Aspects of the present application relate to electronic communications.
Existing communications methods and systems are overly power hungry and/or spectrally inefficient. Further limitations and disadvantages of conventional and traditional approaches will become apparent to one of skill in the art, through comparison of such approaches with some aspects of the present method and system set forth in the remainder of this disclosure with reference to the drawings.
Methods and systems are provided for signal reception using non-linearity-compensated, partial response feedback, substantially as illustrated by and/or described in connection with at least one of the figures, as set forth more completely in the claims.
As utilized herein the terms “circuits” and “circuitry” refer to physical electronic components (i.e. hardware) and any software and/or firmware (“code”) which may configure the hardware, be executed by the hardware, and or otherwise be associated with the hardware. As used herein, for example, a particular processor and memory may comprise a first “circuit” when executing a first one or more lines of code and may comprise a second “circuit” when executing a second one or more lines of code. As utilized herein, “and/or” means any one or more of the items in the list joined by “and/or”. As an example, “x and/or y” means any element of the three-element set {(x), (y), (x, y)}. As another example, “x, y, and/or z” means any element of the seven-element set {(x), (y), (z), (x, y), (x, z), (y, z), (x, y, z)}. As utilized herein, the term “exemplary” means serving as a non-limiting example, instance, or illustration. As utilized herein, the terms “e.g.,” and “for example” set off lists of one or more non-limiting examples, instances, or illustrations. As utilized herein, circuitry is “operable” to perform a function whenever the circuitry comprises the necessary hardware and code (if any is necessary) to perform the function, regardless of whether performance of the function is disabled, or not enabled, by some user-configurable setting.
The mapper 102 may be operable to map bits of the Tx_bitstream to be transmitted to symbols according to a selected modulation scheme. The symbols may be output via signal 103. For example, for an quadrature amplitude modulation scheme having a symbol alphabet of N (N-QAM), the mapper may map each Log2(N) bits of the Tx_bitstream to single symbol represented as a complex number and/or as in-phase (I) and quadrature-phase (Q) components. Although N-QAM is used for illustration in this disclosure, aspects of this disclosure are applicable to any modulation scheme (e.g., amplitude shift keying (ASK), phase shift keying (PSK), frequency shift keying (FSK), etc.). Additionally, points of the N-QAM constellation may be regularly spaced (“on-grid”) or irregularly spaced (“off-grid”). Furthermore, the symbol constellation used by the mapper may be optimized for best bit-error rate performance that is related to log-likelihood ratio (LLR) and to optimizing mean mutual information bit (MMIB). The Tx_bitstream may, for example, be the result of bits of data passing through a forward error correction (FEC) encoder and/or an interleaver. Additionally, or alternatively, the symbols out of the mapper 102 may pass through an interleaver.
The pulse shaper 104 may be operable to adjust the waveform of the signal 103 such that the waveform of the resulting signal 113 complies with the spectral requirements of the channel over which the signal 113 is to be transmitted. The spectral requirements may be referred to as the “spectral mask” and may be established by a regulatory body (e.g., the Federal Communications Commission in the United States or the European Telecommunications Standards Institute) and/or a standards body (e.g., Third Generation Partnership Project) that governs the communication channel(s) and/or standard(s) in use. The pulse shaper 104 may comprise, for example, an infinite impulse response (IIR) and/or a finite impulse response (FIR) filter. The number of taps, or “length,” of the pulse shaper 104 is denoted herein as LTx, which is an integer. The impulse response of the pulse shaper 104 is denoted herein as hTx. The pulse shaper 104 may be configured such that its output signal 113 intentionally has a substantial amount of inter-symbol interference (ISI). Accordingly, the pulse shaper 104 may be referred to as a partial response pulse shaping filter, and the signal 113 may be referred to as a partial response signal or as residing in the partial response domain, whereas the signal 103 may be referred to as residing in the symbol domain. The number of taps and/or the values of the tap coefficients of the pulse shaper 104 may be designed such that the pulse shaper 104 is intentionally non-optimal for additive white Gaussian noise (AWGN) in order to improve tolerance of non-linearity in the signal path. In this regard, the pulse shaper 104 may offer superior performance in the presence of non-linearity as compared to, for example, a conventional near zero positive ISI pulse shaping filter (e.g., root raised cosine (RRC) pulse shaping filter). The pulse shaper 104 may be designed as described in one or more of: the United States patent application titled “Design and Optimization of Partial Response Pulse Shape Filter,” the United States patent application titled “Constellation Map Optimization For Highly Spectrally Efficient Communications,” and the United States patent application titled “Dynamic Filter Adjustment For Highly-Spectrally-Efficient Communications,” each of which is incorporated herein by reference, as set forth above.
