Instantaneous companding of quantized signals

Abstract

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 of Panter and Dite permits the calculation of the quantizing error power as a function of the degree of companding, the number of quantizing steps, the signal volume, the size of the “equivalent dc component” in the signal input to the compressor, and the statistical distribution of amplitudes in the signal. It appears, from Bennett's spectral analysis, that the total quantizing error power so calculated may properly be studied without attention to the detailed composition of the error spectrum, provided the signal is complex (such as speech or noise) and is sampled at the minimum information‐theoretic rate. These calculations lead to the formulation of an effective process for choosing the proper combination of the number of digits per code group and companding characteristic for quantized speech communication systems. An illustrative application is made to the planning of a hypothetical PCM system, employing a common channel compandor on a time division multiplex basis. This reveals that the calculated companding improvement, for the weakest signals to be encountered in such a system, is equivalent to the addition of about 4 to 6 digits per code group, i.e., to an increase in the number of uniform quantizing steps by a factor between 24 = 16 and 26 = 64. Comparison with the results of related theoretical and experimental studies is also provided.

Bounds on performance of optimum quantizers

“…Gish and Pierce [23] (whose preprint received in the fall of 1967 introduced me to the Panter and Dite reference after my first presentation of some of these results) show that uniform, one-dimensional quantization is asymptotically (large K, small error) optimum in minimizing distortion for given entropy for a large class of measures, including mean rth-power difference measures but not the geometric mean. Wood [33] independently reaches the same conclusion as Gish and Pierce in the mean-square one-dimensional case, citing Roe but not Panter and Dite or Algazi and missing (2). The multidimensional minimal-entropy quantization problem is considered by Schutzenberger [24], who gives inequalities with unknown coefficients on trading relations between entropy and mean rth-power difference measures, and by Zador [4], who gives an asymptotic (large K, small error) result with unknown coefficient for this case too.…”

“…Max [7], Bruce [8], and Bluestein [9], looking for algorithms for finding optimum quantizers (for signal plus noise in [9]) miss the reference and the result; Algazi [6], finding simpler suboptimal algorithms also misses the reference but cites the quantizer from Roe and rediscovers (2). As Smith [2] notes, Sheppard [101 was the first to give the effect of a uniform quantizer (i.e., grouping statistical data in uniform intervals) on variance in 1898. Sheppard's correction, missed by Clavier et al [11] in the first paper on PCM distortion, is rederived by Bennett [12] and by Oliver et al [13].…”

“…He cites Bennett [12] and Widrow [16] on uniform quantization, and the numerical analysis literature on roundoff error in the numerical solution of differential equations. Tou [17], in 1963, devotes a chapter to design of optimal quantizers in control systems, includes nonuniform spacing of levels in his formulation, is concerned with effects of quantization on the dynamics of the systems as well as the static considerations dealt with here, references only Bertram on quantization per se and misses (2). This paper, and [1]- [9] and [17] deal with minimization of a measure of the average distortion introduced by a quantizer by varying the location of K -I quantizing levels, given a fixed probability distribution.…”

System Design, Implementation, and Evaluation of the Optical Broadband Correlator

“…Each of these methods can normalize by a user-supplied value, by the peak amplitude in a block, or by other methods based on the "local" statistics of the data. A modified mu-law method 16 that we have developed compresses the amplitude logarithmically but preserves the phase. All of the methods preserve the phase as much as possible, since, for our waveforms, it is the phase information that provides most of the signal processing gain.…”

Reducing transmission error effects using a self-organizing network

“…The quantization levels are most closely spaced at low amplitudes, where most speech samples lie$. Smith [3] gives the quantization noise for a p-law quantizer as…”