Efficient source decoding over memoryless noisy channels using higher order Markov models

Exploiting the residual redundancy in a source coder output stream during the decoding process has been proven to be a bandwidth-efficient way to combat noisy channel degradations. This redundancy can be employed to either assist the channel decoder for improved performance or design better source decoders. In this work, a family of solutions for the asymptotically optimum minimum mean-squared error (MMSE) reconstruction of a source over memoryless noisy channels is presented when the redundancy in the source encoder output stream is exploited in the form of a /spl gamma/-order Markov model (/spl gamma//spl ges/1) and a delay of /spl delta/,/spl delta/>0, is allowed in the decoding process. It is demonstrated that the proposed solutions provide a wealth of tradeoffs between computational complexity and the memory requirements. A simplified MMSE decoder which is optimized to minimize the computational complexity is also presented. Considering the same problem setup, several other maximum a posteriori probability (MAP) symbol and sequence decoders are presented as well. Numerical results are presented which demonstrate the efficiency of the proposed algorithms.     

Minimum mean squared error equalization using a priori information

A number of important advances have been made in the area of joint equalization and decoding of data transmitted over intersymbol interference (ISI) channels. Turbo equalization is an iterative approach to this problem, in which a maximum a posteriori probability (MAP) equalizer and a MAP decoder exchange soft information in the form of prior probabilities over the transmitted symbols. A number of reduced-complexity methods for turbo equalization have been introduced in which MAP equalization is replaced with suboptimal, low-complexity approaches. We explore a number of low-complexity soft-input/soft-output (SISO) equalization algorithms based on the minimum mean square error (MMSE) criterion. This includes the extension of existing approaches to general signal constellations and the derivation of a novel approach requiring less complexity than the MMSE-optimal solution. All approaches are qualitatively analyzed by observing the mean-square error averaged over a sequence of equalized data. We show that for the turbo equalization application, the MMSE-based SISO equalizers perform well compared with a MAP equalizer while providing a tremendous complexity reduction     

A combined multiple-candidate likelihood decoding and error concealment scheme for compressed video transmission over noisy channels

This paper presents a joint forward error correction (FEC) and error concealment (EC) scheme to enhance the quality of a compressed video signal transmitted over a noisy channel. A multiple candidate likelihood (MCL) channel decoding strategy is used in conjunction with redundancy in the compressed video (syntax validity and spatial discontinuity) to select the best-detected signal.

Simulation results on both objective and subjective performance measures indicate a significant improvement provided by the proposed scheme.     

A sequence-based approximate MMSE decoder for source coding overnoisy channels using discrete hidden Markov models

In previous work on source coding over noisy channels it was recognized that when the source has memory, there is typically “residual redundancy” between the discrete symbols produced by the encoder, which can be capitalized upon by the decoder to improve the overall quantizer performance. Sayood and Borkenhagen (1991) and Phamdo and Farvardin (see IEEE Trans. Inform. Theory, vol.40, p.186-93, 1994) proposed “detectors” at the decoder which optimize suitable criteria in order to estimate the sequence of transmitted symbols. Phamdo and Farvardin also proposed an instantaneous approximate minimum mean-squared error (IAMMSE) decoder. These methods provide a performance advantage over conventional systems, but the maximum a posteriori (MAP) structure is suboptimal, while the IAMMSE decoder makes limited use of the redundancy. Alternatively, combining aspects of both approaches, we propose a sequence-based approximate MMSE (SAMMSE) decoder. For a Markovian sequence of encoder-produced symbols and a discrete memoryless channel, we approximate the expected distortion at the decoder under the constraint of fixed decoder complexity. For this simplified cost, the optimal decoder computes expected values based on a discrete hidden Markov model, using the wellknown forward/backward (F/B) algorithm. Performance gains for this scheme are demonstrated over previous techniques in quantizing Gauss-Markov sources over a range of noisy channel conditions. Moreover, a constrained delay version is also suggested     