It should be noted that a partial response signal (or signals in the “partial response domain”) is just one example of a type of signal for which there is correlation among symbols of the signal (referred to herein as “inter-symbol-correlated (ISC) signals”). Such ISC signals are in contrast to zero (or near-zero) ISI signals generated by, for example, raised-cosine (RC) or root-raised-cosine (RRC) filtering. For simplicity of illustration, this disclosure focuses on partial response signals generated via partial response filtering. Nevertheless, aspects of this disclosure are applicable to other ISC signals such as, for example, signals generated via matrix multiplication (e.g., lattice coding), and signals generated via decimation below the Nyquist frequency such that aliasing creates correlation between symbols.
The timing pilot insertion circuit 105 may insert a pilot signal which may be utilized by the receiver for timing synchronization. The output signal 115 of the timing pilot insertion circuit 105 may thus comprise the signal 113 plus an inserted pilot signal (e.g., a sine wave at ¼×fbaud, where fbaud is the symbol rate). An example implementation of the pilot insertion circuit 105 is described in the United States patent application titled “Timing Synchronization for Reception of Highly-Spectrally-Efficient Communications,” which is incorporated herein by reference, as set forth above.
The transmitter front-end 106 may be operable to amplify and/or upconvert the signal 115 to generate the signal 116. Thus, the transmitter front-end 106 may comprise, for example, a power amplifier and/or a mixer. The front-end may introduce non-linear distortion and/or phase noise (and/or other non-idealities) to the signal 116. The non-linearity of the circuit 106 may be represented as FnlTx which may be, for example, a polynomial, or an exponential (e.g., Rapp model). The non-linearity may incorporate memory (e.g., Voltera series).
The channel 107 may comprise a wired, wireless, and/or optical communication medium. The signal 116 may propagate through the channel 107 and arrive at the receive front-end 108 as signal 118. Signal 118 may be noisier than signal 116 (e.g., as a result of thermal noise in the channel) and may have higher or different ISI than signal 116 (e.g., as a result of multi-path).
The receiver front-end 108 may be operable to amplify and/or downconvert the signal 118 to generate the signal 119. Thus, the receiver front-end may comprise, for example, a low-noise amplifier and/or a mixer. The receiver front-end may introduce non-linear distortion and/or phase noise to the signal 119. The non-linearity of the circuit 108 may be represented as FnlRx which may be, for example, a polynomial, or an exponential (e.g., Rapp model). The non-linearity may incorporate memory (e.g., Voltera series).
The timing pilot recovery and removal circuit 110 may be operable to lock to the timing pilot signal inserted by the pilot insertion circuit 105 in order to recover the symbol timing of the received signal. The output 122 may thus comprise the signal 120 minus (i.e., without) the timing pilot signal. An example implementation of the timing pilot recovery and removal circuit 110 is described in the United States patent application titled “Timing Synchronization for Reception of Highly-Spectrally-Efficient Communications,” which is incorporated herein by reference, as set forth above.
