Relationship to Convolution
Suppose an LTI system has impulse response h(n) and frequency
response H(w ). We have seen that if
the input to this system is ejw n,
then the output is H(w )ejw
n. Suppose the input is instead a signal 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(w
). An integral, after all, is summation over a continuum. Each term in
the summation is X(w )ejw
n. If this term were an input by itself, then the output would
be H(w )X(w
)ejw n. Thus, by linearity,
if the input is x, the output should be
.
Comparing to the inverse DTFT relation for y(n), we see
that
Y(w ) = H(w
)X(w ).
This is the frequency-domain version of convolution
y(n) = (h *
x)(n).
Exercise: Show that if two discrete-time systems with frequency responses
H1(w ) and H2(w
) are connected in cascade, that the DTFT of the output is given by Y(w
) = H1(w )H2(w
)X(w ), where X(w
) is the DTFT of the input.