Nyquist ISI criterion

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In any kind of modulation that uses several different symbols to carry the information (such as ASK, QAM, PSK, etc.), those symbols are sent modulating the amplitude of a certain signal called carrier. These carriers can not be ideal impulses, so they will last a non-null time, that can be quite long compared to the speed of the transmission.

Because of this reason, when receiving a single carrier, its amplitude (thus the information it has) can be influenced by other non-null carriers carrying other symbols. This is called intersymbol interference (ISI); if the carrier will be null in the right instants, then the ISI can be completely removed, even if the signal will not be limited in time. From the point of view of the frequency domain, it is possible to show that any signal that becomes a constant in the frequency domain after being sampled can remove completely the ISI. This is called Nyquist ISI criterion and will be proved in the following part of the article.

[edit] Proof

Let the function h(t) be the carrier, v[n] the symbols to be sent, separated by T_s seconds; the received signal z(t) will be in the form:

$z(t) = n (t) + \sum_{n = -\infty}^{\infty} v[n] \cdot h(t - n T_s)$

where n(t) is the amount of noise. After sampling each T_s seconds it will be:

$z[k] = n [k] + \sum_{n = -\infty}^{\infty} v[n] \cdot h[k - n]$

that can also be expressed as:

$z[k] = n [k] + v[k] \cdot h [0] + \sum_{n \neq k} v[n] \cdot h[k - n]$

from this it is clear that, if h[n] is so that:

$h[n] = \begin{cases} 1; & n = 0 \\ 0; & n \neq 0 \end{cases}$

all the other symbols will disappear after sampling, thus removing any ISI.

Expressing this property in the analog domain, we need a h(t) that:

$h(n T_s) = \begin{cases} 1; & n = 0 \\ 0; & n \neq 0 \end{cases}$

Let us observe that, if we multiply such a h(t) by a sum of dirac delta function (impulses) $δ(t)$ separated by T_s seconds, then the result will be simply one impulse in the origin:

$h(t) \cdot \sum_{k = -\infty}^{+\infty} \delta (t - k T_s) = \delta (t)$