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Discrete-Time Fourier Transform Properties

Module by: Don Johnson. E-mail the author

Summary: Gives various Fourier transform properties

Note: You are viewing an old version of this document. The latest version is available here.

Figure 1: Discrete-time Fourier transform properties and relations.
Discrete-Time Fourier Transform Properties
Sequence Domain Frequency Domain
Linearity a1⁢s1⁢n+a2⁢s2⁢n a1 s1 n a2 s2 n a1⁢S1⁢ei⁢2⁢π⁢f+a2⁢S2⁢ei⁢2⁢π⁢f a1 S1 2f a2 S2 2f
Conjugate Symmetry s⁢n sn real S⁢ei⁢2⁢π⁢f=S⁢e−(i⁢2⁢π⁢f)¯ S 2f S 2f
Even Symmetry s⁢n=s⁢−n sn sn S⁢ei⁢2⁢π⁢f=S⁢e−(i⁢2⁢π⁢f) S 2f S 2f
Odd Symmetry s⁢n=−s⁢−n sn s n S⁢ei⁢2⁢π⁢f=−S⁢e−(i⁢2⁢π⁢f) S 2f S 2f
Time Delay s⁢n−n0 s n n0 e−(i⁢2⁢π⁢f⁢n0)⁢S⁢ei⁢2⁢π⁢f 2 f n0 S 2f
Complex Modulation ei⁢2⁢π⁢f0⁢n⁢s⁢n 2 f0n sn S⁢ei⁢2⁢π⁢(f−f0) S 2 f f0
Amplitude Modulation s⁢n⁢cos2⁢π⁢f0⁢n sn 2 f0n S⁢ei⁢2⁢π⁢(f−f0)+S⁢ei⁢2⁢π⁢(f+f0)2 S 2 f f0 S 2 f f0 2
s⁢n⁢sin2⁢π⁢f0⁢n sn 2 f0n S⁢ei⁢2⁢π⁢(f−f0)−S⁢ei⁢2⁢π⁢(f+f0)2⁢i S 2 f f0 S 2 f f0 2
Multiplication by n n⁢s⁢n n sn 1−(2⁢i⁢π)⁢dS⁢ei⁢2⁢π⁢fdf 1 2 f S 2f
Sum ∑n=−∞∞s⁢n n sn S⁢ei⁢2⁢π⁢0 S 20
Value at Origin s⁢0 s0 ∫−1212S⁢ei⁢2⁢π⁢fdf f 12 12 S 2f
Parseval's Theorem ∑n=−∞∞|s⁢n|2 n sn 2 ∫−1212|S⁢ei⁢2⁢π⁢f|2df f 12 12 S 2f 2

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