OFDM DUOBINARY TRANSMISSION

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application is a continuation in part of U.S. Patent Application Serial No. 15/833,832, filed December 6, 2017, which claims the benefit of and priority to U.S. Provisional Patent Application Serial No. 62/447,661, filed January 18, 2017, U.S. Provisional Patent Application Serial No. 62/447,665, filed January 18, 2017, U.S. Provisional Patent Application Serial No. 62/447,702, filed January 18, 2017, U.S. Provisional Patent Application Serial No. 62/469,962, filed March 10, 2017, and also to U.S. Provisional Patent Application Serial No. 62/512,913, filed May 31, 2017. This application also claims the benefit of and priority to U.S. Provisional Patent Application Serial No. 62/447,584, filed January 18, 2017, U.S. Provisional Patent Application Serial No. 62/453,710, filed February 2, 2017, and U.S. Provisional Patent Application Serial No. 62/546,119, filed August 16, 2017. The disclosures of all of these prior applications are herein incorporated by reference in their entireties.

BACKGROUND

[0002] The field of the disclosure relates generally to digital transmission systems, and more particularly, to multi-carrier wired, wireless, and optical digital transmission systems.

[0003] Conventional digital transmission systems typically include both linear and non-linear distortion. However, for the purposes of the following discussion, use of the term "distortion" is generally intended to refer to linear distortion only. Conventional digital transmission systems also utilize symbols with coefficients, either in the time domain (TD) or frequency domain (FD), which are generally complex-value sequences. That is, the coefficients of the complex symbols typically include both a real component and an imaginary component, or alternatively, a magnitude and a phase value. Time and frequency domains are related, and the two domains are duals of each other. That is, for a plot or a sequence of numerical values, it must be known whether to observe the plotted numbers as time domain or frequency domain values. [0004] This distinction is of particular significance when considering multi-carrier (MC) digital transmissions, such as with orthogonal frequency division multiplexing (OFDM) and orthogonal frequency division multiple access (OFDMA) transmissions. OFDM symbols, for example, when plotted, appear as discrete values in the frequency domain, but look like random noise in the time domain. In contrast, if a transmission is single carrier (SC), its symbols can be viewed as discrete values in the time domain, but look like random noise in the frequency domain. Thus, multi-carrier and single carrier transmissions are viewed in different domains.

[0005] One type of distortion that is known to severely affect digital transmissions is multipath. Multipath linear distortion is sometimes referred to as "echoes," "ghosts," or "dispersion." An example of an echo distortion is shown in FIG. 1A. FIG. 1A is a schematic illustration depicting a conventional transmission system 100. System 100 includes a transmitter 102 with a transmitting antenna 104 and a receiver 106 with a receiving antenna 108. Transmitter 102 modulates a baseband signal onto a carrier, thereby forming an RF signal 110 for communication over a signal path 112 between transmitter 102 and receiver 106. Data over signal path 112 includes telephony, computer code, file data, network data, internet, world wide web (WWW), entertainment video, video phone, security alarm signals, etc. The data may be widely broadcast, or intended for only the single receiver 106. Signal path 112 includes (i) a direct path portion 114, and (ii) an echo path portion 116 that reflects off of a reflecting object 118. Data traveling over direct path portion 114 results in a direct signal portion 120 at receiver 106, and data traveling over echo path portion 116 results in an echo signal portion 122 at receiver 106. Direct signal portion 120 and echo signal portion are collectively received by receiver 106 as a received signal 124.

[0006] Thus, the presence of echoes from reflections (e.g., from reflecting object 118) causes received signal 124 to be distorted by the presence of echo signal portion 122 combined with a direct signal portion 120. In some cases, direct path portion 114 may be blocked by an obstruction 126 between transmitter 102 and receiver 106, and the only signal reception possible will be from echo signal portion 122 over echo path portion 116. In many cases, the received signal includes multiple echo signals. If the frequency response of the signal path 112 is measured, the presence of a strong echo will cause ripples in the magnitude of the response, where the reciprocal of the frequency difference between peaks is the time delay of the echo from path 116, relative to the direct signal 120.

[0007] On wired signal paths, echoes may alternatively occur from impedance mismatches within coaxial networks. With multipath linear distortion, one or more copies of the original signal - typically with a delay and attenuation - are added to the original signal. On wireless signal paths, the multipath linear distortion, and therefore the attenuated and delayed copies, may be caused by signals reflecting from structures, such as buildings, water towers, etc. In the example of radiated transmissions such as received broadcast analog television pictures, the multipath linear distortion may appear as additional fainter images superimposed on the intended image, along with a typical delay relative to the intended image received over a direct transmission path.

[0008] On single mode fiber optic cable using coherent optical signals, the transmission characteristics are different relative to RF (sub-lOOGHz) wired and wireless signal paths. For example, a 1550nm wavelength laser has an optical frequency of 193el2Hz. In single mode glass fiber, an impairment called Chromatic Dispersion (CD) is encountered, which is similar to group delay encountered, typically from filters, at much lower wired and wireless frequencies. Signals at different frequencies (optical wavelengths) travel at different speeds down the fiber optic cable and this linear distortion needs to be equalized to minimize inter-symbol interference (ISI). This impairment becomes more severe with increasing bandwidth and more severe with longer fiber optic cable spans. This type of signal interference is different from echoes, which are not typically encountered on fiber optic transport media. Coaxial cables and free space, on the other hand, experience virtually no group delay (chromatic dispersion). Thus, CD is created by different wavelengths of a signal traveling at different speeds, whereas echoes are reflected copies of the same signal.

[0009] On optical fiber, different polarizations can be used for simultaneous transmission of two different signals, but because fiber cores sometimes have elliptical shapes, as opposed to round shapes, signals with different polarizations travel at different rates, resulting in an impairment that must be removed at the receiver. Satellites and microwave towers also use vertical and horizontal polarizations for signals propagating in free space, but free space propagation does not experience polarization distortions. In some conventional OFDMA and single-carrier frequency-division multiple-access (SC- FDMA) systems, overlapping signals from multiple transmitters are combined from multiple transmitters at RF frequencies. At optical frequencies, however, these overlapping signal techniques experienced optical beat interference (OBI) problem unless the optical signal phase purity is high. Therefore, echo problems are generally encountered on wired and wireless channels, but not on fiber optic cable (e.g., due to use of angled connectors).

[0010] In the case of digital transmissions, mild multipath distortion will increase the bit error rate (BER) in the presence of random noise or other additive impairments, and may thus be corrected, whereas un-equalized severe multipath distortion, on the other hand, may render the received data useless for further processing at the receiver end. Other types of linear distortions, such as a non-flat frequency response or a group delay, may also affect or impair digital transmissions. These types of distortions may result from imperfect filters and/or amplifier tilt.

[0011] Some conventional techniques have been utilized to eliminate, or at least equalize, multipath linear distortion in the time domain or frequency domain. One conventional time domain technique utilizes an adaptive equalizer to correct for multipath distortion. The adaptive equalizer sums a received distorted signal with a delayed version of the distorted signal, in order to cancel the received echoes. This cancellation process is referred to as equalization, or "de-ghosting." Conventional adaptive equalization schemes sometimes employ a filter architecture such as a finite impulse response (FIR) filter, which utilizes a number of taps (either hardware or software) that are configured to execute a multiply-accumulate operation, and programs coefficients in the adaptive equalizer to cancel received echoes. Such programming may utilize a particular reference signal, also referred to as a training signal or "ghost-canceling" signal, so that the programmed coefficients are computed as an inverse of the channel response. The frequency and the impulse response of the channel may also be determined as intermediate steps, and, in the time domain, tap coefficients are computed as the reciprocal of the impulse response. In some instances, the adaptive equalizer is programmed using blind equalization techniques. An exemplary FIR filter architecture is disclosed in U.S. Patent No. 5,886,749, which is incorporated by reference herein.

[0012] Echoes on cable lines can come in two varieties: single recursion and multiple (infinite) recursion. An infinite recursion occurs, for example, by an echo tunnel as a signal bounces back and forth in the tunnel, getting weaker on each pass. FIG. IB is a graphical illustration depicting examples of conventional time domain plots 128 for a multiple recursion 130 and a single recursion 132 on an echo signal path, and respective adaptive equalizer solutions (impulse response) 134 implemented to correct the encountered recursion 130 or 132. As shown in illustration 128, the adaptive equalizer solution 134 _M that corrects multiple recursion echo 130 though, is itself a single recursion. In contrast, where the main signal path experiences single recursion 132 (e.g., as can happen with two signal paths), the adaptive equalizer solution 134s that corrects single recursion echo 132 though, is a multiple recursion. In the conventional systems, multiple recursion solution 134 _M requires more taps, or more time, than would single recursion solution 134s to remove an echo. If the echo is stronger, even more taps, with significant energy, are required.

[0013] Conventional adaptive equalization techniques, however, are significantly limited with respect to complex signals that require processing of a large number of coefficients (e.g., greater than 8 or 64). Because the FIR filter utilizes linear convolution, the rate of required computations increases as the product of the number of taps, multiplied by the clocking speed of the taps. For hardware-implemented taps, a large number of coefficients will significantly increase the cost and size of the physical structure that performs the adaptive equalization. For software-implemented taps, an exponential increase will significantly increase the required processing speed of the programming that performs computations. Large quantities of taps are particularly necessary where, for example, the clock frequency of the equalizer is high, and where a received echo is long.

[0014] Some conventional techniques attempt to solve the multiple- computational problems associated with adaptive equalization by taking a received distorted signal in the time domain, separating the time domain signal into blocks, and then transforming the blocks into the frequency domain, such as through use of a fast Fourier transform (FFT). The transformed blocks thus become sets of frequency domain subcarriers, and each frequency domain subcarrier may then be multiplied by a single complex coefficient to remove the associated linear distortion. This process is known as frequency domain equalization (FDE).

[0015] Once the linear distortion is removed, the frequency domain blocks are converted back into time domain blocks. A problem occurs, however, if the blocks are contaminated with foreign or extraneous energy, such as might occur from echo energy transporting distortion from a previous block. Conventional systems address this energy transportation problem through use of cyclic prefixes (CP), which are a set of time domain symbols copied from the end of the block and pasted onto the front of the block.

[0016] FIG. 2A is a graphical illustration of a time domain signal 200 of a conventional OFDM transmission 202 having a cyclic prefix 204, utilized in accordance with transmission system 100, FIG. 1A. Typically, an OFDM carrier signal is the sum of one or more OFDM symbols, each symbol made up of a plurality of orthogonal subcarriers, and with baseband data on each subcarrier being independently modulated. In an embodiment, OFDM transmission 202 is a carrier signal transmitted using technology such as the Data Over Cable Service Interface Specification (DOCSIS), version 3.1, or one or more of many known and burgeoning wireless standards. As described above, OFDM implements a plurality of different subcarriers, all of which are harmonics of a fundamental, to obtain orthogonality.

[0017] DOCSIS specifications conventionally utilize OFDM for downstream signals and OFDMA for upstream signals. OFDM and OFDMA are complimentary. OFDM is typically used in the downstream transmission where there is one transmitter (e.g., a Cable Modem Termination System (CMTS)) sending information to multiple receivers (e.g., cable modems). OFDMA is typically used in the upstream transmission where there are multiple transmitters (e.g., the cable modems) transmitting to one receiver (e.g., the CMTS). The cyclic prefix is therefore commonly used in both OFDM and OFDMA. [0018] The cyclic prefix functions as a "guard time" that separates data bursts, and allows any micro-reflection from one burst to die out before the next burst is received, thereby eliminating interference from one block to the next. The cyclic prefix is therefore considered particularly essential over an HFC network, where reflections frequently occur. Reflections are often created by impedance mismatches on the HFC network and result from a number of issues, including manufacturing tolerances of passive hardware such as taps, power inserters, and splitter, active hardware such as amplifiers, and connectors. Combined multiple micro-reflections therefore, create linear distortions, which cause a number of impairments to signal transmission, including amplitude ripple (standing waves), group delay ripple, inter-symbol interference, and degraded modulation error ratio (MER) on digital signals transmitted on the HFC network. For a conventional multicarrier equalization processes, the cyclic prefix should be longer than any echo in the signal path. Accordingly, various durations of cyclic prefixes are utilized to accommodate a variety of echo delays in the HFC network, thereby significantly increasing the overhead of the network, but without carrying any useful customer information in the transmission of the cyclic prefix.

[0019] OFDM transmission 202 includes, for example, first harmonic subcarrier 206, second harmonic subcarrier 208, third harmonic subcarrier 210, and fourth harmonic subcarrier 212. For ease of explanation, only four such subcamers of OFDM transmission 202 are illustrated, but typically, many more subcarriers exist. Each of the harmonically-related subcarriers 206, 208, 210, 212 may have different magnitude and phase values. When all four subcarriers 206, 208, 210, 212 are combined, or summed, for transmission, the summed result is a single composite signal 214.

[0020] Orthogonality allows each of the original subcarriers to be separated at the receiver (e.g., receiver 106, FIG. l), such as with utilization of Fourier processes. That is, a discrete Fourier transform (e.g., an FFT) converts a set of time domain values into frequency domain values, and an inverse discrete Fourier transform (e.g., an IFFT) converts frequency domain values into time domain values. The larger the transform, the greater is the efficiency improvement by using an FFT, as opposed to a simple discrete Fourier transform. The FFT block sizes may be efficiently implemented with a block size of radix 2 (2 ^A n, where n is an integer), such as 256, or 1024, but other efficient transform block sizes are possible. Cyclic prefix 204 is created by copying symbols from an end region 216 (shown shaded in gray) of composite signal 214, and pasting the copied signals onto the beginning region thereof as a guard interval. Cyclic prefix 204 thus allows circular convolution or FDE to be performed on OFDM transmission 202, without suffering interference from a previous OFDM block if there is an echo on the channel. However, for this example, it is assumed that the echo is shorter than cyclic prefix 204.

[0021] FIG. 2B is a graphical illustration of a frequency domain signal 218 of OFDM transmission 202, FIG. 2A. That is, frequency domain signal 218 represents a spectral plot of OFDM transmission 202 in the frequency domain. Frequency domain signal 218 may be obtained by performing a discrete Fourier transform or an FFT on composite time domain signal 214, FIG. 2 A. As illustrated in FIG. 2B, each of time domain harmonic subcarriers 206, 208, 210, 212 has a respective frequency domain component subcarrier 220, 222, 224, 226, each having both a magnitude and a phase value. Under these conventional techniques, OFDM transmissions may be viewed in either the time domain or the frequency domain. However, composite signal 214 appears noise-like, or random in the time domain, which is problematic for multiple-carrier symbols. In comparison, some single-carrier symbols, such as single-carrier frequency-division multiple-access (SC-FDMA) can also be viewed either in the time domain or the frequency domain, but such single-carrier symbols resemble noise when viewed in the frequency domain. Accordingly, it is desirable to develop systems and methods capable of receiving, equalizing, and utilizing the same multiple-carrier or multiple-access symbols in both the time domain and the frequency domain.

[0022] Cyclic prefixes provide block-to-block isolation, and thus transmission of digital information is often performed with blocks of data. Some conventional systems use linear code, such as a Reed-Solomon code or Low-Density Parity Check (LDPC) codes, for purposes of forward error correction (FEC), which allows a percentage of erred symbols to be corrected by the code. Use of cyclic prefixes, however, requires additional resources to transmit the extra data that constitutes the cyclic prefix. The required cyclic prefix data reduces the bandwidth efficiency of transmissions, thereby limiting the amount of data that can be transmitted within a given frequency band, while also requiring additional power and decreasing the battery life of system components. Moreover, as described above, cyclic prefixes are not completely effective in the case where the cyclic prefix portion of time domain data is shorter than the length, or duration, of a received echo.

BRIEF SUMMARY

[0023] In an embodiment, a signal equalizing receiver is configured to capture a plurality of OFDM symbols transmitted over a signal path adding linear distortion to the plurality of OFDM symbols, and form the plurality of captured OFDM symbols into an overlapped compound data block. The compound data block includes at least one pseudo-extension in addition to payload data from at least one of the plurality of OFDM symbols. The receiver is further configured to process the overlapped compound data block with one of (i) a circular convolution having an inverse channel response in the time domain, and (ii) a frequency domain equalization in the frequency domain, to produce an equalized compound block, and discard at least one end portion of the equalized compound block to produce a narrow equalized block. The at least one end portion corresponds with the at least one pseudo-extension, and the narrow equalized block corresponds with the payload data. The receiver is further configured to cascade two or more narrow equalized blocks to form a de-ghosted signal stream of the plurality of OFDM symbols. The plurality of OFDM symbols includes one or more of an OFDM transmission and an OFDMA transmission. The plurality of OFDM symbols further includes one or more of a cyclic prefix and no cyclic prefix. A length of the at least one pseudo extension is different than a length of the cyclic prefix.

