# Can two nonzero signals $x[n]$ and $y[n]$ give a zero convolution

Suppose $x[n]$ and $y[n]$ are two nonzero signals(i.e., $x[n] \neq 0$ for at least one value of n and similarly for $y[n]$).Can the convolution between $x[n]$ and $y[n]$ result in an identically zero signal? In other words, is it possible that $\displaystyle\sum_{k = -\infty}^{k = +\infty}x[k]y[n-k] = 0$ for all n.

Yes, for example let

$$x[k]=1$$

for all $k$ and

$$y[k] = \begin{cases}1 & k=0\\-1 & k=1\\0 & otherwise \end{cases}$$

It is easy to see that in case of a convolution, the result will be zero for all values of $n$.

• It is worth noting that this result only holds if the signal $x[k]$ has infinite length. In practice you will see non-zero convolution tails. Oct 24, 2013 at 22:40