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Course by: Mark A. Davenport. E-mail the author

# The Discrete-Time Fourier Transform

Module by: Mark A. Davenport. E-mail the author

## The discrete-time Fourier transform

The (non-normalized) DTFT is simply a special case of the zz-transform for the case z=1z=1, i.e., z=ejωz=ejω for some value ω[-π,π]ω[-π,π]

X ( e j ω ) = n = - x [ n ] e - j ω n . X ( e j ω ) = n = - x [ n ] e - j ω n .
(1)

The picture you should have in mind is the complex plane. The zz-transform is defined on the whole plane, and the DTFT is simply the value of the zz-transform on the unit circle, as illustrated below.

This picture should make it clear why the DTFT is defined only for ω[-π,π]ω[-π,π] (or why it is periodic). Using the normalization above, we also have the inverse DTFT formula:

x [ n ] = 1 2 π - π π X ( e j ω ) e j ω n d ω . x [ n ] = 1 2 π - π π X ( e j ω ) e j ω n d ω .
(2)

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