[0024] In an embodiment, a digital transmission receiver has a processor and a memory. The receiver is configured to receive a digital signal transmission from a signal path including a plurality of data blocks having linear distortion, determine, from the signal path of the digital signal transmission, a duration of at least one reflection of the digital signal transmission on the digital signal path, and attach a pseudo-extension to a first data block of the plurality of data blocks. The length of the pseudo-extension in the time domain is greater than the duration of the at least one reflection. The receiver is further configured to process the first data block, together with the pseudo-extension attached thereto, to remove linear distortion from the first data block, and discard the processed pseudo-extension from the processed first data block after the linear distortion has been removed.

[0025] In an embodiment, a digital transmission system includes a transmitter configured to transmit orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto, a receiver for receiving the transmitted OFDM symbols from the transmitter, and a signal path for communicating the transmitted OFDM symbols from the transmitter to the receiver. The OFDM symbols received by the receiver include linear distortion from the signal path, and the receiver is configured to process the received OFDM symbols and linear distortion using an overlapped circular convolution function to produce equalized OFDM symbols.

[0026] In an embodiment, a digital transmission system includes a transmitter configured to transmit orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto, a receiver for receiving the transmitted OFDM symbols from the transmitter, and a signal path for communicating the transmitted OFDM symbols from the transmitter to the receiver. The OFDM symbols received by the receiver include linear distortion from the signal path, and the receiver is configured to process the received OFDM symbols and linear distortion by an overlapped Fourier transform function to produce equalized OFDM symbols. The overlapped Fourier transform function is configured to (i) overlap individual ones of the distorted OFDM symbols with overlapped time energy from respectively adjacent ones of the distorted OFDM symbols, (ii) transform the overlapped individual distorted OFDM symbols into distorted frequency domain symbols, (iii) perform complex multiplication of the distorted frequency domain symbols by equalization coefficients to equalize the distorted frequency domain symbols, (iv) remove the overlapped time energy from a time domain component of the equalized frequency domain symbols, and (v) produce undistorted frequency domain symbols from a frequency domain component of the time domain component with the overlapped time energy removed.

[0027] In an embodiment, a digital transmission system includes a transmitter configured to transmit (i) a series of orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto, and (ii) at least one constant amplitude zero autocorrelation waveform sequence (CAZAC) sequence. The system further includes a receiver for receiving the transmitted series of OFDM symbols and the CAZAC sequence from the transmitter, and a signal path for communicating the transmitted series of OFDM symbols and CAZAC sequence from the transmitter to the receiver. The series of OFDM symbols and the CAZAC sequence are received by the receiver with linear distortion from the signal path, and the receiver is configured to utilize the received CAZAC sequence as a reference signal for equalizing the received series of OFDM symbols.

[0028] In an embodiment, a method of equalizing a transmitted digital signal includes steps of receiving, in the time domain, a sequential series of first, second, and third data blocks of the transmitted digital signal, forming a compound block in the time domain from the second data block including an end portion of first data block and a leading portion of the third data block, performing circular convolution on the compound block using a set of equalization coefficients to equalize the compound block in the time domain, extracting from the equalized compound block a narrow block corresponding to equalized time domain data of the second data block, converting the narrow block from the time domain into frequency domain data, and reading frequency domain symbols relating to the second data block from the converted narrow block.

[0029] In an embodiment, a method of equalizing a transmitted digital signal includes steps of receiving, in the time domain, a sequential series of time domain samples of the transmitted digital signal, forming the received sequential series of time domain samples into a separate sub-series of overlapping compound time domain blocks, wherein each compound time domain block of the sub-series includes a pseudo-prefix comprising information from an immediately preceding block, determining an echo delay on the signal path, converting the compound blocks into the frequency domain to form compound frequency domain blocks, equalizing the compound frequency domain blocks to form equalized frequency domain blocks, converting the equalized frequency domain blocks into the time domain to form equalized time domain compound blocks, discarding, from the equalized time domain compound blocks, overlapping time domain energy portions corresponding to respective equalized pseudo-prefixes, to form narrow equalized blocks, pasting the narrow equalized blocks together to form a composite equalized time domain signal, and converting the composite equalized time domain into the frequency domain and read equalized frequency domain symbols therefrom.

[0030] In an embodiment, a method of modulating a series of input digital symbols of a first modulation scheme is provided. The method is implemented by a transmitter and includes a step of receiving a sequential series of samples of the digital symbols in a first domain of the first modulation scheme. The first domain is one of the time domain and the frequency domain. The method further includes a step of determining a dual of the first modulation scheme. The dual has a second modulation scheme in a second domain that is different from the first domain the second domain is the other of the time domain and the frequency domain. The method further includes steps of applying a 90 degree rotational operation to the second modulation scheme to generate a rotational modulation format, modulating the series of digital symbols with the generated rotational modulation format, and outputting the modulated series of digital symbols to a receiver.

[0031] In an embodiment, a digital transmission system includes a transmitter configured to transmit an input series of complex symbols, a duobinary encoder disposed within the transmitter, and configured to filter the input series of complex symbols and output a partial response signaling (PRS) signal, a converter disposed within the transmitter, and configured to convert the PRS signal output into the time domain, and a receiver for receiving the time domain-converted PRS signal from the transmitter over a signal path.

[0032] In an embodiment, a method of modulating an input digital signal transmission is provided. The method is implemented by a transmitter and includes steps of receiving the input digital signal having a first time-frequency order on the time- frequency axis, rotating the time-frequency axis by 90 degrees, modulating the input digital signal according to the rotated time-frequency axis, and outputting the modulated digital signal to a receiver.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] These and other features, aspects, and advantages of the present disclosure will become better understood when the following detailed description is read with reference to the following accompanying drawings, in which like characters represent like parts throughout the drawings.

[0034] FIG. 1A is a schematic illustration depicting a conventional transmission system.

[0035] FIG. IB is a graphical illustration depicting examples of conventional time domain plots for a multiple recursion and a single recursion echoes on a signal path, and respective adaptive equalizer solutions implemented to correct the encountered recursion.

[0036] FIG. 2A is a graphical illustration of the time domain signal of a conventional OFDM transmission having a cyclic prefix, utilized in accordance with the example depicted in FIG. 1.

[0037] FIG. 2B is a graphical illustration of the frequency domain signal of the conventional OFDM transmission depicted in FIG. 2A.

[0038] FIG. 3 is a schematic illustration depicting an exemplary multi- carrier transmission system, according to an embodiment.

[0039] FIG. 4 is a schematic illustration depicting an exemplary equalization scheme for an OFDM symbol without a cyclic prefix, utilizing overlapped circular convolution, in an embodiment.

[0040] FIG. 5 is a flow chart diagram of an exemplary circular convolution process for the embodiment depicted in FIG. 4.

[0041] FIG. 6A is a schematic illustration depicting an exemplary equalization scheme utilizing an overlapped Fourier transform, in an alternative embodiment.

[0042] FIG. 6B is a graphical illustration of a histogram depicting a distribution of echo duration for a large population of signal paths. [0043] FIG. 7 is a flow chart diagram of an exemplary overlapped Fourier transform process for the embodiment depicted in FIG. 6.

[0044] FIG. 8 is a graphical illustration of a plot of a transmitted signal having in-phase and quadrature components, according to an embodiment.

[0045] FIG. 9 is a computer program listing demonstrating an exemplary coding for implementing the Zadoff Chu sequence depicted in FIG. 8.

[0046] FIG. 10 is a graphical illustration depicting a frequency domain spectral plot of the Zadoff Chu sequence depicted in FIG. 8.

[0047] FIG. 11 is a flow chart diagram of an exemplary process for simulating an idealized transmitter-receiver chain for an OFDM transmission having no cyclic prefix, according to an embodiment.

[0048] FIG. 12 is a flow chart diagram of an alternative process for a simulated transmitter-receiver chain for an OFDM transmission having no cyclic prefix, according to an embodiment.

[0049] FIG. 13 A is a flow diagram depicting an alternative process for operating the receiver depicted in FIG. 3.

[0050] FIG. 13B is a graphical illustration depicting alternative frequency plots of subcarrier schemes that may be implemented with the process depicted in FIG. 13A.

[0051] FIG. 14 is a graphical illustration depicting a comparative overlay of an OFDM sine function implementing a duobinary technique, according to an embodiment.

[0052] FIG. 15 is a graphical illustration depicting a comparative overlay of a duobinary OFDM block transmission with a conventional OFDM block transmission.

[0053] FIG. 16 is a graphical illustration depicting a block time-frequency plot. [0054] FIG. 17A is a graphical illustration depicting a time domain plot of a duobinary block transmission.

[0055] FIG. 17B is a graphical illustration depicting a frequency domain plot of the duobinary block transmission depicted in FIG. 17A.

[0056] FIG. 18A is a graphical illustration depicting an impulse response of a single carrier transmission.

[0057] FIG. 18B is a graphical illustration depicting a spectral response of the single carrier transmission depicted in FIG. 18 A.

[0058] FIG. 18C is a graphical illustration depicting an impulse response of a duobinary transmission.

[0059] FIG. 18D is a graphical illustration depicting a spectral response of the duobinary transmission depicted in FIG. 18C.

[0060] FIG. 19A is a graphical illustration of a constellation depicting relative power calculations for error thresholds of the single carrier transmission depicted in FIGS. 18A-B.

[0061] FIG. 19B is a graphical illustration of a constellation depicting relative power calculations for error thresholds of the duobinary transmission depicted in FIGS. 18C-D.

[0062] FIG. 20 is a graphical illustration depicting a diagram of single carrier voltage versus time.

[0063] FIG. 21 is a graphical illustration depicting a timing diagram for a spread spectrum signal.

[0064] FIG. 22 is a schematic illustration depicting an exemplary block diagram of a system for direct sequence spread transmission and reception.

[0065] FIG. 23 is a graphical illustration depicting a received OFDM signal in the frequency domain. [0066] FIG. 24 is a graphical illustration depicting time and frequency relationships between common transmission impairments.

[0067] FIG. 25 is a graphical illustration depicting an OFDM modulation matrix.

[0068] FIG. 26A is a graphical illustration depicting a Hamming window in the time domain, according to an embodiment.

[0069] FIG. 26B is a graphical illustration depicting an inverse Hamming function in the time domain, according to an embodiment.

[0070] FIG. 26C is a graphical illustration depicting an implementation of a Hamming window on a time domain waveform, according to an embodiment.

[0071] FIG. 27A is a graphical illustration depicting a Hamming impulse response in the frequency domain, according to an embodiment.

[0072] FIG. 27B is a graphical illustration depicting an inverse convolution impulse response in the frequency domain, according to an embodiment.

[0073] FIG. 28A is a graphical illustration depicting an unequalized constellation on which a Hamming window has been implemented.

[0074] FIG. 28B is a graphical illustration depicting an equalized constellation on which a Hamming window has been implemented.

[0075] FIG. 29A is a graphical illustration depicting an implementation of a raised cosine voltage function on a time domain waveform, according to an embodiment.

[0076] FIG. 29B is a graphical illustration depicting an unequalized constellation on which a raised cosine function has been implemented.

[0077] FIG. 29C is a graphical illustration depicting a constellation, after equalization, on which a raised cosine function has been implemented. [0078] Unless otherwise indicated, the drawings provided herein are meant to illustrate features of embodiments of this disclosure. These features are believed to be applicable in a wide variety of systems including one or more embodiments of this disclosure. As such, the drawings are not meant to include all conventional features known by those of ordinary skill in the art to be required for the practice of the embodiments disclosed herein.

DETAILED DESCRIPTION

[0079] In the following specification and the claims, reference will be made to a number of terms, which shall be defined to have the following meanings.

[0080] The singular forms "a," "an," and "the" include plural references unless the context clearly dictates otherwise.

[0081] "Optional" or "optionally" means that the subsequently described event or circumstance may or may not occur, and that the description includes instances where the event occurs and instances where it does not.

[0082] Approximating language, as used herein throughout the specification and claims, may be applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term or terms, such as "about," "approximately," and "substantially," are not to be limited to the precise value specified. In at least some instances, the approximating language may correspond to the precision of an instrument for measuring the value. Here and throughout the specification and claims, range limitations may be combined and/or interchanged; such ranges are identified and include all the sub-ranges contained therein unless context or language indicates otherwise.

[0083] As used herein, the terms "processor" and "computer" and related terms, e.g., "processing device", "computing device", and "controller" are not limited to just those integrated circuits referred to in the art as a computer, but broadly refers to a microcontroller, a microcomputer, a programmable logic controller (PLC), an application specific integrated circuit (ASIC), and other programmable circuits, and these terms are used interchangeably herein. In the embodiments described herein, memory may include, but is not limited to, a computer-readable medium, such as a random access memory (RAM), and a computer-readable non-volatile medium, such as flash memory. Alternatively, a floppy disk, a compact disc - read only memory (CD-ROM), a magneto- optical disk (MOD), and/or a digital versatile disc (DVD) may also be used. Also, in the embodiments described herein, additional input channels may be, but are not limited to, computer peripherals associated with an operator interface such as a mouse and a keyboard. Alternatively, other computer peripherals may also be used that may include, for example, but not be limited to, a scanner. Furthermore, in the exemplary embodiment, additional output channels may include, but not be limited to, an operator interface monitor.

[0084] Further, as used herein, the terms "software" and "firmware" are interchangeable, and include any computer program storage in memory for execution by personal computers, workstations, clients, and servers.

[0085] As used herein, the term "non-transitory computer-readable media" is intended to be representative of any tangible computer-based device implemented in any method or technology for short-term and long-term storage of information, such as, computer-readable instructions, data structures, program modules and sub-modules, or other data in any device. Therefore, the methods described herein may be encoded as executable instructions embodied in a tangible, non-transitory, computer readable medium, including, without limitation, a storage device and a memory device. Such instructions, when executed by a processor, cause the processor to perform at least a portion of the methods described herein. Moreover, as used herein, the term "non-transitory computer- readable media" includes all tangible, computer-readable media, including, without limitation, non-transitory computer storage devices, including, without limitation, volatile and nonvolatile media, and removable and non-removable media such as a firmware, physical and virtual storage, CD-ROMs, DVDs, and any other digital source such as a network or the Internet, as well as yet to be developed digital means, with the sole exception being a transitory, propagating signal. [0086] Furthermore, as used herein, the term "real-time" refers to at least one of the time of occurrence of the associated events, the time of measurement and collection of predetermined data, the time for a computing device (e.g., a processor) to process the data, and the time of a system response to the events and the environment. In the embodiments described herein, these activities and events occur substantially instantaneously.

[0087] As described herein, signal equalization in the time domain may be performed at a receiver by capturing symbols of the received symbol into an overlapped compound block and circularly convolving the compound block with an inverse channel response to produce an equalized compound block. End portions of the equalized compound block may then be discarded to produce a narrow equalized block, and a plurality of such narrow equalized blocks may further then be cascaded to form a de- ghosted signal stream. Alternatively, the overlapped compound block may be processed in the frequency domain using frequency domain equalization. In an exemplary embodiment, cyclic prefixes are not sent on OFDM or OFDMA multicarrier transmissions. In alternative embodiments, the present techniques may be implemented on any type of linearly distorted modulation transmission without a cyclic prefix (also known as guard intervals or cyclic extensions), or transmissions using relatively short cyclic prefixes.

