Quantizer Output Research Articles

A hierarchical dynamic programming approach to fixed-rate, entropy-coded quantization

In quantization of any source with a nonuniform probability density function, the entropy coding of the quantizer output can result in a substantial decrease in bit rate. A straightforward entropy coding scheme faces us with the problem of variable data rate. A solution in a space of dimensionality N is to select an appropriate subset of elements in the N-fold Cartesian product of a scalar quantizer and represent its elements with codewords of the same length. The drawback is that the search/addressing of this scheme can no longer be achieved independently along the one-dimensional subspaces. A reasonable rule is to select the N-fold symbols of the highest probability. For a memoryless source, this is equivalent to selecting the N-fold symbols with the lowest additive self-information. In this case, due to the additivity property of the self-information, the selected subset has a high degree of structure which can be used to substantially decrease the search/addressing complexity. A dynamic programming approach is used to exploit this structure. We build our recursive structure required for the dynamic programming in a hierarchy of levels. This results in several benefits over the conventional trellis-based approaches. Using this structure, we develop efficient rules (based on aggregating the states) to substantially reduce the search/addressing complexities while keeping the degradation in performance negligible.

Rate-distortion performance in coding bandlimited sources by sampling and dithered quantization

The rate-distortion characteristics of a scheme for encoding continuous-time band limited stationary sources, with a prescribed band, is considered. In this coding procedure the input is sampled at Nyquist's rate or faster, the samples undergo dithered uniform or lattice quantization, using subtractive dither, and the quantizer output is entropy-coded, The rate-distortion performance, and the tradeoff between the sampling rate and the quantization accuracy is investigated, utilizing the observation that the coding scheme is equivalent to an additive noise channel. It is shown that the mean-square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant, while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the Nyquist rate. Finally, as the lattice quantizer dimension becomes large, the equivalent additive noise channel of the scheme tends to be white Gaussian, and both the rate and the distortion performance become invariant to the sampling rate. >

Effect of additive dither on the resolution of ideal quantizers

The effect of adding dither to a given signal before its quantization is investigated. To this aim the relationship between the statistical description of the dither and the mean value of the quantization error and of the quantizer output is derived. The presented results allow the determination of the resolution limits that can be achieved by low-pass filtering a set of quantized samples of a slowly varying dithered signal. Discrete, uniform, sinusoidal, and Gaussian dithering are analyzed, thus providing a quantitative basis for choosing the most appropriate dither signal for a given application. >

Simple high-quality lossy image coding scheme

A simple yet efficient image data compression method is presented. This method is based on coding only those segments of the image that are perceptually significant to the reconstruction of the image. Sequences of image pixels whose gray-level differences from the pixels of the previous row exceed two prespecified thresholds are considered significant. These pixels are coded using a differential pulse code modulation scheme that uses a 15-level recursively indexed nonuniform quantizer for the first pixel in a segment and a 7-level recursively indexed nonuniform quantizer for all other pixels in the segment. The quantizer outputs are Huffman coded. Simulation results show that this scheme can obtain subjectively satisfactory reconstructed images at low bit rates. It is also computationally very simple, which makes it amenable to fast implementation.

Approximation of optimum quantizer-detector using a uniform quantizer and a coder

Abstract A suboptimum quantization-detection system in which the locally optimum nonlinearity is replaced by a uniform quantizer and a coder for the uniform quantizer output levels is proposed. The proposed detection system does not require iterations to obtain the parameters of the quantizer and is easily implementable in practice. We develop a design procedure of the suboptimum quantizer and obtain the parameters of the suboptimum quantizer for the generalized Gaussian noise density. We also compare the performance of the suboptimum quantization detector with that of other detectors.

On quantization of noisy signals

Quantized noise distributions derived from continuous signals with additive noise are studied. Two noise sources are considered, quantized image noise derived from the continuous input noise source and noise due to quantization roundoff error. These are treated as statistically independent sources. An analytic solution for the quantized noise probability density is obtained. The analytic solution is estimated by two expressions valid for normally distributed noise over different ranges of variance. The estimates have excellent agreement in the region of overlapping validity. Quantized noise variance is related to the continuous noise variance from normally distributed noise using these expressions. A table and plots of useful values are included. These results are helpful in choosing a quantization interval for a particular application. They can also be used to determine quantizer output noise level and signal-to-noise ratio in digital applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Derivation of a class of training algorithms

A novel derivation is presented of T. Kohonen's topographic mapping training algorithm (Self-Organization and Associative Memory, 1984), based upon an extension of the Linde-Buzo-Gray (LBG) algorithm for vector quantizer design. Thus a vector quantizer is designed by minimizing an L(2) reconstruction distortion measure, including an additional contribution from the effect of code noise which corrupts the output of the vector quantizer. The neighborhood updating scheme of Kohonen's topographic mapping training algorithm emerges as a special case of this code noise model. This formulation of Kohonen's algorithm is a specific instance of the robust hidden layer principle, which stabilizes the internal representations chosen by a network against anticipated noise or distortion processes.

