Fourier Transform Tables

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ENTS 699R: Lecture 1d support

ENTS 699R Lecture 1d support: Fourier Transform Tables
Alejandra Mercado June, 2013

1 Transform Pairs
The following is a table of basic transform pairs that can be used as building blocks to derive more complicated transform pairs: Time domain function, with dummy variable t 1 2 3 4 5 6 7
F F F

Frequency domain function, with dummy variable f

δ(t) ⇐⇒ 1 1 ⇐⇒ δ(f ) δ(t − t0 ) ⇐⇒ e−j2πf t0 sin (2πf0 t + φ) ⇐⇒
F F j −jφ δ(f 2 [e 1 −jφ δ(f 2 [e

+ f0 ) − ejφ δ(f − f0 )]

cos (2πf0 t + φ) ⇐⇒ + f0 ) + ejφ δ(f − f0 )] 1 |t| ≤ T F t 2 rect ( T ) = ⇐⇒ T sinc(f T ) = T sin(πf T ) πf T 0 o.w. sinc(βt) ⇐⇒
F 1 β f · rect ( β ) = 1 β

·1

|f | ≤ o.w.

β 2

0

Page 1

ENTS 699R: Lecture 1d support

2 Properties
For the table of Fourier Transform properties, assume that we already know that: g(t) ⇐⇒ G(f ) h(t) ⇐⇒ H(f ) and that α, β, T, φ, f0 , t0 are all arbitrary constants. Time domain function, with dummy variable t A B C D E F F G H I
F F F F F

Frequency domain function, with dummy variable f

Property name time/frequency reversal duality time shift frequency shift linearity

g(−t) ⇐⇒ G(−f ) G(t) ⇐⇒ g(−f ) g(t − t0 ) ⇐⇒ g(t)ej2πf0 t
F F F

e−j2πf t0 G(f )

⇐⇒ G(f − f0 )
1 2 j 2

αg(t) + βh(t) ⇐⇒ αG(f ) + βH(f ) g(t) cos(2πf0 t) ⇐⇒ g(t) sin(2πf0 t) ⇐⇒
F F F

(G(f − f0 ) + G(f + f0 )) modulation (G(f + f0 ) − G(f − f0 )) modulation multipl. in time domain convolution in time domain time scaling

g(t) × h(t) ⇐⇒ G(f ) ∗ H(f ) g(t) ∗ h(t) ⇐⇒ G(f ) × H(f ) g(αt) ⇐⇒
F f 1 |α| G( α )

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