User:Porceberkeley
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[edit] Table of important Fourier transforms
The following table records some important Fourier transforms. G and H denote Fourier transforms of g(t) and h(t), respectively. g and h may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.
[edit] Functional relationships
Signal | Fourier transform unitary, angular frequency |
Fourier transform unitary, ordinary frequency |
Remarks | |
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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 |
[edit] Square-integrable functions
Signal | Fourier transform unitary, angular frequency |
Fourier transform unitary, ordinary frequency |
Remarks | |
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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 exp( − αt2) is its own Fourier transform. For this to be integrable we must have Re(α) > 0. | |||
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. | ||||
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Un (t) is the Chebyshev polynomial of the second kind |
[edit] Distributions
Signal | Fourier transform unitary, angular frequency |
Fourier transform unitary, ordinary frequency |
Remarks | |
---|---|---|---|---|
15 | δ(ω) 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: cos(at) = (eiat + e − iat) / 2. | |||
19 | Also from 1 and 17. | |||
20 | Here, n is a natural number. δn(ω) is the n-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 sgn(ω) 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 u(t) is the Heaviside unit step function; this follows from rules 1 and 21. | |||
u(t) is the Heaviside unit step function and a > 0. | ||||
25 | The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time. |