Optimization of signal sets for partial-response channels. I.Numerical techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: a power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint, problem is equivalent to packing N points in an ellipsoid in min(K,N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise     

Improved coding techniques for preceded partial-response channels

A coset of a convolutional code may be used to generate a zero-run length limited trellis code for a 1-D partial-response channel. The free squared Euclidean distance, d_free², at the channel output is lower bounded by the free Hamming distance of the convolutional code. The lower bound suggests the use of a convolutional code with maximal free Hamming distance, d_max(R,N), for given rate R and number of decoder states N. In this paper we present cosets of convolutional codes that generate trellis codes with d_free ²>d_max(R,N) for rates 1/5⩽R⩽7/9 and (d _free²=d_max(R,N) for R=13/16,29/32,61/64, The tabulated convolutional codes with R⩽7/9 were not optimized for Hamming distance. Instead, a computer search was used to determine cosets of convolutional codes that exploit the memory of the 1-D channel to increase d_free² at the channel output. The search was limited by only considering cosets with certain structural properties. The R⩾13/16 codes were obtained using a new construction technique for convolutional codes with free Hamming distance 4. Newly developed bounds on the maximum zero-run lengths of cosets were used to ensure a short maximum run length at the 1-D channel output     

Coding techniques for partial-response channels

A coding technique for improving the reliability of digital transmission over noisy partial-response channels with characteristics (±D^m), m=1, 2, where the channel input symbols are constrained to be ±1, is presented. In particular, the application of a traditional modulation code as an inner code of a concentrated coding scheme in which the outer code is designed for maximum (free) Hamming distance is considered. A performance comparison is made between the concentrated scheme and a coding technique presented by Wolf and G. Ungerboeck (see ibid., vol. COM-34, p.765-773, Aug. 1986) for the dicode channel with transfer function (1- D)     

Iterative reduced-search decoding for coded partial-response channels

The full-complexity soft-input/soft-output (SISO) detector based on the BCJR algorithm for coded partial-response channels has a computational complexity growing exponentially with channel memory length. In this letter, we propose a low complexity soft-output channel detector based on the Chase decoding algorithm, which was previously applied to decode turbo product codes. At each iteration, the proposed detector forms a candidate list using all possible combinations of bit patterns in the weakest indices based on tentative hard estimates and a priori information fed back from the outer decoder. To demonstrate the performance/complexity tradeoff of the proposed detector, simulation results over rate-8/9 turbo-coded EPR4 and ME/sup 2/PR4 channels are presented, respectively. It is shown that the proposed detector can significantly reduce the computational complexity with only a small performance loss compared to the BCJR algorithm.     

Sequence estimation for class-IV partial-response channels withjitter

An analysis is presented of the output of class-IV partial-response channels corrupted simultaneously by jitter and additive white noise. The authors analyze the behavior of the least-squares (LS) decoder for this channel. They introduce the projection-minimization (P-M) algorithm and demonstrate that, under favorable conditions, its performance, even when not acceptable on its own, could possibly be beneficial when used in conjunction with the LS decoder in enhancing the reliability of the channel. Implementation of the P-M decoder is rather expensive, and it is not at all clear that combining the P-M algorithm with LS decoding and traditional error control schemes is the best way of dealing with channels corrupted by jitter. It may be possible to avoid the P-M algorithm and instead utilize a more aggressive traditional error correction scheme. Alternatively, it may be possible to choose modulation schemes which utilize pulses whose derivatives at the sampled values are small, and perhaps obtain better results than with partial-response class-IV signaling     

Trellis coding with multidimensional QAM signal sets

Trellis coding using multidimensional quadrature amplitude modulation (QAM) signal sets is investigated. Finite-size 2D signal sets are presented that have minimum average energy, are 90° rotationally symmetric, and have from 16 to 1024 points. The best trellis codes using the finite 16-QAM signal set with two, four, six, and eight dimensions are found by computer search (the multidimensional (multi-D) signal set is constructed from the 2-D signal set). The best moderate complexity trellis codes for infinite lattices with two, four six, and eight dimensions are also found. The minimum free squared Euclidean distance and number of nearest neighbors for these codes were used as the selection criteria. Many of the multi-D codes are fully rotationally invariant and give asymptotic coding gains up to 6.0 dB. From the infinite lattice codes, the best codes for transmitting J, J+1/4, J+1/3, J+1/2, J+2/3, and J+3/4 b/sym (J an integer) are presented     