The input filter 109 may be operable to adjust the waveform of the partial response signal 119 to generate partial response signal 120. The input filter 109 may comprise, for example, an infinite impulse response (IIR) and/or a finite impulse response (FIR) filter. The number of taps, or “length,” of the input filter 109 is denoted herein as LRx, an integer. The impulse response of the input filter 109 is denoted herein as hRx. The number of taps, and/or tap coefficients of the pulse shaper 109 may be configured based on: a non-linearity model, F{circumflex over (n)}l, signal-to-noise ratio (SNR) of signal 120, the number of taps and/or tap coefficients of the Tx partial response filter 104, and/or other parameters. The number of taps and/or the values of the tap coefficients of the input filter 109 may be configured such that noise rejection is intentionally compromised (relative to a perfect match filter) in order to improve performance in the presence of non-linearity. As a result, the input filter 109 may offer superior performance in the presence of non-linearity as compared to, for example, a conventional near zero positive ISI matching filter (e.g., root raised cosine (RRC) matched filter). The input filter 109 may be designed as described in one or more of: the United States patent application titled “Design and Optimization of Partial Response Pulse Shape Filter,” the United States patent application titled “Constellation Map Optimization For Highly Spectrally Efficient Communications,” and the United States patent application titled “Dynamic Filter Adjustment For Highly-Spectrally-Efficient Communications,” each of which is incorporated herein by reference, as set forth above.
As utilized herein, the “total partial response (h)” may be equal to the convolution of hTx and hRx, and, thus, the “total partial response length (L)” may be equal to LTx+LRx−1. L may, however, be chosen to be less than LTx+LRx−1 where, for example, one or more taps of the Tx pulse shaper 104 and/or the Rx input filter 109 are below a determined level. Reducing L may reduce decoding complexity of the sequence estimation. This tradeoff may be optimized during the design of the system 100.
The equalizer and sequence estimator 112 may be operable to perform an equalization process and a sequence estimation process. The equalizer may utilize a least mean square (LMS) algorithm that attempts minimize the least mean square of an error between the partial response feedback signal 203 and the equalizer output 222. Details of an example implementation of the equalizer and sequence estimator 112 are described below with respect to
The de-mapper 114 may be operable to map symbols to bit sequences according to a selected modulation scheme. For example, for an N-QAM modulation scheme, the mapper may map each symbol to Log2(N) bits of the Rx_bitstream. The Rx_bitstream may, for example, be output to a de-interleaver and/or an FEC decoder. Alternatively, or additionally, the de-mapper 114 may generate a soft output for each bit, referred as LLR (Log-Likelihood Ratio). The soft output bits may be used by a soft-decoding forward error corrector (e.g. a low-density parity check (LDPC) dedecoder). The soft output bits may be generated using, for example, a Soft Output Viterbi Algorithm (SOVA) or similar. Such algorithms may use additional information of the sequence decoding process including metrics levels of dropped paths and/or estimated bit probabilities for generating the LLR, where LLR (b)=log(Pb/1−Pb), where Pb is the probability that bit b=1.
In an example implementation, components of the system upstream of the pulse shaper 104 in the transmitter and downstream of the equalizer and sequence estimator 112 in the receiver may be as found in a conventional N-QAM system. Thus, through modification of the transmit side physical layer and the receive side physical layer, aspects of the invention may be implemented in an otherwise conventional N-QAM system in order to improve performance of the system in the presence of non-linearity as compared, for example, to use of RRC filters and an N-QAM slicer.
The equalizer 202 may be operable to process the signal 122 to reduce ISI caused by the channel 107. The output 222 of the equalizer 202 is a partial response domain signal. The ISI of the signal 222 is primarily the result of the pulse shaper 104 and the input filter 109 (there may be some residual ISI from multipath, for example, due to use of the least means square (LMS) approach in the equalizer 202). The error signal, 201, fed back to the equalizer 202 is also in the partial response domain. The signal 201 is the difference, calculated by combiner 204, between 222 and a partial response signal 203 that is output by non-linearity modeling circuit 236a. An example implementation of the equalizer is described in the United States patent application titled “Feed Forward Equalization for Highly-Spectrally-Efficient Communications,” which is incorporated herein by reference, as set forth above.
The carrier recovery circuit 208 may be operable to generate a signal 228 based on a phase difference between the signal 222 and a partial response signal 207 output by the non-linearity modeling circuit 236b. The carrier recovery circuit 208 may be as described in the United States patent application titled “Coarse Phase Estimation for Highly-Spectrally-Efficient Communications,” which is incorporated herein by reference, as set forth above.
