Nyquist Theorem

A slecture by ECE student Robert Stein

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

The Nyquist Theorem states that it is possible to reproduce a signal from sampled version of that signal given that the sampling frequency is greater than twice the greatest frequency component of the original signal.

# Proof

Let's begin by looking at X(f) and $ X_{s} $(f):

Observe that $ X_{s} $(f) consists of $ (1/T_{s}) $*X(f) repeated every $ 1/T_{s} $.

If we use a low-pass filter with gain $ T_{s} $ and cutoff frequency between $ f_{m} $ and $ 1/T_{s} - f_{m} $ on $ X_{s} $(f), we can obtain the original signal if the repetitions don't overlap.

For this case to be met, $ 1/T_{s} - f_{m} $ must be greater than $ f_{m} $.

In other words,

$ \frac{1}{T_{s}} > 2f_{m} $

Note that satisfying the Nyquist condition is not necessary to perfectly reconstruct a signal from its sampling. However, if the Nyquist condition is satisfied, perfect reconstruction will be possible.

## Questions and comments

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