[0088] In the exemplary embodiment, systems and methods herein are configured to transmit OFDM and OFDMA signals through wired and wireless channels without considering, or having to create, cyclic prefixes. In some embodiments, overlapped circular convolutions are performed at the receiver end of the system to eliminate linear distortion, such as echoes. In such embodiments, the overlapped circular convolution may be programmed with equalization coefficients obtained, for example, from pilot subcarriers or from a training-synchronization signal, such as a constant amplitude zero autocorrelation (CAZAC) signal. Each received frequency coefficient may then be corrected by a single complex multiplication to correct the respective magnitude and phase. In at least one embodiment, a Zadoff Chu sequence functions as the training- synchronization signal. [0089] In other embodiments, overlapped Fourier transforms and inverse Fourier transforms are performed at distinct signal conversion stages, and the pseudo- prefixes are determined at the receiver end of the system according to predetermined pilot/training signals, or from other signal energy received from the transmitter. For example, irrespective of whether a transmitter creates a new cyclic prefix to serve as a guard interval between transmitted data, the present systems and methods utilize the receiver to determine a pseudo-prefix from random or noise-like portions of the transmission (by performing an autocorrelation), or from training signals, or pilots . Therefore, the present systems and methods are equally effective for receiving digital transmissions that include, or do not include, cyclic prefixes from the transmitter. Accordingly, by eliminating (or ignoring) the need for cyclic prefixes, the bandwidth efficiency of transmissions may be increased to allow a frequency band to transport more data within a time-frequency resource, and thereby reduce power consumption and increase the battery life of system components.

[0090] The embodiments herein are of also of particular use with respect to OFDM signals, which are sent in blocks, and include a number of harmonics that are orthogonal to each other because of their integer relationship to a fundamental frequency. By varying the phase and the magnitude of the harmonics, information can be transmitted while preserving the orthogonality between each of the harmonics. As described in greater detail below, echoes may affect OFDM signals, but can be corrected in the frequency domain. Conventional systems only correct for the effects of inter-block distortion by using a time domain cyclic prefix. Correction schemes using cyclic prefixes, however, are only fully effective if the delay of the echo is shorter than the duration of the cyclic prefix/guard interval. In multi-carrier transmission schemes though, the length of echo delay can vary from receiver to receiver, or according to the respective distances from reflection points in a signal path. The present embodiments address and solve this multicarrier echo variance problem by pseudo-prefix determination at the receiver end.

[0091] As described herein, transmission efficiency is increased by eliminating the cyclic prefix from transmitted OFDM or OFDMA blocks. Alternatively, reception efficiency may also be increased by configuring the receiver such that the cyclic prefix may be disregarded, if a CP is included in a digital transmission from the transmitter. In an exemplary embodiment, equalization is accomplished in the time domain through use of an overlapped circular convolution process, which overlaps a selected block with surrounding information to form a larger compound, or "fat," block. The compound block is then circularly-convolved with programmed time domain coefficients to remove distortion, and thus equalize the compound block. After equalization, the compound block is "trimmed" into an equalized narrow block, which is then converted into the frequency domain. OFDM symbols may then be determined from the equalized, narrow, converted frequency domain block.

[0092] In an alternative embodiment, the cyclic prefix is eliminated in the frequency domain through use of an overlapped Fourier transform. In this alternative process, a large compound time domain block of an OFDM carrier is transformed into the frequency domain, and then equalized by complex multiplication FDE sub-process. This equalized compound frequency domain block is transformed back into the time domain, and then trimmed to form an equalized, time domain, narrow block. In an exemplary embodiment, because the original signal is an OFDM transmission, and additional Fourier transform (e.g., an FFT) is performed on the narrow block so that the OFDM symbols may be determined.

[0093] Use of cyclic prefixes is known in the art to provide more than only block-to-block isolation. For example, if a received signal is cross-correlated with a delayed copy of itself, a phase shift on a resulting correlation peak will reveal any offset frequency between the transmitter and the receiver, and correction may then be applied. However, as described further below, this additional functionality from the cyclic prefix is advantageously replaced by a constant amplitude zero autocorrelation (CAZAC) function signal, such as a Zadoff Chu sequence. In some embodiments, the Zadoff Chu sequence is implemented together with channel characterization for further equalization purposes, as well as block timing information.

Overlapped Circular Convolution

[0094] FIG. 3 is a schematic illustration depicting an exemplary multi- carrier transmission system 300. System 300 includes a first transmitter 302 having a first transmitting antenna 304 and a second transmitter 306 having a second antenna 308. First and second transmitters 302, 306 transmit first and second carrier signals 310, 312, respectively, over communication channels 314 to a first receiver 316 and a second receiver 318. In the example shown in FIG. 3, two receivers and two transmitters are illustrated for simplicity of explanation. In operation, many more transmitters and/or receivers may be implemented within system 300.

[0095] Alternatively, the embodiments described herein advantageously operate in the case where multiple receivers are receptive to different carrier signals from multiple transmitters and also in the case where a multiple transmitters deliver a number of different carrier signals to a plurality of antennas (e.g., multiple-input/multiple-output (MIMO). Accordingly, communication channel 314 may include wired signal paths, wireless signal paths, or combination of both. In the example shown in FIG. 3, first receiver 316 receives a composite signal 320. In at least some embodiments, composite signal 320 further includes at least one echo 322 of first carrier signal 310(1), reflected off of a reflecting object 324. For ease of explanation, potential reflections of first carrier signal 310(2) and second carrier signal 312 are not shown.

[0096] FIG. 4 is a schematic illustration depicting an exemplary equalization scheme 400 for equalizing an OFDM symbol without a cyclic prefix, utilizing overlapped circular convolution. In an exemplary embodiment, equalization scheme 400 is implemented by a receiver or a processor thereof (e.g., first receiver 316, FIG. 3), and equalization is performed in the time domain. A received time domain OFDM carrier signal 402 includes a series of time domain blocks 404, 406, 408, which are labeled herein as "W" block 404, "X" block 406, "Y" block 408, respectively. Each of W block 404, X block 406, and Y block 408 contains potential distortion, and includes OFDM data symbols, but no cyclic prefixes. For ease of explanation, only three blocks are shown in this example; however, OFDM carrier signal 402 may include significantly more blocks in the series, or be continuous in the time domain.

[0097] In operation, the receiver/processor selects X block 406, and forms a compound block 410, which includes all of X block 406, an end portion 412 of W block 404, and a beginning portion 414 of Y block 408. In this example, end portion 412 and beginning portion 414 are illustrated to be one fraction (k) of W block 404 and a different fraction (m) of Y block 408, respectively. Nevertheless, a person of ordinary skill in the art will understand that end portion 412 and beginning portion 414 may constitute larger or smaller portions of the respective origin blocks, equal portions, or portions of unequal length. End portion 412 thus functions as a pseudo-prefix, in the time domain, for X block 406. Pseudo-prefixes function differently from conventional cyclic prefixes. Whereas a conventional cyclic prefix for X block 406 would require a repetition of information from the trailing portion of the X block itself, the present pseudo-prefix is based on information from the preceding W block 404, and therefore requires no repetitive data. In a similar manner, beginning portion 414 of the following Y block 408 functions as a "pseudo-suffix" in the time domain for X block 406. Pseudo-prefixes and pseudo- suffixes are generally referred to herein as "pseudo-extensions." In this example, blocks W, X, and Y are all subject to a same linear distortion.

[0098] In further operation, compound block 410 is convolved with equalization coefficients 416 (eight coefficients, C0-C7, are illustrated; however much larger numbers of coefficients, e.g., 64 or more, may be processed according to this embodiment) to form an equalized convolution block 418, which remains in the time domain. That is, each term in compound block 410 is multiplied by the respective coefficient 416 determine beneath it, and the products are summed. A circular shift (to the right, as illustrated in FIG. 4) of the top row (i.e., compound block 410) is performed, in the process is repeated until the circular convolution is complete. Data from equalized convolution block 418 is subjected to one or more circular convolutions. Nevertheless, the size of equalized convolution block 418 remains the same before and after the circular convolution process(es). After circular convolution, symbols from end portion 412 and beginning portion 414 (associated with W block 404 and Y block 408, respectively) are trimmed from equalized convolution block 418 to form a narrow equalized block 420, or X' block 420, which represents a de-ghosted version of X block 406. That is, the time domain pseudo-prefix and pseudo-suffix information is removed, and X' block 420 may then be converted into the frequency domain, after which the OFDM symbols may be determined (step not illustrated). [0099] In an exemplary embodiment, equalization scheme 400 is implemented sequentially for each of the series of distorted time domain blocks 404, 406, 408. That is, after the formation of X' block 418, equalization scheme 400 similarly forms a compound block (not shown) from Y block 408 (e.g., using portions from preceding X block 406 and a following (Z) block, also not shown), and performs circular convolution thereupon to create an equalized Y' block (not shown) nominally using the same coefficients. In an alternative embodiment, equalization scheme 400 may be performed on individual blocks of OFDM carrier signal 402 non-sequentially, that is, Y block 408 may be equalized before W block 404. In at least one embodiment, some blocks of OFDM carrier signal 402 may be equalized, while equalization is not performed other blocks according to equalization scheme 400. Conventional (non-overlapped) multi-carrier technology of OFDM encounters problems in the implementation of circular convolutions, in that extraneous energy from one block contaminates adjacent blocks. According to the embodiments described herein, this inter-symbol interference is eliminated.

[00100] According to equalization scheme 400, which uses an overlapped circular convolution, linear distortion may be removed from any received signal in the time domain irrespective of whether the received signal contains a CP, and irrespective of the underlying modulation used. The received signal may be viewable in the time domain (e.g., single-carrier or SC-FDMA), or in the frequency domain (e.g., OFDM and OFDMA), spread-spectrum or wavelet based. In some embodiments, pseudo-extensions 412, 414 may be or include quiet time, another type of signal suffering the same linear distortion, an unused cyclic prefix, or a training signal. The signal may even be a baseband, such as an audio signal having an echo. In the exemplary embodiment, the pseudo-extensions are longer than the relevant encountered echo and all recursions in the echo solution. That is, the respective pseudo-prefix is longer than the trailing echo(es), and the respective pseudo- suffix is longer than the leading echo(es) plus recursions.

[00101] FIG. 5 is a flow chart diagram of an exemplary circular convolution process 500 for equalization scheme 400, FIG. 4. In the exemplary embodiment, process 500 is implemented by a receiver or the processor thereof (e.g., first receiver 316, FIG. 3), and the equalization similarly occurs in the time domain with an overlapped circular convolution. Process 500 begins at step 502, in which a digital transmission signal (e.g., OFDM carrier signal 402, FIG. 4) is received, including a series of sequential OFDM W, X, Y blocks (e.g., time domain blocks 404, 406, 408, FIG. 4). In step 504, a compound block (e.g., compound block 410, FIG. 4) is created from the entirety of the X block, an end portion of the W block, and the beginning/lead portion of the Y block. In step 506, circular convolution is performed on the compound block and its respective coefficients (e.g., coefficients 416, FIG. 4). In step 508, an equalized narrow block (e.g., narrow block 418, FIG. 4) is extracted from the compound block by trimming/discarding the W (pseudo-prefix) and Y (pseudo- suffix) portions of the convolved compound block. In step 510, the equalized narrow block is converted into the frequency domain (e.g., by an FFT), and the OFDM symbols are read from the frequency- converted narrow block. In step 512, steps 502 through 510 are optionally repeated for the next block in the sequence. That is, in step 512, a compound block is formed from the entirety of the Y block, together with portions of the unequalized X block and "Z" block, to create an equalized Y' block. Alternatively, in step 512, a distorted block other than the Y block may be equalized. If step 510 is optionally eliminated, and the signal being received is single-carrier, the narrow block will contain the equalized time domain symbols.

[00102] The present embodiments thus overcome the known problems associated with the use of cyclic prefixes, described above. By implementing the present processing techniques at the receiver end, a digital signal transmission system is able to overcome failures that would occur in a multi-carrier scenario when a cyclic prefix is too short, that is, has a shorter duration than an echo/reflection in the time domain. When the cyclic prefix is too short, the conventional receiver may fail. In contrast, by implementing a pseudo-prefix that is adjustable in time and adjustable according to echo duration, the receiver according to the present systems and methods will continue to function irrespective of the length - or existence - of the cyclic prefix.

Overlapped Fourier Transform

[00103] FIG. 6A is a schematic illustration depicting an alternative equalization scheme 600 utilizing an overlapped Fourier transform. Similar to equalization scheme 400, FIG. 4, equalization scheme 600 may be implemented in accordance with system 300, FIG. 3, and eliminates the need for cyclic prefixes, implementing frequency domain equalization for computational efficiency on large transforms.

[00104] In operation, a receiver (e.g., first receiver 316, FIG. 3) of scheme 600 receives a series of OFDM symbols 602 displayed in the time domain. As described above, in conventional OFDM transmission, the energy transportation problem between blocks of data, also known as inter-block interference (IBI) or "leakage," is addressed by the use of cyclic prefixes. In single-carrier optical implementations, overlap frequency domain equalization (OFDE) has been proposed to eliminate the cyclic prefixes. However, such single-carrier OFDE implementations have not accounted for the variable length of echoes, and particularly the type of echo variability and dispersion that can occur in a multi-carrier system. The conventional OFDE techniques set arbitrary values to the amount of time domain overlap (CP length) based on anticipated chromatic dispersion, and thus may not obtain leakage-free transforms because the echo may be longer than anticipated. Echo energy, for example, may persist longer than the discarded portion 412.

[00105] In contrast to these conventional techniques, scheme 600 forms a compound block 604 from OFDM time domain symbols 602 such that compound block 604 is of sufficient size to allow an encountered echo (plus significant recursions, if any) to die out within portions of compound block 604 that may be subsequently discarded after equalization has been performed. Specifically, each compound block 604 of time domain information includes a data portion 606, a pseudo-prefix portion 608, and a pseudo-suffix portion 610. In the exemplary embodiment, the respective sizes of extension portions 608, 610 may be determined independently from the size of data portion 606. Where cyclic prefixes are utilized, the conventional system is required to optimize the trade-off between the package size of the data portion and the amount of the transmission utilized to include the cyclic prefix. According to the present system, on the other hand, this trade-off is eliminated. The pseudo-extensions may be set to any length that is needed to address an encountered echo. The pseudo-extensions may utilize portions, or the entirety, of the data portions of adjacent blocks, and may even include data from more than one adjacent block if such length is necessary. [00106] Furthermore, the size/duration of pseudo-prefix portion 608 may be determined independently from the size of pseudo-suffix portion 610. That is, in some cases, a trailing echo might be of a significantly different duration than a leading echo. A receiver configured according to the present embodiment is thus advantageously configured to be capable of dynamically adjusting each extension portion 608, 610 according to the actual echoes encountered in real time for the respective data portion 606. Extension portions 608 and 610 thus form the overlapping regions of the compound block and, although these portions are not themselves cyclic prefixes, these overlapping portions are further capable of advantageously performing the same functional guard purpose as the cyclic prefix, but without requiring additional transmission time. The overlapping portions 608, 610 represent an extension period of time for echoes to die out that need not be cyclic/cyclical.

[00107] In further operation, after OFDM time domain symbols 602 have been formed into compound blocks 604, compound blocks 604 are converted into the frequency domain (e.g., by an FFT), to form compound frequency domain blocks 612, upon which frequency domain equalization (FDE) is then performed, for example, using a complex multiplication on each subcarrier with an inverse channel coefficient. Equalized compound frequency domain blocks 612 are then transformed back into the time domain (e.g., by an IFFT) to form equalized compound time domain blocks 614. Each equalized compound time domain block 614 includes equalized data 616, an equalized pseudo-prefix 618, and an equalized pseudo-suffix 620. The respective equalized pseudo-prefixes 618 and equalized pseudo-suffixes 620 are cut from the equalized compound time domain blocks 614 to extract equalized data 616. Blocks of equalized data 616 may then be pasted together to form a single equalized time domain signal 622, if desired.

[00108] The single equalized time domain signal 622 may then be transformed into the frequency domain to form a composite stream of equalized OFDM data, constituted of individual OFDM blocks 624, 626, and 628. That is, the frequency domain composite stream is made up of individual frequency domain blocks (e.g., block 624), which each correspond with a respective data portion (e.g., portion 606). In an exemplary embodiment, individual frequency domain blocks 624, 626, and 628 are pasted together as one sequenced composite block. In an alternative embodiment, each individual frequency domain block 624, 626, 628, etc. is processed separately to determine OFDM signals therefrom. In the exemplary embodiment, equalization scheme 600 considers each compound block 604 in the order of transmission, and thus repeats the process on an ongoing overlapped basis for each following block of the received OFDM time domain signals.