The phase adjust circuit 206 may be operable to adjust the phase of the signal 222 to generate the signal 226. The amount and direction of the phase adjustment may be determined by the signal 228 output by the carrier recovery circuit 208. The signal 226 is a partial response signal that approximates (up to an equalization error caused by finite length of the equalizer 202, a residual phase error not corrected by the phase adjust circuit 206, non-linearities, and/or other non-idealities) the total partial response signal resulting from corresponding symbols of signal 103 passing through pulse shaper 104 and input filter 109.
The buffer 212 buffers samples of the signal 226 and outputs a plurality of samples of the signal 226 via signal 232. The signal 232 is denoted PR1, where the underlining indicates that it is a vector (in this case each element of the vector corresponds to a sample of a partial response signal). In an example implementation, the length of the vector PR1 may be Q samples.
Input to the sequence estimation circuit 210 are the signal 232, the signal 228, and a response ĥ. Response ĥ is based on h (the total partial response, discussed above). For example, response ĥ may represent a compromise between h (described above) and a filter response that compensates for channel non-idealities such as multi-path. The response ĥ may be conveyed and/or stored in the form of LTx+LRx−1 tap coefficients resulting from convolution of the LTx tap coefficients of the pulse shaper 104 and the LRx tap coefficients of the input filter 109. Alternatively, response ĥ may be conveyed and/or stored in the form of fewer than LTx+LRx−1 tap coefficients—for example, where one or more taps of the LTx and LRx is ignored due to being below a determined threshold. The sequence estimation circuit 210 may output partial response feedback signals 205 and 209, a signal 234 that corresponds to the finely determined phase error of the signal 120, and signal 132 (which carries hard and/or soft estimates of transmitted symbols and/or transmitted bits). An example implementation of the sequence estimation circuit 210 is described below with reference to
The non-linear modeling circuit 236a may apply a non-linearity function F{circumflex over (n)}l (a model of the non-linearity seen by the received signal en route to the circuit 210) to the signal 205 resulting in the signal 203. Similarly, the non-linear modeling circuit 236b may apply the non-linearity function F{circumflex over (n)}l to the signal 209 resulting in the signal 207. F{circumflex over (n)}l may be, for example, a third-order or fifth-order polynomial. Increased accuracy resulting from the use of a higher-order polynomial for F{circumflex over (n)}l may tradeoff with increased complexity of implementing a higher-order polynomial. Where FnlTx is the dominant non-linearity of the communication system 100, F{circumflex over (n)}l modeling only FnlTx may be sufficient. Where degradation in receiver performance is above a threshold due to other non-linearities in the system (e.g., non-linearity of the receiver front-end 108) the model F{circumflex over (n)}l may take into account such other non-linearities
For each symbol candidate at time n, the metrics calculation circuit 304 may be operable to generate a metric vector Dn1 . . . DnM×Su×P based on the partial response signal PR1, the signal 303a conveying the phase candidate vectors PCn1 . . . PCnM×Su×P and the signal 303b conveying the symbol candidate vectors SCn1 . . . SCnM×Su×P, where underlining indicates a vector, subscript n indicates that it is the candidate vectors for time n, M is an integer equal to the size of the symbol alphabet (e.g., for N-QAM, M is equal to N), Su is an integer equal to the number of symbol survivor vectors retained for each iteration of the sequence estimation process, and P is an integer equal to the size of the phase alphabet. In an example implementation, the size of phase alphabet is three, with each of the three symbols corresponding to one of: a positive shift, a negative phase shift, or zero phase shift, as further described below with respect to
The candidate selection circuit 306 may be operable to select Su of the symbol candidates SCn1 . . . SCnM×Su×P and Su of the phase candidates PCn1 . . . PCnM×Su×P based on the metrics Dn1 . . . DnM×Su×P. The selected phase candidates are referred to as the phase survivors PSn1 . . . PSnSu. Each element of each phase survivors PSn1 . . . PSnSu may correspond to an estimate of residual phase error in the signal 232. That is, the phase error remaining in the signal after coarse phase error correction via the phase adjust circuit 206. The best phase survivor PSn1 is conveyed via signal 307a. The Su phase survivors are retained for the next iteration of the sequence estimation process (at which time they are conveyed via signal 301b). The selected symbol candidates are referred to as the symbol survivors SSn1 . . . SSnSu. Each element of each symbol survivors SSn1 . . . SSnSu may comprise a soft-decision estimate and/or a hard-decision estimate of a symbol of the signal 232. The best symbol survivor SSn1 is conveyed to symbol buffer 310 via the signal 307b. The Su symbol survivors are retained for the next iteration of the sequence estimation process (at which time they are conveyed via signal 301a). Although, the example implementation described selects the same number, Su, of phase survivors and symbol survivors, such is not necessarily the case. Operation of example candidate selection circuits 306 are described below with reference to FIGS. 5D and 6A-6B.
