Multidimensional companding quantization of the Gaussian source

Multidimensional companding quantization is analyzed theoretically for the case of high-resolution and mean-squared-error distortion. The optimality of choosing the expander function to be the inverse of the compressor function is established first. Heuristic derivations of the point density and moment of inertia of companding are given, which combined result in the distortion formula by Bucklew. Further, the interaction between the lattice quantizer (LQ) and compressor function is studied. Optimality is achieved with a second moment optimal LQ, shaped by a compressor-dependent linear transform. High rate theory for a radial compander and spherically symmetric sources is reviewed. The result is used to evaluate the performance on an independent and identically distributed (i.i.d.) Gaussian source. The radial compander clearly outperforms both a product scalar quantizer and a spherical quantizer for dimensions higher than two. Radial companding is also generalized to the correlated Gaussian source. Finally, a comparison of theory and practical companders is made in a Gaussian framework.     

New error bounds and optimum quantization for multisensordistributed signal detection

The binary signal detection problem is considered, when a distributed system of sensors operates in a decentralized fashion. Local processing at each sensor is performed. Using Chernoff's large deviation theorems, the author considers as a criterion the rate of convergence of the error probability to zero. It is shown that the optimum quantizer of blocks of data under the above criterion is the likelihood ratio quantizer. A lower bound to the error probability is also developed. The question of how many coarsely quantized sensors can replace the infinitely quantized one is also answered. The main result given is the structure of the optimum quantizer, consisting of the calculation of the likelihood ratio concatenated by a scalar quantizer     

Soft-decoding vector quantizer using reliability information fromturbo-codes

The optimum soft-decoding vector quantizer using the reliability information from turbo-codes is derived for combined source-channel coding. The encoder and decoder of the quantizer are optimized iteratively. For a four-dimensional vector quantizer having a rate of 1 bit/sample transmitted through a noisy channel, the soft-decoding channel-optimized quantizer can achieve about 3-3.7 dB performance improvement over conventional source-optimized quantizer     

Optimal quantizer design for noisy channels: An approach to combined source - channel coding

We present an analysis of the zero-memory quantization of memoryless sources when the quantizer output is to be encoded and transmitted across a noisy channel. Necessary conditions for the joint optimization of the quantizer and the encoder/decoder pair are presented, and an iterative algorithm for obtaining a locally optimum system is developed. The performance of this locally optimal system, obtained for the class of generalized Gaussian distributions and the binary symmetric channel, is compared against the optimum performance theoretically attainable (using rate-distortion theoretic arguments), as well as against the performance of Lloyd-Max quantizers encoded using the natural binary code and the folded binary code. It is shown that this optimal design could result in substantial performance improvements. The performance improvements are more noticeable at high bit rates and for broad-tailed densities.     

Uniform Spherical Coordinate Quantization of Spherically Symmetric Sources

For signal quantization with the minimum mean square error criterion, optimum spherical coordinate quantizers have been shown to outperform optimum rectangular coordinate qnantizers for many spherically symmetric sources. In this paper, spherical coordinate quantizers are designed and analyzed under the added constraint that each scalar quantizer is a uniform (equal step-size) quantizer. It is shown that for small block length, uniform spherical coordinate quantizers outperform the uniform rectangular coordinate quantizer as well as the uniform qnantizer consisting of a tesselation of the minimum inertia, space filling polytope.     

Uniform Quantizers for Noisy Channels

We consider optimum uniform data quantization for noisy channels. We present a general formulation for natural encoding that results in simple expressions for the mean-square error. Specifically, we show that the optimum location of the center of the quantizer is at the mean of the distribution for all error rates. The optimum levels for quantization and the corresponding mean-square error are presented for Gaussian and uniform data. For the latter the width of the optimum quantizer for noisy channels is shown to be smaller than the entire range of probability distribution.     

On the amount of statistical side information required for lossydata compression

Consider a vector quantizer that is equipped with N side information bits of an arbitrary representation of the statistics of the input source. We investigate the minimum value of N such that rate-distortion performance of this quantizer would be essentially the same as the optimum quantizer for the given source     

Properties of minimum mean squared error block quantizers (Corresp.)

Two results in minimum mean square error quantization theory are presented. The first section gives a simplified derivation of a well-known upper bound to the distortion introduced by ak-dimensional optimum quantizer. It is then shown that an optimum multidimensional quantizer preserves the mean vector of the input and that the mean square quantization error is given by the sum of the component variances of the input minus the sum of the variances of the output.     

