Receiver with Joint Equalization and Phase-noise Estimation and

Corresponding Method

Field of the Invention

Technical field in which the, invention falls

The current invention refers to wireless digital communications systems (SC-FDE)., particularly, to receivers with joint equalization and phase-noise estimation and corresponding iterative receiver methods for the detection in wireless communications systems employing the, single-carrier (SC) transmission technique. The received signal is corrupted by phase-noise arid/or carrier frequency offsets (CFO) . The receiver uses frequency-domain equalization (FDE) .

State-of-the-art

Emerging wireless systems tend to use the transmission technique based on orthogonal frequency division multiplexing (OFDM) , thus reaching high data throughput with increased performance.. Due to the high peak-to-mean power ratio (PMPR), the use of this transmission technique in mobile terminals (uplink) is impractical. For that reason, the SC-FDE based transmission technique is an alternative to OFDM, allowing to address the power limitation and keep the capability of making use of the benefits resulting from employing. the frequency-domain equalization. Interestingly enough, the OFDM based transmission technique was selected by the specifications of the downlink (from the base station to the mobile terminal) of the 'Long Term Evolution' (LTE) release 8 of the 'Third Generation Partnership Project' (3GPP) , which is the · fourth generation of cellular communications. This transmission technique replaces the code division multiple access (CDMA, ) employed on the third generation of cellular communications. Due to the foregoing reasons, the SC-FDE based transmission technique was also selected as an option by the specifications of the uplink (from the mobile terminal to the base station) of LTE, in the release 8 of the 3GPP.

Wireless digital communications systems are, more and more, immersed in highly populated environments, coexisting with several radio frequency devices of various kinds. The requirements for these systems are increasingly demanding, both in terms of transmission rates as well as in terms of spectral occupation, exacerbating the adverse effects of the common channel.

Moreover, in high-bandwidth systems, due to . the multiple reflections which may affect the signals during their paths, multiple copies of the same signal may reach the receiver, with different delays, attenuations, and phase offsets. This phenomenon is known as frequency-selective fading and causes inter-symbol interference (IS!) . This interference is even more severe the higher the transmission rate is, thus limiting increasing bit rates. The level of signal at the receiver is seen as the sum of independently distorted signals, resulting from the multiple paths.

Under these conditions a signal transmitted over the radio channel is radically altered, which makes it necessary an adequate treatment at receiver to reverse the channel effects, an operation typically known as equalization. The inclusion of a cyclic prefix (CP) in each data block is designed to absorb the delayed signal replicas. In fact, this procedure allows to eliminate inter-block interference (IBI) (nevertheless remaining the interference within the block) provided that the CP duration is longer than the channel impulse response (CIR) .

The SC-FDE systems proposed in [Sari94] present themselves as a possible solution to the equalization problem, revealing a good performance/complexity ratio, in addition to superior power efficiency, if compared to the OFDM systems. This equalization system has a structurally simple transmitter. To the modulated signal one adds the cyclic prefix and, after this procedure, the signal is transmitted. Given that the value of the signal envelope fluctuation, in most cases, is enough to generate distortions, it becomes necessary to use a linear amplification process at the transmitter.

In order to improve the equalization performance it is advisable to use an SC-FDE scheme with iterative feedback (IB-DFE) [Benvenuto02 ] , [BenvenutolO] . This receiver uses the values of the estimated signal to improve, in each iteration, the estimation of the data being treated. Concerning the estimation procedure, the use of the decision block based on soft-decisions reaches increased precision and consequently a better performance .

Typically, within the IB-DFE receivers equalization and channel decoding are performed separately, i.e., the feedback chain supplies the equalizer output and not the channel decoder output. However, it is possible to reach higher performance gains if these procedures are performed jointly. An effective way of reaching this is through the implementation of a turbo equalization scheme. In this kind of structure the equalization and decoding procedures are repeated, iteratively, with some sort of soft information being exchanged between them. Although initially proposed for time-domain implementations, turbo equalizers can also be implemented in the frequency-domain. Namely, the turbo FDE schemes based on the IB-DFE [Benvenuto05 ] , [Gusmao07], as in the case of this invention. Since the feedforward, and feedback coefficients, depended on the channel frequency response (CFR) , a reliable channel estimate must be provided to the receiver. Typically, these channel estimates are obtained with the aid of training symbols multiplexed with the data symbols, either in the time-domain or in the frequency-domain [Lam06] . Although both options may be used with any modulation scheme, usually pilots in the frequency-domain are employed with OFDM modulations while the time-domain pilots are employed with SC modulations. Typically, the estimation of the channel corresponds to an increased consumption of the available bandwidth, particularly with channels presenting high variability and/or in bursty communications, i.e., short and intense transmissions. Since the bandwidth available to the communications systems is limited, it must be used wisely by limiting the bandwidth used estimating the channel. A promising solution to this problem is the use of implicit pilots, also known as superimposed pilots. With this solution the training sequence is added to the data sequence instead of being multiplexed. This means that it is possible to increase significantly the density of pilots [Dinis08] . Alternatively, one can use training sequences [CoelholO] .

