462740817e-01 , +6.878367295965062e-01 , +7.214176027060971 e-01 , +7.538887973483771 e-01 , +7.850546571907628e-01 , +8.147397447696774e-01 , +8.427819363777799e-01 , +8.690376742017057Θ-01 , +8.933935477349644e-01 , +9.157483563218768e-01 , +9.360270196617569e-01 , +9.541731 142261065e-01 , +9.701635474343885e-01 , +9.840036439809510e-01 , +9.957199420334376e-01 , +1.005374268639838e+00, +1 .013046655758663e+00, +1 .018843380560658e+00, +1.022896948293643e+00, +1.025355286710874e+00, +1.026382881625701 e+00, +1.026155530733488e+00, + 1.024853974580724e+00, +1 .022664602721801 e+00, +1.019779396547454e+00, + 1 .016391686789653e+00, +1 .012697033320358e+00, +1.008885191761748e+00, +1 ,005378742804807e+00, +1 .001563778373068e+00, +9.982531564931281e-01 , +9.954346644968789e-01 , +9.930950268060122e-01 , +9.91217091 1359961 e-01 , +9.897805192546195e-01 , +9.887624937408933e-01 , +9.881383235740961 e-01 , +9.878819413827574e-01 , +9.879662130250981 e-01 , +9.883630508181326e-01 , +9.890434070785485e-01 , +9.899772316163624e-01 , +9.91 1334564321237e-01 , +9.924800441092685e-01 , +9.939841207305906e-01 , +9.956121471675398e-01 , +9.973300590248015e-01 , +9.991033633647473e-01 , +1.000897441314013e+00, +1.002677088643863e+00, +1 .004407190937699e+00, +1.006052289109999e+00, + 1.007576934100958e+00, +1.008945862447015e+00, +1.010124241309341 e+00, +1 .01 1077969726137e+00, +1 .0 1773962181442e+00, +1.012180362866919e+00, +1.0 2266707295288e+00, + .012004064757857e+00, + .011365223023975e+00, +1.01032499685 905e+00, +1.008860731864438e+00, +1.006952983357691 e+00, + 1 .004586273379809Θ+00, +1.001749900308864e+00, +9.9843866321 16344e-01 , +9.94650033290 397e-0 , +9.895756853352 72e-01 , +9.838303 27859 96e-0 ,

+9.769999155793757Θ-01 +9.689141 159310996e-01 +9.594038121639412e-01 +9.483086322505029Θ-01 +9.354860218216989Θ-01 +9.208101305030523e-01 +9.041732260327581 e-01 +8.854882249661838e-01 +8.646864947605046e-01 +8.41723746771 1 145Θ-01 +8.165875713256009e-01 +7.892986353718001 e-01 +7.599171886893816e-01 +7.28547451541 1827e-01 +6.953282935906302Θ-01 +6.604334017809461 e-01 +6.240661431421666e-01 +5.864461424698465e-01 +5.478160663871 147e-01 +5.084499758302218e-01 +4.686361426418982e-01 +4.286789889246253e-01 +3.889032719013045e-01 +3.496431418636314e-01 +3.1 12360816586544e-01 +2.740128472224535e-01 +2.382847225401666e-01 +2.043379825955252Θ-01 +1 .724305860483632e-01 +1.427939789949265Θ-01 +1.156385879569741e-01 +9.1 15821766571995e-02 +6.952749039054593e-02 +5.088975408628225e-02 +3.533430192568954Θ-02 +2.286680405144430e-02 +1.338005016725895Θ-02 +6.640506529168652Θ-03, +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00. A numerical example of windowing function w _lzQ , for frame size N=120, is herewith provided by the 240 entries for n = 0...2N-1 , As explained above, the last values may be zeros.