Performance of SDM/COFDM system in the presence of nonlinear power amplifier

The performance of Space Division Multiplexing/ Companded Orthogonal Frequency Division Multiplexing (SDM/COFDM) system will be evaluated in the presence of nonlinear amplifier. The evaluation comes from decreasing high dynamic range that is resulted from characteristics of OFDM and nonlinearity of power amplifier. The high dynamic range means high Peak to Average Power Ratio (PAPR). The reduction of dynamic range or PAPR is made by using a compander in this system, which is effective that is because of Gaussian distribution of OFDM signal where large OFDM signal only occurs infrequently. System simulation models are employed using Rapp??s nonlinear power amplification model. The simulation results show that the compander can provide better performance in comparison to a system that does not employ the compander, i.e., SDM/OFDM system. The effect of main parameter for the compander will be studied. Comparisons between the performances of our system and SDM/OFDM system that is used clipping technique for reduction of PAPR will be made. All these results are made at different modulation techniques and different sub-carriers     

A robust CMOS compander

A robust CMOS compander circuit meeting all of the requirements for analog cellular telephony and using an improved sigma-delta compander topology is presented. Rather than digitizing and reconstructing the input signal using a sigma-delta modulator as has been done previously, only the amplitude path is digitized while the voice path remains analog. The amplitude information is obtained digitally, and is reduced to a single bit using a first-order sigma-delta modulator. Performing this function digitally eliminates problems due to analog offsets and in implementing the long time constant required. The output signal is formed by gating the analog input signal under control of the amplitude signal. The expander and compressor circuits each consist of a single op amp and 2000 gates of digital logic, and have been implemented on 0.8-μm CMOS processes. The ADC for the amplitude path uses a compact switched-capacitor second-order sigma-delta modulator implemented using a single amplifier. No external components are required. Tracking error for the compressor was measured to be less than 0.3 dB over a 60-dB input range when operating on a 3.0-V supply. The test time, when compared to conventional compander implementations, is considerably reduced     

On optimal quantization of noisy sources

Problems in optimal multidimensional quantization of sources corrupted by noise are addressed. Expressions for the optimum quantizer values and the optimum quantization rule for the weighted squared error distortion measure are found and calculated for the Gaussian signal in additive independent Gaussian noise problem. Some properties of the optimum quantizer, and its relations with the optimal estimator for the general problem, are derived     

Joint design of a channel-optimized quantizer and multicarriermodulation

We present an algorithm for design of a joint source-channel coder using a channel-optimized quantizer and multicarrier modulation. By changing the power of each subchannel in the multicarrier modulation system, different degrees of error protection can be provided for different bits according to their importance. The algorithm converges to a locally optimum system design. Compared to a Lloyd-Max scalar quantizer or a LBG vector quantizer using single-channel transmission, our optimized code can yield substantial performance improvements. The performance improvements are most pronounced at low channel signal-to-noise ratios     

On the support of fixed-rate minimum mean-squared error scalar quantizers for a Laplacian source

This correspondence shows that the support growth of a fixed-rate optimum (minimum mean-squared error) scalar quantizer for a Laplacian density is logarithmic with the number of quantization points. Specifically, it is shown that, for a unit-variance Laplacian density, the ratio of the support-determining threshold of an optimum quantizer to 3//spl radic/2lnN/2 converges to 1, as the number N of quantization points grows. Also derived is a limiting upper bound that says that the support-determining threshold cannot exceed the logarithmic growth by more than a small constant, e.g., 0.0669. These results confirm the logarithmic growth of the optimum support that has previously been derived heuristically.     

Optimum design of vector-quantized subband codecs

This article provides an approach for representing an optimum vector quantizer by a scalar nonlinear gain-plus-additive noise model. The validity and accuracy of this analytic model is confirmed by comparing the calculated model quantization errors with actual simulation of the optimum Linde-Buzo-Gray (1980) vector quantizer. Using this model, we form an MSE measure of an M-band filter bank codec in terms of the equivalent scalar quantization model and find the optimum FIR filter coefficients for each channel in the M-band structure for a given bit rate, filter length, and input signal correlation model. Specific design examples are worked out for four-tap filters in the two-band paraunitary case. These theoretical results are confirmed by extensive Monte Carlo simulation     

Quantization issues for soft-decision decoding of linear blockcodes

In general, a channel quantizer for a communication system subject to additive white Gaussian noise (AWGN) is designed based on the cutoff rate. This criterion is good if the scheme considered performs close to the theoretical performance corresponding to the cutoff rate, as for error control systems employing convolutional codes. However, it is no longer true for systems using low complexity suboptimum decoding algorithms for block codes. We illustrate this point and present three examples for which we compare the optimum quantizer and the quantizer based on the cutoff rate for Q=4 quantization levels     