The presence of carrier frequency offsets (CFO) and phase-noise in digital communications systems seriously compromises the reception quality of the signal. These phenomena are ubiquitous to the communications systems and cannot be avoided, only mitigated. CFO originates from the frequency misalignment of the transmitter and receiver oscillators and/or in the relative motion of the mobile terminals which results in the Doppler Effect. As for the phase-noise, it results from imperfections in the local oscillators, which results in instability phenomena in the oscillator clock frequency.

The signal-to-noise ratio (SNR) performance loss caused by CFO and phase-noise in OFDM, and SC-FDE modulations, is evaluated in [Pollet95] . There, it is proved that this performance degradation is due to, for OFDM modulations, two different, contributions: the phase, error common to all subcarriers, and the inter-carrier interference (ICI). With respect to the SC modulations, the performance is only influenced by the common phase error. Pollet et al . also argue that OFDM is several times more sensitive to CFO and phase-noise than SC modulations.

While the CFO presents a linear behavior in time, phase-noise is described by a random process. One may consider a Wiener-Levy or an Orstein-Uhlenbeck process depending on whether the phase- noise results from a free running or a phase locked-loop (PLL) oscillator, respectively. On PLL driven oscillators the closed- loop control mechanism tracks the variations of the carrier frequency, and consequently, the phase-noise has limited variance. As for the free-running oscillators, the generated phase noise results from the accumulation of random frequency deviations leading to unlimited variance.

Several solutions to the phase-noise estimation problem _. are present in the literature. However, only [Sabbaghian08 ] proposes a joint equalization and phase-noise estimation solution _t Nevertheless, the solution of Sabbaghian and Falconer is limited to phase-defined constellations, e.g., M-PSK. Differently, the solution proposed in this invention solves the phase estimation problem even for bi-dimensional signal constellations (i.e., amplitude and phase-defined) , e.g., M-QAM.

From a Bayesian perspective, phase-noise estimation based on past observations of the equalizer output requires determining the state posterior probability density function (PDF), i.e., the value of the phase-noise conditioned on all past observations, thus enabling the computation of the optimal phase-noise estimate with respect to any criterion, e.g.,. minimum mean-square error (MMSE) . Typically, to determine such a posterior PDF is extremely hard. A notable exception occurs when state and observations are described by linear models, and the observation noise is Gaussian. For this particular case the posterior PDF can be determined optimally by the Kalmart Filter.

This invention uses a stochastic recursive filtering solution in order to propagate the "a posteriori" PDF based on the fact that phase-noise is characterized by the Wiener model, acting as prior, and that one eliminates the dependence of the observation factor relatively to the data symbols with a marginalization procedure. These two features allow an effective Bayesian recursive algorithm.

The present invention considers an iterative block decision- feedback equalizer for SG-FD.E modulations combined with a phase- noise estimator. This receiver, employing direct and feedback filtering, presents better performance results than receivers based in non-iterative methods, as shown in [Benvenuto02 ] , [Dinis03] and [BenvenutolO] .

Summary of the invention

The current invention concerns wireless telecommunications systems, particularly, joint equalization and phase-noise estimation receivers and respective methods.

It combines an IB-DFE receiver for SC-FDE modulations with a recursive Bayesian algorithm for phase-noise estimation in highly dispersive and highly time-variable channels due. to the presence of phase-noise and/or carrier frequency offsets. This invention may find applications ^' in digital radio receivers, namely, 3GPP LTE-A cellular radio communications and future cellular radio communications standards. Other possible applications are satellite communications, and underwater acoustic communications. Detailed description of the invention

The present invention, concerning a wireless telecommunications system, particularly, a joint equalization and phase-noise estimation receiver structure and respective methods, will be described in detail in the following using the simplified scheme depicted in Figure 1 and in Figure 2. In these figures the identification of each particular element in discussion is made using a number where its most significant digit is equal to the number of the figure in which the element is found. For instance, the element block for the inclusion of the cyclic prefix (102) is found on Figure 1.