[00109] In some embodiments, data portion 606 is anchored according to the center time of the frequency duration equalization signal (e.g., blocks 612), and a de- ghosted time domain signal (e.g., equalized time domain signal 622) is formed by combining the equalized-and-transformed odd and even time domain blocks corresponding to data window portions 606. That is, after frequency domain equalization, the trimmed odd and even blocks of equalized data can be put back together to form an equalized time domain signal. For single carrier, such as pulse amplitude modulation (PAM) signals, the final time domain symbol sequence is considered "clean," and may then be subject to "slicing" and forward error correction (FEC). For OFDM symbols, the final equalized time domain symbol sequence is converted into the frequency domain before slicing and optionally implementing FEC. In DOCSIS implementations, elimination of the cyclic prefix allows for faster data transfer, which would be approximately a 15% improvement for a symbol period of 20 and a cyclic prefix of 3 μβ.

[00110] The techniques described herein may additionally be implemented by the receiver irrespective of what type of signal is, or signals are, received from the transmitter(s). The pseudo-prefixes and pseudo-suffixes are made sufficiently long relative to the duration of the longest echo, and its recursions, and thus the linear distortion from the echo(es) can be completely removed. As described above, because leading and trailing echoes are not necessarily of equal length, the present embodiments realize the additional advantage of allowing the receiver to dynamically adjust the pseudo- prefix independently of the pseudo-suffix to address the actual distortion encountered, and without sacrificing the size of the data package. In some embodiments, an optional windowing function, such as a raised cosine window, may be placed on pseudo-prefix portion 608 and/or pseudo-suffix portion 610 prior to performing FDE. [00111] In at least one embodiment, the size of the pseudo-extensions may be set to a predetermined threshold value that represents a duration of the longest echo expected to be encountered among a system including multiple transmissions from different transmitters (OFDMA). The echo may be determined, for example, from the channel response associated with the respective signal path. In an alternative embodiment, the size of one or more pseudo-extension is set to be greater than a size necessary to remove the distortion from an echo dynamically measured along the path by the receiver during signal characterization. In a further alternative embodiment, the receiver is configured to implement pseudo-extensions of sufficient size to eliminate the distortion from expected echoes on the signal path, and also measure the transmitted signals and dynamically adjust the predetermined size of the pseudo-extensions when encountering an echo having an unexpected length greater than the predetermined threshold.

[00112] In at least one embodiment, a frequency domain equalizer is programmed to utilize pilot signals for training prior to frequency domain equalization. In this embodiment, magnitude and phase correction are calculated to render the pilot signals sufficient for use. In one example, continuous pilots are used for synchronization, and timing for the start of a block may be determined by subjecting a set of captured pilots to an IFFT, with a zero value inserted for all data subcarriers, and a resulting time domain impulse response will indicate early or late timing. In at least one alternative embodiment, the time domain signal equalizer is programmed blindly, or using conventional training signals prior to frequency domain equalization. Interpolation of channel response between subcarrier pilots may be used, provided that the pilot spacing is closer than a predetermined minimum value. When the echo duration is longer, for example, the frequency domain ripple will have a shorter period, and should therefore implement closer pilot spacing. In some embodiments, as explained further below in greater detail, cyclic prefixes, pilot signals, and training signals may be completely eliminated through a novel use of constant amplitude zero autocorrelation (CAZAC) function. In the exemplary embodiment, the channel response is determined prior to slicing and FEC. The final equalized and trimmed frequency domain blocks back together to determine the OFDM symbols need not be pasted together (e.g., element 622). [00113] FIG. 6B is a graphical illustration of a histogram 630 depicting a distribution of echo duration for an increasing population of receivers. More particularly, histogram 630 illustrates, for a large population of receivers, the considerations that must be addressed for echoes of increasing time delays for the receivers to sufficiently function. The distribution curve of histogram 630 might be encountered, for example, in a broadcast signal distribution network, or in the case where a large number of receivers are disposed within a relatively small geographic distance from one another (e.g., a large number of mobile telephones brought into a football stadium or public event). Histogram 630 depicts a first echo time interval 632, a second echo time interval 634, and a third echo time interval 636.

[00114] First time interval 632 represents a time delay that could be addressed by a typical conventional cyclic prefix. Second time interval 634 represents the duration of a time delay that could be addressed by a cyclic prefix having a compromise value of increased duration according to the trade-off optimization discussed above. The length of the compromise cyclic prefix might be increased, but at the sacrifice of the size of payload data and the transmission. Beyond the length of second time interval 634, the trade-off between the payload and the guard interval is too great to make further increases in the guard interval practical. Second time interval 634 thus represents the maximum time delay that a conventional system is capable of addressing utilizing a cyclic prefix. Third time interval 636 therefore represents the worst-case echo delay that a system is likely to encounter for receivers in the network, and which require equalization for all of the receivers to properly function. A significant percentage of receivers in the network (represented by hashed area 638 in FIG. 6B) will encounter echoes beyond the maximum capability, i.e. compromise interval length 634, of the conventional system, and will therefore experience equalization difficulties.

[00115] Systems and methods according to the present embodiments though, are capable of fully equalizing all of the receivers in the network, and without requiring any consideration of the cyclic prefix length, or the trade-off of the cyclic prefix length against length of the data package. That is, in the conventional system, if a transmitted cyclic prefix is made to sufficiently long enough to equalize the longest echo that can be encountered in the network, the size of the transmitted payload will be too small to make the transmission efficient. As the length of the cyclic prefix is shortened for better transmission efficiency, the number of receivers that will encounter equalization problems increases. Conventional OFDM/OFDMA systems therefore are required to strike a balance between the number of reception sites and poor transmission efficiency. According to the present systems and methods, on the other hand, this balance may be completely ignored. Although the pseudo-extensions of the present embodiments may perform the functions of cyclic prefixes, the pseudo-extensions are not bound by any of the cyclic prefix limitations. The pseudo-extensions are determined independently from the payload/data package, and may even utilize adjacent blocks of payload data as the pseudo-extensions (but discarded after equalization of the desired block).

[00116] Some conventional systems have been proposed to eliminate cyclic prefixes from single carrier optical transmissions. These single carrier proposals, however, are unable to cope with channels that fade to zero. That is, division by zero cannot be performed where channel response is zero. In wireless channels, signals from multiple antennas can be combined to produce a composite signal without frequency response nulls. That is because, it two antennas are spaced apart, it is very unlikely that a complete fade will occur at a same frequency for both antennas. Moreover, conventional techniques of eliminating cyclic prefixes only been proposed for single carrier transmissions exhibiting chromatic dispersion on fiber optic lines. Fiber optics, however, do not typically encounter discrete echoes, as described above. Furthermore, OFDM transmission is multi-carrier, and may be implemented on both wired and wireless networks, where echoes may be commonly encountered.

[00117] FIG. 7 is a flow chart diagram of an exemplary overlapped Fourier transform process 700 for equalization scheme 600, FIG. 6A. In the exemplary embodiment, process 700 is implemented by a receiver or the processor thereof (e.g., first receiver 316, FIG. 3). Process 700 begins at step 702, in which a digital transmission signal (e.g., OFDM carrier signal 602, FIG. 6) is received, including a series sequential OFDM time domain samples, and formed into overlapping series of compound time domain blocks (e.g., blocks 604, FIG. 6A). In some embodiments, process 700 is particularly useful if the received time domain symbols are from a single carrier transmission, or from a direct sequence spread spectrum transmission. In an optional embodiment, the narrow blocks, such as data portions 606, are centered in the respective compound block 604, and are of sufficient size to allow echoes to die out (e.g., go to zero) in between bursts.

[00118] In step 704, the compound time domain blocks are converted into the frequency domain (e.g., by an FFT). In step 706, the frequency domain blocks are equalized using frequency domain equalization. In step 708, the equalized compound frequency domain blocks are converted into the time domain (e.g., by an IFFT) to create equalized compound time domain blocks (e.g., blocks 614, FIG. 6). In step 710, the "early" (e.g., pseudo-prefix 618, FIG. 6A) and "late" (e.g., pseudo-suffix 620, FIG. 6A) time symbol ends are cut/discarded from the equalized compound time domain blocks to extract narrow equalized time domain blocks (e.g., blocks 616, FIG. 6).

[00119] Step 712 is a decision step. In step 712, a processor of the receiver determines whether received input symbols are single carrier or multicarrier (OFDM). If the received time domain symbols are determined to be single carrier, process 700 proceeds to step 714, where the narrow equalized time domain blocks are pasted together to form an equalized time domain signal (e.g., signal 622, FIG. 6) and, for single carrier signals (or the equivalent) then sliced and forward error corrected. In at least one example of step 714, where the input symbols are, for example, a direct sequence spread spectrum (not shown), step 714 further includes an optional sub-step of de-spreading the symbols. If, however, in step 712, the processor of the receiver determines that the input signals are multicarrier signals (e.g., OFDM/OFDMA), process 700 proceeds to step 716, where the equalized time domain signal is converted into the frequency domain, after which slicing/FEC may be implemented, and/or OFDM symbols are read therefrom. In some embodiments, narrow blocks may include multiple OFDM transforms, which are separated prior to performing the FFT of step 716. In other embodiments, the narrow blocks may alternatively or additionally include partial OFDM transforms, which are combined to perform the FFT of step 716.

[00120] In an alternative embodiment, process 700 further includes optional step 718. Step 718 is implemented in the case where compound time domain blocks form a continuous stream, as opposed to burst mode reception. In such cases, optional step 718 proceeds to the next block in sequential time order and repeats process 700 for the next block. According to process 700, frequency domain equalization can be performed on multiple blocks the same time, and across block boundaries. Alternatively, or additionally, a large block can also be broken into smaller sub-blocks for separate equalization.

[00121] According to this advantageous process, a receiver may be configured or programmed to dynamically adjust the size of the compound block according to the length of an encountered echo on the signal path. For example, if a very long echo is encountered in the signal path, the size of the overlapping compound blocks can be enlarged at the receiver end to effectively create longer pseudo-prefixes having a greater duration than the encountered echo. Such dynamic adjustability is particularly advantageous with respect to terrestrial broadcast signals, which are known to suffer from very long echoes in the signal path. The equalization processes described herein are thus effectively modulation-indifferent to the type of data that is being linearized, and thus is fully and simultaneously adaptable to single carrier, multi-carrier, or spread spectrum transmissions. According to the embodiments herein, the data may be continuous, or formed into individual blocks.

[00122] Similarly, the size of the transform block may also be dynamically made larger or smaller as desired, and each separate path may be converted by the transform of a different size than that implemented on a different path. This ability to dynamically alter the transform size provides a receiver with superior versatility over conventional OFDE techniques, which implement a one-size-fits-all transform approach. In conventional OFDM transmissions, for example, FFT length is made large in order to prevent the overhead of the cyclic prefix from becoming too burdensome as a percentage of the total transmission time. However, phase noise becomes more unstable as the length of the transform increases. The present embodiments address this problem by both (i) rendering the transform size dynamically adjustable, and (ii) eliminating the need for cyclic prefix transmission at the transmitter end.

Zadoff-Chu Sequence as a Pseudo-Prefix [00123] FIG. 8 is a graphical illustration of a plot 800 of an exemplary transmitted signal 802 having in-phase (I) components 804 (black) and quadrature (Q) components 806 (gray). Signal 802 may be used, for example, as a demonstration signal. Plot 800 depicts linear voltage over time of captured signal 802. The I and Q components are also sometimes referred to as the real and imaginary samples, respectively, and orthogonal to one another. In this example, plot 800 is obtained from a test system using a software-defined radio (SDR, not shown) as a transmitter, which is an Ettus (National Instruments) model B200. Another B200 SDR is used as a receiver (also not shown), which will introduce frequency error due to the different oscillators of the SDRs. The test system receiver captures at 8 million samples per second (i.e., an 8 MHz channel), and transmitted signal 802 includes a group of 15 contiguous OFDM blocks 812, each 64 symbols wide (i.e., subcarrier spacing of 125 kHz). In the example illustrated in FIG. 8, testing was performed at a center frequency of 840 MHz and 2.4GHz, with bandwidths of both 8 MHz and 16 MHz.

[00124] In the test system this example, transmitted signal 802 further includes a first CAZAC function 808 and a second CAZAC function 810. As illustrated in FIG. 8, first and second CAZAC functions 808, 810 are Zadoff Chu sequences placed on either end of the 15 OFDM data sequences 812 therebetween. A Zadoff Chu sequence is a particular type of CAZAC function that has no crest factor, and operates as a constant envelope time domain waveform. In operation, the Zadoff Chu sequence waveforms 808, 810 are captured from the antenna periodically from transmitted signal 802, which then allow the determination of the offset frequency, the start of the OFDM block, and the channel characterization.

[00125] More particularly, the Zadoff Chu sequences 808, 810 operate as a sync/timing signal to establish an exact offset frequency difference between the transmitter and receiver by a time domain cross-correlation process. Cross-correlation provides exact timing and establishes the precise start of each of the 15 OFDM blocks 808. Accordingly, the use and placement of the Zadoff Chu sequences 808, 810 serves a threefold function: (i) offset frequency measurement; (ii) start of block detection; and (iii) channel response determination for equalization coefficients. [00126] In operation, plot 800 is representative of either a wireless or wired transmission, and measurement of the frequency differences between the transmitter and the receiver carriers is performed by cross-correlating the time domain Zadoff Chu sequences 808, 810 to produce a cross-correlation with real and imaginary components measured at peak values. Accordingly, a time domain phase error between Zadoff Chu sequences 808, 810 is the arctangent of the imaginary value divided by the real value. Frequency error may then be eliminated by de-rotating the captured complex samples, and then the OFDM block group 812 may be parsed to create 15 complex time domain blocks, which, as described with respect to the embodiments above, may include the selected OFDM block, part of a previous OFDM block, and part of the subsequent OFDM block. As also described above, such portions of the previous and subsequent blocks may include other OFDM data signals themselves, all or part of the Zadoff Chu sequences 808, 810, or no energy (quiet time) etc. In an alternative embodiment, data sequences 812 include one or more of single carrier blocks, dead-air (quiet) time, direct-sequence spread spectrum, or any other type of mixed signals experiencing the same linear distortion. That is, linear distortion can be removed from any signal transmitted in the block allocated to data sequences 812.

[00127] Similar to the processes described above (see e.g. FIGS. 6-7), all 15 of the compound time domain blocks may then be further converted into the frequency domain, equalized, converted back into the time domain, trimmed to become narrow, equalized, time domain blocks, and finally (e.g., for OFDM signals, as described above) converted into the frequency domain where the component OFDM symbols may be determined. This example, frequency domain equalization coefficients may be determined by performing a Fourier transform/FFT on first Zadoff Chu sequence 808, and then performing a complex division by a stored complex coefficient for each symbol (in frequency). In some embodiments, the compound blocks will require more coefficients than the particular frequency domain equalization coefficients provided in the Zadoff Chu sequence (64 in this example), and therefore the equalization solution is interpolated to create at least twice the number (e.g., 128) of frequency domain coefficients. [00128] In an exemplary embodiment, the combination of the Zadoff Chu sequence 808 and OFDM block group 812 forms a trained block group (not separately numbered). In this example, a continuous transmission may therefore be created by transmitting a series of such trained block groups. Alternatively, a burst mode transmission may be created by transmitting a single trained block group, followed by a single ZC sequence for frequency offset estimation. In at least one embodiment, a plurality of quiet time symbols (e.g., 8 symbols) are placed on either side of both Zadoff Chu sequences 810, 812, to further prevent energy from some data symbols contaminating the channel characterization results.

[00129] In at least some embodiments, because an initial Zadoff Chu sequence (e.g., sequence 808) is effectively repeated (e.g., sequence 810) for a single block of OFDM signals, the CAZAC functions/Zadoff Chu sequences according to plot 800 may perform further advantageous utility as a substitute for cyclic prefix, or pseudo-prefix, for transmission implementations that still intend to utilize cyclic prefixes. That is, the Zadoff Chu sequence may be utilized as a substitute cyclic prefix (because it is repeated energy), in addition to the functionality described above, namely, offset frequency measurement, block start detection, and use as a training signal. Accordingly, implementation of the present Zadoff Chu sequence techniques allows for not only the substitution for other types of transmissions, but also for the improvement of OFDM transmissions that still desire to utilize cyclic prefixes.