Mean-Squared Error Sampling and Reconstruction in the Presence of Noise

One of the main goals of sampling theory is to represent a continuous-time function by a discrete set of samples. Here, we treat the class of sampling problems in which the underlying function can be specified by a finite set of samples. Our problem is to reconstruct the signal from nonideal, noisy samples, which are modeled as the inner products of the signal with a set of sampling vectors, contaminated by noise. To mitigate the effect of the noise and the mismatch between the sampling and reconstruction vectors, the samples are linearly processed prior to reconstruction. Considering a statistical reconstruction framework, we characterize the strategies that are mean-squared error (MSE) admissible, meaning that they are not dominated in terms of MSE by any other linear reconstruction. We also present explicit designs of admissible reconstructions that dominate a given inadmissible method. Adapting several classical estimation approaches to our particular sampling problem, we suggest concrete admissible reconstruction methods and compare their performance. The results are then specialized to the case in which the samples are processed by a digital correction filter     

A Competitive Mean-Squared Error Approach to Beamforming

We treat the problem of beamforming for signal estimation where the goal is to estimate a signal amplitude from a set of array observations. Conventional beamforming methods typically aim at maximizing the signal-to-interference-plus-noise ratio (SINR). However, this does not guarantee a small mean-squared error (MSE), so that on average the resulting signal estimate can be far from the true signal. Here, we consider strategies that attempt to minimize the MSE between the estimated and unknown signal waveforms. The methods we suggest all maximize the SINR but at the same time are designed to have good MSE performance. Since the MSE depends on the signal power, which is unknown, we develop competitive beamforming approaches that minimize a robust MSE measure. Two design strategies are proposed: minimax MSE and minimax regret. We demonstrate through numerical examples that the suggested minimax beamformers can outperform several existing standard and robust methods, over a wide range of signal-to-noise ratio (SNR) values. Finally, we apply our techniques to subband beamforming and illustrate their advantage in estimating a wideband signal.     

Frequency-Domain Channel Estimation for SC-FDE in UWB Communications

Recently, single-carrier block transmission with frequency-domain equalization (SC-FDE) has been shown to be a promising candidate for ultra-wideband (UWB) communications. In this paper, we address the channel estimation problem for SC-FDE transmission over UWB channels. A mean-square error (MSE) lower bound for the frequency-domain linear minimum mean-squared error (LMMSE) channel estimator is derived, and the optimal pilot sequence that achieves this lower bound is obtained. Further simplification leads to a frequency-domain channel estimator with reduced computational complexity. The performance of the simplified estimator for SC-FDE over UWB channels is evaluated and compared with that with perfect channel state information. The effects of nonoptimal and optimal pilot symbols are also investigated. Our results show that the proposed frequency-domain channel estimator performs well over UWB channels with only small performance degradation, compared with that with perfect channel estimation     

Quantizers with uniform decoders and channel-optimized encoders

Scalar quantizers with uniform decoders and channel-optimized encoders are studied for a uniform source on [0,1] and binary symmetric channels. Two families of affine index assignments are considered: the complemented natural code (CNC), introduced here, and the natural binary code (NBC). It is shown that the NBC never induces empty cells in the quantizer encoder, whereas the CNC can. Nevertheless, we show that the asymptotic distributions of quantizer encoder cells for the NBC and the CNC are equal and are uniform over a proper subset of the source's support region. Empty cells act as a form of implicit channel coding. An effective channel code rate associated with a quantizer designed for a noisy channel is defined and computed for the codes studied. By explicitly showing that the mean-squared error (MSE) of the CNC can be strictly smaller than that of the NBC, we also demonstrate that the NBC is suboptimal for a large range of transmission rates and bit error probabilities. This contrasts with the known optimality of the NBC when either both the encoder and decoder are not channel optimized, or when only the decoder is channel optimized.     

