Least squares quantization in PCM

  title={Least squares quantization in PCM},
  author={Stuart P. Lloyd},
  journal={IEEE Trans. Inf. Theory},
  • S. P. Lloyd
  • Published 1 March 1982
  • Computer Science
  • IEEE Trans. Inf. Theory
It has long been realized that in pulse-code modulation (PCM), with a given ensemble of signals to handle, the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall. It has been shown by Panter and Dite that, in the limit as the number of quanta becomes infinite, the asymptotic fractional density of quanta per unit voltage should vary as the one-third power of the probability density per unit voltage of signal amplitudes. In this… 

Figures and Tables from this paper

The Effect of Quantization on the Performance of Sampling Designs
It is shown that the rate of convergence of the mean-square error is reduced from n/sup -2k-2/ to n/Sup -2/ when the samples are quantized, which is a very significant reduction.
New Results on Robust Quantization
The robust design of robust block quantizers when the number of quantization levels is large is formulated as a two-person game, and it is shown that for convex families of signal probability density functions there is a saddle point solution.
How to use noise to reduce complexity in quantization
It is shown in this paper how noise can be used to reduce the number of unique values required in the encoding stage by allowing the noise to effectively make all thresholds independent random variables, the end result being a stochastic quantization.
Vector quantization
  • R. Gray
  • Computer Science
    IEEE ASSP Magazine
  • 1984
During the past few years several design algorithms have been developed for a variety of vector quantizers and the performance of these codes has been studied for speech waveforms, speech linear predictive parameter vectors, images, and several simulated random processes.
Asymptotic distribution of the errors in scalar and vector quantizers
High-rate (or asymptotic) quantization theory has found formulas for the average squared length and the probability density of the length of the error and, in certain special cases, for the probabilitydensity of the multidimensional error vector, itself, which can be used to analyze the distortion of two-stage vector quantization.
On embedded scalar quantization
  • G. Sullivan
  • Computer Science
    2004 IEEE International Conference on Acoustics, Speech, and Signal Processing
  • 2004
It is shown that any rational number can be maintained as a stable dead-zone ratio, and performance is explored primarily in the context of the generalized Gaussian pdf using the squared-error distortion measure, but should also apply in other contexts.
Efficient scalar quantization of exponential and Laplacian random variables
The memoryless property of the exponential distribution is used to develop a new noniterative algorithm for obtaining the optimal quantizer design, which needs only a single sequence of solutions to one-dimensional nonlinear equations.
Analysis on Optimal Quantization of Signals for System Identification and the Effect of Noise
In this paper, the property of the optimal quantization of signals used for system identification is analysed and the necessary information to attain the optimal identification errors is given as a function of the entropy of the regressor vector.
Vector quantization in speech coding
This tutorial review presents the basic concepts employed in vector quantization and gives a realistic assessment of its benefits and costs when compared to scalar quantization, and focuses primarily on the coding of speech signals and parameters.
On the structure of optimal entropy-constrained scalar quantizers
It is proved that under rather general conditions there exists an "almost regular" optimal ECSQ for any entropy constraint, and for the squared error distortion measure and sources with piecewise-monotone and continuous densities, the existence of a regular optimal E CSQ is shown.


Instantaneous companding of quantized signals
Instantaneous companding may be used to improve the quantized approximation of a signal by producing effectively nonuniform quantization. A revision, extension, and reinterpretation of the analysis
The Philosophy of PCM
This paper shows in a general way some of the advantages of PCM, and distinguishes between what can be achieved with PCM and with other broadband systems, such as large-index FM.
Quantization distortion in pulse-count modulation with nonuniform spacing of levels
It is shown that the distortion introduced in a pulse-count-modulation system due to quantization can be minimized by nonuniform spacing of levels. Equations are derived for an arrangement of
This note gives a method of proof of the sampling theorem, both for the case where the interval I is centered at the origin and where it is not, which is somewhat simpler than the previously given proofs, and at the same time is more rigorous, and yields several useful generalizations to functions of several variables and random functions.
On the Generalized Sampling Theorem (非線型振動理論の研究会報告集)
Being based on the generalized sampling theorem ('‘) presented by the authors, some series-expansions of functions are given as its special examples, including Krolls expressions@). The discussions
What Philosophy Is
Quantization distortion in pulsecount modulation with nonuniform snacing of levels Instantaneous companding of quantized signals
  • Mathematical Method of Statistics
  • 1953
Editor's Note: This paper first appeared as a Bell Laboratories Technical Memorandum in May 1964. The principal theorem of this paper was
  • Editor's Note: This paper first appeared as a Bell Laboratories Technical Memorandum in May 1964. The principal theorem of this paper was
Contribution aux applications statistiques de la theorie de l'information, " Pub. de I'lnst. de Stutis. de I'Universite de Paris
  • Contribution aux applications statistiques de la theorie de l'information, " Pub. de I'lnst. de Stutis. de I'Universite de Paris
  • 1954