[00130] Therefore, cyclic prefix elimination provides greater data throughput to the transmission system as a whole, and the utilization of the Zadoff Chu sequences instead of the cyclic prefixes provides a more accurate offset frequency estimation. Additionally, utilization of the high-energy, no crest factor Zadoff Chu sequences instead of pilot signals provides a cleaner, less noisy channel model. Accordingly, systems and methods according to the present embodiments realize significantly improved performance over conventional OFDM transmission schemes that utilize cyclic prefixes and/or pilots, because the Zadoff Chu sequence has a zero dB crest factor (as compared with the 10-16 dB factor of OFDM), and therefore allows a stronger signal that can be used for timing, characterization, and frequency offset, thereby resulting in longer battery life, greater range, and greater throughput of system receivers, which is particularly important for portable handheld devices (e.g., cellular phones, tablets, portable computers, etc.). Moreover, elimination of the cyclic prefix is of particular importance at the transmitter and as well, because it will result in significant power savings, thereby delaying the need for plant upgrades, while allowing for improved service tiers.

[00131] FIG. 9 is a computer program listing 900 demonstrating an exemplary coding for implementing the Zadoff Chu sequence depicted in FIG. 8. According to listing 900, a computer code generates 63 complex Zadoff Chu values. A 64 ^th Zadoff Chu point is created by repeating the 63 ^rd Zadoff Chu value. Accordingly, a radix 2 FFT operation may be implemented as listed. The person of ordinary skill in the art will understand though, that in FFT operation may be implemented for other base values, or prime numbers, without departing from the scope of the embodiments described. In an exemplary embodiment, computer program listing 900 is based on a C code programming language, but other programming languages, including Matlab may also or alternatively be used.

[00132] The Zadoff Chu sequences herein represent complex- valued mathematical sequences which, when applied to radio signals, give rise to an electromagnetic signal of constant amplitude. Cyclically shifted versions of the Zadoff Chu sequence, imposed on a transmitted signal, thereby result in zero correlation with one another at the receiver. A generated Zadoff Chu sequence that has not been shifted is referred to as a "root sequence".

[00133] The Zadoff Chu sequences described herein exhibit a further useful property namely, that cyclically-shifted versions of the sequences are orthogonal to one another, provided, that is, that each cyclic shift, when viewed within the time domain of the transmitted signal, is greater than the combined propagation delay and multi-path delay-spread of that transmitted signal between the transmitter and receiver. In wireless implementations, such as MIMO, equalization of several signals, received from several antennas, is performed to produce one or more equalized data streams. Such MIMO operations are more advantageously performed utilizing the Zadoff Chu sequence embodiments described herein, since application of the present Zadoff Chu sequences to different signals from the different MIMO antennas will assist in matrix construction. [00134] In the exemplary code listed in FIG. 9, the particular set of values used for Nzc, u, nl, and n2 (e.g., C code) produces a complex signal with a constant magnitude in the time domain, and nearly-constant values in the frequency domain. Accordingly, Zadoff Chu sequences further operate similarly to a useful training signal that can be used in place of the pilots that are conventionally used in OFDM. Because of the constant amplitude property, the Zadoff Chu sequences, the resultant time domain signal of the sequences has a crest factor of OdB, as described above. The present embodiments are therefore of particularly advantageous use with respect to power-limited transmitters, such as found in cellular phones, because the radiated energy for characterizing the signal path is much greater, while the resulting noise contamination of the channel estimate is significantly reduced.

[00135] FIG. 10 is a graphical illustration depicting a frequency domain spectral plot 1000 of the Zadoff Chu sequence depicted in FIG. 8, and generated according to computer program listing 900, FIG. 9. Plot 1000 depicts magnitude, real, and imaginary values (y-axis) over frequency (x-axis) of captured spectral data of an FFT of a generated time domain Zadoff Chu sequence. Plot 1000 includes a real subplot 1002, and imaginary subplot 1004, and a magnitude subplot 1006. As described above, such as Zadoff Chu sequences have ideal autocorrelation functions, low crest factors, and relatively flat spectral energy. That is, magnitude subplot 1006 remains relatively flat over significant frequency range, thereby rendering the Zadoff Chu sequence particularly advantageous as a channel characterization signal. The Zadoff Chu sequence therefore may be used, according to the embodiments described above, as a substitute for conventional OFDM pilot signals and training signals. In some embodiments, the spectral energy of the transformed time domain Zadoff Chu sequence can be created from other values of Nzc, including 127, 255, 511, etc., where Ncz = 2 ^A n -1. As described above, OFDM blocks 814 and 816 were included as portions of the previous and subsequent signal, respectively, and each illustrate the time and shape of OFDM time domain energy in a single block. It may be further noted that the OFDM energy has a large crest factor.

[00136] FIG. 11 is a flow chart diagram of an exemplary process 1100 for simulating an idealized transmitter-receiver chain for an OFDM transmission having no cyclic prefix (or pilot signals), including any of the embodiments described above. Process 1100 includes a configuration subprocess 1102, and a frequency domain equalization subprocess 1104. Process 1100 begins at step 1106 of configuration subprocess 1102.

[00137] In step 1106, process 1100 performs system configuration, including without limitation the symbol time (e.g., 20 μβ), the FFT size (e.g., 4096), sub symbol times, the number of sequential OFDM symbols to simulate (in some cases, process 1100 will discard the first and last simulated OFDM symbols). The modulation order (e.g., 1024 QAM) is set. In step 1108, process 1100 performs channel configuration, including without limitation the channel type (e.g., AWGN channel with a single echo), signal to noise ratio, echo amplitude relative to the direct signal amplitude, and/or echo time delay (e.g., in seconds). In step 1110, process 1100 calculates subcarrier values, including without limitation the number of unique complex symbol values, the average symbol energy, and/or the average noise energy per subcarriers symbol.

[00138] In step 1112, process 1100 generates the transmitted signal. The generated transmitted signal includes generated random symbols for the real component, and random generated symbols for the imaginary component. In some embodiments, step 1112 includes the further substeps of combining the real and imaginary components, performing an IFFT of each symbol to convert to the time domain, and reorganizing into a one-dimensional time sequence. Step 1114, process 1100 generates the channel, including without limitation a conversion of the echo (in dB) into a linear quantity, and a conversion of the echo delay into a sub-symbol index.

[00139] In step 1116, process 1100 establishes the channel impulse response. In an exemplary embodiment, establishing the channel impulse response includes substeps of starting with all zero values, adding a 1 value at the zero-lag tap, and adding the echo at a desired or appropriate tap. In step 1118, process 1100 establishes the noise sequence (e.g., AWGN), and in step 1120, process 1100 calculates the received signal. In an exemplary embodiment, step 1120 includes substeps of convolving the transmitted signal with a channel impulse response, and trimming the resulting convolution down to its original length and adding the noise sequence thereto. [00140] In step 1122, process 1100 displays the resulting signal as if no equalization had been performed thereupon. In an exemplary embodiment, step 1122 includes the additional substeps of performing an FFT to convert the signal to the frequency domain, and plotting the constellation with associated formatting. Step 1124, process 1100 calculates the average subcarrier symbol error energy. In an exemplary embodiment, step 1124 includes the additional substep of calculating the MER with no equalization. After step 1124, process 1 100 proceeds to frequency domain equalization subprocess 1104.

[00141] Frequency domain equalization subprocess 1104 begins at step 1126. In step 1126, process 1100 pads the channel impulse response to equal two times more than the FFT size of the OFDM transmission. In an exemplary embodiment, step 1126 limits the echo delay to no more than two times the FFT size of the OFDM. In step 1128, process 1100 converts the channel response into the frequency domain.

[00142] In step 1130, process 1100 generates overlapping FFT blocks. In an exemplary embodiment, step 1130 generates overlapping FFT blocks that are two times the FFT size of the OFDM transmission. In step 1132, the overlapping blocks are converted into the frequency domain. In step 1134, the channel response is equalized. In step 1136, the equalized channel response is converted back to the time domain. In step 1138, the overlapping block portions are discarded. In step 1140, the equalized time sequence is converted to the frequency domain.

[00143] In step 1142, the resulting constellation is plotted and displayed. In step 1144, additional statistics are displayed with the resulting constellation, including without limitation the MER, which is calculated for each OFDM symbol. According to the example he process of FIG. 11, a transmitter-receiver chain for a digital transmission system is successfully simulated to demonstrate implementation, for optimization purposes, of the several embodiments described above.

[00144] FIG. 12 is a flow chart diagram of an alternative process 1200 for utilizing Zadoff Chu sequences according to the embodiments described above. Process 1200 begins at step 1202 in which the receiver (e.g., receiver 316 or 318, FIG. 3) captures a wireless signal (e.g., signal 800, FIG. 8) and stores captured time domain symbols into a memory (not shown in FIG. 3) of the receiver. In step 1204, process 1200 determines the frequency offset between the transmitter and receiver from the detected Zadoff Chu sequences (two Zadoff Chu sequences in the example depicted in FIG. 8), and cross- correlates the Zadoff Chu sequences to produce a peak, and the phase between real and imaginary components identifies the offset frequency.

[00145] In step 1206, the captured symbols are de-rotated in the time domain to remove the frequency error from the captured symbols. That is, the received complex time domain samples are de-rotated to remove the frequency offset. In step 1208, the first of the two Zadoff Chu sequences is processed to determine the channel response. In at least one example of step 1208, process 1200 further interpolates the determined channel response in the frequency domain. In step 1210, process 1200 utilizes the timing of the cross-correlation peak of the Zadoff Chu sequences to identify, or locate, the boundaries of each of the 15 OFDM blocks (e.g., blocks 812, FIG. 8).

[00146] In step 1212, overlapped compound blocks are formed (e.g., 15 in the example illustrated in FIG. 8). In at least one example of step 1212, the first and last blocks of the compound blocks include portions of the respective Zadoff Chu sequences, which may be later discarded as needed pseudo-extensions, according to the embodiments described above. In step 1214, the compound blocks (e.g., 15) are converted into the frequency domain by a Fourier transform. In step 1216, frequency domain equalization is performed on each the 15 compound blocks, and then the FDE-equalized blocks are converted into the time domain by an inverse Fourier transform. In step 1218, overlapping portions of the equalized, compound time domain blocks are discarded to extract 15 narrow, equalized blocks in the time domain. In step 1220, the extracted narrow blocks are converted into the frequency domain by Fourier transform, after which, the symbols thereof may be sliced and read.

[00147] Similar to the embodiments described above, upon completion of step 1220, process 1200 may return to step 1202, in which processing is repeated on the next captured burst, or the next group of previously captured symbols, stored in the receiver memory. As also described above, and an optional embodiment, process 1200 may be performed out of sequence, and/or in near simultaneity, on two or more symbols stored in the memory.

[00148] According to the advantageous systems and methods herein, receiver-based processing techniques may be implemented such that the transmitter may eliminate cyclic prefixes from multi-carrier digital signal transmissions. These techniques are thus applicable for equalizing/de-ghosting not only OFDM and OFDMA signals, but also for a variety of other digital transmissions, including without limitation SC-FDMA, single-carrier transmissions, spread spectrum signals, MFMO, and wavelet-based signals.

[00149] The techniques of the present embodiments are also particularly applicable to transmission systems that utilize pre-distortion, such as upstream DOCSIS 3.1. That is, in such transmission systems, the cyclic prefix may be entirely eliminated from the pre-distorted transmission, which would not generally be processed by the CMTS receiver. DOCSIS 3.0, for example, utilizes pre-distortion for single carrier transmissions, and DOCSIS 3.1 utilizes pre-distortion for OFDMA transmissions. In conventional examples of these transmission systems, the transmissions are pre-distorted at the cable modem (CM), and after passing through the linear distortion of an upstream cable network, arrive at a CMTS receiver fully equalized. In one illustrative example, a human eye lens pre-distorts an image such that the image will be in generally perfect focus on the eye retina. In further examples, pre-distortion coefficients are determined in a training process, referred to as "ranging," using pilots. According to the embodiments herein, for OFDMA transmissions, the CP may be eliminated at the CM end, thereby providing up to an additional 25% more upstream throughput. Further to this example, no additional equalization processing would be necessary at the CMTS receiver, utilizing the techniques described herein.

[00150] As described above, the several embodiments use different types of overlapped extensions as pseudo-prefixes/pseudo-suffixes, and these pseudo-extensions may include one or more of several other types of transmissions, such as training signals, pilots, signals with other modulation formats, quiet time, unused or too-short cyclic prefixes, or CAZAC functions/sequences. These pseudo-extensions are multi-functional, and may substitute for cyclic prefixes or other types of guard transmissions and pilot signals.

[00151] Systems and methods according to the present embodiments represent further significant improvements over conventional transmission schemes by providing dynamically adaptive equalization schemes that allow for different transform sizes to be applied to different signals traveling along different signal paths of varying lengths. That is, size of an FFT is adjustable for longer echo/reflection delays, as opposed to the shorter path of the direct signal. Similar advantages apply to the dynamically adjustable lengths of the pseudo-extensions as well.

[00152] In the case of OFDMA transmissions where multiple transmitters, having different respective signal paths, contribute to a composite received signal, the frequency domain symbols of each transmitter may be equalized with frequency domain coefficients specifically configured to correct for the signal path of that respective transmitter. An example of relevant equalization processing is described below with respect to FIGS. 13A-13B.

[00153] FIG. 13A is a flow chart diagram of an alternative process 1300 for operating a receiver, e.g., first receiver 316, FIG. 3, that receives approximately simultaneous signals (e.g., composite signal 320) from two different signal paths, e.g., first carrier signal 310(1) and second carrier signal 312(1). The respective signal paths of carrier signals 310, 312 may be different for a number of reasons, such as due to a reflector in one path (e.g., reflecting object 324) that is not in the other path. In this example, first transmitter 302 and second transmitter 306 are OFDMA transmitters, no cyclic prefixes are included in the respective carrier signals.

[00154] FIG. 13B is a graphical illustration of alternative frequency plots 1302 utilizing an odd-and-even subcarrier scheme 1304 and an upper-and-lower frequency band scheme 1306, respectively. In the embodiments illustrated in FIGS. 13A and 13B, receiver 316 is pre-programmed to know which subcarriers (A and B, respectively, each OFDMA transmitter 302, 306 is using. In subcarrier scheme 1304, first transmitter 302 utilizes odd-numbered subcarriers A and second transmitter 306 utilizes even-numbered subcarriers B. In alternative subcarrier scheme 1306, first transmitter 302 utilizes the lower half of the OFDMA frequency band, and second transmitter 306 utilizes the upper half of the OFDMA frequency band.

[00155] Referring back to FIGS. 3 and 13A, process 1300 begins at step 1308. In step 1308, receiver 316 forms a combined (e.g., from composite signal 320) compound overlapped block that includes subcarriers from at least two separate transmitters, that is, first transmitter 302 and second transmitter 306, in this example. In step 1310, the combined compound overlapped block is converted from the time domain into the frequency domain (e.g., by a Fourier transform or an FFT), and the respective frequency domain subcarriers A and B are separated according to which transmitter sent the respective subcarriers. The separated frequency domain subcarrier symbols are then processed similarly, but separately, as follows.

[00156] In step 1312, FDE is applied to the symbols from both of the A and B subcarriers using equalization coefficients corresponding to each of the respective signal paths 310(1) and 312(1). In step 1314, the equalized A and B symbols are converted, separately, into the time domain (e.g., by an inverse Fourier transform or IFFT). In an embodiment, for the A subcarriers, a value of zero may be inserted for all subcarriers from B, and similarly, for the B subcarriers, a value of zero may be inserted for all subcarriers from A. In step 1316, still in the time domain, the overlapped portions (pseudo- extensions) of the equalized and converted composite A and B blocks are discarded to create two separate narrow A and B blocks. Step 1318 is an optional step, which may be implemented in the case where the processed symbols are from an OFDMA transmission. In an example of step 1318, the narrow A and B blocks are transformed again into the frequency domain (e.g., by Fourier transform, FFT) and the symbols are read.

[00157] The techniques of FIGS. 13A-B are further advantageous with respect to SC-FDMA transmissions that do not utilize cyclic prefixes. In such cases, where the input signal is SC-FDMA, optional step 1318 would not be needed. The present embodiments are particular useful in the case of multiple transmissions of different types. That is, for example, where first transmitter 302 implements an OFDMA transmission and second transmitter 306 implements an SC-FDMA transmission, receiver 316 is configured to receive a composite signal containing different A and B subcarriers, and then separately process the respective symbols thereof. In this example, optional step 1318 would be implemented to read the symbols of the narrow A blocks (OFDMA), but would not need to be implemented for the narrow B blocks (SC-FDMA). This could happen, for example, if B blocks came from a battery-powered device, and A blocks came from a device connected to the AC power supply, and was not therefore power-constrained.