Tied mixture continuous parameter modeling for speech recognition

The acoustic-modeling problem in automatic speech recognition is examined with the goal of unifying discrete and continuous parameter approaches. To model a sequence of information-bearing acoustic feature vectors which has been extracted from the speech waveform via some appropriate front-end signal processing, a speech recognizer basically faces two alternatives: (1) assign a multivariate probability distribution directly to the stream of vectors, or (2) use a time-synchronous labeling acoustic processor to perform vector quantization on this stream, and assign a multinomial probability distribution to the output of the vector quantizer. With a few exceptions, these two methods have traditionally been given separate treatment. A class of very general hidden Markov models which can accommodate feature vector sequences lying either in a discrete or in a continuous space is considered; the new class allows one to represent the prototypes in an assumption-limited, yet convenient way, as tied mixtures of simple multivariate densities. Speech recognition experiments, reported for two (5000- and 20000-word vocabulary) office correspondence tasks, demonstrate some of the benefits associated with this technique.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spectral analysis of quantization noise in a single-loop sigma-delta modulator with DC input

An exact discrete-time analysis of the moments and spectra of the quantization noise of a discrete-time single-loop sigma-delta modulator with a DC input is presented. An exact difference equation for the discrete-time nonlinear system is used to evaluate the first- and second-order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single-loop sigma-delta encoder with a DC input. It is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption, but that the autocorrelation and power spectrum are not consistent with the white noise assumption. The results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used. >

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.

Analysis of adaptive differential PCM of a stationary Gauss - Markov input

An adaptive matched differential pulse-code modulator (AMDPCM) is analyzed. The adaptation of the symmetric uniform quantizer parameter \Delta_{n} is performed by fixed multipliers assigned to the quantizer output levels. The input is stationary first-order Gauss-Markov. The correlation of the samples is used as the leakage parameter in the matched integrator, with the predictive reconstruction similarly matched. For a 4 -level quantizer and multipliers (\gamma^{-1}, \gamma) the limiting joint distribution of the prediction error and \Delta_{n} is derived and the asymptotic sample-point and time-averaged mean-square error (rose) and mean and variance of \Delta_{n} as functions of \gamma \in (1,2] are computed and plotted. It is found that the asymptotic performance of AMDPCM does not depend on the choice of \Delta_{0} , that the increase in mse incurred by using A(M)DPCM instead of (M)DPCM with \Delta_{opt} is small, with mse(A(M)DPCM) \downarrow \min_{\Delta} mse ((M)DPCM) as \gamma \downarrow 1 , and that the signal-to-noise ratio of AMDPCM does not depend on the input power.

量子化出力誤差が小さいDDCアルゴリズム

量子化出力誤差が小さいDDCアルゴリズム

Speech analysis and synthesis apparatus

ABSTRACT OF THE DISCLOSURE Disclosed is a speech analysis and synthesis apparatus having an analysis unit adapted to quantize the characterizing parameters of a speech signal at mutually different quantization steps depending on whether the speech signal represent a voiced sound or an unvoiced sound, and a synthesis unit adapted to decode the quantization output from the analysis unit to repro-duce the characterizing parameters. The unit subtracts a short-term mean value of environment noise from the short-term mean value of the sum of the ambient noise and the speech signal to provide the short-term mean value of the speech signal. The discrimination between the voiced and unvoiced sounds employed in the analysis unit nonlinearly converts decision parameters which have extremely different variance of occurrence rate dis-tribution characteristics between voiced and unvoiced sound used for the voiced-unvoiced decision equation, and analyzes the mixture of the environment-voice-representing signal and known voiced or unvoiced sound to determine the coefficient and thresh-old values of the voiced-unvoiced decision equation, thereby to determine the voiced or unvoiced sound based on the coefficients and threshold values. The apparatus requires reduced amount of transmission information without degrading the quality of synthesized speech sound.

