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Discrete-Time Fourier Transform Properties

Module by: Don Johnson

Summary: Gives various Fourier transform properties

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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=s-n sn sn Sⅇⅈ2πf=Sⅇ-ⅈ2πf S 2f S 2f
Odd Symmetry sn=-s-n sn s n Sⅇⅈ2πf=-Sⅇ-ⅈ2πf S 2f S 2f
Time Delay sn−n0 s n n0 ⅇ-ⅈ2πfn0Sⅇⅈ2πf 2 f n0 S 2f
Complex Modulation ⅇⅈ2πf0nsn 2 f0n sn Sⅇⅈ2πf−f0 S 2 f f0
Amplitude Modulation sncos2πf0n sn 2 f0n Sⅇⅈ2πf−f0+Sⅇⅈ2πf+f02 S 2 f f0 S 2 f f0 2
snsin2πf0n sn 2 f0n Sⅇⅈ2πf−f0−Sⅇⅈ2πf+f02ⅈ S 2 f f0 S 2 f f0 2
Multiplication by n nsn n sn 1-2ⅈπddfSⅇⅈ2πf 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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