Nyquist ISI criterion

In communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel (including responses of transmit and receive filters), result in no intersymbol interference or ISI. It provides a method for constructing band-limited functions to overcome the effects of intersymbol interference.

When consecutive symbols are transmitted over a channel by a linear modulation (such as ASK, QAM, etc.), the impulse response (or equivalently the frequency response) of the channel causes a transmitted symbol to be spread in the time domain. This causes intersymbol interference because the previously transmitted symbols affect the currently received symbol, thus reducing tolerance for noise. The Nyquist theorem relates this time-domain condition to an equivalent frequency-domain condition.

The Nyquist criterion is closely related to the Nyquist-Shannon sampling theorem, with only a differing point of view.

If we denote the channel impulse response as ${\displaystyle h(t)}$, then the condition for an ISI-free response can be expressed as:

for all integers ${\displaystyle n}$, where ${\displaystyle T_{s}}$ is the symbol period. The Nyquist theorem says that this is equivalent to:

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