[00158] Thus, in the case of a composite signal having multiple transmissions, which may or may not individually include cyclic prefixes, the cyclic prefixes become irrelevant. The significant factor will be the duration of the reflections (e.g., the longest echo) among the multiple transmissions. More importantly, where cyclic prefixes are implemented, the duration of a cyclic prefix for particular transmission would not be expected to change for the particular carrier in which it is implemented. Echoes and reflections, on the other hand, are subject to change from path to path, and over time within a single path. According to the present embodiments though, a receiver can be configured to dynamically adjust, in real-time, the length of the overlap/pseudo-extensions according to the conditions that are actually encountered.

[00159] U.S. Patent No. 5,886,749 describes a system where repeated energy (an NTSC horizontal sync signal) is used to enable an overlapped transform with FDE. The present embodiments advantageously utilize a periodically repeated ZC sequence instead of the NTSC sync signal to more reliably enable an overlapped transform with FDE. Nevertheless, as described above, the present embodiments avoid the need for repeated (e.g., cyclic) energy. The present systems and methods advantageously utilize energy adjacent to the target block, such as an adjacent block itself, which does not affect the equalized target block because the adjacent signal portions are discarded after processing.

[00160] Further advantages of the present embodiments become readily apparent in the case where one transmitter's signal arrives early or late relative to the other transmitter's signal. If there was a timing offset between the start of the A block and the B block, this timing offset would be automatically corrected by the FDE subprocess, because the time shift will appear as a rotation in the frequency domain correction coefficients. [00161] The present embodiments offer still further advantages with respect to the use of the present Zadoff Chu sequences as training signals. That is, equalization of conventional OFDM or OFDMA blocks is more effectively executed by substituting a CAZAC functions/Zadoff Chu sequences for pilot subcarriers. According to the present embodiments, even when the substitute (ZC) training signal is shorter in duration than the compound block that is intended to be transformed and equalized in the frequency domain, the frequency domain coefficients may nevertheless be interpolated in order to equalize the overlapped frequency domain extensions.

[00162] In some embodiments, in the case of a series of OFDMA block transmissions, the present systems and methods advantageously allow for use of the cyclic prefix (or quiet time) on the first block on the first transmitter, so that the first transmitter is protected from a previous signal echo from another signal path. Successive blocks from this first transmitter may then advantageously omit the use of the cyclic prefix, thereby saving significant transmission time for substantive data.

[00163] The present embodiments that eliminate cyclic prefixes from digital transmission signals are still further useful beyond simply reducing transmission time to improve system efficiency. Indeed, the present embodiments are particularly valuable for implementations where the length of echoes on a signal path render the use of cyclic prefix impractical, thereby limiting the types of transmission schemes that may be utilized on such signal paths. For example, in the case of a particular signal path having a duration of 50 for its longest echo, and for a maximum allowed overhead of 10%, the OFMD symbol period would have to be greater than 500 μβ. A 500 symbol, however, would require an inordinately expensive precision local oscillator. Such precision local oscillators become even more costly at very high frequencies, such as millimeter wave frequencies (e.g., 60 GHz). According to the present systems and methods though, by eliminating the need for cyclic prefixes, OFDM transmission technology may be implemented on this exemplary signal path without requiring such costly hardware outlays. Duobinary Modulation for OFDM Transmission

[00164] Duobinary modulation is a transmission scheme for transmitting N baud using a bandwidth of less than capital N/2 Hz. However, since the minimum bandwidth required of a transmitted pulse is N/2 Hz, adjacent duobinary pulses experience ISI. A data communication system that implements duobinary modulation includes a duobinary encoder, which implements the duobinary code from original symbols, and a duobinary decoder, which recovers the original symbols from the duobinary signal.

[00165] The duobinary decoding process is prone to error propagation, since an estimate of a given sample relies on the estimate of previous sample. One conventional technique to mitigate this error propagation implements a precoder before the duobinary encoder at the transmitter.

[00166] However, when duobinary coding is applied in the frequency domain, conventional encoding that is designed for sequential time-domain duobinary signals are known to experience compromised efficiency. Accordingly, it is desirable to create an improved encoder design for use with frequency domain duobinary OFDM transmissions.

[00167] In an exemplary embodiment of duobinary transmission, a data communication system has a total number N of OFDM subcarriers. Because of the duobinary modulation scheme, the number of independent subcarriers carrying original symbols will then be N-l . At the transmitter side of the communication system, the duobinary encoding can be expressed as: y = Ax (Eq. 1)

[00168] Where x = (xi, x ₂ , ... ΧΝ-Ι) ^Γ and represents a vector including the N-l original or precoded symbols, y = (yi, _y ₂ , · · · yN) ^T and represents a vector including the N duobinary OFDM subcarriers, and A is a Nx(N-l) matrix. Matrix A can be further expressed according to:

(VK;J (Eq. 2) [00169] Where I _N -i constitutes a (N-l)x(N-l) identical matrix.

[00170] In one example, where the total number of subcarriers is 4, three of the subcarriers are independent, and will contain QPSK symbols 1+j, 1-j, and ^~ l+j . Thus, the duobinary encoding that will correspond to (Eq. 1) can be expressed as:

[00171] FIG. 14 is a graphical illustration depicting a comparative overlay

1400 of an OFDM sine function implementing a duobinary technique according to the present embodiments. In the example illustrated in FIG. 14, one subcarrier 1402 is shown, and represents a sine function (sampling function) in the frequency domain. Implementing a duobinary operation on subcarrier 1402, a copy 1404 of subcarrier 1402 is obtained, but having a frequency shift of pi (π). A combination subcarrier 1406 is then obtained by combining subcarrier 1402 with copy 1404. Combination subcarrier 1406 can then be seen to have significantly reduced sidelobes, as compared with subcarrier 1402 and copy 1404, because the respective sidelobes values of subcarrier 1402 and copy 1404 substantially cancel each other out at all sidelobes. This example is particularly illustrative of how the present embodiments implement duobinary modulation on OFDM transmissions to significantly reduce the OOB leakage in comparison with a conventional OFDM transmission. The significant improvement realized by the present embodiments over the conventional techniques is further illustrated with respect to FIG. 15.

[00172] Duobinary OFDM represents an innovative modulation technique that has a characteristic of low adjacent channel interference with respect to neighboring frequencies. As described further below with respect to FIGS. 17A-B, the energy of a conventional single symbol is spread between two adjacent symbols. In the time domain, this spread creates an OFDM symbol with a half-cosine envelope. In the frequency domain though, the spectrum is rectangular, with low OOB interference. In an exemplary embodiment, this duobinary OFDM symbol is transmitted without a cyclic prefix, and then be demodulated with a CP-elimination receiver (described above). In an alternative embodiment, the duobinary OFDM symbol may be demodulated by a conventional OFMD receiver, and a CP can be optionally added.

[00173] In the case where a CP is added to the duobinary OFDM symbol, the time domain waveform envelope takes on the shape of a "fish" (described further below with respect to FIGS. 17A-B), and some of the OOB performance will then be compromised by the abrupt drop of the "tail" of the fish. Duobinary processing is conventionally done with a filter having an impulse response extending over two symbols. In at least one embodiment, OOB leakage of the OFMD transmission may be reduced utilizing raised cosine tapering in the time domain, however, this alternative method will result in an increased transmission time. Raised cosine tapering is used, for example, in the DOCSIS 3.1 modulation standard.

[00174] FIG. 15 is a graphical illustration depicting a comparative overlay 1500 of a duobinary OFDM block transmission 1502 with a conventional OFDM block transmission 1504. As can be seen from overlay 1500, an amount of adjacent channel interference 1506 (dark shaded area) from duobinary OFDM block transmission 1502 is significantly reduced with respect to an amount of adjacent channel interference 1508 (light shaded area) of conventional OFDM. The present duobinary techniques thus provide a dramatic improvement with respect to conventional OFDM transmissions. In the example shown in FIG. 15, the vertical scale per division is 10 dB.

[00175] In some embodiments, the crest factor (see FIG. 17A, below) may be reduced by interleaving the upper and lower sidebands in the time domain. That is, while the lower sideband is going to zero, the upper sideband is cresting, and vice versa. Additionally, the principles described herein are not limited to only the example shown, but may be also advantageously implemented for modulation formats without uniform subcarrier amplitudes, including but not limited to 64-QAM.

[00176] Furthermore, the duobinary modulation techniques of the present embodiments represent only one example of the new modulation method that is created by interchanging time and frequency axes of signal duals. In an exemplary embodiment, adaptation techniques from single carrier to multicarrier utilize a 90 degree rotation in a time-frequency plot. [00177] A duality between time and frequency exists, which can be observed from discrete Fourier transform and discrete inverse Fourier transform equations (see e.g., Eqs. 6 and 7, below). Differences between such equations typically involve only a scale factor and a negative sign in front of the complex exponential. Substantively, the equations are quite similar. Given a set a transform pairs, it is often difficult to identify which graph of a plotted signal is a time domain plot, and which is a frequency domain plot.

[00178] For example, with a single carrier signal, such as a pulse amplitude modulation (PAM), each symbol is considered to be short in time, but wide in bandwidth, and having a next symbol occurring sequentially in time. An OFDM subcarrier, on the other hand, is considered to be narrow in frequency, but long in duration. Additionally, many OFDM subcarriers operate simultaneously in time. An illustrative comparison of these two exemplary signal types is shown below with respect to FIG. 16.

[00179] FIG. 16 is a graphical illustration depicting a block time- frequency plot 1600. In this example, plot 1600 is illustrated as a 32x32 block of 32 PAM single carrier symbols 1602 (a single PAM symbol illustrated for ease of explanation) extending in the time direction (vertical) and 32 multicarrier OFDM symbols 1604 (a single OFDM symbol also illustrated). That is, all 32 PAM symbols 1602 are time domain symbols in the row direction, and can be transmitted in time over the duration of a single OFDM symbol 1604. Similarly, all 32 OFDM symbols 1604 are frequency domain symbols in the column direction, and can fit within the bandwidth of a single PAM symbol 1602. As can therefore be seen from FIG. 16, a 90 degree rotation of plot 1600 (visually, about the plot "center point") demonstrates the present modulation technique that effectively turns the PAM transmission into an OFDM transmission, and vice versa.

[00180] By this modulation technique, a 32x1 symbol can be mathematically rotated to become a 1x32 symbol, and vice versa. This principle may be expanded, for example, to rotate a plurality of time division multiple access (TDMA) single carrier sequential transmissions, from a plurality of transmitters, to effectively obtain a plurality of OFDMA simultaneous transmissions from the plurality of transmitters. Furthermore, although OFDM (without time domain tapering, e.g., raised cosine) exhibits OOB energy splatter, this characteristic is analogous to the sin(x)/x response (in time) of PAM signals if the channel rolloff factor (alpha) is small, or zero (the "brick wall"). Therefore, according to this embodiment, an OFDM transmission utilizing time domain tapering will perform an operation analogous to a PAM transmission utilizing a roll-off factor. This rotational illustration of the present modulation techniques demonstrates still further advantages that may be realized over conventional duobinary transmissions, as illustrated below with respect to FIGS. 17A-B.

[00181] FIG. 17A is a graphical illustration depicting a time domain plot 1700 of a duobinary block transmission, and FIG. 17B is a graphical illustration depicting a frequency domain plot 1702 of the duobinary block transmission depicted in FIG. 17A. As illustrated in FIG. 17 A, time domain plot 1700 generally includes an envelope shaped like a half-cosine. When time domain plot 1700 is rotated 90 degrees, a "conventional" duobinary result is produced, and the time axis can then be relabeled as "frequency.". This rotational modulation technique, when performed on a conventional duobinary transmission, is referred to herein as "FD duobinary" or "duobinary OFDM." As illustrated in FIG. 17B, frequency domain plot 1702 is substantially flat, and exhibits an abrupt drop of OOB energy, which is desirable to reduce interference with the neighboring channels.

[00182] In some embodiments, the present FD duobinary modulation techniques may be implemented with respect to an OFDM transmission utilizing cyclic prefixes. An optional cyclic prefix 1704 is illustrated in FIG. 17A, and represents the front portion of the envelope of time domain plot 1700 being utilized as a CP. When cyclic prefix 1704 is so utilized, the sine-shaped envelope of the time domain duobinary signal resembles the shape of a fish, with cyclic prefix 1704 observable as the "tail" of the "fish." As described further below, the use of cyclic prefixes with the present FD duobinary modulation techniques is optional, and unnecessary when implementing a "no-CP receiver" according to the embodiments described above. When a cyclic prefix is not so implemented, the "fishtail" of time domain plot 1700 disappears, but the frequency domain plot 1702 improves, as the abrupt drop of the fishtail causes some OOB leakage. [00183] As can be seen with respect to frequency domain plot 1702, three different power levels are visible: (i) the peak subcarrier level (flat portion); (ii) an intermediate subcarrier level (drop off portions); and (iii) a zero-power subcarrier level (where the energy drops to the origin). In conventional time domain duobinary implementations, the occupied bandwidth of a QPSK signal, relative to partial response signaling (PRS, or 9-PRS), is greater according to the channel roll-off factor, alpha. For DOCSIS single carrier modulations, this bandwidth increase is approximately 5% greater for the downstream transmission and approximately 25% greater for the upstream transmission. According to the present FD duobinary techniques though, if the number of subcarriers is the same, with equal subcarrier spacing, the occupied bandwidth will remain the same, and not experience this increase. This bandwidth advantage occurs as result of the FD duobinary techniques producing the more abrupt drop of OOB energy (see e.g., FIG. 15), which thereby enables closer carrier spacing.

[00184] The FD duobinary modulation techniques of the present embodiments demonstrate a time-frequency swapping technique that advantageously relates OFDM and single carrier modulation transmissions as duals of one another, after the respective time and frequency axes are rotated and "relabeled." This time-frequency swapping technique allows new modulation candidates to be created, such as duobinary (e.g., PRS) OFDM, which has valuable properties for the cable plant.

[00185] Conventional modulation techniques are used at carrier frequencies to send digital data over a distance, either by wires, wireless, or optically. Three known modulation techniques include single carrier, multi-carrier, and code division multiple access (CDMA). All three techniques have been used on cable networks at one time or another.

[00186] The orthogonality between signals is a property that allows one signal, which includes a plurality of symbols, to be clearly received without interference from the symbols of another signal. This orthogonality can be expressed, for example, according to the following equation: x(n) * y(n) = 0 (Eq. 7) [00187] Where the variables, x and y, are orthogonal over a range if the sum of all x*y products are equal to zero over a range where y≠ x. Where the signals represent complex numbers, for example, (Eq. 7) would consider the sum of all x*conj(y) products. Different modulation techniques are known to achieve orthogonality by other means or calculations. A comparative example of impulse and resulting spectral responses are illustrated below with respect to FIGS. 18A-D, for a single carrier QPSK transmission and a duobinary 9-PRS transmission

[00188] FIG. 18A is a graphical illustration depicting an impulse response 1800 of the single carrier transmission. FIG. 18B is a graphical illustration depicting a spectral response 1802 of the single carrier transmission depicted in FIG. 18 A. FIG. 18C is a graphical illustration depicting an impulse response 1804 of a conventional duobinary transmission (e.g., an impulse response of 1.0 for two symbols, in this example). FIG. 18D is a graphical illustration depicting a spectral response 1806 of the duobinary transmission depicted in FIG. 18C. As illustrated in FIG. 18 A, a basic modulation technique, such as BPSK, can be created by connecting a periodic series of positive or negative impulses to a lowpass filter (not shown) having a sine(x)/x impulse response (see also FIG. 14, above). A raised cosine frequency response on the modulated signal may then be produced therefrom, as represented by spectral response 1802 of FIG. 18B. The abruptness of the frequency domain roll-off, as illustrated in FIG. 18B, represents the "alpha" factor, and is affected by damping applied to the sine(x)/x waveform.

[00189] In contrast, duobinary modulation employees a different impulse response, as illustrated by duobinary impulse response 1804 of FIG. 18C. In comparison with impulse response 1800 of the single carrier transmission (FIG. 18 A), duobinary impulse response can be seen to last over two symbol periods, as opposed to one symbol period, as illustrated in FIG. 18C. Moreover, the frequency domain of the resultant spectral response 1806, FIG. 18D, has a cosine shape, as opposed to the raised cosine shape of spectral response 1802 of the single carrier transmission.

