Non-uniform quantization using uniform quantizer; μ-law

Non-uniform quantization using a uniform quantizer

A non-uniform quantizer is equivalent to a compressor, followed by a uniform quantizer, with an expander at the receiver. This is called companding.

x(t)->[Compressor]--y-->[Uniform  ]-->[Encoder]-->channel
       y = c(x)          quantizer
channel->[Decoder]-->[Expander x = c^-1(y)]-->x^(t)

Technique:

1. The input $x$ (normalised to $|x| \le 1$) is passed through a non-linear compressor $y = c(x)$ with large slope near zero and small slope near full scale.
2. A uniform quantizer with step $\Delta$ operates on $y$. Step $\Delta$ in $y$ corresponds to a step in $x$ of

$$\Delta_x \approx \frac{\Delta}{c'(x)}$$

so where $c'(x)$ is large (small $x$) the effective step is fine, and where $c'(x)$ is small (large $x$) the step is coarse. 3. At the receiver, the expander $c^{-1}(y)$ undoes the compression. Compressor + expander together have a linear overall response. 4. If $c(x)$ is logarithmic, $\Delta_x \propto |x|$, so the noise is proportional to the signal and SQNR is nearly constant for all levels.

In practice $c(x)$ is implemented digitally as a piecewise-linear segmented characteristic, so standard uniform ADC/DACs can be used.

μ-law

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

• $\mu = 0$: uniform (linear). Standard $\mu = 255$ for 8-bit PCM (North America, Japan).
• Small $|x|$ ($\mu|x| \ll 1$): $y \approx \frac{\mu x}{\ln(1+\mu)}$, linear with large gain (46 for $\mu = 255$).
• Large $|x|$ ($\mu|x| \gg 1$): $y \approx \frac{\ln(\mu|x|)}{\ln(1+\mu)}$, logarithmic.

Example ($\mu = 255$): $x = 0.01 \to 0.228$, $x = 0.1 \to 0.591$, $x = 0.5 \to 0.876$, $x = 1 \to 1$.

 y
1.0|              ____----   mu = 255
   |        __----      __-- mu = 5
   |    _-'      ___---
   |  /    __---    .  mu = 0 (y = x)
   | / _--     .
   |/-    .
   +---------------------> x
   0                   1.0

Larger $\mu$ gives more compression and a wider dynamic range with nearly flat SQNR (about 38 dB over 40 dB input range for 8-bit $\mu = 255$).