The device presented in this invention includes a block for iterative frequency-domain . equalization (201), a block for the estimation and compensation of the phase-noise (202) , and a block for the. estimation of the channel parameters and the computation of the equalization coefficients (203) .

The method for joint equalization and phase-noise estimation is carried out by the receiver structure in this invention through the following steps: transform the frequency-domain phase-noise affected equalized signal samples into time-domain using the. block for the inverse Fourier transform (201a) and obtain the corresponding time-domain phase-noise affected equalized signal samples; compensate for the phase-noise presence in the time- domain phase-noise affected equalized signal samples at the input of the block for counter-rotation (202b) and obtain the corresponding time-domain phase-noise compensated equalized signal samples; compute the symbol likelihoods with respect to the time-domain phase-noise compensated equalized signal samples at the input of the block for symbol decision (201b) and obtain the time-domain phase-noise compensated soft-decisions signal samples; re-introduce the phase-noise in the time-domain phase- noise compensated soft-decisions signal samples at the input of the block for rotation (202c) and obtain the time-domain phase- noise affected soft-decisions signal samples; delay by one iteration the time-domain phase-noise affected soft-decisions signal samples using the block for delay (201c) ; transform the delayed time-domain phase-noise affected soft-decisions signal samples into frequency-domain by using the block for the direct Fourier transform at the feedback chain of the equalizer (201d) ; estimate the channel parameters by using the block for the estimation of the channel parameters (203a); compute the filtering coefficients by using the block for the computation of the filtering coefficients . (203b) and compute the feedback coefficients using the block for the computation of the feedback coefficients (203c) .

It should be noted that the block diagrams presented do not include the block for radiofrequency processing which is independent of the present invention.

A block diagram representing a simplified transmission/reception chain for SC-FDE modulations is depicted in Figure 1. This figure has the following elements:

A block for mapping (101) which maps the data to transmit using a mapping technique, e.g., Gray coding. At the block for mapping (101), {s _n ; n = 0, 1, ... , N - 1} corresponds to the transmitted data block, where each symbol s _n is the signal sampled at the instant n. N is the size of the data block. The data to transmit are illustrated by the sequence ...101100....

The block for the inclusion of the cyclic prefix (102) appends the cyclic prefix to the data block. The data signal with a cyclic prefix is given by: where N _cp is the size of the cyclic prefix. The block representing the channel (103) represents the radio channel. In the block representing the channel (103) the signal is transformed and distorted due to the unwanted characteristics of the multi-path channel and the phase-noise. The resulting signal can be described as the convolution between the signal at the channel input and the channel impulse response rotated by the corresponding phase-noise value plus the channel noise, i . e . , by :

Y = ^je "∑ h^ + v _n , n = 0, 1, ... , N - 1 . (eq. 2)

i-o · where {θ _η ;. n = 0, 1, ... , N - 1} ^■ is the phase-noise,

{h _x ; 1 = 0, 1, ... , N _h - 1} is the channel impulse response - (naturally and. in order to avoid ISI, N _cp > N _h - 1) and {v _n ; n = 0, 1, ... , N - 1} is complex zero-mean Gaussian noise with variance .

The block for the removal of the cyclic prefix (104) removes the cyclic prefix from the received signal.

The block for the direct Fourier transform at the input of the equalizer (105) , which has as its input the time-domain received signal, {y' _n ; n = 0, 1, ... , N - 1}, and as its output the corresponding signal in the frequency-domain obtained using the discrete Fourier transform (DFT) , i.e.,

{Y _k ; k = 0, 1, ... , N - 1} = DFT {y^; n = 0, 1, ... _. , N - 1} ;

The block for joint equalization and phase-noise estimation (106) which is presented in detail in Figure 2;

The block for hard-decisions (107), which performs an hard- decision (HD) on the value of the equalized sample, i.e.,

{s^; n = 0, 1, ... , N - 1} = HDis^; n = 0, 1, ... , N - 1} ;

The block for demapping (108) which, based on the hard-decisions {s ⁽ _n ^l! ; n = 0, 1, ...,N— 1}, returns the received data;. In Figure 2 one may observe a block diagram representing the processing chain. of a joint iterative frequency-domain equalizer and phase-noise estimation and compensation. This figure corresponds to the block for joint equalization and phase-noise estimation (106) and consists in the following elements: the block for iterative frequency-domain equalization (201) , the block for the estimation and compensation of the phase-noise

(202) , and the block for the estimation of the channel parameters, and the computation of the equalization coefficients

(203) .