Content-based multiple bitstream image transmission over noisy channels

We propose a novel combined source and channel coding scheme for image transmission over noisy channels. The main feature of the proposed scheme is a systematic decomposition of image sources so that unequal error protection can be applied according to not only bit error sensitivity but also visual content importance. The wavelet transform is adopted to hierarchically decompose the image. The association between the wavelet coefficients and what they represent spatially in the original image is fully exploited so that wavelet blocks are classified based on their corresponding image content. The classification produces wavelet blocks in each class with similar content and statistics, therefore enables high performance source compression using the set partitioning in hierarchical trees (SPIHT) algorithm. To combat the channel noise, an unequal error protection strategy with rate-compatible punctured convolutional/cyclic redundancy check (RCPC/CRC) codes is implemented based on the bit contribution to both peak signal-to-noise ratio (PSNR) and visual quality. At the receiving end, a postprocessing method making use of the SPIHT decoding structure and the classification map is developed to restore the degradation due to the residual error after channel decoding. Experimental results show that the proposed scheme is indeed able to provide protection both for the bits that are more sensitive to errors and for the more important visual content under a noisy transmission environment. In particular, the reconstructed images illustrate consistently better visual quality than using the single-bitstream-based schemes.     

Hadamard-based soft decoding for vector quantization over noisychannels

We present an estimator-based, or soft, vector quantizer decoder for communication over a noisy channel. The decoder is optimal according to the mean-square error criterion, and Hadamard-based in the sense that a Hadamard transform representation of the vector quantizer is utilized in the implementation of the decoder. An efficient algorithm for optimal decoding is derived. We furthermore investigate suboptimal versions of the decoder, providing good performance at lower complexity. The issue of joint encoder-decoder design is considered both for optimal and suboptimal decoding. Results regarding the channel distortion and the structure of a channel robust code are also provided. Through numerical simulations, soft decoding is demonstrated to outperform hard decoding in several aspects     

Joint design of precoder and receiver in cognitive radio networks using an MSE criterion

We consider a single-user multiple-input multiple output (MIMO) transmission from secondary base station (SBS) in cognitive radio network where there is one primary user to which spectrum is licensed. It is assumed that coexistence of secondary network is allowed if interference power from the secondary network measured at primary user (PU) can be kept below a given threshold. In this paper, joint design problem of precoder and receiver is treated using mean-squared error (MSE) criterion between transmitted signal and its estimate at the receiver. Due to constraint on interference power measured at the PU, it is not possible to find an optimal solution in closed form. Therefore, a suboptimal solution with low complexity is proposed in closed form. Simulation results show that the MSE and bit error rate (BER) performances of the proposed scheme are 1.5-4.0 dB better than those of a conventional capacity-maximization scheme when the received signal-to-noise ratio (SNR) is high, and they converge to their lower bounds, i.e., those of an MSE-based scheme without considering the PU interference constraint, as the received SNR increases.     

DPCM picture transmission over noisy channels with the aid of aMarkov model

The problem of DPCM picture transmission over noisy channels is considered. It is well known that DPCM systems are very sensitive to channel errors. The goal in this work is to build robustness against channel errors. Three methods are proposed in this paper and are obtained by modeling the encoded signal as a Markov sequence. First, an optimum method for decoding correlated sequences is derived, and it is shown to require Viterbi decoding. Then, a modified MAP method (MMAP) for Markov sequences is described. A maximal signal-to-noise (MSNR) receiver for DPCM systems is also developed that minimizes the distortion power due to channel errors. The appropriate cost matrix for this receiver is computed. These methods are applied to DPCM picture transmission over noisy channels and are compared with a another method. The SNR graphs, as well as subjective examination of the received pictures, demonstrate that the proposed procedures are quite effective and superior to that method. Among the proposed methods, the MSNR receiver was found to be more effective than the others for a given order of the Markov model. It is observed that the proposed methods are most beneficial for low detail pictures.     