[00190] FIG. 19A is a graphical illustration of a constellation 1900 depicting relative power calculations for error thresholds of the QPSK single carrier transmission depicted in FIGS. 18A-B, and FIG. 19B is a graphical illustration of a constellation 1902 depicting relative power calculations for error thresholds of the 9-PRS duobinary transmission depicted in FIGS. 18C-D. The 9-PRS signal may be produced, for example, by passing a two-level complex (i.e., I and Q) signal through a duobinary filter (not shown in FIG. 19B).

[00191] As illustrated in FIG. 19A, the QPSK signal has four equally probable states, whereas, as illustrated in FIG. 19B, the 9-PRS signal has a single state (middle of plot) having a probability of 0.25, four high-power corner states with a combined probability of 0.25 (i.e., 0.0625 each), and four intermediate power levels between the corners having a combined probability of 0.5 (i.e., 0.125 each). Thus, if the voltage difference between points A and B on constellation 1902 is assumed to be 1.0V, the power of the 9-PRS constellation will be 0.25*0 + 0.5* 1.0 + 0.25* 1.414 ^Λ 2 = 1 watt. In comparison, if the voltage difference on the QPSK constellation 1900 is set to be 0.707V between points C and D, the QPSK power it will also be 1 watt. Accordingly, a noise vector that would be required to make a slicing error on the QPSK signal will be 0.707V, whereas 0.5V would be required for the 9-PRS signal, resulting in a difference of 3dB between the two constellations. This difference is realized for the duobinary transmission over the QPSK signal by taking advantage of the fact that not all states are equally probable for the duobinary signal, despite the fact that both signals have a same RF power.

[00192] FIG. 20 is a graphical illustration depicting a diagram 2000 of single carrier voltage versus time. Diagram 2000 is comparable to overlay 1400, FIG. 14, and impulse response 1800, FIG. 18 A. In cable systems, single carrier modulation, e.g., 64-QAM and 256-QAM, has been used extensively on downstream signal paths, and upstream signal paths have utilized advanced time division multiple access (ATDMA), which is essentially a burst-mode single carrier transmission technique. Single carrier modulation includes a time series of voltage impulses (symbols) that have been filtered to limit interference with other frequency bands. Diagram 2000 thus illustrates five different sine(x)/x impulses with uniform time shifts. That is, five different symbols represented on diagram 2000 have a same value, but are shifted in time with respect to one another. [00193] In operation, the symbols of diagram 2000 may have any positive or negative values, and/or have real-only values or include complex values. The vertical lines appearing in diagram 2000 represent five separate sampling instants. The five separate waveforms are not considered to interfere with each other because, at each sampling instant, a particular symbol reaches its peak the value (1), while the other symbols are passing through zero. Accordingly, orthogonality is maintained throughout diagram 2000. In further operation, the system represented by diagram 2000 may be optimized to remove linear distortions, such as echoes, prior to sampling (e.g., using an adaptive equalizer). Without such optimization, the responses from the other respective symbols may not be zero at the particular sampling instant. In such instances, the non-zero symbols may contribute to distortion energy to the selected symbol.

[00194] FIG. 21 is a graphical illustration depicting a timing diagram 2100 for a spread spectrum signal. Direct sequence spread spectrum (DSSS) technology has been used in military applications to hide or mask communications and radar signals by making the signals appear noise-like. A related technology, known as synchronous code division multiple access (S-CDMA), has been used in cable transmission systems to provide upstream noise immunity, and for multiple access implementations. In an exemplary embodiment, timing diagram 2100 includes a low speed data input 2102, a pseudo-noise (PN) sequence 2104, an output 2106, and a chip rate 2108.

[00195] In the example illustrated in FIG. 21, multiple orthogonal codes are assigned to one or more users, and simultaneous transmissions may thus occur on different codes without interference. In an exemplary embodiment, the DSSS/S-CDMA technique of timing diagram 2100 further utilizes equalized signals to prevent loss of orthogonality between codes. In operation, a low speed data input 2102 is clocked against high-speed PN sequence 2104 (e.g., by use of an exclusive-OR gate, shown below with respect to FIG. 22) to produce output 2106, which appears noise-like. Further operation of timing diagram 2100 is explained further below with respect to FIG. 22.

[00196] FIG. 22 is a schematic illustration depicting an exemplary block diagram of a system 2200 for direct sequence spread transmission and reception. System 2200 includes a transmitter portion 2202 and a receiver portion 2204, and is configured to implement operation of timing diagram 2100, FIG. 21. In operation of system 2200, a random PN sequence (i.e., PN sequence 2104) is created using a plurality of cascading shift registers 2206 and an exclusive-OR gate 2208. The clocking rate of the several components represents chip rate 2108. A signal to be transmitted, i.e., output 2106, is generated by a data source inverting or not inverting, i.e., through implementation of exclusive-OR gate 2208, the pseudo-random output (PN sequence 2104) of shift registers 2206. At receiver portion 2204, a PN generator 2210 synchronizes with transmitter portion 2202, and also uses the same code (PN sequence 2104) to reproduce the signal of low speed data input 2102. Where system 2200 includes S-CDMA DOCSIS functionality, each circular time shift by shift registers 2206 (excluding the initial chip, which would not the shifted) produces another basis function that is orthogonal to all other shifts. In at least one embodiment, system 2200 further implements a pre-coder (not shown in FIG. 22), as described above, to eliminate error propagation at receiver portion 2204, as well as the FD duobinary OFDM techniques also described above. In some embodiments, receiver portion 2204 is configured to consider only one received symbol at a time.

[00197] Referring back to FIG. 2A-B, OFDM is used in DOCSIS 3.1 technology, as well as in many wireless standards. Some OFDM implementations obtain orthogonality through further utilization of different subcarriers that are all harmonics of the same fundamental. FIG. 2A is an illustrative example of an OFDM waveform with only four such subcarriers. Each of the harmonically-related subcarriers depicted in the illustration has a different magnitude and phase value from each other. When all four subcarriers are combined (i.e., summed) for transmission, the result is a single composite signal. However, orthogonality allows the original subcarriers to be separated at the receiver, typically using an FFT. Accordingly, FIG. 2B as an illustrative example of the same OFDM signal of FIG. 2 A, but shown in the frequency domain.

[00198] FIG. 23 is a graphical illustration depicting a plot 2300 of a received OFDM signal in the frequency domain. Specifically, plot 2300 represents a spectral plot of an OFDM signal that is affected by a deep frequency-selective fade. This fading phenomenon occurs frequently in wireless channels where a sum of echo components cancels the signal entirely, or at least at some subcarrier frequencies (e.g., the second harmonic carrier HC2, in the example illustrated in FIG. 23). Accordingly, the techniques and principles of the present embodiments are particularly applicable to such OFDM environments, where faded subcarriers that are essentially lost in the noise floor may be recovered using forward error correction (FEC) operations.

[00199] FIG. 24 is a graphical illustration of a plot 2400 depicting time and frequency relationships between common transmission impairments. Plot 2400 is particularly illustrative when considered together with time-frequency plot 1600, FIG. 16, above. Plot 2400 illustrates the relative effects from common transmission impairments in both the time domain and the frequency domain. As illustrated in FIG. 24, plot 2400 enables the visual understanding of the various effects from the different impairments on the different types of modulation.

[00200] For example, plot 2400 illustrates how random thermal, or Gaussian, noise is present at all frequencies and all times, and thus there is no present modulation technique that has a relative advantage to address additive white Gaussian noise (AWGN). The maximum data capacity in a channel having AWGN can be determined, for example, according to the Shannon Hartley Theorem. If the noise is not white noise (i.e., the spectrum of the noise is not flat), the maximum capacity of the channel may be determined according to the "water-pour" method of transmit power distribution. In cable systems, having a non-flat S R may result from cable loss varying with frequency, as well as nonlinear distortion products, which are random noise-like if the distortion was created by digital carriers.

[00201] Plot 2400 further illustrates that burst noise come on the other hand, occurs locally in time, but often has a wide spectrum. Single carrier modulation, with FEC, may therefore be effective to address burst noise in order to correct corrupted time domain symbols. However, it should be noted that a typical OFDM receiver will perform an FFT on the corrupted sequence, and thus spread the burst noise contamination to all related frequency domain symbols. In contrast, plot 2400 illustrates that a continuous wave (CW) interference source may be continuous in time, but relatively localized in frequency. OFDM modulation techniques having FEC might be utilized to repair such localized damage, to a limited number of subcarriers. However, use of single carrier modulation with a CW interferer will affect all symbols, thereby turning a constellation point (see e.g., FIGS. 19A-B) into a "donut" shape. Plot 2400 still further illustrates that a deep frequency-selective fade may be addressed in a similar manner as with CW interference, namely, by OFDM modulation with FEC.

[00202] There are other important considerations for selecting a modulation technique for a particular RF signal path, such as (i) tolerance to frequency offsets and tolerance to phase noise, both of which increase the cost of local oscillators, and (ii) peak-to-average power ratio, which makes transmitters consume more power, thereby decreasing battery life. Additionally, some receiver designers are known to further implement a number of design tricks, sometimes referred to as "secret sauces," to mitigate the effects of impairments, such as noise cancelers.

[00203] In consideration of the embodiments described above, a signal E for transmission, includes a plurality of individual component symbols e _n , and may be represented as follows:

[00204] For this signal E, a modulation matrix C may be formulated. The modulation matrix C includes rows and columns, and is optimally configured such that the respective rows are orthogonal to one another. Accordingly, in this mathematical representation, each of the three modulation techniques described above can be seen as essentially representing merely a different set of row functions, which are also referred to as orthogonal basis functions. For single carrier modulation, the modulation matrix C may constitute simply an identity matrix having a single diagonal row of Is and 0s elsewhere. (See e.g., Eq.5, above). For a DSSS signal modulation, the respective rows may be provided from a Walsh matrix, implementing a single circular shift between rows. [00205] The modulation matrix C may also include complex components, which may be represented as follows:

[00206] The general principle of orthogonality between rows may then be restated, for all rows where x≠ y, according to:

∑n=k

c(x, n) * c(y, ri) = 0 (Eq. 10)

71=1

[00207] Accordingly, the unmodulated signal E may be transmitted as a modulated signal F by multiplying the input sequence of the unmodulated signal E by the modulation matrix C, represented by:

F = E * C = \f ₁ , f ₂ , f ₃ , f ₄ ... fj\ (Eq. 11)

[00208] With respect to OFDM modulation, the respective rows of the modulation matrix C may also represent complex exponentials (e.g., sine and cosine waves), where the first row would represent, for example, the first harmonic, the second row would represent the second harmonic, etc., as illustrated below with respect to FIG. 25.

[00209] FIG. 25 is a graphical illustration depicting an OFDM modulation matrix 2500. FIG. 25 is a graphical illustration depicting an OFDM modulation matrix 2500. In the example illustrated in FIG. 25, modulation matrix 2500 is an eight-row matrix of signs and cosines for an OFDM modulation, where the cosine waves are depicted as solid lines, and the sine waves are depicted as dashed lines. Further in this example, row X[l] is depicted to form an upper sideband, and row X[7] is depicted to form a matching lower sideband. [00210] The innovative principles of the embodiments described above advantageously coordinate the time and frequency relationships between transmitted symbols to effectively render single carrier modulation and multicarrier OFDM modulation as separate aspects of the same modulation technique, wherein the separate aspects are distinguished by a 90-degree rotation in the time-frequency plot.

[0021 1] The duality between time and frequency according to the present embodiments can be further observed in consideration of the following discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) equations:

W-1

F[k] = ^ f [k]e ^~j ^ ^nk (Eq. 13)

[00212] That is, the essential differences between these two equations are the scale factor and the polarity sign in front of the complex exponential. The equations are otherwise very similar. Observing only a plotted set of transform pairs, for example, it would be difficult to identify which plot represents time and which represents frequency, as illustrated above in time-frequency plot 1600, FIG. 16. Without knowing the rotational frame of reference, that is, which axis represents time and which axis represents frequency, the PAM transmission is essentially indistinguishable from the OFDM transmission.

[00213] In some cases, non-orthogonal signals are utilized for communication transmissions. For example, a non-orthogonal spread-spectrum signal may operate in the presence of other signal types by taking advantage of spreading gain. In such instances, some level of interference will be experienced with the non-orthogonal signal, however, such interference is optimally kept within tolerable levels. [00214] According to the rotational principles of the modulation techniques described above, the same symbol may be observed as either a 32x1 symbol or as a 1x32 symbol, depending on the rotational frame of reference. It should be noted though, that dispersion, or other linear distortion, occurs along the time axis, but not along the frequency axis. Dispersion along the frequency axis would be instead indicate nonlinear distortion. Thus, according to the embodiments described herein, performing a 90 degree rotation on a particular sequence is effectively the same as performing a FFT on that sequence. Similarly, performance of a -90 degree rotation on the particular sequence effectively accomplishes an IFFT on that sequence. Thus, the rotational operations of the present embodiments may be implemented using FFTs and IFFTs (or DFTs and IDFTs, respectively).

[00215] Referring back to FIGS. 17A-B, the desirable characteristic of a gentle rise and fall in transmit power level is a result of duobinary modulation summing each subcarrier with in an adjacent subcarrier, where that the adjacent subcarrier has a same magnitude component. The duobinary OFDM modulation techniques of the present embodiments are of particular advantage with respect to very narrow bandwidth OFDM transmissions having a relatively small number of subcarriers, e.g., Internet of Things (IoT), ham radio operation, etc. The techniques described herein effectively resolve the spectral splatter experienced in such environments, which cause adjacent channel interference. The present embodiments are also of particular use in signaling operations that utilize a small number of bits in a narrow bandwidth, such as "acks" or acknowledgements.

[00216] The present embodiments may also be implemented with respect to conventional pre-coding techniques (see e.g., "Digital Telephony," 3d Ed, by John Bellamy), which allow allows a symbol to be decoded without referencing a preceding symbol (e.g., correlative level encoding). The duobinary OFDM techniques herein are described, by way of example, for 2-level I and Q signals (QPSK), which are illustrated to be converted into a 3x3 constellation. The person of ordinary skill in the art though, will understand that these examples are provided for illustration purposes, and are not intended to be limiting. According to the principles described herein, the present embodiments may be implemented with respect to other modulation orders. A 16-QAM modulation order, for example, when duobinary-filtered, will create a 49-point constellation.

[00217] The duobinary modulation techniques of this description may also utilize impulse responses other than those with 2 adjacent symbols. For example, an impulse response according to the present embodiments include more than 2 symbols. In other instances, where two symbols are utilized, the 2 symbols need not be on adjacent subcarriers

[00218] As described above, time-frequency duals are conventionally known, but the modulation techniques thereof are conventionally considered to be separate from one another. That is, OFDM is considered to be the dual of the single carrier, and OFDMA is considered to be the dual of single carrier TDMA, but the conventional modulation techniques therefor are considerably different from one another.

Windowed OFDM to Reduce ISI and OOB Emissions

[00219] As described above, a raised cosine function may be produced on the spectral response of some of the embodiments. In an exemplary embodiment, an alternative process applies a time domain window function, such as a Hamming window, to reduce ISI and OOB emissions.

[00220] FIG. 26A is a graphical illustration depicting a Hamming window function 2600 in the time domain. FIG. 26B is a graphical illustration depicting an inverse Hamming function 2602 in the time domain. FIG. 26C is a graphical illustration depicting an implementation of a Hamming window on a time domain waveform 2604. In the exemplary embodiment, the time domain window function utilizes Hamming window functions according to the examples shown in FIGS. 26A-C.

[00221] In operation, the time domain window function is applied to an OFDM or OFDMA waveform, and the windowed waveform therefrom is then transmitted. The implementation of the window function will create some ISI, but reduce the OOB emissions. However, the window function may be selected, as described below, to cancel the created ISI in the frequency domain with a frequency domain convolution, as described below with respect to FIGS. 27A-B. Implementation of the windowing/window function operation may be performed at either or both of the transmitter and the receiver of the system.

