Waveform Quantization and Compression - Digital Signal Processing: Fundamentals and Applications - page 501

11.2 m -LAW COMPANDING
In this section, we will study analog m -law companding, which takes an analog input signal; and digital
m -law companding, which deals with linear pulse code modulation (PCM) codes.

11.2.1 Analog -Law Companding

To reduce the number of bits required to encode each speech datum, m -law companding, called log-
PCM coding, is applied. m -law companding (Roddy and Coolen, 1997; Tomasi, 2004) was first used in
the United States and Japan in the telephone industry (G.711 standard). m -law companding is
a compression process. It explores the principle that the higher amplitudes of analog signals are
compressed before ADC and expanded after digital-to-analog conversion (DAC). As studied in the
linear quantizer, the quantization error is uniformly distributed. This means that the maximum
quantization error stays the same no matter how big or small the speech samples are. m -law com-
panding can be employed to make the quantization error smaller when the sample amplitude is smaller
and to make the quantization error bigger when the sample amplitude is bigger, using the same number
of bits per sample. It is described in Figure 11.3 .
As shown in Figure 11.3 , x is the original speech sample, which is the input to the compressor,
while y is the output from the m -law compressor; then the output y is uniformly quantized. Assuming
that the quantized sample y q is encoded and sent to the m -law expander, the expander will perform the
reverse process to obtain the quantized speech sample

m

x q . The compression and decompression

processes cause the maximum quantization error jx q xj max to be small for the smaller sample

amplitudes and large for the larger sample amplitudes.
The equation for the m -law compressor is given by

ln 1 þ m jxj
j x j max

y ¼ signðxÞ (11.4)

ln ð 1 þ mÞ

where jxj max is the maximum amplitude of the inputs, while m is a positive parameter to control the
degree of the compression. m ¼ 0 corresponds to no compression, while m ¼ 255 is adopted in the
industry. The compression curve with m ¼ 255 is plotted in Figure 11.4 . Note that the sign function
signðxÞ shown in Equation (11.4) is defined as

( 1

x 0

signðxÞ¼ (11.5)

1 x < 0

x y q x q

y

-law
compressor

µ Uniform

quantizer

-law
expander

µ

FIGURE 11.3

Block diagram for m-law compressor and m-law expander.