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##

Shifts and phase changes

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 have corresponding identities
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:

(The third line is just the second one with the terms summed in a
different order). We therefore get the Time Shift Formula for Fourier
Transforms:

The Fourier transform of is a phase term times the Fourier transform
of . The phase is changed by , a
linear function 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:

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Miller Puckette
2006-12-30