### Shift Theorem for the DTFT

We define the shift operator for sampled signals by

 (3.18)

where is any integer ( ). Thus, is a right-shift or delay by samples.

The shift theorem states3.5

 (3.19)

or, in operator notation,

 (3.20)

Proof:

Note that is a linear phase term, so called because it is a linear function of frequency with slope equal to :

 (3.21)

The shift theorem gives us that multiplying a spectrum by a linear phase term corresponds to a delay in the time domain by samples. If , it is called a time advance by samples.

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Symmetry of the DTFT for Real Signals