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A Delta modulator is used to encode speech signal band-limited to $3 \mathrm{KHz}$ with sampling frequency $100 \mathrm{KHz}$. For $\pm 1$ volt peak signal voltage, find,..

(a) Minimum step size to avoid slope overloading.

(b) Signal-to-quantization noise ratio if speech is assumed to have nonuniform probability density function (PDF).

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Solution:

For the DM system, if input signal $f(t)=b \cos \omega_m t$.

(a) Minimum step size to avoid slope overloading.

$\therefore\left|\frac{d f(t)}{d t}\right|_{\max }=b 2 \pi f_m$ if step size is used in DM system=a

$ \begin{aligned}\\ &\therefore f_s \geq \frac{2 \pi f_m}{a / b} \\\\ &\therefore a \geq \frac{2 \pi f_m b}{f_s} \\\\ &a \geq \frac{2 \pi * 3 * 10^3 * 1}{100 * 10^3}=0.188 \mathrm{~V} \text { minimum step size. }\\ \end{aligned} $

(b) Signal-to-quantization noise ratio if speech is assumed to have a nonuniform probability density function (PDF).

$ \begin{aligned}\\ \frac{S}{N} &=\frac{3}{8 \pi^2}\left(\frac{f_s}{f_m}\right)^3 \\\\ &=\frac{3}{8 \pi^2}\left(\frac{100}{3}\right)^3\\ \end{aligned}\\ $

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