Notice that each of these blocks consists in different sub- blocks referenced by the number of block to which they, belong followed by a letter, for instance, the block for rotation (202c) is a sub-block of the block for the estimation and compensation of the phase-noise (202). Below follows a detailed description of the blocks constituting Figure 2.

The block for iterative frequency-domain, equalization (201) consists in four different sub-blocks. Namely, the block for the inverse Fourier transform (201a) , the block for symbol decision (201b), the block for delay (201c), and the block for the direct Fourier transform at the feedback chain of the equalizer (201d) .

The block for the inverse Fourier transform (201a), which has as its input the frequency-domain equalized signal and as its output the corresponding time-domain signal obtained by using the inverse Fourier transform (IDFT), i.e.,

{s n = 0, 1, ... , N - 1} - IDFT{S' _k ^(i> ; k = 0, 1, ... , N - 1} . Notice that, at the input of the processing chain of the iterative equalizer one multiplies the filtering coefficients {F _k ; k = 0, 1, ... , N - 1} ^" and then subtracts the feedback factor, {S^ ^-1 ^ ¹ ; k = 0, 1, ... , - 1} .

The block for the inverse Fourier transform (201a) operates over the samples given by: Sf = Y _k 'Ef - S' ¹ , k = 0, 1, ... , N - 1 (eq. 3) where the received signal samples, {Y _k ; k = 0,1,....,N -1} _/ is impaired by ISI and phase-noise. The received signal samples, {Y _k ; k = 0, 1, ... , N - 1} , are described by:

Y^ = S' _k H _k + N _k , k = 0, 1, ... , N — 1 (eq. 4) where {S _k ; k = 0, 1, ... , N - 1} are frequency-domain signal . samples corresponding to the transmitted data impaired by phase-noise (the presence of phase-noise affecting the signal samples is indicated by the use of the prime) , {H _k ; _. k = 0, 1, ... , N - 1} the channel frequency response and {N _k ; k = 0, 1, ... , N - 1} complex zero- mean Gaussian noise with variance. .

The block for symbol decision (2.01b) carries out a soft-decision (SD) operation on the value of the equalized samples, i.e., s ; n = 0, 1, ... , N - 1} = SO[s ; n = 0, 1, ... , N - 1} .. These symbol decisions ' are _. used afterwards in the reconstruction of the transmitted signal with interference in order to use it in the canceling of the residual ISI in the next iteration and to estimate the channel parameters. Alternatively,, one can use in the feedback chain of the block for iterative frequency-domain equalization (201) the output of a soft-input soft-output (SISO) channel decoder, thus integrating with the IB-DFE the channel decoding procedure and reaching the performance known to turbo FDE implementations by observing the following steps: a) Demapping of the equalized signal samples within the block for soft-demapping (301) resulting in the log- likelihood ratios (LLR) s of each data bit;

b) Bit de-interleaving within the block for de-interleaving (302);

c) Channel decoding within the block for SISO decoding (303) resulting in the refined log-likelihood ratios; d) Bit re-interleaving within the block for interleaving (304) ;

e) Bit remapping in coded data symbols withi the block for soft-remapping (305) .

The block for delay (201c) which delays the signal samples at its input by one iteration, i.e.,

{s ^'1 ; n = 0, 1, ... , N - 1} = Delay{s >; n = 0, 1, ... , N - 1} , where (i-1) denotes the delayed signal samples relative to the (i) -th iteration;

The block for the direct Fourier transform at the feedback chain of the equalizer (201d) transforms the time-domain delayed signal samples in the feedback chain of the equalizer,

{s^ ^1_1) ; n = 0, 1, ... , N - 1} , into their frequency-domain counterparts,

{S^ ^1_1) ; k = 0, 1, ... , N - 1} , using the discrete Fourier transform, i.e., {s n = 0, 1, ... , N - 1} - DFT{¾ ^1_1> ; k = 0, 1, ... , N - 1} . The block for the direct Fourier transform at the feedback chai of the equalizer (201d) has its output multiplied by the feedback coefficient {B^; k = 0, 1, ... , N - 1} .

