Companding and its types

Companding (compressing + expanding) is a way to obtain non-uniform quantization: the signal is compressed by a non-linear amplifier before a uniform quantizer at the transmitter, and expanded by the inverse characteristic at the receiver.

m(t)->[Compressor]->[Uniform Q]->channel->[Expander]->m(t)
       y = c(x)                          x = c^-1(y)

Why it is necessary

• With uniform quantization, noise $\Delta^2/12$ is fixed, so weak signals have poor SQNR.
• Speech has a large dynamic range and small amplitudes are most frequent.
• The compressor boosts small amplitudes and reduces large ones, so effectively small inputs see fine steps and large inputs coarse steps. The SQNR stays nearly constant (within a few dB) over about 40 dB of input range with only 8 bits.

μ-law companding (North America, Japan)

$$y = \frac{\ln(1 + \mu|x|)}{\ln(1+\mu)}\,\text{sgn}(x), \qquad 0 \le |x| \le 1$$

• $\mu = 0$ gives uniform quantization; the standard value is $\mu = 255$ (8-bit PCM).
• For small $x$ it is nearly linear ($y \approx \mu x/\ln(1+\mu)$); for large $x$ it is logarithmic.
• Example with $\mu = 255$: $x = 0.01 \to y = 0.228$; $x = 0.1 \to y = 0.591$; $x = 0.5 \to y = 0.876$.

A-law companding (Europe, India, Nepal)

$$y = \begin{cases} \dfrac{A|x|}{1+\ln A}\,\text{sgn}(x), & 0 \le |x| \le \dfrac{1}{A} \\[2mm] \dfrac{1+\ln(A|x|)}{1+\ln A}\,\text{sgn}(x), & \dfrac{1}{A} \le |x| \le 1 \end{cases}$$

• Standard value $A = 87.6$. It is linear for small signals and logarithmic for large ones. Both A-law and μ-law are specified in ITU-T G.711.
• In practice both laws are realised by a 13-segment (A-law) or 15-segment (μ-law) piecewise-linear approximation.

 y (out)
 1 |            . . . .   <- compressed
   |      . .
   |   .          ....  <- uniform (y = x)
   | .       ....
   |.   ....
   |. ..
   +-------------------- x (in)
   0                  1