The candidate generation circuit 302 may be operable to generate phase candidates PCn1 . . . PCnM×Su×P and symbol candidates SCn1 . . . SCnM×Su×P from phase survivors PSn-11 . . . PSn-1Su and symbol survivors SSn-11 . . . SSn-1Su, wherein the index n−1 indicates that they are survivors from time n−1 are used for generating the candidates for time n. In an example implementation, generation of the phase and/or symbol candidates may be as, for example, described below with reference to
The symbol buffer circuit 310 may comprise a plurality of memory elements operable to store one or more symbol survivor elements of one or more symbol survivor vectors. The phase buffer circuit 312 may comprise a plurality of memory elements operable to store one or more phase survivor vectors. Example implementations of the buffers 310 and 312 are described below with reference to
The combiner circuit 308 may be operable to combine the best phase survivor, PSn1, conveyed via signal 307a, with the signal 228 generated by the carrier recovery circuit 208 (
The phase adjust circuit 314 may be operable to adjust the phase of the signal 315a by an amount determined by the signal 234 output by phase buffer 312, to generate the signal 205.
The circuit 316a, which performs a convolution, may comprise a FIR filter or IIR filter, for example. The circuit 316a may be operable to convolve the signal 132 with response ĥ, resulting in the partial response signal 315a. Similarly, the convolution circuit 316b may be operable to convolve the signal 317 with response ĥ, resulting in the partial response signal 209. As noted above, response ĥ may be stored by, and/or conveyed to, the sequence estimation circuit 210 in the form of one or more tap coefficients, which may be determined based on the tap coefficients of the pulse shaper 104 and/or input filter 109 and/or based on an adaptation algorithm of a decision feedback equalizer (DFE). Response ĥ may thus represent a compromise between attempting to perfectly reconstruct the total partial response signal (103 as modified by pulse shaper 104 and input filter 109) on the one hand, and compensating for multipath and/or other non-idealities of the channel 107 on the other hand. In this regard, the system 100 may comprise one or more DFEs as described in one or more of: the United States patent application titled “Decision Feedback Equalizer for Highly-Spectrally-Efficient Communications,” the United States patent application titled “Decision Feedback Equalizer with Multiple Cores for Highly-Spectrally-Efficient Communications,” and the United States patent application titled “Decision Feedback Equalizer Utilizing Symbol Error Rate Biased Adaptation Function for Highly-Spectrally-Efficient Communications,” each of which is incorporated herein by reference, as set forth above.
Thus, signal 203 is generated by taking a first estimate of transmitted symbols, (an element of symbol survivor SSn1), converting the first estimate of transmitted symbols to the partial response domain via circuit 316a, and then compensating for non-linearity in the communication system 100 via circuit 236a (
The circuit 404, which performs a convolution, may comprise a FIR filter or IIR filter, for example. The circuit 404 may be operable to convolve the symbol candidate vectors SCn1 . . . SCnM×Su×P with ĥ. The signal 405 output by the circuit 404 thus conveys vectors SCPRn1 . . . SCPRnM×Su×P, each of which is a candidate partial response vector.