The block for the estimation and compensation of the phase-noise (202) consists in three different sub-blocks. Namely, the block for the estimation of the phase-noise (202a) , the block for counter-rotation (202b) , and the block for rotation (202c) .

The block for the estimation of the phase-noise (202a) estimates the value of the phase-noise in phase and amplitude defined constellations using a recursive Bayesian filter.

The block for counter-rotation (202b) performs the phase-noise compensation. One compensates the phase-noise in time-domain using the phase-noise estimates, i.e.,

{sjf = § ^!i! e- ^j'¾l '; n = 0,...,N - lj , where ^' {s ^K , n = 0, .... , N - l} are the time-domain equalized signal samples at the output of the block for iterative frequency-domain equalization (201) .

The block for rotation (202c) re-introduces the phase-noise in the equalizer feedback chain, i.e., {s ^*u) = s^W ⁶ "'; n = 0, 1, ...,N - 1}.

The block for the estimation of the phase-noise (202a) estimates the phase-noise in the time-domain by describing the output of the block for the inverse Fourier transform (201a) by means of an equivalent channel, described as an additive zero-mean Gaussian channel. This characterization is possible because the output of the block for the inverse Fourier transform (201a) only shows two terms: the signal of interest and an additive element comprising channel noise and residual interference.

The block for the estimation of the channel parameters and the computation of the equalization coefficients (203) consists in three different sub-blocks. Namely, the block for the estimation of the channel parameters (203a) , the block for the computation of the filtering coefficients (203b) and the block for the computation of the feedback coefficients (203c) .

The block for the estimation of the channel parameters (203a) estimates the channel, frequency response, {Η^'; k = 0, 1, ... , N— 1} , and the reciprocal of the SNR, {ά} , using the output of the block for symbol decisions (201b) and samples at the input . of the equalizer, {Y _k '; k = 0, 1, ... , N - 1} .

The block for the computation of the filtering coefficients (203b) has as its inputs the channel frequency response estimates, k = 0, 1, ... , N - ^■ 1} , and the estimate of the reciprocal of the SNR, {a}, and as its output the filtering coefficients, {E ¹ ; k = 0, 1, ... , N - 1} . The block for the computation of the feedback coefficients (203c) has as its inputs channel frequency response estimates, k = 0, 1, ... ; N - 1} , and the estimate of the reciprocal of the

SNR, {ά} , and as its output the feedback coefficients,

{Bj^-k = 0, 1, ...,N - 1},

Since the device presented in Figure 2 is iterative, the result of each symbol decision performed within the block for symbol decision (201b) , is fed back to the block for the inverse Fourier transform (201a.) . This takes place after re-introducing the phase-noise estimate within the block for rotation (202c) , delaying it by one iteration within the block for delay (201c) , and performing the operations corresponding to the block for the direct Fourier transform at the feedback chain of the equalizer (201d) . Thereby, one ^■ proceeds to the canceling of the interference still remaining in the data. This procedure allows a significant performance improvement at each iteration since the canceling of the interference improves successively as the estimates of the transmitted become more precise, i.e., with less errors in the symbol, decisions.

Computation of the filtering coefficients

The direct filtering coefficients- and negative feedback coefficients are {F^t k ^' - 0, 1, ... , N - 1} and {B^; k = 0, 1, ... , N— 1} , respectively. The optimal values for the feedback coefficients are given by:

^B k' = ^F _* - ^k = 0, 1, ...,N - 1. (eq. 5)

The optimal values for the filtering coefficients are given by:

F ^li!

F _k ^w = k = 0, 1, ... , N - 1, (eq. 6) with

=;<i) k = 0, 1, ... , N - 1 (eq. 7;

a + (1 - (p' ¹ - ¹¹ ) ² ) I H _k ' ² where

.eq.

k=0 and

E[ 1 N _k I ² ]

a = _|2l , ⁽ eq.

E[| S _k ] where Ε [ · ] is the expected value and | · | the absolute value. Notice that, typically, the values of {H _k ; k = 0, 1, ... , N - 1}, and a are unknown being required in this case to use their estimates _/ {H _k ; k = 0, 1, ... , N - 1} and ά .

