Fourier Transform Pairs

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This section gives a list of Fourier Transform pairs. That is, we present several functions and there corresponding Fourier Transforms. The derivation can be found by selecting the image or the text below. For convenience, we use both common definitions of the Fourier Transform, using the (standard for this website) variable f, and the also used "angular frequency" variable angular frequency in fourier transform.


Table of Fourier Transforms

g(t)
fourier transform symbol
fourier transform of time function
fourier transform with omega
box or rectangle function
Square Pulse (Box Function)
sinc function
sinc function with omega
triangle function
Triangle Function
sinc squared
sinc squared
gaussian or normal function
The Gaussian
fourier transform of gaussian
fourier transform of gaussian
constant function
Constant
fourier transform of constant
fourier transform of constant
dirac delta functional
The Dirac-Delta Impulse
fourier transform of delta
fourier transform of delta
cosine function
Cosine
fourier transform of cosine
fourier transform of cosine
sine function
Sine
fourier transform of sine
fourier transform of sine
step function
Step Function
fourier transform of step function
fourier transform of step function
signum function
Signum Function
fourier transform of the signum function
fourier transform of the signum function
left sided exponential
Right-Sided Decaying Exponential
fourier transform of the left sided decaying exponential
fourier transform of the left sided decaying exponential
left sided exponential
Left-Sided Decaying Exponential
fourier transform of the left sided decaying exponential
fourier transform of the left sided decaying exponential
two sided decaying exponential
Two-Sided Decaying Exponential
fourier transform of the two sided decaying exponential
fourier transform of the two sided decaying exponential

g(t)
fourier transform symbol
fourier transform of time function
fourier transform with omega
shah function
Shah Function (Bed of Nails)
fourier transform of the bed of nails function
fourier transform of the bed of nails function
let's do Fourier Transforms
Truncated Cosine
fourier transform of the truncated cosine
fourier transform of the truncated cosine
right sided cosine for Fourier analysis
Right-Sided Cosine
fourier transform of the right sided cosine
fourier transform of the right-sided cosine
right sided sine for Fourier analysis
Right-Sided Sine
fourier transform of the right sided sine
fourier transform of the right-sided sine
quadratically varying sinusoid
Complex Gaussian
fourier transform of the complex gaussian

k > 0

fourier transform of quadratically varying sinusoid

k > 0

quadratically varying cosine
Quadratic Cosine
fourier transform of the quadratic cosine
fourier transform of quadratically varying cosine
quadratically varying cosine
Quadratic Sine
fourier transform of the quadratic sinusoid

k > 0

fourier transform of quadratically varying sinusoid

k > 0

g(t)
fourier transform symbol
fourier transform of time function
fourier transform with omega
fourier transforms of x*f(x)
t multiplied by arbitrary function h(t) with Fourier Transform H(f)
fourier transform of t*f(t)
angular frequency fourier x*f(x)
x^n*f(x) fourier transform
t^n * h(t)
fourier transform of t^n*f(t)
angular frequency fourier x^n*f(x)
bessel function of first kind of order 0
Absolute Value Function
fourier transform of t*f(t)
angular frequency fourier x*f(x)
1/(1+t^2)
Inverted Polynomial
fourier transform of inverted polynomial
angular frequency fourier transform of inverted polynomial
inverse square root of t or 1/sqrt(x)
Inverse Square Root
fourier transform of the inverse square root
angular frequency fourier sqrt(t)^-0.5
bessel function of first kind of order 0
Bessel Function of the First Kind (order 0)
fourier transform of the bessel function
angular frequency fourier transform J0(t)
Fourier Transforms Table

Note that the following two inversion formulas are used for the definition of the inverse Fourier Transform:

inverse fourier transform definitions


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