User:Porceberkeley

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Table of important Fourier transforms[edit]

The following table records some important Fourier transforms. and denote Fourier transforms of and , respectively. and may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

Functional relationships[edit]

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks






1 Linearity
2 Shift in time domain
3 Shift in frequency domain, dual of 2
4 If is large, then is concentrated around 0 and spreads out and flattens
5 Duality property of the Fourier transform. Results from swapping "dummy" variables of and .
6 Generalized derivative property of the Fourier transform
7 This is the dual to 6
8 denotes the convolution of and — this rule is the convolution theorem
9 This is the dual of 8

Square-integrable functions[edit]

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks






10 The rectangular pulse and the normalized sinc function
11 Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12 tri is the triangular function
13 Dual of rule 12.
14 Shows that the Gaussian function is its own Fourier transform. For this to be integrable we must have .
common in optics
a>0
the transform is the function itself
J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.

 


 

Un (t) is the Chebyshev polynomial of the second kind

Distributions[edit]

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks






15 denotes the Dirac delta distribution. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of a constant function.
16 Dual of rule 15.
17 This follows from and 3 and 15.
18 Follows from rules 1 and 17 using Euler's formula:
19 Also from 1 and 17.
20 Here, is a natural number. is the -th distribution derivative of the Dirac delta. This rule follows from rules 7 and 15. Combining this rule with 1, we can transform all polynomials.
21 Here is the sign function; note that this is consistent with rules 7 and 15.
22 Generalization of rule 21.
23 The dual of rule 21.
24 Here is the Heaviside unit step function; this follows from rules 1 and 21.
is the Heaviside unit step function and .
25 The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.