The variable p ^(1_11 corresponds to the . correlation factor obtained during the (i-l)th iteration and it is given by:

The variable p ^{il _1>} can be regarded as a blockwise reliability parameter on the estimates {s ^! _n ^1_1) ; n = 0, 1, ... , N - 1} . This means that one may define an "average symbol" with respect to the data block {s - ¹¹ = p' ^{1 "1} ^; n = 0, 1, ... , N - 1} .

In fact, the performance of the iterative receiver can. be improved if one replaces the "average, symbols" in terms of the "data block" with "average symbols" in terms of the "data symbol". To clarify this last point assume that the transmitted symbols are taken from a QPSK constellation with Gray coding. Define ±1 ± j = sl + js°, with 5n = Re{sJ = ±1 and s° = Im {sj = ±1, n=0, 1, N-l . Where Re{-} and Im{-} are respectively the real part and the imaginary part of a complex number. Similar definitions can be obtained for s _n = _. s + js ^® ,

The LLRs for the in-phase and quadrature bits associated to s I _n (i) and s° ⁽¹⁾ , respectively, are given by: Hi) _ 5 H. i)

(eq. 11) and

Q(i)

λ' -.Q(i)

_2 (eq. 12)

where σ- = - E[ | sn - EC I

Considering that the signal samples {s^ ¹ ; n = 0, 1,.,.,Ν — 1} are Gaussian distributed, results that the average value of s _n is: s _n (i) = tanh |eq. 13)

The HD = ±1, and s° ⁽ⁱ⁾ = ±1, are defined according to the signs of ^! , and ^! , respectively. Therefore, = pf ^! s ' + jp 's ^! , where

(eq. 14)

and (eq. 15) p _n and can be regarded as a reliability parameter associated to the in-phase and quadrature bits of the n-th symbol (naturally, 0 < pf < 1 and 0 < p° ⁽ⁱ⁾ < 1). During the first iteration one has " ¹¹ = _. p° ^J = 0 and s^ ¹ ' = 0; after some iterations and/or when the SNR is high, typically, one has ' « 1 and p° ^(i> « 1, resulting ¾ ^l! « s ⁽ " . The filtering coefficient is still given by (eq. 6) - (eq . 7) but the correlation factor (eq. 10) is now given by:

Notice that (eq. 16) allows the computation of the. correlation factor p ^U) without the explicit knowledge of the transmitted symbols s _n , contfarily to what happens with (eq. 10).

These filtering Coefficients are for QPSK constellations. For phase and amplitude-defined constellations, e.g., M-QAM, the procedure is the following:

Assume that the transmitted data symbol s _n belongs to a given alphabet L, i.e., to a determined constellation, size #L=M, and that this same $ymbol s _n is selected in accordance with its corresponding bits β^' , m = 1, 2, ... , μ , where μ = log ₂ (M) , i.e., s _n = f(b ^l _n ^l] , b ⁽ _n ²⁾ , ...,b^), with b' _n ^m) = - 1. where it is assumed that β^' is the m-th bit associated to the n-th symbol, and that b ⁽ _n ^m) is the corresponding polar representation, i.e., β'" ¹ = {0,1} and b ⁽ _n ^m) = {-1, +1} . - For 4-PAM constellations and Gray coding one has s _rt = 2b ⁽ _n ^2> + bj bj . For 8-PAM constellations and Gray coding one has s _n = 4b ⁽ _n ³⁾ + 2b ⁽ _n ^3! b ⁽ _n ² ' 4- bfbfb? . If the transmitted symbols were to be selected from a QAM constellation with Gray coding, the M-QAM constellation is described as the sum of two PAM constellations, each one of them of size M , one corresponding to the in-phase component, i.e., the real part, and the other to the quadrature component, i.e., the imaginary part. Therefore, for 16-QAM one has : s _n = 2b? + b¾ + j(2b? + bW, (eq. 17) and for 64-QAM, one has:

(6) A6), (5)

4b ⁽ _n ³⁾ + 2b ^l „ -' + b¾¾ + j(4b ^l _n ⁶ ' + 2b*>h > + b^'bW). .(eq. 18)

To determine the value of the average symbol conditioned by the equalizer output, s _n , one is required to obtain the value of the average bit conditioned by the equalizer output, b'™' . These relate with the corresponding LLR in the following way: bT = tanh (eq. 19)

2 using the mapping rules (eq. 17) -(eq. 18) results s _n The LLR of the m-th bit of the n-th symbol is given

where Ψ'^' and Ψ^' are the subsets of L where β' ^*1 = 1 or 0, respectively. Obviously, Ψ[ ^η Ψ'"" = L and n = 0.

