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Fourier Transform Properties

Module by: Don Johnson. E-mail the author

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Summary: A table of commonly seen transforms, for reference.

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Table 1: Short Table of Fourier Transform Pairs
s(t) S(f)
ⅇ-atut a t u t 1ⅈ2πf+a 1 2 f a
ⅇ-a|t| a t 2a4π2f2+a2 2 a 4 2 f 2 a 2
pt=1if|t|<Δ20if|t|>Δ2 p t 1 t Δ 2 0 t Δ 2 sinπfΔπf f Δ f
sin2πWtπt 2 W t t Sf=1if|f|<W0if|f|>W S f 1 f W 0 f W
Table 2: Fourier Transform Properties
Time-Domain Frequency Domain
Linearity a 1 s 1 t+ a 2 s 2 t a 1 s 1 t a 2 s 2 t a 1 S 1 f+ a 2 S 2 f a 1 S 1 f a 2 S 2 f
Conjugate Symmetry st∈ℝ s t Sf=S-f¯ S f S f
Even Symmetry st=s-t s t s t Sf=S-f S f S f
Odd Symmetry st=-s-t s t s t Sf=-S-f S f S f
Scale Change sat s a t 1|a|Sfa 1 a S f a
Time Delay st−τ s t τ ⅇ-ⅈ2πfτSf 2 f τ S f
Complex Modulation ⅇⅈ2π f 0 tst 2 f 0 t s t Sf− f 0 S f f 0
Amplitude Modulation by Cosine stcos2π f 0 t s t 2 f 0 t Sf− f 0 +Sf+ f 0 2 S f f 0 S f f 0 2
Amplitude Modulation by Sine stsin2π f 0 t s t 2 f 0 t Sf− f 0 −Sf+ f 0 2ⅈ S f f 0 S f f 0 2
Differentiation ddtst t s t ⅈ2πfSf 2 f S f
Integration ∫-∞tsαdα α t s α 1ⅈ2πfSf 1 2 f S f if S0=0 S 0 0
Multiplication by tt tst t s t 1-ⅈ2πddfSf 1 2 f S f
Area ∫-∞∞stdt t s t S0 S 0
Value at Origin s0 s 0 ∫-∞∞Sfdf f S f
Parseval's Theorem ∫-∞∞|st|2dt t s t 2 ∫-∞∞|Sf|2df f S f 2

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