Optimal detection of discrete Markov sources over discretememoryless channels-applications to combined source-channel coding

The authors consider the problem of detecting a discrete Markov source which is transmitted across a discrete memoryless channel. Two maximum a posteriori (MAP) formulations are considered: (i) a sequence MAP detection in which the objective is to determine the most probable transmitted sequence given the observed sequence and (ii) an instantaneous MAP detection which is to determine the most probable transmitted symbol at time n given all the observations prior to and including time n. The solution to the first problem results in a “Viterbi-like” implementation of the MAP detector (with Large delay) while the latter problem results in a recursive implementation (with no delay). For the special case of the binary symmetric Markov source and binary symmetric channel, simulation results are presented and an analysis of these two systems yields explicit critical channel bit error rates above which the MAP detectors become useful. Applications of the MAP detection problem in a combined source-channel coding system are considered. Here, it is assumed that the source is highly correlated and that the source encoder (a vector quantizer (VQ)) fails to remove all of the source redundancy. The remaining redundancy at the output of the source encoder is referred to as the “residual” redundancy. It is shown, through simulation, that the residual redundancy can be used by the MAP detectors to combat channel errors. For small block sizes, the proposed system beats Farvardin and Vaishampayan's channel-optimized VQ by wide margins. Finally, it is shown that the instantaneous MAP detector can be combined with the VQ decoder to form an approximate minimum mean-squared error decoder     

噪声信道中基于残留冗余的联合信源信道编码.

对信源编码中的残留冗余在联合编码中的作用进行了研究，提出了一个在噪声信道中对可变长信源编码码流传输提供有效差错保护的联合信源信道编码方法，该方法利用信源编码器输出中的残留冗余为传输码流提供差错保护。与Sayood K提出的系统相比，该方法是基于改进的联合卷积软解码以及采用非霍夫曼码的通用可变长码，更接近于一般的信源和信道编码方法，并且信源符号集的大小也不受限制。仿真表明，所提出的联合编码方法可获得比传统的分离编码方法更高的性能增益。     

噪声信道中基于残留冗余的联合信源信道编码

对信源编码中的残留冗余在联合编码中的作用进行了研究,提出了一个在噪声信道中对可变长信源编码码流传输提供有效差错保护的联合信源信道编码方法,该方法利用信源编码器输出中的残留冗余为传输码流提供差错保护。与SayoodK提出的系统相比,该方法是基于改进的联合卷积软解码以及采用非霍夫曼码的通用可变长码,更接近于一般的信源和信道编码方法,并且信源符号集的大小也不受限制。仿真表明,所提出的联合编码方法可获得比传统的分离编码方法更高的性能增益。     

Sequence MAP decoding of trellis codes for Gaussian and Rayleighchannels

This paper considers the use of sequence maximum a posteriori (MAP) decoding of trellis codes. A MAP receiver can exploit any “residual redundancy” that may exist in the channel encoded signal in the form of memory and/or a nonuniform distribution, thereby providing enhanced performance over very noisy channels, relative to maximum likelihood (ML) decoding. The paper begins with a first-order two-state Markov model for the channel encoder input. A variety of different systems with different source parameters, different modulation schemes, and different encoder complexities are simulated. Sequence MAP decoding is shown to substantially improve performance under very noisy channel conditions for systems with low-to-moderate redundancy, with relative gain increasing as the rate increases. As a result, coding schemes with multidimensional constellations are shown to have higher MAP gains than comparable schemes with two-dimensional (2-D) constellations. The second part of the paper considers trellis encoding of the code-excited linear predictive (CELP) speech coder's line spectral parameters (LSPs) with four-dimensional (4-D) QPSK modulation. Two source LSP models are used. One assumes only intraframe correlation of LSPs while the second one models both intraframe and interframe correlation. MAP decoding gains (over ML decoding) as much as 4 dB are achieved. Also, a comparison between the conventionally designed codes and an I-Q QPSK scheme shows that the I-Q scheme achieves better performance even though the first (sampler) LSP model is used     