+5.087227626168386e-04 +9.959086585790517e-04 +1.682208006328800e-03 +2.609697259047744e-03 +3.8 9056787237678e-03 +5.3493 9592933909e-03 +7.243906383895192e-03 +9.540832613229890e-03 +1.227637642543709e-02 +1.548950238899404e-02 +1.921659800166160e-02 +2.349369619441617e-02 +2.835199581667961 e-02 +3.382719081038548e-02 +3.994939538719628e-02 +4.674775238543380e-02 +5.424831667522354e-02 +6.247770776443612e-02 +7.145835917501348e-02 +8.12077766877561 Oe-02 +9.174400412319896e-02 +1 .030764959637497e-01 + 1 .152171887125930Θ-0Ί +1.281665713944242e-01 +1 .41926438 068653e-01 + 1 .564942331034909e-01 +1.718593189799504e-01 +1 .880134254543744Θ-01 +2.049363422022628e-01 +2.226123055761096e-01 +2.410151242797736e-01 +2.601 166575816199e-01 +2.798871008989962e-01 +3.002880135563586e-01 +3.212814164616093e-01 +3.428208463088390e-01 +3.648596557863134e-01 +3.873472997948746e-01 +4.102294951869188e-01 +4.334494534591082e-01 +4.569497078592333e-01 +4.806696403251 166e-01 +5.045473815014847e-01 +5.285192958868393e-01 +5.525 96099932443e-01 +5.764872452085427e-01 +6.003522489375573e-01 +6.240509872809882e-01 +6.475182586093196e-01 +6.706896380227332e-01 +6.935029068990036e-01 +7.158927516396895e-01 +7.378044458510402e-01 +7.591787241845952e-01 +7.799586608897265e-01 +8.000925313431716e-01 +8.195318652294690e-01 +8.382288957404715e-01 +8.561409184410547e-01 +8.732316951214179e-01 +8.89470202217083 e-01 +9.048272294524792e-01 +9.192736375782965e-01 +9.327940405054362e-01 +9.453685031730190e-01 +9.569883933538136e-01 +9.676486424195593e-01 +9.773507430600533e-01 +9.86 027831072527e-01 +9.939122412655677e-01 +1 .000800872826561 e+00, +1.00678781 1971719e+00,

+1 .01 901269172423e+00, +1 .01617 5901 12097e+00, +1.019636414864842e+00, +1.022336613864005e+00, +1 .024315247630982e+00, +1.025621299895396e+00, +1.026303439275662e+00, +1.026415431432931 e+00, +1.026007933174836e+00, +1.025137435167917e+00, +1.023858366571912e+00, +1.022226936424625e+00, + 1.020300550334848e+00, +1.018135705524407e+00, +1.015792146756340e+00, +1.013325966774524e+00, +1.010794822557756e+00, +1.008265131568879e+00, +1.006046874304407e+00, +1.003406509762925e+00, +1.000977398831985e+00, +9.987704535700208e-01 , +9.967831265986109e-01 , +9.9501 18905889862e-01 , +9.934523971504882e-01 , +9.920995520917141 e-01 , +9.909475998606236e-01 , +9.899902426925508e-01 , +9.892206942816891 e-01 , +9.886318043013834e-0 , +9.882160904669929e-01 , +9.879658322200813e-01 , +9.878730767519871 e-01 , +9.879296932443894e-01 , +9.881273531631907e-01 , +9.884575535474619e-01 , +9.8891 15869213529e-01 , +9.894805541465801 e-01 , +9.901553455166457e-01 , +9.9092665629 3843e-0 , +9.9 78499 6000535e-0 , +9.927206838643636e-0 , +9.937239208721489e-01 , +9.947847580943504e-01 , +9.958931493776203e-0 , +9.970389567617592e-01 , +9.9821 966930 160e-01 , +9.994020338838508e-01 , +1.000598323893564e+00, +1.001791235858836e+00, +1.002969837054169e+00, + 1.0041237863971 11e+00, +1.005242583245485e+00, +1.006315717067918e+00, + .007332693127034e+00, +1 .008283053756130e+00, +1.009156423082384e+00, +1.009942535308151 e+00, +1.010631281038659e+00, +1.01 1212744622770e+00, +1.01 1677230257499e+00, +1 .012015300253356e+00, +1.012217779097186e+00, +1.012275790821 109e+00, +1.012180753005270e+00, +1.01 1924425888915e+00, +1.01 1498917644724e+00, +1 ,010896765282633e+00, +1.0101 10965619444e+00, +1.009135094671655e+00, +1 ,007963362035220e+00, +1.006590756505588e+00, +1.0050131 15379014e+00, +1 .003227255072391 e+00, +1.001231060075500e+00, +9.990235555436858e-01 , +9.966050551498514e-01 , +9.9398947061 13089e-01 , +9.904539200261 149e-01 , +9.868284225039941 e-01 , +9.827716736909488e-01 , +9.782206672373213e-01 , +9.731250287581631 e-01 , +9.674323528812744e-01 , +9.610947043524248e-01 +9.540636479502398Θ-01 +9.462952991 190324e-01 +9.377489107516087e-01 +9.283864275822276e-01 +9. 81762606422500e-01 +9.070861558801854Θ-01 +8.950916858157935e-01 +8.821696237804294e-01 +8.683025287048570e-01 +8.534769362643825e-01 +8.376852006833730e-01 +8.209275259764013e-01 +8.032090930429980Θ-01 +7.845450482523652e-01 +7.649554851899686e-01 +7.444735201251689e-01 +7.231348066419057Θ-01 +7.009860555207412e-01 +6.780787033699450e-01 +6.544686506489734e-01 +6.302212149502727e-01 +6.053970453856138e-01 +5.800715766089168e-01 +5.543 29276657669Θ-01 +5.282077505750727e-01 +5.018369724442092Θ-01 +4.752902962082383e-01 +4.486552956056652e-01 +4.220281 1 18338883Θ-01 +3.955057965950340e-01 +3.691875990296320e-01 +3.431732847389720e-01 +3.175633015043183e-01 +2.924566408966782e-01 +2.679463783886042e-01 +2.441231331518492e-01 +2.2107185371 10466e-01 +1.988719153219592e-01 +1.775967625327044e-01 +1 .573148583944310Θ-01 +1.380903364946733Θ-01 +1 .199837497591550e-01 +1.030525757797769Θ-01 +8.73508501 1789188e-02 +7.29281 1584897502e-02 +5.982732244758056e-02 +4.808178837444506Θ-02 +3.771 135297837851 e-02 +2.871831923385135e-02 +2.108352028641225e-02 + 1 .476289412849005e-02 +9.683884928956495e-03, +5.642168789286858e-03 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w ₁₆₀ , for frame size N=160, is herewith provided by the 320 entries for n = 0...2N-1. As explained above, the last values may be 0.

