Subject: Priniciples of Communication Engineering

Difficulty : Medium

Marks : 10

Statement:

Continuous time band limited signal x(t) can be completely represented in its sampled form and recovered back from sampled form if sampling frequency $f_s$ is greater than or equal to twice the maximum frequency (w) of continuous time signal x (t)

i.e. $f_s$ ≥ 2w

In order to arrive at sampling theorem, we model the sample output as

$x_s(t)$ = x (t) *p (t)

Where p(t) = $\sum_{n=-\infty}^{\infty}$ $\delta$ ( t – n T ) is periodic impulse train.

Sampled signal is considered to be product of continuous time signal x (t) and impulse train p (t) and hence it is referred to as impulse modulation model for sampling operation.

P(w) = $\frac{2π}{T} \sum_{n=-\infty}^{\infty} \delta( w- \frac{2πn}{T})$ = $\frac{2π}{T} \sum_{n=-\infty}^{\infty} \delta( w-nws)$

Hence $x_{s(w)}$ = $\frac{1}{2π} x (w) * p (w)$

= $\frac{1}{2π}$ $\int_{-\infty}^{\infty} x (\sigma)$ p (w-$\sigma$) d $\sigma$

= $\frac{1}{T}$ $\sum_{n=-\infty}^{\infty} x ( w – n w s )$

The signals x(t), p(t) and $x_s$ (t) are depicted together with their magnitude spectra.

$x_s(t)$ is a sampled version of continuous time signal x(t) consists of impulses spaced T seconds apart. The spectrum $x_s(w)$ of sampled signal is obtained by convolution of spectrum x(w) with impulse train p(w) and hence consists of periodic repetition at intervals ws of x(w). If we pass sampled signal $x_s(t)$ through ideal IPF which passes only those frequencies contained in x(t). Thus to recover x(t) we pass $x_s(t)$ through a filter with frequency response.

$ f(x) = \begin{cases} T, & |w| \lt w_b \\ 0, &otherwise\end{cases}$

= $T \ rect ( \frac{w}{2WB})$

This filter is called ideal reconstruction filter. As sampling frequency is reduced the different components in the spectrum of $x_s (w)$ start coming close together and eventually will overlap signal x(t) can be recovered from its samples only if

$Ws \gt 2 WB$

The minimum permission value of $W_S$ is called Nyquist rate.