The cost function circuit 406 may be operable to generate metrics indicating the similarity between one or more of the partial response vectors PR2n1 . . . PR2nM×Su×P and one or more of the vectors SCPRn1 . . . SCPRnM×Su×P to generate error metrics Dn1 . . . DnM×Su×P.
In an example implementation, the error metrics may be Euclidean distances calculated as shown below in equation 1.
Dni=|(SCPRni)−(PR2ni)|2 EQ. 1
for 1≦i≦M×Su×P.
Referring to
Referring to
Referring to
Referring to
Shown are the contents of example symbol buffer 310 at times n−3, n−2, n−1, and n. At time n−3, a symbol survivor having values α,β,χ,δ is stored in the symbol buffer 310. Accordingly, as shown in
In
In block 966, a vector of one or more symbols of the symbol buffer (e.g., L symbols beginning at index q2) is output for generation of the first feedback signal. In block 968, a value of the phase buffer (e.g., at index q3=q2) is output for generation of the first feedback signal. In block 970, the vector output in block 966 is convolved (e.g., by circuit 316a) with tap coefficients (e.g., tap coefficients corresponding to response h) to generate a partial response signal (e.g., signal 315a). In block 972, a phase correction is applied (e.g., by circuit 314) to the partial response signal generated in block 970 based on the value output from the phase buffer in block 968. The result of block 970 is a phase-adjusted partial response signal (e.g., signal 205). In block 974, a non-linearity model (e.g., modeling the nonlinearity of the non-linear circuit via which the symbols were processed in block 1002) is applied to the partial response signal generated in block 918, resulting in a non-linearized partial response signal. In block 976, the non-linearized partial response signal generated in block 974 is subtracted from the equalizer output to generate the first partial response feedback signal.
In an example implementation, a receiver may receive an inter-symbol correlated (ISC) signal (e.g., signal 122). The receiver may equalize, via an equalizer (e.g. 202), the received ISC signal to generate an equalized signal (e.g., 222). The equalization may be based on a first inter-symbol correlated (ISC) feedback signal (e.g., 203). The receiver may correct, via a phase adjuster (e.g., 206) based on a second ISC feedback signal (e.g., 207), a phase error of the equalized signal to generate a phase-corrected equalized signal (e.g., 226). One or both of the first ISC feedback signal and the second ISC feedback signals may be partial response signals generated via a convolution with tap coefficients. Generation of the first ISC feedback signal may comprise convolution of an estimated symbol vector (e.g., L symbols, beginning at index q2, from the symbol buffer 310) with a plurality of tap coefficients, the convolving resulting in a first estimated ISC signal (e.g., 315a). Generation of the first ISC feedback signal may comprise phase adjustment of the first estimated ISC signal to generate a second estimated ISC signal. Generation of the first ISC feedback signal may comprise application (e.g., in circuit 316a) of a model of a non-linear circuit (e.g., 106 and/or 108) through which the received signal passed to the second estimated ISC signal. The tap coefficients may be based on tap coefficients of a partial response filter (e.g., 104 and/or 109). Generation of the second ISC feedback signal may comprise convolution of an estimated symbol vector (e.g., L symbols, beginning at index q1, from the symbol buffer 310) with a plurality of tap coefficients, the convolution resulting in a third estimated ISC signal (e.g., 317). Generation of the second ISC feedback signal may comprise application (e.g., in circuit 316a) of a model of a non-linear circuit (e.g., 106 and/or 108) through which the received signal passed to the third estimated ISC signal. Circuitry of the receiver may control, during run-time of the equalizer, a bandwidth of one or both of a first feedback loop comprising the first ISC feedback signal, and a second feedback loop comprising the second ISC feedback signal, the controlling being based on a measured performance indicator (e.g., an indicator of channel conditions and/or receiver performance such as noise levels, signal to noise ration (SNR), symbol error rate (SER), value of the signal 228, etc.). The control may comprises adjustment one or more buffer index values (e.g., q1, q2, and/or q3) wherein a first one of the buffer index values (e.g., q2) corresponds to a buffer element that is read for generating the first ISC feedback signal, and a second one of the buffer index values (e.g., q1) corresponds to a buffer element that is read for generating the second ISC feedback signal.