+4.595886345493055e-04 +7.919323614002698e-04 +1.227927169310031 e-03 +1 .783653266717233e-03 +2.479549413444207e-03 +3.329799454594261 e-03 +4.353535478916468e-03 +5.564965156664018e-03 +6.986108359341676e-03 +8.629882322202329e-03 +1.051343406844975e-02 +1.265082642578719e-02 +1 .506090447446532e-02 +1.775591229287213e-02 +2.075475983187825e-02 +2.406813715401559e-02 +2.771207863541604e-02 +3.169933248543932e-02 +3.604609640533871 e-02 +4.076128638095439e-02 +4.586038120884381 e-02 +5.135136676471998e-02 +5.724780220726930e-02 +6.355854744461048e-02 +7.029450733434550e-02 +7.745987198268531 e-02 +8.506635369887924e-02 +9.31 1641620512773e-02 +1.016162955027316e-01 +1.105690806271684e-01 +1.199789286645804e-01 +1 .298417294090302e-01 +1.401623800497866e-01 +1 .509371564593891 e-01 +1.62 632295622287e-01 +1.738354123649302e-01 +1.859520359191026e-01 +1.985008828937603e-01 +2.1 14778554475382e-01 +2.248732557074316e-01 +2.386763947872762e-01 +2.528729453658238e-01 +2.674547009618951 e-01 +2.824031465430401 e-01 +2.977050145264297e-01 +3.133419120661713e-01 +3.292976696294886e-01 +3.455490160824131 e-01 +3.620795045342974e-01 +3.788648665671841 e-01 +3.958851576591690e-01 +4.131 143794748322e-01 +4.305308301005456e-01 +4.481076715576617e-01 +4.658227790464821 e-01 +4.836466393241829e-01 +5.015564851667653e-01 +5.195228071 176610e-01 +5.375197039843709e-01 +5.555183841040963e-01 +5.734957812557457e-01 +5.914186654649489e-01 +6.092622887527459e-01 +6.269981 160888640e-01 +6.446002007776794e-01 +6.620384583071039e-01 +6.792906550106088e-01 +6.963256426589250e-01 +7.131 194393772130e-01 +7.296469905863920e-01 , +7.458864594794676e-01 , +7.618094719403713e-01 , +7.773958448163656e-01 , +7.926208751337592e-01 , +8.074666387233143e-01 , +8.219101564897180e-01 , +8.359343163788637e-01 , +8.495180470826319e-01 , +8.626485837105826e-01 , +8.753083234662220e-01 , +8.874884715160425e-01 , +8.991737724042251 e-01 , +9.103527429187326Θ-01 , +9.210144133066616e-01 , +9.311556192776946Θ-01 , +9.407644740241826e-01 , +9.498382236872068e-01 , +9.583732599601223e-01 , +9.663690412284377e-01 , +9.738235617865406e-01 , +9.807442506043361 e-01 , +9.871297972052695e-01 , +9.929872268444632e-01 , +9.983241398929388e-01 , +1.003150760219063e+00, +1.007473713377193e+00, +1.01 1309151636166e+00, +1.014666681083198e+00, +1.017563337333301 e+00, +1.020014681326785e+00, +1.022039872150903e+00, +1.023654257342442e+00, +1.024881624147540e+00, +1.025739288978437e+00, +1.026250709375593e+00, + 1.026436666375082e+00, +1.026320857404224e+00, +1.025922917798664e+00, +1.025269979527211 e+00, +1.024382188798244e+00, +1.023284940887058e+00, +1.022000829220643e+00, +1.020555973231408e+00, +1.018971390778550e+00, +1.017275179369116e+00, +1.0154891291 1 1694e+00, +1.013639356938881 e+00, +1.011747750709711 e+00, +1.009840844244693e+00, +1.007939764480188e+00, +1.006407400915498e+00, +1.004374825095777e+00, +1.002469814737132e+00, +1.000689073754539e+00, +9.990346001249977e-01 , +9.975024904153303e-01 , +9.960941547576162e-01 , +9.948051243621099e-01 , +9.936362728142866e-01 , +9.925826537087717e-01 , +9.916447007525191 e-01 , +9.908170758245324e-01 , +9.900998445795673e-01 , +9.894873685512386e-01 , +9.889794323427195e-01 , +9.885701787626714e-01 , +9.882591911282058e-01 , +9.880404423341358e-01 , +9.879133688360181 e-01 , +9.878718098237022e-01 , +9.879150762106034e-01 , +9.88036893884661 Oe-01 , +9.882364564839506e-01 , +9.885073687439192e-01 , +9.888487088987707e-01 , +9.892539488627546e-01 , +9.897220412447528e-01 , +9.902463287269285e-01 , +9.908256340476208e-01 , +9.91453 811725067e-01 , - -

