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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 a1s1n+a2s2n a1 s1 n a2 s2 n a1S1ei2πf+a2S2ei2πf a1 S1 2f a2 S2 2f
Conjugate Symmetry sn sn real Sei2πf=Se(i2πf)¯ S 2f S 2f
Even Symmetry sn=sn sn sn Sei2πf=Se(i2πf) S 2f S 2f
Odd Symmetry sn=sn sn s n Sei2πf=Se(i2πf) S 2f S 2f
Time Delay snn0 s n n0 e(i2πfn0)Sei2πf 2 f n0 S 2f
Complex Modulation ei2πf0nsn 2 f0n sn Sei2π(ff0) S 2 f f0
Amplitude Modulation sncos2πf0n sn 2 f0n Sei2π(ff0)+Sei2π(f+f0)2 S 2 f f0 S 2 f f0 2
snsin2πf0n sn 2 f0n Sei2π(ff0)Sei2π(f+f0)2i S 2 f f0 S 2 f f0 2
Multiplication by n nsn n sn 1(2iπ)dSei2πfdf 1 2 f S 2f
Sum n=sn n sn Sei2π0 S 20
Value at Origin s0 s0 1212Sei2πfdf f 12 12 S 2f
Parseval's Theorem n=|sn|2 n sn 2 1212|Sei2πf|2df f 12 12 S 2f 2

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