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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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