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 paper the corresponding result for any finite number of quanta is derived; that is, necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy. The optimization criterion used is that the average quantization noise power be a minimum. It is shown that the result obtained here goes over into the Panter and Dite result as the number of quanta become large. The optimum quautization schemes for 2^{b} quanta, b=1,2, \cdots, 7 , are given numerically for Gaussian and for Laplacian distribution of signal amplitudes.

/pdf/least-squares-quantization-in-pcm-56gjks6m10.pdf

Least squares quantization in PCM

Least Squares Quantization in PCM

A method is developed for representing any communication system geometrically Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise Some of the properties of "ideal" systems which transmit at this maximum rate are discussed The equivalent number of binary digits per second for certain information sources is calculated

/pdf/communication-in-the-presence-of-noise-p5l6l0cm8w.pdf

Communication in the presence of noise

Image processing techniques find applications in many areas, chief among which are image enhancement, pattern recognition, and efficient picture coding. Some aspects of image processing are discussed--specifically: the mathematical operations one is likely to encounter, and ways of implementing them by optics and on digital computers; image description; and image quality evaluation. Many old results are reviewed, some new ones presented, and several open questions are posed.

Image processing

Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

Recent papers describe experiments in transmitting speech by PCM (pulse code modulation). 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. The intent is to explain the various points simply, rather than to elaborate them in detail. The paper is for those who want to find out about PCM rather than for those who want to design a system. Many important factors will arise in the design of a system which are not considered in this paper.

B.M. Oliver

Papers

The Philosophy of PCM