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Discrete-Time Fourier Transform(DTFT)

Module by: Don Johnson. E-mail the author

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

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

S⁢ei⁢2⁢π⁢f=∑n=−∞∞s⁢n⁢e−(i⁢2⁢π⁢f⁢n) 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: S⁢e−(i⁢2⁢π⁢f)=S⁢ei⁢2⁢π⁢f¯ 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: S⁢ei⁢2⁢π⁢(f+1)=S⁢ei⁢2⁢π⁢f S 2 f 1 S 2 f . Derive this property from the definition of the DTFT.

Solution

S⁢ei⁢2⁢π⁢(f+1)=∑n=−∞∞s⁢n⁢e−(i⁢2⁢π⁢(f+1)⁢n)=∑n=−∞∞e−(i⁢2⁢π⁢n)⁢s⁢n⁢ei⁢2⁢π⁢f⁢n=∑n=−∞∞s⁢n⁢ei⁢2⁢π⁢f⁢n=S⁢ei⁢2⁢π⁢f 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 −12 12 12 12 . When the signal is real-valued, we can further simplify our plotting chores by showing the spectrum only over 0 12 0 12 ; the spectrum at negative frequencies can be derived from positive-frequency spectral values.

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