Multiplying Signals
We have seen that convolution in the time domain corresponds to multiplication
in the frequency domain. It turns out that this relationship is symmetric,
in that multiplication in the time domain corresponds to a peculiar form
of convolution in the frequency domain. That is, given two discrete-time
signals x and p with DTFTs X and P, if we multiply
them in the time domain,
y(n) = x(n) p(n)
then in the frequency domain,
Y(w ) = X(w
) Ä P(w
),
where the symbol "Ä " indicates circular
convolution, defined by
.
To verify this, we can substitute into the above integral the definitions
for the DTFTs X(w ) and P(w
) to get
where the last equality follows from the observation that the integral
in the middle expression is zero except when m = k, when
it has value one. Thus, X(w ) Ä
P(w ) is the DTFT of x(n)
p(n), as we claimed.