# Echo

An **echo effect** can be obtained for audio signals by realizing the following
difference equation,

*y*(*n*) = *x*(*n*) + *a y*(*n*-*N*),

where the (discrete-time) audio input is *x*, the output is *y*,
and 0 <= *a* < 1 is a real constant and *N* is an integer constant
in the set *Naturals*. Together, *a* and *N* determine how long
the echo lasts and what it sounds like. Here are some examples, where *x*
is an audio signal with 8000 samples per second:

The impulse response *h* can be obtained by simply letting the input be
an impulse, *x* = δ,
and finding the output *y* = *h*. The result is, for all *n*
in the integers,

*h*(*n*) = ∑_{(k=0
to infinity)} *a ^{k}* δ(

*n*-

*kN*).

The impulse response is plotted below: