Mumbai University > Computer Engineering > Sem 3 > Electronic Circuits and Communication Fundamentals

Marks: 10 Marks

Year: May 2014

Sampling Theorem

1. Sampling theorem was introduced to the communication theory in 1949 by Shannon. Therefor this theorem is also called as “Shannon’s sampling theorem”
2. The statement of sampling theorem in time domain for the bandlimited signals of finite energy as

Statement

• If a finite energy signal x(t) contains no frequencies higher than ‘W’ Hz (i.e. it is a band limited signal) then it is completely determined by specifying its values at te instants of time which are spaced (1/2W) seconds apart.
• If a finite energy signal x(t) contains no frequencies higher than ‘W’ Hz then it may be completed recovered from its samples which are spaced (1/2W) seconds apart.
• Combined Statement: A continuous time signal x(t) can be completely represented in its sampled form and recovered back from the sampled form if the sampling frequency fs ≥ 2W where ‘W’ is the maximum frequency of the continuous time signal x(t).

Let us assume that the highest frequency component in the signal is fmax.

We know that, frequency spectra of realistic signals are not zero above some arbitrary frequency. A real signal will have a spectrum that gets smaller and smaller at higher and higher frequencies, but which only approach zero asymptotically.

To reproduce a signal with a highest frequency component, fmax, the sampling frequency (the frequency at which samples are taken) must be at least twice the highest frequency component.

In other words, the signal cannot be reproduced accurately unless the sampling frequency is at least 2fmax - a frequency that is referred to as the Nyquist frequency for the signal. If the sampling frequency is lower than the Nyquist frequency, that is referred to as under sampling.