Nyquist ISI Criterion - Derivation

Derivation

To derive the criterion, we first express the received signal in terms of the transmitted symbol and the channel response. Let the function h(t) be the channel impulse response, x the symbols to be sent, with a symbol period of T_s; the received signal y(t) will be in the form (where noise has been ignored for simplicity):

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Sampling this signal at intervals of T_s, we can express y(t) as a discrete-time equation:

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If we write the h term of the sum separately, we can express this as:

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and from this we can conclude that if a response h satisfies

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only one transmitted symbol has an effect on the received y at sampling instants, thus removing any ISI. This is the time-domain condition for an ISI-free channel. Now we find a frequency-domain equivalent for it. We start by expressing this condition in continuous time:

for all integer . We multiply such a h(t) by a sum of Dirac delta function (impulses) separated by intervals T_s This is equivalent of sampling the response as above but using a continuous time expression. The right side of the condition can then be expressed as one impulse in the origin:

Fourier transforming both members of this relationship we obtain:

and

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This is the Nyquist ISI criterion and, if a channel response satisfies it, then there is no ISI between the different samples.