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Section 7.2 showed how time-shifting a signal changes the
phases of its sinusoidal components, and section 8.4.3
showed how multiplying a signal by a complex sinusoid shifts its component
frequencies. These two effects each correspond to an identity
involving the Fourier transform.
First we consider a time shift. If
, as usual, is a complex-valued
signal that repeats every
samples, let
be
delayed
samples:
which also repeats every
samples since
does. We can reduce the Fourier
transform of
this way:
We therefore get the TIME SHIFT FORMULA FOR FOURIER TRANSFORMS:
So the Fourier transform of
is a phase term times the Fourier transform
of
. The phase is changed by
, a multiple of the frequency
.
Now suppose instead that we change our starting signal
by multiplying
it by a complex exponential
with angular frequency
:
The Fourier transform is:
We therefore get the PHASE SHIFT FORMULA FOR FOURIER TRANSFORMS:
Next: Fourier transform of a
Up: Properties of Fourier transforms
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Miller Puckette
2005-04-01