# Relationship to Convolution

Suppose an LTI system has impulse response*h*(

*n*) and frequency response

*H*(ω ). We have seen that if the input to this system is

*e*, then the output is

^{jω n}*H*(ω )

*e*. Suppose the input is instead a signal

^{jω n}*x*with DTFT

*X*. Using the inverse DTFT relation, we know that for all

*n*,

View this as a summation of exponentials, each with weight *X*(ω
). An integral, after all, is summation over a continuum. Each term in
the summation is *X*(ω )*e ^{jω
n}*. If this term were an input by itself, then the output would
be

*H*(ω )

*X*(ω )

*e*. Thus, by linearity, if the input is

^{jω n}*x*, the output should be

Comparing to the inverse DTFT relation for *y*(*n*), we see
that

*Y*(ω ) =

*H*(ω )

*X*(ω ).

This is the frequency-domain version of convolution

*y*(

*n*) = (

*h**

*x*)(

*n*).

*H*

_{1}(ω ) and

*H*

_{2}(ω ) are connected in cascade, that the DTFT of the output is given by

*Y*(ω ) =

*H*

_{1}(ω )

*H*

_{2}(ω )

*X*(ω ), where

*X*(ω ) is the DTFT of the input.