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.