Asymptotic performance and complexity of a coding scheme for memoryless channels

The purpose of this paper is to show that decoding complexity need not grow exponentially with the code block length at rates close to channel capacity and also to show the expediency of the approach of imbedding codes in each other. It is demonstrated that it is possible to communicate over a memoryless channel of capacityCat any rateR < Cwith a probability of error of less than2^{-E(R)nu}, E(R) > 0, per block of a length approximately proportional tonu^{2}and with a computational decoding complexity per digit which is asymptotically proportional tonu^{alpha}whennuis large,nu^{alpha}being finite forR < C.(alpha rightarrow mbox{as} R rightarrow C, alpha rightarrow 2 mbox{as} R rightarrow 0).     

Precoding technique for partial-response channels with applicationsto HDTV transmission

An equivalent partial-response (PR) channel 1-Z^-k arises in the envisioned terrestrial over-the-air broadcasting of digital high-definition television (HDTV) signals when a comb filter is used by an HDTV receiver to reduce the NTSC cochannel interference. The design of signal constellations and their associated precoders for this PR channel is considered. Besides PAM and square QAM, it is shown that generalized square and hexagonal constellations can also be used. Coded modulation and graceful degradation in the received signal quality are discussed. The results extended to a more general PR channel     

A 50 MHz eight-tap adaptive equalizer for partial-response channels

A new architecture for digital implementation of the adaptive equalizer in Class IV partial-response maximum likelihood (PRML) channels employing parallelism and pipelining is described. The architecture was used in a prototype integrated circuit in a 1.2 μm CMOS technology to implement an eight-tap adaptive equalizer and Viterbi sequence detector which consumes a total of 70 mW from a 3.3 V supply operating at an input sampling rate of 50 MHz     

Constrained block codes for class-IV partial-response channels withmaximum-likelihood sequence estimation

Significant improvements in magnetic storage densities have been made feasible by the application of partial-response signaling combined with maximum-likelihood sequence estimation. To enhance the performance of this technique when applied to the class-IV partial-response channel, which is recognized as being appropriate to model the magnetic recording channel, it is often required to bound the number of consecutive zeros in the recorded data sequence and its odd and even subsequences. We investigate block codes that satisfy such a constraint. In particular, we look for a set of maximal number of fixed-length sequences such that any pair of them can be concatenated without violating the constraint. In many cases, depending on the constraint and the length of the sequences, we determine such a set, and in the remaining cases, we determine at most three candidates for it. These results are used to study the best possible constrained block codes     

Optimization of nonlinear signal constellations for real-world MIMO channels

The performance of spatial multiplexing (SM) multiple-input multiple-output (MIMO) communication systems is highly dependent on the richness of scattering, the presence of dominant components, and the interelement spacings. In this paper, a new interpretation of the impact of transmit correlation on the performance of SM is given based on a so-called "symbol-related array factor." Nonlinear signal constellations for SM over real-world fading channels are then designed by minimizing an estimate of the average symbol error rate under an average transmit power constraint. The new transmission scheme exploits the spectral efficiency advantage of SM and the robustness of eigen-beamforming. Through simulations, it is shown to be more robust against fading correlations and high Ricean K-factors than SM using the classical phase shift keying (PSK) and quadrature amplitude modulation (QAM) constellations. The symbol error rate performance of this scheme is not affected by a change in the propagation environment or the interelement distance. Furthermore, if the scheme is used on the uplink, no explicit rate-consuming feedback link from the base station to the mobile station is required.     

An approach to Ungerboeck coding for rectangular signal sets (Corresp.)

A method of finding good Ungerboeck codes for large rectangular [quadrature amplitude modulation (QAM)] signal sets is described. Using the concept of Euclidean weights due to Ungerboeck, we prove that a2^{n}point basic constellation may be employed to determine exactly the free distance for an Ungerboeck-coded rectangular2^{m}point set, whenm-n-1bits are uncoded and the remaining bits pass through a rate(n-1)/nconvolutionai encoder. It is shown that rate2/3encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered.     