Nonlinear quantization effects in the frequency domain complex scalar LMS adaptive algorithm

Nonlinear quantization effects in the frequency domain complex scalar LMS adaptive algorithm are analyzed by using conditional expectations. The probability density function of the quantizer input, conditioned on the weight, is derived. The density is applied to finding the conditional characteristic function and the Mth conditional moment at the quantizer output. The first and second conditional moments of the quantizer output are used to derive difference equations that approximate the dynamical behavior of the first and second weight moments. These difference equations are solved numerically and compare favorably to simulation results. A model of the quantizer as an additive noise source is of no analytical value since the quantizatian noise has negligible effect on the mean square error when the model is valid. Finally, a design approach is proposed for selecting the number of bits in the weight accumulator. The moment equations are also used to determine the algorithm mean square error for different quantizer step sizes and the optimum algorithm step size μ when a fixed amount of input data is available for adaptation.

Design of optimal quantizers with an entropy constraint

Source encoding of transform domain coefficients can be used to obtain pictures of very good quality at low bit rates. Such a scheme requires that the entropy of the quantizer output be fixed at a prespecified value. The design of optimal quantizers subject to an entropy constraint is discussed in this paper and their specifications for three different input distributions associated with the transform coefficients is given.

Quantization for minimum intermodulation distortion

Intermodulation due to quantization is analysed, and it is shown that the intermodulation distortion can be calculated by multiplying each quantization output level by a corresponding coefficient and then summing the resulting products over all the quantization steps. This coefficient can be obtained irrespective of the properties of the quantizer, provided that the ratio of each amplitude of the input signals and the output frequency of interest are given. Using this approach, a new technique is proposed for optimum quantization by which the input dynamic range of the quantizer is maximized without letting intermodulation distortion exceed the allowable level given by a system. Starting with the smallest input, once the optimum step sizes for the first two steps are obtained, merely increasing a certain factor gives the proceeding step sizes. For the allowable value of intermodulation distortion of − 40 dB, dynamic ranges realized by the quantizer presented and the uniform quantizer are 33 dB and 12·5dB ...

Conditional Variable-Length Coding for Gray-Level Pictures

This paper presents a new variable-length coding algorithm for bit-rate reduction of digital images. The coding scheme is based on the fact that by taking into account the local conditional statistics of the picture, the coding efficiency can be improved considerably. Results of computer simulations are presented in terms of picture quality and required bit rate. Simulations show that for various Differential Pulse Code Modulation (DPCM) coded pictures, coding efficiency is such that a bit rate below the entropy of the DPCM quantizer output is attained. For a 3-bit DPCM encoded picture, a bit rate of 1.64 bits/picture element is achieved.

Transmission Errors and Forward Error Correction in Embedded Differential Pulse Code Modulation

We have derived formulas for the combined effects of quantization and transmission errors on the performance of embedded Differential Pulse Code Modulation (DPCM), a source code that can be used for variable-bit-rate speech transmission. Our analysis is more general and more precise than previous work on transmission errors in digital communication of analog signals. Special cases include conventional DPCM and Pulse Code Modulation (PCM). Our main result is a signal-to-noise ratio formula in which the effects of source characteristics (input signal, codec design parameters) and the effects of transmission characteristics (modulation, channel, forward error correction) are clearly distinguishable. We also present, in computationally convenient forms, specialized formulas that apply to uncoded transmission through a random-error channel, transmission through a slowly fading channel, and transmission with part or all of the DPCM signal protected by an error-correcting code. Numerical results show how channel coding can have different effects on conventional and embedded DPCM. They also show how the binary-number representation of quantizer outputs influences performance.

A DPCM System Based on a Composite Image Model

This paper presents a DPCM system for storage or transmission (via noiseless channel) of a moving (3-D) TV image. A fully nonlinear(adaptive) DPCM system based on a composite model of the image fragment is described. The description takes into consideration mutual influence of the prediction and quantization parts of the system. New results are presented concerning the probability density functions of the image luminance and the prediction error. The paper also addresses two types of system optimization: 1) minimization of the mean square quantization error (MSQE); 2) minimization of the MSQE when the entropy of the quantizer output is below a specified level. The results of the optimization determine the parameters of the system.

Ultrafast Feedback A/D Conversion Made Possible by a Nonuniform Error Quantizer

The paper presents a means of increasing the speed of a tracking (feedback) A/D converter by replacing the single comparator with an error quantizer having exponentially spaced representative levels. The technique is particularly suited to digitizing the class of signals which exhibit a large covariance between temporally adjacent samples. An n bit converter realized in this way incorporates a quantizer which has a maximum of 2n comparators. Data compression may be simply realized by encoding the quantizer output to produce a DPCM word stream.