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# Fourier Transform Properties

Module by: Don Johnson. E-mail the author

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

Table 1: Short Table of Fourier Transform Pairs
s(t) S(f)
e(at)ut 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={1  if  |t|<Δ20  if  |t|>Δ2 p t 1 t Δ 2 0 t Δ 2 sinπfΔπf f Δ f
sin2πWtπt 2 W t t Sf={1  if  |f|<W0  if  |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 stR s t Sf=Sf¯ S f S f
Even Symmetry st=st s t s t Sf=Sf S f S f
Odd Symmetry st=st s t s t Sf=Sf S f S f
Scale Change sat s a t 1|a|Sfa 1 a S f a
Time Delay stτ s t τ e(i2πfτ)Sf 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 dd t st t s t i2πfSf 2 f S f
Integration tsαd α α t s α 1i2πfSf 1 2 f S f if S0=0 S 0 0
Multiplication by tt tst t s t 1(i2π)dSfd f 1 2 f S f
Area std t t s t S0 S 0
Value at Origin s0 s 0 Sfd f f S f
Parseval's Theorem |st|2d t t s t 2 |Sf|2d f f S f 2

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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