Mario Malcangi
Fall 1995
© IEEE Computer Society Press.

Introduction

In telephony systems, two companding/expanding laws are used to reduce quantization noise in speech coding:

These conversion technique differ only on some points, but the common purpose is to enhance signal-to-quantization noise ratio preserving the 64 kbit/s standard bit rate (8 kHz sampling rate and 8 bit coding).
The speech intellegibility constraints lead to at least a 12 bit word lenght when linear quantization is used, so 96 kbit/s (12 bits x 8000 sample/s) bit rate is required, much higher then the 64 kbit/s bit rate required in telecom applications.
8 bit non-uniform (logarithmic) quantization has been defined to match both the bit-rate and the intellegibility constrants. It can be proven that the signal to quantization-noise ratio becames independent from the signal level if the quantization function is a logaritmic function like:

y=1+(log(x)/k)

The size of the quantization step increases with the input signal value: small amplitudes are coded with fine granularity (moore steps) then larger amplitudes. Signal to quantization-noise ratio becames constant because each signal level is coded with the same relative precision (fig. 1). As when x --> 0, log x becames infinite, this function can not be used for small signal levels, so an approximation of this curve is necessary for small signal levels:

Fig. 1 - Logarithmic compression Signal-to-Noise ratio.

- for the A-law, the curve is approximated with its tangent near the origin, so the qantization is linear for small signals. The slope of the linear part, called the compensation rate, has been fixed to 16 and is given by:

C=A/(1+ln(A))=16

which makes A = 87.6

- for the Mu-law, the function taken is quasi-linear for x small and quasi-logarithmic for x large. The compression rate near the origin is given by:

C=Mu/ln(1+Mu)=16

which makes Mu = 255.

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