PCM, DPCM and DM comparison; SQNR of PCM

Comparison of PCM, DPCM and DM

SQNR of PCM

Let $m(t)$ have peak value $m_p$ and be quantized uniformly into $L = 2^n$ levels.

Step size: $\Delta = \dfrac{2m_p}{L}$

Noise power: the error $q$ is uniform in $(-\Delta/2, \Delta/2)$:

$$N_q = \int_{-\Delta/2}^{\Delta/2}\frac{q^2}{\Delta}\,dq = \frac{\Delta^2}{12} = \frac{m_p^2}{3L^2}$$

SQNR: with signal power $P$,

$$\text{SQNR} = \frac{P}{N_q} = \frac{3L^2P}{m_p^2} = \frac{3P}{m_p^2}\,2^{2n}$$

For a full-scale sinusoid, $P = m_p^2/2$:

$$\text{SQNR} = 1.5\times 2^{2n} \quad\Rightarrow\quad \text{SQNR}_{dB} = 1.76 + 6.02\,n$$

Each extra bit adds about 6 dB. Example: 8-bit PCM gives $1.76 + 48.16 = 49.92$ dB.