The reliability of the estimates to be used on the equalizer feedback chain, for 16-QAM, are given by: and

4 I b' ⁴¹ I + I b

P° = ⁽ (eq. 22) where, for 64-QAM, they are given by: and

1.6 1 b' _n ⁶ ' I +4 I 5 ¹ 1 + I Έξ ^ I

[eq. 24)

21,

Turbo equalization or IB-DFE with coding

As an alternative to the data transmission without channel coding one can use Turbo Equalization schemes or, otherwise known, IB-DFE with channel coding.

In this case the feedback chain of the block for iterative frequency—domain equalization (201) uses the output of a channel decoder. This means that the block for symbol decision (201b) is replaced by the chain of blocks presented in Figure 3. For Turbo equalization to be possible it is required that, prior to its transmission, the data bits be coded, interleaved, and mapped. Already in the receiver, the feedback element of the block for iterative frequency-domain equalization (201) will have to include a block for soft-demapping (301) , a block for de-interleaving (302), a block for SISO decoding (303), a block for interleaving (304) and a block for soft-remapping (305).

The set of equalized samples {§„'; n = 0, 1, ... , N - 1} _{are first} demapped using the block for soft-demapping (301) which returns the LLRs of each bit. Then follows a block for de-interleaving (302) and a block for SISO decoding (303) . The latter returns the fine LLR resorting to the properties of the error correcting codes [TuchlerOl], [Tuchler02] . Finally follows the block for interleaving (304) and ,a block for soft-remapping (305).

Characterization of the equalizer output

The set of samples . {sjj ¹¹ ; n = 0, 1, ...,N - 1} can be described by: L ⁾ = ^s _n e ^e " + vjf , n = 0,1, ...,N - 1 (eq. 25) where the set of samples {v ^l _n ^L) ; n = 0,1, ...,N - 1} is zero-mean Gaussian noise plus residual inter-symbolic interference. To this description of the equalizer output one calls the characterization of the equalizer output by an equivalent Gaussian channel.

In fact, it is possible to determine the power of the equivalent channel noise. Assuming that the set of samples in the time- domain { ^; n = 0, 1, ...,N- 1} has the corresponding set of samples in the frequency-domaih given by {V^ ¹¹ ; k = 0,1, ...,N - 1}, i.e.,

{V* ¹¹ ; k = 0, 1, ... , N - 1} = D iv^ ¹ ; n = 0, 1, ... , N - 1} , then. the equivalent channel noise power is given by: E[ I I ² ] = E[ I F H _k - 1 - (p ^{i - ^l, ) ² B^ | ² ]E[ | S _k | ² ]

+E[| B, ¹¹ I ² ] (p ^(i_11 ) ^≥ (l - (p ^<i"1) ) ² )E[ I S _k I ² ] (eq. 26)

+E[ I | ² ]E[ I V _k I ² ]

State-space model

The estimation of the phase-noise can be regarded as a nonlinear filtering problem. This kind of problems consists in estimating the state of a non-linear stochastic process, based on a set of noisy observations. Many of these problems are described by a pair of equations known as the state-space model. This model includes the state variable dynamics, and the observation or measurement of that state variable. Typically the observation is a noisy and transformed version of the state variable .

While estimating the phase-noise, the state variable is θ _η with distribution ρ(θ _η | 0 _n _ _j ), and the observation is s ¹ ' with observation factor p^ ¹ ' I θ _η ) . In fact, the phase-noise dynamics can be described by a Brownian motion, given by: θ _η = θ _η _ _! + w _n (eq. 27)

As for the observations, these correspond to the equalizer output characterized by the equivalent Gaussian channel , (eq. 25) .

