### 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* states^{3.5}

(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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Convolution Theorem for the DTFT

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Symmetry of the DTFT for Real Signals