Quantization; SQNR for linear quantization

Quantization

Quantization is the process of approximating each continuous-amplitude sample of a signal by the nearest of a finite set of allowed levels. With $L$ levels spaced $\Delta$ apart it turns a sampled (discrete-time, continuous-amplitude) signal into a discrete-amplitude signal that can be coded with $n = \log_2 L$ bits. The difference between the true sample and the quantized value is the quantization error (quantization noise).

SQNR for linear (uniform) quantization

Let the message $m(t)$ lie in the range $-m_p$ to $+m_p$ and let the quantizer have $L$ equally spaced levels.

1. Step size

$$\Delta = \frac{2m_p}{L}$$

2. Error range: each sample is replaced by the nearest quantization level (the centre of its step), so the error $q$ lies between $-\Delta/2$ and $+\Delta/2$.
3. Error pdf: for a busy signal with many levels, $q$ is assumed uniformly distributed, $f_Q(q) = 1/\Delta$ for $|q| \le \Delta/2$.
4. Quantization noise power (mean square error):

$$\begin{aligned} N_q = \overline{q^2} &= \int_{-\Delta/2}^{\Delta/2} q^2 \frac{1}{\Delta}\,dq = \frac{1}{\Delta}\left[\frac{q^3}{3}\right]_{-\Delta/2}^{\Delta/2} \\ &= \frac{\Delta^2}{12} = \frac{(2m_p/L)^2}{12} = \frac{m_p^2}{3L^2} \end{aligned}$$

5. SQNR: with signal power $S = \overline{m^2(t)}$,

$$\text{SQNR} = \frac{S}{N_q} = 3L^2\,\frac{\overline{m^2(t)}}{m_p^2}$$

6. Sinusoidal message $m(t) = m_p\cos\omega_m t$ (full load): $S = m_p^2/2$, so

$$\begin{aligned} \text{SQNR} &= \frac{m_p^2/2}{m_p^2/(3L^2)} = \frac{3}{2}L^2 = 1.5 \times 2^{2n} \\ \text{SQNR}_{dB} &= 10\log_{10}1.5 + 20n\log_{10}2 \approx 1.76 + 6.02n\ \text{dB} \end{aligned}$$

Key results

• SQNR rises as $L^2$, i.e. by about 6 dB for each extra bit.
• For a signal whose power equals $m_p^2$ (e.g. a square-like signal) the result is $3L^2$, i.e. $4.77 + 6.02n$ dB.
• Example: 8-bit PCM with a full-scale sine gives $1.76 + 48.16 \approx 49.9$ dB.
• Weak signals give a much lower SQNR because $N_q$ is fixed while $S$ falls; this is why non-uniform quantization (companding) is used for speech.