A pyramid vector quantizer

The geometric properties of a memoryless Laplacian source are presented and used to establish a source coding theorem. Motivated by this geometric structure, a pyramid vector quantizer (PVQ) is developed for arbitrary vector dimension. The PVQ is based on the cubic lattice points that lie on the surface of anL-dimensional pyramid and has simple encoding and decoding algorithms. A product code version of the PVQ is developed and generalized to apply to a variety of sources. Analytical expressions are derived for the PVQ mean square error (mse), and simulation results are presented for PVQ encoding of several memoryless sources. For large rate and dimension, PVQ encoding of memoryless Laplacian, gamma, and Gaussian sources provides rose improvements of5.64, 8.40, and2.39dB, respectively, over the corresponding optimum scalar quantizer. Although suboptimum in a rate-distortion sense, because the PVQ can encode large-dimensional vectors, it offers significant reduction in rose distortion compared with the optimum Lloyd-Max scalar quantizer, and provides an attractive alternative to currently available vector quantizers.     

Vector Quantizer Design for Memoryless Gaussian, Gamma, and Laplacian Sources

Locally optimum vector quantizer (VQ) designs are presented for memoryless Gaussian, gamma, and Laplacian sources. For Gaussian sources, low (2-6) dimensional vector quantization provides relatively little improvement in mean-squared error (MSE) compared to the minimum mean-squared error (MMSE) scalar quantizer. For Laplacian or gamma sources, however, significant improvement in MSE is available with vector quantization. The Laplacian and gamma 6 bit, sixdimensional vector quantizers achieve, respectively, improvements of 2 and 4.5 dB over the corresponding scalar MMSE quantizer distortions.     

Fixed-Tap ADPCM System Divergence and a Bound on the Robust Quantizer Overload Point

The divergence of ADPCM systems with fixed, multipletap predictors and a Jayant quantizer is investigated. It is shown that system divergence occurs due to excessive quantization noise in the feedback loop coupled with the infinite quantizer memory. Further, divergence may result for even finer quantization if the predictor is poorly matched with the system input. New insight into quantizer/ predictor interaction is provided by a demonstration that for all average speech data available in the literature and more than one feedback tap, the system that describes the quantization noise evolution is unstable whenever the predictor is stable. It is noted that robust quantizer designs originally proposed for transmission error suppression are also effective in preventing the ADPCM system divergence problem discussed here, and a bound on the robust quantizer overload point is derived which illustrates the effect of the finite quantizer memory. Simulation results which validate the bound are presented.     

高效短时DFT压扩器的实现

压扩技术作为提高伟输信号信噪比的有效手段，在通信系统中得到了广泛应用。短时DFT压扩器根据瞬时谱直接在频域对信号压扩而无需峰值检测，对降低衰落噪声、提高信号传输质量比传统压扩器更为有效，但短时DFT压扩器运算量大，计算效率低。本文通过对传输信号的序号模运算和引人循环相关，构筑了高速化短时DFT压扩器，在保证提高信噪比的前提下，有效减少了运算量和提高了计算效率，具有较大的理论意义和实用价值。     

Causal coding of stationary sources and individual sequences with high resolution

In a causal source coding system, the reconstruction of the present source sample is restricted to be a function of the present and past source samples, while the code stream itself may be noncausal and have variable rate. Neuhoff and Gilbert showed that for memoryless sources, optimum performance among all causal source codes is achieved by time-sharing at most two memoryless codes (quantizers) followed by entropy coding. In this work, we extend Neuhoff and Gilbert's result in the limit of small distortion (high resolution) to two new settings. First, we show that at high resolution, an optimal causal code for a stationary source with finite differential entropy rate consists of a uniform quantizer followed by a (sequence) entropy coder. This implies that the price of causality at high resolution is approximately 0.254 bit, i.e., the space-filling loss of the uniform quantizer. Then, we consider individual sequences and introduce a deterministic analogue of differential entropy, which we call "Lempel-Ziv differential entropy." We show that for any bounded individual sequence with finite Lempel-Ziv differential entropy, optimum high-resolution performance among all finite-memory variable-rate causal codes is achieved by dithered scalar uniform quantization followed by Lempel-Ziv coding. As a by-product, we also prove an individual-sequence version of the Shannon lower bound.