[00222] FIG. 27A is a graphical illustration depicting a Hamming impulse response 2700 in the frequency domain. FIG. 27B is a graphical illustration depicting an inverse convolution impulse response 2702 in the frequency domain. In operation, by reversing the Hamming window (i.e., inverting in time domain, for example, by inverse Hamming function 2602, FIG. 26B) any noise-plus-interference that occurs may be multiplied while a transmitted signal is weak. In an exemplary embodiment, this problem may be alleviated by implementing a maximum likelihood estimation for the frequency domain symbols, based upon the knowledge of what the original Hamming impulse response (e.g., impulse response 2700) looks like, as illustrated below with respect to FIGS. 28A-B. In this exemplary embodiment, ISI created utilizing a Hamming impulse response 2700 may be corrected through frequency domain convolution with an inverse filter H(f), representing inverse convolution impulse response 2702.

[00223] FIG. 28A is a graphical illustration depicting an unequalized constellation 2800 on which a Hamming window (e.g., Hamming window function 2600, FIG. 26A) has been implemented. FIG. 28B is a graphical illustration depicting an equalized constellation 2802 on which a Hamming window has been implemented. Unequalized constellation 2800 demonstrates the effect of the created ISI and OOB emissions on the transmission from applying only the time domain window function without further correction. Equalized constellation 2802 demonstrates how the created ISI and OOB emissions can be corrected through the coordinated frequency domain equalization techniques described above, which implement the inverse impulse response of the selected time domain window function, or Hamming window.

[00224] FIG. 29A is a graphical illustration depicting an implementation of a raised cosine voltage function on a time domain waveform 2900. FIG. 29B is a graphical illustration depicting an unequalized constellation 2902 on which a raised cosine function has been implemented. FIG. 29C is a graphical illustration depicting a constellation 2904, after equalization, on which a raised cosine function has been implemented. FIGS. 29A-C illustrate, by way of comparison, the significant improvement in windowing presented by the present techniques, as compared with conventional raised cosine operations. The resulting constellations produced according to the present techniques may be much more carefully controlled than with implementation of the raised cosine window, which, although exhibiting more uniform power, will also experience the "division-by-zero" problem in its inverse time response.

[00225] According to these embodiments, ISI damage to the waveform can be more easily repaired, and performed with significantly improved precision. These advantageous techniques have particular applicability to cable transmission operations, where, for example, one OFDM signal may be cresting, while another OFDM signal is passing through a minimum power. OFDM signals, for example, constitute orthogonal basis functions, which include sines and cosines, each having an integer number of cycles. The basis functions must be orthogonal to prevent energy leakage into neighboring signals. The present embodiments advantageously implement a time domain Hamming function on the time domain waveform of the OFDM signal, but in the frequency domain, utilize a Hamming impulse response, and inverse impulse response for convolution.

[00226] In an exemplary embodiment, the Hamming window is implemented on a QPSK OFDMA transmission exhibiting ISI. Implementation of the time domain windowing operation/Hamming window results in significant reduction in an amount of leakage from adjacent blocks, while also reducing OOB splatter. In at least one embodiment, implementation of the Hamming window is made more uniform through utilization of a second carrier out of phase.

[00227] Although specific features of various embodiments may be shown in some drawings and not in others, this is for convenience only. In accordance with the principles of the systems and methods described herein, any feature of a drawing may be referenced or claimed in combination with any feature of any other drawing. For example, the following list of example claims represents only some of the potential combinations of elements possible from the systems and methods described herein. [00228] a(i). A signal equalizing receiver, configured to: capture a plurality of OFDM symbols transmitted over a signal path adding linear distortion to the plurality of OFDM symbols; form the plurality of captured OFDM symbols into an overlapped compound data block, wherein the compound data block includes at least one pseudo-extension in addition to payload data from at least one of the plurality of OFDM symbols; process the overlapped compound data block with one of (i) a circular convolution having an inverse channel response in the time domain, and (ii) a frequency domain equalization in the frequency domain, to produce an equalized compound block; discard at least one end portion of the equalized compound block to produce a narrow equalized block, wherein the at least one end portion corresponds with the at least one pseudo-extension, and wherein the narrow equalized block corresponds with the payload data; and cascade two or more narrow equalized blocks to form a de-ghosted signal stream of the plurality of OFDM symbols, wherein the plurality of OFDM symbols includes one or more of an OFDM transmission and an OFDMA transmission, wherein the plurality of OFDM symbols includes one or more of a cyclic prefix and no cyclic prefix, and wherein a length of the at least one pseudo-extension is different than a length of the cyclic prefix.

[00229] a(ii). A digital transmission receiver having a processor and a memory, configured to: receive a digital signal transmission from a signal path including a plurality of data blocks having linear distortion; determine, from the signal path of the digital signal transmission, a duration of at least one reflection of the digital signal transmission on the digital signal path; attach a pseudo-extension to a first data block of the plurality of data blocks, wherein the length of the pseudo-extension in the time domain is greater than the duration of the at least one reflection; process the first data block, together with the pseudo-extension attached thereto, to remove linear distortion from the first data block; and discard the processed pseudo-extension from the processed first data block after the linear distortion has been removed.

[00230] b(ii). The receiver of claim a(ii), wherein the signal path includes a first signal subpath and second signal subpath different from the first signal subpath.

[00231] c(ii). The receiver of claim b(ii), wherein the first signal subpath is a wired subpath and the second signal subpath is a wireless subpath. [00232] d(ii). The receiver of claim b(ii), wherein the first signal subpath is a direct signal path and the second signal subpath is an indirect signal path.

[00233] e(ii). The receiver of claim b(ii), wherein the reflection is transmitted along the second signal subpath.

[00234] f(ii). The receiver of claim b(ii), wherein the second signal subpath is longer than the first signal subpath.

[00235] g(ii). The receiver of claim b(ii), wherein the receiver is further configured to receive (i) a real component of the digital signal transmission from the first signal subpath, and (ii) an imaginary component of the digital signal transmission from the second signal subpath.

[00236] h(ii). The receiver of claim a(ii), wherein the receiver is further configured to process the first data block using an overlapped circular convolution process.

[00237] i(ii). The receiver of claim a(ii), wherein the receiver is further configured to process the first data block using an overlapped Fourier transform process.

[00238] j(ii). The receiver of claim a(ii), wherein the pseudo-extension comprises at least one of a pseudo-prefix obtained from a second data block preceding the first data block, and a pseudo-suffix obtained from a third data block succeeding the first data block.

[00239] k(ii). The receiver of claim a(ii), wherein the receiver is further configured to process the digital signal transmission from one or more digital transmission schemes, including orthogonal frequency-division multiplexing (OFDM), orthogonal frequency-division multiple-access (OFDMA), data over cable service interface specification (DOCSIS), multiple input/multiple output (MTMO), and single-carrier frequency-division multiple-access (SC-FDMA).

[00240] a(iii). A digital transmission system, comprising: a transmitter configured to transmit orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto; a receiver for receiving the transmitted OFDM symbols from the transmitter; and a signal path for communicating the transmitted OFDM symbols from the transmitter to the receiver, wherein the OFDM symbols received by the receiver include linear distortion from the signal path, and wherein the receiver is configured to process the received OFDM symbols and linear distortion using an overlapped circular convolution function to produce equalized OFDM symbols.

[00241] a(iv). A digital transmission system, comprising: a transmitter configured to transmit orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto; a receiver for receiving the transmitted OFDM symbols from the transmitter; and a signal path for communicating the transmitted OFDM symbols from the transmitter to the receiver, wherein the OFDM symbols received by the receiver include linear distortion from the signal path, wherein the receiver is configured to process the received OFDM symbols and linear distortion by an overlapped Fourier transform function to produce equalized OFDM symbols, and wherein the overlapped Fourier transform function is configured to (i) overlap individual ones of the distorted OFDM symbols with overlapped time energy from respectively adjacent ones of the distorted OFDM symbols, (ii) transform the overlapped individual distorted OFDM symbols into distorted frequency domain symbols, (iii) perform complex multiplication of the distorted frequency domain symbols by equalization coefficients to equalize the distorted frequency domain symbols, (iv) remove the overlapped time energy from a time domain component of the equalized frequency domain symbols, and (v) produce undistorted frequency domain symbols from a frequency domain component of the time domain component with the overlapped time energy removed.

[00242] a(v). A digital transmission system, comprising: a transmitter configured to transmit (i) a series of orthogonal frequency-division multiplexing (OFDM) symbols having no cyclic prefix attached thereto, and (ii) at least one constant amplitude zero autocorrelation waveform sequence (CAZAC) sequence; a receiver for receiving the transmitted series of OFDM symbols and the CAZAC sequence from the transmitter; and a signal path for communicating the transmitted series of OFDM symbols and CAZAC sequence from the transmitter to the receiver, wherein the series of OFDM symbols and the CAZAC sequence are received by the receiver with linear distortion from the signal path, and wherein the receiver is configured to utilize the received CAZAC sequence as a reference signal for equalizing the received series of OFDM symbols.

[00243] b(v). The system of claim a(v), wherein the CAZAC sequence comprises at least one Zadoff Chu sequence.

[00244] c(v). The system of claim b(v), wherein the at least one Zadoff Chu sequence comprises a first Zadoff Chu sequence preceding the series of OFDM symbols in the time D and a second Zadoff Chu sequence succeeding the series of OFDM symbols in the time domain.

[00245] d(v). The system of claim a(v), wherein the receiver is further configured to determine from the received CAZAC sequence at least one of a channel characterization, an offset frequency, and a start of one or more of the OFDM symbols in the time domain.

[00246] a(vi). A method of equalizing a transmitted digital signal, comprising the steps of: receiving, in the time domain, a sequential series of first, second, and third data blocks of the transmitted digital signal; forming a compound block in the time domain from the second data block including an end portion of first data block and a leading portion of the third data block; performing circular convolution on the compound block using a set of equalization coefficients to equalize the compound block in the time domain; extracting from the equalized compound block a narrow block corresponding to equalized time domain data of the second data block; converting the narrow block from the time domain into frequency domain data; and reading frequency domain symbols relating to the second data block from the converted narrow block.

[00247] b(vi). The method of claim a(vi), further comprising a step of forming a compound block in the time domain from the third data block including an end portion of second data block and a leading portion of a fourth data block immediately succeeding the third data block. [00248] c(vi). The method of claim a(vi), wherein the transmitted digital signal is an orthogonal frequency-division multiplexing (OFDM) signal, and wherein the frequency domain symbols are OFDM symbols.

[00249] a(vii). A method of equalizing a digital signal transmitted over a signal path, comprising the steps of: receiving, in the time domain, a sequential series of time domain samples of the transmitted digital signal; forming the received sequential series of time domain samples into a separate sub-series of overlapping compound time domain blocks, wherein each compound time domain block of the sub-series includes a pseudo-prefix comprising information from an immediately preceding block; determining an echo delay on the signal path; converting the compound blocks into the frequency domain to form compound frequency domain blocks; equalizing the compound frequency domain blocks to form equalized frequency domain blocks; converting the equalized frequency domain blocks into the time domain to form equalized time domain compound blocks; discarding, from the equalized time domain compound blocks, overlapping time domain energy portions corresponding to respective equalized pseudo-prefixes, to form narrow equalized blocks; pasting the narrow equalized blocks together to form a composite equalized time domain signal; and converting the composite equalized time domain into the frequency domain and read equalized frequency domain symbols therefrom.

[00250] b(vii). The method of claim a(vii), wherein the digital signal is an orthogonal frequency-division multiplexing (OFDM) signal, and wherein the equalized frequency domain symbols are OFDM symbols.

[00251] c(vii). The method of claim a(vii), wherein the step of determining comprises one of (i) performing signal characterization for the signal path and (ii) assigning a pre-determined threshold value for the echo delay.

[00252] a(viii). A method of modulating, by a transmitter, a series of input digital symbols of a first modulation scheme, comprising the steps of: receiving a sequential series of samples of the digital symbols in a first domain of the first modulation scheme, wherein the first domain is one of the time domain and the frequency domain; determining a dual of the first modulation scheme, wherein the dual has a second modulation scheme in a second domain that is different from the first domain, and wherein the second domain comprises the other of the time domain and the frequency domain; applying a 90 degree rotational operation to the second modulation scheme to generate a rotational modulation format; modulating the series of digital symbols with the generated rotational modulation format; and outputting the modulated series of digital symbols to a receiver.

[00253] b(viii). The method of claim a(viii), wherein the first modulation scheme comprises a single carrier modulation scheme.

[00254] c(viii). The method of claim b(viii), wherein the single carrier modulation scheme comprises quadrature amplitude modulation.

[00255] d(viii). The method of claim b(viii), wherein the single carrier modulation scheme comprises pulse amplitude modulation.

[00256] e(viii). The method of claim d(viii), wherein the second modulation scheme comprises orthogonal frequency division multiplexing modulation.

[00257] f(viii). The method of claim b(viii), wherein the orthogonal frequency division multiplexing modulation comprises partial response signaling.

[00258] g(viii). The method of claim a(viii), wherein the first modulation scheme comprises orthogonal frequency division multiple access modulation, and wherein the second modulation scheme comprises time division multiple access modulation.

[00259] h(viii). The method of claim a(viii), wherein the first modulation scheme comprises a multicarrier modulation format, and wherein the second modulation scheme comprises a single carrier modulation format.

[00260] i(viii). The method of claim a(viii), wherein the first modulation scheme comprises one of a spread spectrum modulation format and a code division multiple access format.

[00261] j(viii). The method of claim a(viii), further comprising a step of equalizing the series of digital symbols prior to the step of modulating. [00262] k(viii). The method of claim a(viii), further comprising a step of precoding the series of digital symbols prior to the step of modulating.

[00263] l(viii). The method of claim a(viii), wherein the generated rotational modulation format comprises one of binary phase shift keying and duobinary modulation.

[00264] m(viii). The method of claim a(viii), wherein the step of applying comprises one of a Fourier transform operation and an inverse Fourier transform operation.

[00265] a(ix). A digital transmission system, comprising: a transmitter configured to transmit an input series of complex symbols; a duobinary encoder disposed within the transmitter, and configured to filter the input series of complex symbols and output a partial response signaling (PRS) signal; a converter disposed within the transmitter, and configured to convert the PRS signal output into the time domain; and a receiver for receiving the time domain-converted PRS signal from the transmitter over a signal path.

[00266] b(ix). The system of claim a(ix), wherein the transmitter further comprises a pre-coder configured to pre-code the complex symbols prior to filtering by the duobinary encoder.

[00267] c(ix). The system of claim a(ix), wherein the converter is further configured to perform at least one of a fast Fourier transform, an inverse fast Fourier transform, a discrete Fourier transform, and an inverse discrete Fourier transform.

[00268] d(ix). The system of claim a(ix), wherein the signal path comprises at least one of a cable network, a wired transmission line, a wireless path, and a fiber optic line.

[00269] e(ix). The system of claim a(ix), wherein the complex symbols comprise an orthogonal frequency division multiple access symbols transmitted in an upstream direction of the digital transmission system. [00270] f(ix). The system of claim a(ix), wherein the converter is further configured to construct a duobinary OFDM signal for transmission by splitting a subcarrier of the input series of complex symbols to appear over two adjacent frequency domain subcarriers.

[00271] a(x). A method of modulating, by a transmitter, an input digital signal transmission, comprising the steps of: receiving the input digital signal having a first time-frequency order on the time-frequency axis; rotating the time-frequency axis by 90 degrees; modulating the input digital signal according to the rotated time- frequency axis; and outputting the modulated digital signal to a receiver.

[00272] Some embodiments involve the use of one or more electronic or computing devices. Such devices typically include a processor, processing device, or controller, such as a general purpose central processing unit (CPU), a graphics processing unit (GPU), a microcontroller, a reduced instruction set computer (RISC) processor, an application specific integrated circuit (ASIC), a programmable logic circuit (PLC), a programmable logic unit (PLU), a field programmable gate array (FPGA), a digital signal processing (DSP) device, and/or any other circuit or processing device capable of executing the functions described herein. The methods described herein may be encoded as executable instructions embodied in a computer readable medium, including, without limitation, a storage device and/or a memory device. Such instructions, when executed by a processing device, cause the processing device to perform at least a portion of the methods described herein. The above examples are exemplary only, and thus are not intended to limit in any way the definition and/or meaning of the term processor and processing device.

[00273] This written description uses examples to disclose the embodiments, including the best mode, and also to enable any person skilled in the art to practice the embodiments, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the disclosure is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal language of the claims.