Multidimensional signal sets through the shell construction forparallel channels

The authors derive a procedure to send r bits on M parallel channels. A decomposition of the best constellation in Z^M+(1/2, . . ., 1/2) is given in terms of the cross-products of lower dimensional shells of points. The proposed scheme can be used with good known coset codes to provide an alternate method of coded modulation. The results indicate that one can get good shaping gains for low encoder complexity. The method is also generalized for channels with unequal gains. The authors also find a significant performance advantage at fixed shaping gain, in certain cases, with respect to the recent Voronoi constellations in terms of peak-to-average power and constellation expansion     

Kikuchi approximation method for joint decoding of LDPC codes and partial-response channels

In this letter, we apply the Kikuchi approximation method to the problem of joint decoding of a low-density parity-check code and a partial-response channel. The Kikuchi method is, in general, more powerful than the conventional loopy belief propagation (BP) algorithm, and can produce better approximations to an underlying inference problem. We will first review the Kikuchi approximation method and the generalized BP algorithm, which is an iterative message-passing algorithm based on this method. We will then report simulation results which show that the Kikuchi method outperforms the best conventional iterative method.     

Asymptotic redundancies for universal quantum coding

Clarke and Barren (1990, 1994, 1995) have shown that the Jeffreys' invariant prior of Bayesian theory yields the common asymptotic (minimax and maximin) redundancy of universal data compression in a parametric setting. We seek a possible analog of this result for the two-level quantum systems. We restrict our considerations to prior probability distributions belonging to a certain one-parameter family, q_u,-∞n×2ⁿ (Bayesian density) matrices, ζ _n(u). These matrices are the weighted averages (with respect to q_u) of all possible tensor products of n identical 2×2 density matrices, representing the two-level quantum systems. We propose a form of universal coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown. A sequence of n signals would be projected onto the dominant eigenspaces of ζ_n(u)     

Sliding-block coding for input-restricted channels

Work on coding arbitrary sequences into a constrained system of sequences (called a sofic system) is presented. Such systems model the input constraints for input-restricted channels (e.g., run-length limits and spectral constraints for the magnetic recording channel). In this context it is important that the code be noncatastrophic to ensure that the decoder has limited error propagation. A constructive proof is given of the existence of finite-state invertible noncatastrophic codes from arbitrary n-ary sequences to a sofic system S at constant rate p:q provided only that Shannon's condition (p/q)⩽(h/log n) is satisfied, where h is the entropy of the system S. If strict inequality holds or if equality holds and S satisfies a natural condition called `almost of finite type' (which includes the systems used in practice), a stronger result is obtained, namely, the decoders can be made `state-independent' sliding-block. This generalizes previous results. An example is also given to show that the stronger result does not hold for general sofic systems     

Superposition coding for side-information channels

We present simple, practical codes designed for the binary and Gaussian dirty-paper channels. We show that the dirty-paper decoding problem can be transformed into an equivalent multiple-access decoding problem, for which we apply superposition coding. Our concept is a generalization of the nested lattices approach of Zamir, Shamai, and Erez. In a theoretical setting, our constructions are capable of achieving capacity using random component codes and maximum-likelihood decoding. We also present practical implementations of the constructions, and simulation results for both dirty-paper channels. Our results for the Gaussian dirty-paper channel are on par with the best known results for nested lattices. We discuss the binary dirty- tape channel, for which we present a simple, effective coding technique. Finally, we propose a framework for extending our approach to general Gel'fand-Pinsker channels.     

Convolutional coding for finite-state channels

We propose new decoders for decoding convolutional codes over finite-state channels. These decoders are sequential and utilize the information about the channel state sequence contained in the channel output sequence. The performance of these decoders is evaluated by simulation and compared to the performance of memoryless decoders with and without interleaving. Our results show that the performance of these decoders is good whenever the channel statistics are such that the joint estimate of the channel state sequence and the channel input sequence is good, as, for example, when the channel is bursty. In these cases using even a partial search decoder such as the Fano decoder over the appropriate trellis is nearly optimal. However, when the information between the output sequence and the sequence of channel slates and inputs diminishes, the memoryless decoder with interleaving outperforms even the optimal decoder which knows the channel state     

语音信号的线性预测分析在DSP上的优化实现

语音信号的线性预测分析是语音压缩中的一个重要环节，占据语音压缩计算中的大部分时间，为提高语音压缩的速度，在C54x系列DSP上用汇编语言实现了语音信号的线性预测分析，并对其进行优化。最终可以使代码的运行时间减少到原来的5％左右。