+9.921276814881759e-01 , +9.928422499725458e-01 , +9.935955098307742e-01 , +9.943804814776256e-01 , +9.951957244449919e-01 , +9.960341878404958e-0 , +9.968943831870675e-01 , +9.977692009836100e-01 , +9.986571 134591464e-01 , +9.995509738170480Θ-01 , +1 .000449227898040e+00, +1.001344692310058e+00, +1 .002235786606954e+00, +1 .0031 15291715261 e+00, +1.003981602446902e+00, +1 .004827468041713e+00, +1 .005651275972376e+00, +1.006445772052972e+00, +1 ,007209352772459e+00, +1.007934783656087e+00, +1.008620496650569e+00, +1 .009259314290145e+00, +1.009849742422788e+00, +1.010384692193296e+00, +1 .010862783160582e+00, +1.01 1277044709547e+00, +1.01 1626247430694e+00, +1 .01 1903571699736e+00, +1 .012107954864219e+00, +1 .012232755709885e+00, + 1.012277089047072e+00, +1 .0122345051 14778e+00, +1.012104319978655e+00, +1.01 1880293122688e+00, +1 .01 1561972516341 e+00, +1.01 1 143373963981 e+00, +1.010624321020038e+00, +1 .009999148545101 e+00, +1.009268031808824e+00, + 1.008425698479647e+00, +1 .007472774447058e+00, +1.006404483571931 e+00, +1 .005222003295591 e+00, +1.003921 160689206e+00, +1.002503762756151 e+00, + 1 .000966332772540e+00, +9.9931 1400741 1373e-01 , +9.975362702189898e-01 , +9.956442306333592e-0 , +9.936333924912825e-01 , +9.90867748036 242e-01 ,

+9.882326326262749e-01 +9.853620567056602e-01 +9.822305093671991 e-01 +9.788185853162172e-01 +9.751026333215268e-01 +9.710631852370086e-01 +9.666759668947944e-01 +9.619242192293307e-01 +9.567841986369235e-01 +9.512394303101863e-01 +9.452700238623795e-01 +9.388615698236068e-01 +9.319946435581 106e-01 +9.246592033932568e-01 +9.168383396399868e-01 +9.085218034087421 e-01 +8.99696701 1299613e-01 +8.903562054918268e-01 +8.804877931535187e-01 +8.700884209228057e-01 +8.591492134848259e-01 +8.476686394755906e-01 +8.356428970797861 e-01 +8.230753889817990e-01 +8.099649296155544e-01 +7.963204506324437e-01 +7.821460539775005e-01 +7.674541821769616e-01 +7.522563457568547e-01 +7.365702052057368e-01 - -

