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Discrete-Time Fourier Transform(DTFT)

Module by: Don Johnson

Summary: Computing the Fourier Transform is made simpler by the symmetry of the signal's conjugate and the properties of real-valued signals.

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Now that we have the underpinnings of digital computation, we need to return to signal processing ideas. The most prominent of which is, of course, the Fourier transform. The Fourier transform of a sequence is defined to be

Fourier Transform

Sⅇⅈ2πf=∑n=-∞∞snⅇ-ⅈ2πfn S 2 f n sn 2 f n (1)
Frequency here has no units. As should be expected, this definition is linear, with the transform of a sum of signals equaling the sum of their transforms. Real-valued signals have conjugate-symmetric spectra: Sⅇ-ⅈ2πf=Sⅇⅈ2πf¯ S 2 f S 2 f .

Exercise 1

A special property of the discrete-time Fourier transform is that it is periodic with period one: Sⅇⅈ2πf+1=Sⅇⅈ2πf S 2 f 1 S 2 f . Derive this property from the definition of the DTFT.

Solution

Sⅇⅈ2πf+1=∑n=-∞∞snⅇ-ⅈ2πf+1n=∑n=-∞∞ⅇ-ⅈ2πnsnⅇⅈ2πfn=∑n=-∞∞snⅇⅈ2πfn=Sⅇⅈ2πf S 2 f 1 n s n 2 f 1 n n 2 n s n 2 f n n s n 2 f n S 2 f (2)

Because of this periodicity, we need only plot the spectrum over one period to understand completely the spectrum's structure; typically, we plot the spectrum over the frequency range -1212 12 12 . When the signal is real-valued, we can further simplify our plotting chores by showing the spectrum only over 012 0 12 ; the spectrum at negative frequencies can be derived from positive-frequency spectral values.

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