mu-law

• See How audio is represented on computers for context, but it’s a useful quantization scheme when we have few bits to allocate to each sample.

Uniform quantization is wasteful perceptually:

• Human hearing is roughly logarithmic in amplitude.
• We care more about resolution at low amplitudes than at high amplitudes.

Idea of μ-law / A-law:
Apply a nonlinear compressor before quantization, so that:

• Quiet values are spread out → more quantization levels → less noise.
• Loud values are squashed → fewer levels → more noise, but less audible.

For μ-law (used in North American telephony; $\mu \approx 255$:

For $x\in [-1,1]$, the compressor is:

$F(x)=\mathrm{sgn}(x)\frac{\ln (1+\mu |x|)}{\ln (1+\mu )}$ ​

Then you quantize F(x) uniformly to 8 bits (256 levels).

So the effective mapping is:

$x\stackrel{\text{compress}}{\to }F(x)\stackrel{\text{uniform 8-bit quantizer}}{\to }k\in \{0,\dots ,255\}$

On decode, you do the inverse:

$\hat{x}=\mathrm{sgn}(F)\frac{1}{\mu }\left((1+\mu {)}^{|F|}-1\right)$

So μ-law doesn’t change the fact that audio is a sequence of discrete-time samples; it just uses a smarter nonlinear mapping from amplitude → code.