Uniform vs non-uniform quantization; companding laws

Uniform vs non-uniform quantization

Why non-uniform quantization for speech

• Speech has a large dynamic range (about 40 dB between loud and soft talkers).
• Its amplitude pdf is peaked near zero: small amplitudes occur most often.
• With uniform steps, SQNR $\propto$ signal power, so quiet talkers and soft sounds get poor SQNR.
• Non-uniform steps give small steps where samples are most likely, so SQNR becomes almost independent of level, and 8 bits give quality that would need about 12 bits uniformly.

Companding laws

Companding = compressing at the transmitter + expanding at the receiver. With $x = m/m_p$ normalised input ($|x| \le 1$):

µ-law (North America, Japan; $\mu = 255$):

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

A-law (Europe, ITU-T E1 systems; $A = 87.6$):

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

• $\mu = 0$ or $A = 1$ gives uniform quantization.
• Both laws are nearly linear for small inputs and logarithmic for large inputs.
• In practice they are implemented as 13-segment (A-law) or 15-segment (µ-law) piecewise linear approximations.