+7.204090552899627e-01 +7.037975107157410e-01 +6.867542812151 157e-01 +6.69304 888771051 e-01 +6.514710959179395e-01 +6.33285483282091 e-01 +6.147685389896460e-01 +5.959553778639692Θ-01 +5.768737955463938e-01 +5.575534287167304e-01 +5.380320138068979e-01 +5.183454027643563e-01 +4.985259415650634e-01 +4.786156067459849e-01 +4.586473038370941 e-01 +4.386643656872842Θ-01 +4.187046888325280Θ-01 +3.988123056192917e-01 +3.790262923635886Θ-01 +3.593914828698096e-01 +3.399474132903109e-01 +3.207392420889753e-01 +3.018061 1 13177048e-01 +2.831905952786929e-01 +2.649288369241889e-01 +2.470608550624402Θ-01 +2.296201 1 19084317e-01 +2.126433716126151 Θ-01 +1.961601816145380Θ-01 +1.802035203864437e-01 +1 .647996883470626e-01 +1 .499787548077656Θ-01 +1.35764352299161 1 e-01 +1.221842534547464e-01 +1.092601994264172e-01 +9.701788451015501e-02

+8.547680283183663e-02 +7.465976378295235e-02, +6.458254322751883e-02 +5.526281 189874138e-02 +4.670976978373095e-02, +3.893244425578719e-02 +3.192976013776996e-02 +2.569810636390756e-02, +2.022259265088492e-02 + 1.548317776486452e-02 + 1.144924909653903e-02 , +8.076482660383199e-03 +5.300044080947794e-03 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w ₂₄ o, for frame size N=240, is herewith provided (480 samples):

+4.090106504820579e-04, +6.143388180964179e- ^■ 04, +8.571759876954877e-04,

+1.147015057857495e-03, +1 .489582832987000e- -03, +1.889770382231583e-03,

+2.353000800169909e-03, +2.884104959764029e- -03, +3.488213786635855e-03,

+4.170040431489613e-03, +4.934298832466617e- ^■ 03, +5.787076505403503e-03,

+6.73381 1743137561 e-03, +7.779130464154915e ^■ 03, +8.927044958757816e-03,

+1 .018202888968871 e-02, +1.154910606525086e ^■ 02, +1.303349217699797e-02,

+1 .463951288465963e-02, +1.637155619860352e ^■ 02, +1.823455383898077e-02,

+2.023309488998589e-02, +2.2371 16158648752e -02, +2.465237348403478e-02,

+2.708101935270475e-02, +2.966159685753317e ^■ 02, +3.239884850877327e-02,

+3.529601774976465e-02, +3.835663329277230e -02, +4.158447932459513e-02,

+4.498322421745353e-02, +4.855610986150206e -02, +5.230596475016741 e-02,

+5.623624576084146e-02, +6.035055738891727e -02, +6.465186317477950e-02,

+6.914195749790462e-02 , +7.382288203064732e -02, +7.869709331995660e-02,

+8.376761638427657e-02, +8.90356368721 1 1 18e -02, +9.450199243028472e-02,

+1 .001680 93006426e-01 , +1 .060356225286319e -01 , +1.121060220821844e-01 ,

+1.183788547045326e-01 , +1.248534855777947e- -01 , +1.315302847610869e-01 ,

+1.384103079528939e-01 , +1.454931890869180e- -01 , +1.527772946853750e-01 ,

+1 .602608842337125e-01 , +1.679435556337752e- ^■ 01 , +1 .758245615079801 e-01 ,

+1 .8390201 19821303e-01 , +1.92172862263441 1 e- ^■ 01 , +2.006344295681524e-01 ,

+2.092853879977170e-01 , +2.181238261985594e- -01 , +2.271462264407930e-01 ,

+2.363479205237173e-01 , +2.457259744642953e- ^■ 01 , +2.552771551741 124e-01 ,

+2.649981094228982e-01 , +2.748839432649996e- ^■ 01 , +2.849296444030153e-01 , W - -