Observation factor

A central element in the design of a Bayesian recursive filter is the definition of the observation factor. For an observations model given by equation (eq. 25) results an observation factor given by: where σ ² = E[ | ' | ² ] / N

Defining y' ¹¹ = (I s' _n ^(il | ² + I s _n | ² ) / σ ² , and β'> =1 s 's _n | /σ ² and noticing that I s - s _n e ^je " | ² =| s I ² + I s _n f -2 | ¾¾„ I cos (φ _η + θ _η - r ) , where η^' = arg{s' _n ⁽¹⁾ } and φ _η = arg{s _n } , where the function arg{.-} returns the complex argument, then, (eq. 28) may be re-written as:

P(s I Θ _Β , s _n ) =— exp (2β' _η ¹¹ cos (φ _η + θ _η - η'») - ) (eq. 29)

Notice that (eq. 29) depends on s _n , which is obviously undesirable since, as a rule, the receiver^ which is where the recursive Bayesian filter may be found, does not know the set of samples {s _rt ; n = 0, 1, , N - 1}. In order to remove the dependence of the observation factor relatively to s _n one carries out a marginalization procedure, corresponding to:

∑ P(s i e _n/ s)p(s) = p(s I θ _η ) (eq. 30)

Applying the marginalization procedure (eq. 3.0) to the observation factor (eq. 29) results pisf I θ _η ) = + θ _η - ν ) - y _n (s) ^(il ) (eq. 31) ·

where it is assumed equiprobable symbols, i.e., p(s)=l/M. Notice that now one has, Y _n (s = (| s' _n ^(il f + | s f) I σ ² , P _n (s) ^fi) =| s s | /σ ² , If the equiprobable symbols assumption is not supported one has to consider the likelihoods at the output, of the SISO decoder. In what follows, is exemplified how to proceed if so for the particular case, of QPSK . constellations .

Observation factor supported by soft-decisions Recalling that e ^x "

p(b _n = 1) = = ς _η θχρ( _. λ _η, / 2), (eq.. 32)

1 + e ⁿ and p(b _n = -1) = ¹ _λ = ς _η θχρ(-λ _η / 2) , (eq. ,33)

1 + e "

With ζ _η = e ^K / (1 + e ^K ) results that p(b _n ) = ς _θ ρ(^λ _η / 2) . (eq. 34)

Now, in order to remove the dependence of the observation factor relatively to s _n resorting to the marginaliz tion procedure one has :

where s = b + j is used.

Using the marginalization procedure (eq. 35) and the bit probability (eq. 34) results that the observation factor (eq. 29) is given by:

cosh (V2 _n (s) ^a, (cos (η'^ - θ _η ) + sin (if - θ„) ) + λ^' / 2 + / 2) (eq. 36)

+ cosh (^{s 'icos <τ£' - θ„) - sin (n' ^i! - θ _η ) ) + ' / 2 - λ° ^α > / 2) ]

Figure Description

Figure 1 depicts a block diagram representing the simplified transmission/reception chain for ^" SC-FDE signals. Particularly Figure 1(a) depicts the transmission chain plus channel and Figure 1 (b) depicts the reception chain. In Figure 1 one may see the following elements: block for mapping (101), block for the inclusion of the cyclic prefix (102), block representing the channel (103), block for the removal of the cyclic prefix (104), block for the direct Fourier transform at the input of the equalizer (105), block for joint equalization and phase-noise estimation (106), block for hard-decisions (107), and the block for demapping (108) .

Figure 2 depicts a block diagram of the processing chain of an IB-DFE iterative receiver combined with a phase^noise estimator, carrying out the equalization procedure in the f equency-domain. This figure shows the block for joint equalization and phase- noise estimation (106) in greater detail, comprising the following elements: block for iterative frequency-domain equalization (201) , block for. the inverse Fourier transform (201a) , block for symbol decision (201b) , block for delay (201c) , block for the direct Fourier transform at the. feedback chain of the equalizer (20ld) , block for the estimation and compensation of the phase-noise (202) , block for the estimation of the phase-noise (202a), block for counter-rotation (202b), block for rotation (202c) , block for the estimation of the channel parameters and computation of the. equalization coefficients (203), block for the estimation of the channel parameters (203a) , block for the computation of the filtering coefficients (203b) , block for the computation of the feedback coefficients (203c) .

Figure 3 depicts the elements of the feedback chain of the block for iterative frequency-domain equalization (201) that must be included in case channel coding (Turbo equalization) is intended. This figure comprises: a block for soft-demapping (301), a block for de-interleaving (302), a block for SISO decoding (303), a block for interleaving (304), and a block for soft-remapping (305) .

The present invention may find application in future standards of the following technologies:

- WLAN radio interfaces.

- Digital radio broadcasting systems, e.g., DAB.

- Digital television signal broadcasting systems, e.g., DVB, particularly for mobile systems.

- Personal area networks (PAN) type wireless systems.

- Broadband mobile and cellular networks systems.

References

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May 27 ^th 2015