Companding; delta modulator numerical (5 kHz, 200 kHz)

Companding

Companding = compressing + expanding. At the transmitter the signal is passed through a compressor that amplifies weak signals more than strong ones; it is then uniformly quantized; at the receiver an expander applies the inverse characteristic. The overall effect is non-uniform quantization: small steps for weak signals and large steps for strong signals, so SQNR stays nearly constant over a wide dynamic range. Standard laws: µ-law ($\mu = 255$, North America/Japan) and A-law ($A = 87.6$, Europe/India/Nepal).

Given

$f_m = 5$ kHz, $f_s = 200$ kHz, $A_{max} = 1$, $T_s = 1/f_s$.

a) Minimum step size to avoid slope overload

The staircase can rise at most $\Delta/T_s = \Delta f_s$. For $m(t) = A\sin 2\pi f_m t$ the maximum slope is $2\pi f_m A$. No slope overload requires

$$\begin{aligned} \Delta f_s &\ge 2\pi f_m A_{max} \\ \Delta_{min} &= \frac{2\pi f_m A_{max}}{f_s} = \frac{2\pi \times 5000 \times 1}{200\,000} = 0.157\ \text{V} \end{aligned}$$

b) SQNR for a sinusoidal signal

• Signal power at the limit of slope overload: $S = A^2/2$ with $A = \Delta f_s/(2\pi f_m)$, so $S = \dfrac{\Delta^2 f_s^2}{8\pi^2 f_m^2}$.
• Granular noise is uniform in $(-\Delta, \Delta)$: power $\Delta^2/3$, spread evenly over $0$ to $f_s$; the output LPF (bandwidth $f_M = f_m$) passes the fraction $f_M/f_s$: $N_q = \dfrac{\Delta^2}{3}\dfrac{f_M}{f_s}$.

$$\begin{aligned} \text{SQNR} &= \frac{3}{8\pi^2}\,\frac{f_s^3}{f_m^2 f_M} = \frac{3}{8\pi^2}\left(\frac{f_s}{f_m}\right)^3 \\ &= 0.0380 \times \left(\frac{200}{5}\right)^3 = 0.0380 \times 64\,000 = 2431.7 \\ \text{SQNR}_{dB} &= 10\log_{10}2431.7 = 33.86\ \text{dB} \end{aligned}$$

c) Minimum transmission bandwidth

DM sends one bit per sample, so $R_b = f_s = 200$ kbps. The minimum (Nyquist) bandwidth for binary data is $R_b/2$:

$$B_{T,min} = \frac{R_b}{2} = \frac{200}{2} = 100\ \text{kHz}$$

Answer: $\Delta_{min} = 0.157$ V; SQNR = 2431.7 (33.86 dB); $B_T = 100$ kHz (if rectangular pulses with $B_T = R_b$ are assumed, 200 kHz).