+2.95 306505827265e-01 +3.054824712370942e-01 +3.15979963823894 e-01 +3.266169794538543e-01 +3.373873799614014e-01 +3.482855915638277e-01 +3.593057693987583Θ-01 +3.704415932452488e-01 +3.816862385160692e-01 +3.930329782788047e-01 +4.044749630814483e-01 +4.160051 103939122e-01 +4.276159595085598e-01 +4.393004362003260e-01 +4.510516333434032e-01 +4.6286160461 19925Θ-01 +4.747225454237193e-01 +4.866266705542529e-01 +4.985664508456704e-01 +5.105341492548225e-01 +5.225212793188740Θ-01 +5.345 90505841265e-01 +5.4652019 6422433e-01 +5.5851727695527 1 e-01 +5.705021536899105e-01 +5.824658100332457e-01 +5.943991720381216e-01 +6.062948176207270e-01 +6.181452662624718e-01 +6.299422016714543e-01 +6.416768736044914Θ-01 +6.53341 1462740817e-01 +6.649277540000037e-01 +6.76429270022331 1 e-01 +6.878367295965062e-01 +6.991421467689277e-01 +7.103379606632721 e-01 +7.214176027060971 e-01 +7.323746102405828e-01 +7.432008025932804e-01 +7.538887973483771 e-01 +7.644315495717613e-01 +7.748223151443820e-01 +7.850546571907628e-01 +7.951223518167163e-01 +8.05019386220 107e-01 +8.147397447696774e-01 +8.242774413707643e-01 +8.3362671 14335371 e-01 +8.427819363777799e-01 +8.517386186427548e-01 +8.604920874939698Θ-01 +8.6903767420 7057e-01 +8.773720451249032e-01 +8.854927938913381 e-01 +8.933935477349644e-01 +9.010727088526728e-01 +9.085249398881980e-01 +9.157483563218768e-01 +9.227413839712016e-01 +9.295017469413066e-01 +9.3602701966 7569e-01 +9.423143052164881e-01 +9.483629792665091 e-01 +9.541731 142261065e-01 +9.597438382130120e-01 +9.650738394220176e-01 +9.701635474343885e-01 +9.750143364412617e-01 +9.796277191617885e-01 +9.840036439809510e-01 +9.881426772731259e-01 +9.92047044691 21 e-01 +9.957199420334376Θ-01 , +9.991640812275709e-01 +1 .002381307710643e+00, +1.005374268639838e+00, +1.008146718214831 e+00, +1 .010703123647275e+00, +1.013046655758663e+00, +1.015181271 161417e+00, +1 .0171 1 1643578857e+00, +1.018843380560658e+00, +1.020381713994816e+00, +1.021731 101971518e+00, +1 .022896948293643e+00, +1.023885455671576e+00, +1.024702974608899e+00, +1 .025355286710874e+00, +1.025848243050604e+00, + 1 .026188366261215e+00, +1.026382881625701 e+00, +1.026438102255574e+00, +1.026360125820994Θ+00, +1.026155530733488e+00, +1.025831456886557e+00, +1.025395432284244e+00, +1.024853974580724e+00, +1.024213482124578e+00, +1 .023481 184943025Θ+00, +1 .022664602721801 e+00, +1.021770903480975e+00, +1 .020806917660529e+00, +1.019779396547454e+00, +1.018695995335235e+00, +1 .017564416918053e+00, +1.016391686789653e+00, +1.015184918030332e+00, +1 .013950835315021 e+00, +1 .012697033320358e+00, +1.01 1430749860716e+00, +1.010158346781076e+00, +1 .008885191761748e+00, +1.007592718305943e+00, +1 .006805603478092Θ+00, +1 .005378742804807e+00, +1.0040490517871 12e+00, + 1 .002778356298787e+00, +1 .001563778373068e+00, +1.000404915291105e+00, +9.993014844615166e-01 , +9.982531564931281 e-01 , +9.972595460676951 e-01 ,

