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.

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