Other implementations may provide a non-transitory computer readable medium and/or storage medium, and/or a non-transitory machine readable medium and/or storage medium, having stored thereon, a machine code and/or a computer program having at least one code section executable by a machine and/or a computer, thereby causing the machine and/or computer to perform the processes as described herein.
Methods and systems disclosed herein may be realized in hardware, software, or a combination of hardware and software. Methods and systems disclosed herein may be realized in a centralized fashion in at least one computing system, or in a distributed fashion where different elements are spread across several interconnected computing systems. Any kind of computing system or other apparatus adapted for carrying out the methods described herein is suited. A typical combination of hardware and software may be a general-purpose computing system with a program or other code that, when being loaded and executed, controls the computing system such that it carries out methods described herein. Another typical implementation may comprise an application specific integrated circuit (ASIC) or chip with a program or other code that, when being loaded and executed, controls the ASIC such that is carries out methods described herein.
While methods and systems have been described herein with reference to certain implementations, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the scope of the present method and/or system. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the present disclosure without departing from its scope. Therefore, it is intended that the present method and/or system not be limited to the particular implementations disclosed, but that the present method and/or system will include all implementations falling within the scope of the appended claims.
This patent application is a continuation of U.S. patent application Ser. No. 13/755,014 filed on Jan. 31, 2013 (now U.S. Pat. No. 8,559,496), which in turn, claims priority to U.S. Provisional Patent Application Ser. No. 61/662,085 entitled “Apparatus and Method for Efficient Utilization of Bandwidth” and filed on Jun. 20, 2012. This patent application is also a non-provisional of U.S. Provisional Patent Application Ser. No. 61/726,099 entitled “Modulation Scheme Based on Partial Response” and filed on Nov. 14, 2012, U.S. Provisional Patent Application Ser. No. 61/729,774 entitled “Modulation Scheme Based on Partial Response” and filed on Nov. 26, 2012; and U.S. Provisional Patent Application Ser. No. 61/747,132 entitled “Modulation Scheme Based on Partial Response” and filed on Dec. 28, 2012. Each of the above-identified applications is hereby incorporated herein by reference in its entirety. This patent application also makes reference to: U.S. patent application Ser. No. 13/754,964 (now patented as U.S. Pat. No. 8,582,637), titled “Low-Complexity, Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/754,998, titled “Design and Optimization of Partial Response Pulse Shape Filter,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,001, titled “Constellation Map Optimization For Highly Spectrally Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,008 (now patented as U.S. Pat. No. 8,571,131), titled “Dynamic Filter Adjustment for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,011 (now patented as U.S. Pat. No. 8,559,494), titled “Timing Synchronization for Reception of Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,018 (now patented as U.S. Pat. No. 8,599,914), titled “Feed Forward Equalization for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,021, titled “Decision Feedback Equalizer for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,025, titled “Decision Feedback Equalizer with Multiple Cores for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,026 (now patented as U.S. Pat. No. 8,559,498), titled “Decision Feedback Equalizer Utilizing Symbol Error Rate Biased Adaptation Function for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,028 (now patented as U.S. Pat. No. 8,548,097), titled “Coarse Phase Estimation for Highly-Spectrally-Efficient Communications,” and filed on the same date as this application;U.S. patent application Ser. No. 13/755,039 (now patented as U.S. Pat. No. 8,565,363), titled “Fine Phase Estimation for Highly Spectrally Efficient Communications,” and filed on the same date as this application; andU.S. patent application Ser. No. 13/755,043 (now patented as U.S. Pat. No. 8,605,832), titled “Joint Sequence Estimation of Symbol and Phase with High Tolerance of Nonlinearity,” and filed on the same date as this application. Each of the above stated applications is hereby incorporated herein by reference in its entirety.
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