+9.963202131848272Θ-01 +9.954346644968789e-01 +9.946023543844607e-01 +9.938226881029791 e-01 +9.930950268060122e-01 +9.924186919309267e-01 +9.917929657090736e-01 +9.91217091 1359961 e-01 +9.906902760441473e-01 +9.9021 1700378681 1 e-01 +9.897805192546193e-01 +9.893958602389157e-01 +9.890568243690241 e-01 +9.887624937408933e-01 +9.8851 19364350415e-01 +9.883042028597552e-01 +9.881383235740961 e-01 +9.880133161 159576e-01 +9.879281898513085e-01 +9.878819413827574e-01 +9.878735508484809e-01 +9.879019873841568e-01 +9.879662130250981 e-01 +9.880651778500144e-01 +9.881978162133899e-01 +9.883630508181326e-01 +9.885597957603544e-01 +9.887869526128024e-01 +9.890434070785485e-01 +9.893280319166613e-01 +9.896396899060125e-01 +9.899772316163624e-01 +9.903394929409673e-01 +9.907252968668226e-01 +9.91 1334564321237e-01 +9.915627747807666e-01 +9.920120436855326e-01 +9.924800441092685e-01 +9.929655482846576e-01 +9.934673212537685Θ-01 , +9.939841207305906e-01 , +9.945146969322154e-01 , +9.9505779319421 17e-01 , +9.956121471675398e-01 , +9.961764915534302e-01 , +9.967495543671935e-01 , +9.973300590248015e-01 , +9.979167245036720e-01 , +9.985082643417953e-01 , +9.991033633647473e-01 , +9.997003478609010e-01 , +1 .000299741957418e+00, +1 .000897441314013e+00, +1.001493964257960e+00, +1 ,002087624593489e+00, +1 .002677088643863e+00, +1.003261045483858e+00, +1.003838183774652e+00, +1.004407190937699e+00, +1.004966753528881 e+00, +1 .005515557572658e+00, +1 .006052289109999e+00, +1.006575635258931 e+00, +1 .00708428578161 1e+00, +1.007576934100958e+00, +1.008052277555815e+00, +1 .0085090177181 16e+00, +1.008945862447015e+00, +1.009361528531 177e+00, +1.009754742820913e+00, +1.010124241309341 e+00, +1.010468769795370e+00, +1.010787087537248e+00, +1.01 1077969726137e+00, +1.01 1340205650538e+00, +1 .01 15725971 14216e+00, +1.01 1773962181442e+00, +1.01 1943138906979e+00, +1 .012078982659783e+00, +1.012180362866919e+00, +1.012246166897464e+00, +1.012275305013586e+00, +1.012266707295288e+00, +1.012219319453278e+00, +1.012132107622966e+00, +1.012004064757857e+00, +1.01 1834207632025e+00, +1 .01 1621572933544e+00, +1.01 1365223023975e+00, +1.01 1064253702468e+00, + 1 .010717792733157e+00, +1.010324996851905e+00, +1.009885057526159e+00, +1.009397209381 147e+00, +1 .008860731864438e+00, +1.008274947065247e+00, +1.007639223374887e+00, +1 .006952983357691 e+00, +1.006215708265639e+00, +1.005426938305289e+00, +1.004586273379809e+00, +1.003693377657581e+00, + 1.002747984657666e+00, +1.001749900308864e+00, +1.000699003803502e+00, +9.995952485989262e-01 , +9.9843866321 16344e-01 , +9.972293415774932e-01 ,

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+0.000000000000000Θ+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000Θ+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0 .OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. .OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0 .OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000e+00 +0 .OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0, OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000e+00 +0, .OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0, .OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000Θ+00 +0. OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00 +0. OOOOOOOOOOOOOOOe+OO +0.000000000000000e+00, +0.000000000000000e+00 +0 .OOOOOOOOOOOOOOOe+00 +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w ₃₂₀ , for frame size N=320, is herewith provided (640 samples). As explained above, the last values may be zeros.