A novel product coding and recurrent alternate decoding scheme for image transmission over noisy channels

In this letter, we present a novel product channel coding and decoding scheme for image transmission over noisy channels. Two convolutional codes with at least one recursive systematic convolutional code are employed to construct the product code. Received data are decoded alternately in two directions. A constrained Viterbi algorithm is proposed to exploit the detection results of cyclic redundancy check codes so that both reduction in error patterns and fast decoding speed are achieved. Experiments with image data coded by the algorithm of set partitioning in hierarchical trees exhibit results better than those currently reported in the literature.     

Robust deconvolution of deterministic and random signals

We consider the problem of designing an estimation filter to recover a signal x[n] convolved with a linear time-invariant (LTI) filter h[n] and corrupted by additive noise. Our development treats the case in which the signal x[n] is deterministic and the case in which it is a stationary random process. Both formulations take advantage of some a priori knowledge on the class of underlying signals. In the deterministic setting, the signal is assumed to have bounded (weighted) energy; in the stochastic setting, the power spectra of the signal and noise are bounded at each frequency. The difficulty encountered in these estimation problems is that the mean-squared error (MSE) at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized. Beginning with the deterministic setting, we develop a minimax MSE estimation filter that minimizes the worst case point-wise MSE between the true signal x[n] and the estimated signal, over the class of bounded-norm inputs. We then establish that the MSE at the output of the minimax MSE filter is smaller than the MSE at the output of the conventional inverse filter, for all admissible signals. Next we treat the stochastic scenario, for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties. This filter is designed to minimize the worst case difference between the MSE in the presence of power spectrum uncertainties, and the MSE of the Wiener filter that knows the correct power spectra. The minimax regret filter takes the entire uncertainty interval into account, and as demonstrated through an example, can often lead to improved performance over traditional minimax MSE approaches for this problem     

Design of the Periodic Training Sequence for Joint Channel and Frequency Estimation in MIMO Frequency-Selective Channels

This paper considers the design of a periodic training sequence (TS) for joint channel and frequency estimation in multiple-input, multiple-output (MIMO) frequency-selective channels. The design criterion for the periodic TS is to jointly minimize the mean-squared error (MSE) of the maximum likelihood channel estimation and the asymptotic Cramer-Rao bound (CRB) for the frequency estimation. This paper shows that all the TSs that minimize the MSE of the channel estimation have the same asymptotic CRB for the frequency estimation. Furthermore, they also minimize the asymptotic CRB as long as the channel is i.i.d. Rayleigh fading. The design of low-complexity frequency estimators based on the proposed periodic TS is also investigated. Finally, the performance of the proposed periodic TS is evaluated by simulation results.     

Soft multiuser decoding for vector quantization over a CDMA channel

An approach to optimal soft decoding for vector quantization (VQ) over a code-division multiple-access (CDMA) channel is presented. The decoder of the system is soft in the sense that the unquantized outputs of the matched filters are utilized directly for decoding (no decisions are taken), and optimal according to the minimum mean-squared error (MMSE) criterion. The derived decoder utilizes a priori source information and knowledge of the channel characteristics to combat channel noise and multiuser interference in an optimal fashion. Hadamard transform representations for the user VQs are employed in the derivation and for the implementation of the decoder. The advantages of this approach are emphasized. Suboptimal versions of the optimal decoder are also considered. Simulations show the soft decoders to outperform decoding based on maximum-likelihood (ML) multiuser detection. Furthermore, the suboptimal versions are demonstrated to perform close to the optimal, at a significantly lower complexity in the number of users. The introduced decoders are, moreover, shown to exhibit near-far resistance. Simulations also demonstrate that combined source-channel encoding, with joint source-channel and multiuser decoding, can significantly outperform a tandem source-channel coding scheme employing multiuser detection plus table lookup source decoding