# nyquist ISI criterion

For some filter with impulse response \(h(t)\), when applied (via linear convolution) to a signal, is ISI free if

\begin{equation}
h(nT_s) = \begin{cases}
1 & n = 0 \\
0 & n \ne 0 \\
\end{cases}
\end{equation}

In other words, for at every point right after every symbol (given by symbol duration \(T_s\)), the impulse response is 0, except at the first one.

In the frequency domain, this property is written as

\begin{equation}
\frac{1}{T_s} \sum_{k = -\infty}^{\infty} H(f - k f_s) = 1 \quad \forall f
\end{equation}

In other words, at every frequency an integer multiple of the symbol bandwidth, the frequency should be the same.

Satisfied by the raised cosine filter and sinc filter.