+3.8219929681 16373e-04, +5.337292962876158e-04, +7.010318350640769e-04,

+8.910357994632938e-04 +1.107726421735994e-03 + 1.354305556720234e-03, +1.632968062524617e-03 +1 .944392065549996e-03 +2.291251920978034e-03, +2.676357471 188558e-03 +3.102160324883627e-03 +3.569302917352426e-03, +4.080147004222224e-03 +4.637409387087253e-03 +5.243697651 1 12320e-03, +5.900394401532485e-03 +6.609908124535202e-03 +7.375089966000295e-03, +8.19761041 1215884e-03 +9.078195201482999e-03 +1.001888876231 1 12e-02, +1.102272421949215e-02 +1.209201783148431 e-02 +1.322773320467288e-02, +1.443170227861997e-02 +1 .570673344965518e-02 +1.705481769961258e-02, +1 .8477 280 7 765e-02 + .997546609433320e-02 +2. 55255505248360e-02, +2.320993080187466e-02 +2.494800748775296e-02 +2.676898100961685e-02, +2.867550388692567e-02 +3.06697686591 1597e-02 +3.275241094528637e-02, +3.492474656851959e-02 +3.718932683936013e-02 +3.954748420753697e-02, +4.200013716597904e-02 +4.454866443036003e-02 +4.719547798640356e-02, +4.994161943600844e-02 +5.278755308483513e-02 +5.573471840677637e-02, 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+6.46957354741 1541 e-01 , +6.378629377753962e-01 , +6.286855280775698e-01 , +6.194273058858271 e-01 , +6.100922809170846Θ-01 , +6.006842631336996e-01 , +5.912087588215279e-0t +5.816678990293079e-01 , +5.72065001 1081 139e-01 , +5.624031 153715043e-01 +5.526906296747590Θ-01 , +5.429296335729603e-01 , +5.331253296256703e-01 +5.232805923025862e-01 , +5.134010079652535e-01 , +5.034916218473681e-01 +4.935565059779637e-01 , +4.835996679580253e-01 , +4.736270399539165e-01 +4.636428606356920e-01 , +4.5365191 15376573e-01 , +4.436589520558524e-01 +4.3367021 10092904e-01 +4.236905139305571 e-01 +4.13724 358370227e-01 +4.037764981593496e-01 +3.9385366365994 1 e-01 +3.839608678 21244e-01 +3.741023650337649e-01 +3.642834353302138e-01 +3.545105744959439e-01 +3.447887254766501 e-01 +3.351225758795436e-01 +3.2551741492 6463e-01 +3.159783340225490e-01 +3.0651 18829951374e-01 +2.971215837581699e-01 +2.878124834032541 e-01 +2.785898831431837e-01 +2.694592444601873e-01 +2.604240268886858e-01 +2.5148869091 15454e-01 +2.426587307354934e-01 +2.339385917103926e-01 +2.253316454409216e-01 +2.168420127437019e-01 +2.084745870335012e-01 +2.002332724591 125e-01 +1 .921209969418307e-01 + 1.841414579810840e-01 +1.762991374092108e-01 +1 .685975427 09672e-01 +1.610392647455030Θ-01 +1.5362763874554 1 e-01 +1.463667950599712e-01 +1 .392599476606262e-01 +1.323093893293405e-01 +1.2551821551 1 1708e-01 +1 .188903216873086e-01 +1.124286726877977e-01 +1.06 353122093495Θ-01 +1.000130754353283e-01 +9.406556557413207e-02 +8.829543394466104e-02 +8.270438757849530e-02 +7.729486433686703e-02 +7.206998099643687e-02 +6.703183558090599e-02 +6.218153160076208e-02 +5.752079270262547Θ-02 +5.305187265543024e-02 +4.877591488398974e-02 +4.469299727922282Θ-02 +4.080367800167831 e-02 +3.710883802024244e-02 +3.360819061792393Θ-02 +3.030027537135876e-02 +2.718416076819826e-02 +2.425872983336823e-02 +2.152170070657780e-02 +1.896992105664512e-02 +1 .660053693426304e-02, +1 .441002712765955e-02, +1 .239334766429038e-02, +1 .054549096241532e-02, +8.862485552559351 e-03, +7.329085750247888e-03, +5.896299968476346e-03, +4.91475345978871 1 e-03, +0.000000000000000e+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0 OOOOOOOOOOOOOOOe+00,

+0. OOOOOOOOOOOOOOOe+OO , +0 OOOOOOOOOOOOOOOe+00, +0 OOOOOOOOOOOOOOOe+OO,

+0.000000000000000e+00, +0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0.000000000000000Θ+00, +0 OOOOOOOOOOOOOOOe+00,

+0.000000000000000e+00, +0.000000000000000e+00, +0 OOOOOOOOOOOOOOOe+OO, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w ₄₈₀ , for frame size N=480, is herewith provided (960 samples). As explained above, the last values may be zeros.

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- 140 -

+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _: +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000Θ+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 +0.000000000000000e+00, +0.000000000000000e+00. +0.000000000000000Θ+00 +0.000000000000000e+00, +0.000000000000000e+00 +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 _: +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00 _: +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00 _; +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000Θ+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function, for frame size N=960, is herewith provided 1920 samples). As explained above, the last values may be zeros.

+3.235700349233950e-04, +3.8219929681 16373e-04, +4.346763076430269e-04,

+4.841399974556557e-04, +5.337292962876158e-04, +5.864735900759105e-04,

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8 Bibliography

[1 ] 3GPP, "Codec for Enhanced Voice Services (EVS); Detailed algorith description," [Online]. Available:

http://www.3gpp.org/ftp/Specs/archive/26_series/26.445/26 445-d00.zip. [2] P. P. Julien Faure, "Delay-optimized overlap transform, coding/decoding weighting windows". US Patent US8847795, 28 06 201 1.

[3] Geiger, Audio Coding based on integer transform, llmenau: https://www.db- thueringen.de/receive/dbt_mods_00010054, 2004.