Fourier transforms are mathematical tools used to convert a function from the time domain into the frequency domain. They are widely used in engineering, signal processing, and physics to analyse waveforms, vibrations, and system behaviour.
This page provides a practical reference to Fourier transform formulas, theorems, and commonly used functions used in engineering applications.
The Fourier transform of a function f(x) is given by:
A typical fourier transform pair based on an x value and frequency ( p ) is as shown below .
Not all functions have fourier transforms. The necessary conditions required for a function to be transformable are called the Dirichlet conditions. These are listed as follows
1) The function f(x) and F(p) are square-integrable. That is
is finite which implies f(x) -> 0 as x -> 
2) f( x ) and F ( p ) are single valued. The function as shown below with f(x) having three values at on value of x cannont be transformed
3) f(x) and F ( p ) are piece-wise continuous . The function can be broken down into separate pieces, with isolated discontinuities at the junctions. Between the dicontinuities the function must be continuous.
4) The functions f( x ) and F ( p ) have upper and lower bounds. This requirement has not been proved to be totally necessary but provides sufficient requirement. A function satisfying this requirement is transformable. There are however transformable functions which do not satisfy this requirement
Assuming F1( p ) is the Fourier Transform of f1( x ) and vice versa and F2( p ) is the Fourier Transform of f2( x ) and vice versa . That is..
The addition theorem.
The shift theorem.
x and p scaling. (Time and frequency scaling -For Fourier transforms related to time ( t ) and frequency ( f )
If the Fourier transform of f( x ) is F(p) , then the Fourier transform of
f( kx ) where k is a real constant greater than zero, is obtained by
substituting x' = xk in the Fourier integral equation.
For negative values of k the RHS term changes sign because the limits of integration are interchanged. Therefore, time scaling results in the Fourier transform pair.
p / frequency scaling is a very similar process to x / time( t ) scaling.
1)
The sinc function s defined as sinc( x ) = sin ( x ) / x is of use throughout the application of Fourier transforms.
The plot of the fourier transform is as follows.
Note: Unit Top Hat (Rectangular Function ) Π is 1 unit high and a = 1. This has a Fourier Transform
2)
This function has the value unity at x = 0 and is zero whenever
x = nπ.
3)
G(x) = e -x2 / a2
Note:
Note: The e exponent -( j.2. π p.x + x2 /a 2 ) can be replaced ( by completing the square with ) - ( x / a + j π.p.a) 2 - π 2 .p 2.a2
Making ( x/a + j π p a) = z so that dx = a dz results in
4)
This is accepted as the positive part of the function e-xa The fourier transform is complex.
Quick proof
A variation of the exponential decay function with a ceofficient A.
5)
δ(x) = 0 unless x = 0
at x= 0 δ(0) = 
This function disobeys the Dirichlet condition 4 as it is not bounded at x = 0 .
This is crudely defined as the top hat function with the width aproaching zero but the area remaining constant at unity.
The fourier transform is sinc( pπa ) and as a-> 0 sinc( pπa )
stretches and in the limit is a straight line at unit height about the x axis.
5)
The Heaviside function is a unit step at x = 0 and is shown below
Differentiating the Heaviside function results in the Dirac /Delta function
The Fourier Transform of the Heaviside Function is given by
6)
If two δ-functions are symmetrically positioned on either side of the origin the fourier transform is a cosine wave.
A.δ( x - a ) + A.δ( x + a )
A.e2πjpa + A.e-2πjpa
= 2A. cos (2 πpa)
Similarly
A.j.δ( x - a ) - A.jδ( x + a )
A.j.e2πjpa - A.j.e-2πjpa =
2A sin (2 πpa)
7)
Mathematically, a convolution is defined as the integral over all space of one function at u times another function at x-u . The convolution is a function of a variable x, as shown in the following equations. The * is used to indicate the convolution operation.
If two functions f1( x ) and f2( x ) have relevant Fourier transforms F1( p ) and F2( p ) the the convolution of f1( x ) and f2( x ) has a resultant fourier tranform which is the product of F1(p) and F2(p)
8)
This function is an infinite set of equally spaced δ-functions that is
The Fourier transform of a Dirac comb III a ( x ) is another Dirac comb ( 1/a ) III 1 / a( p )
9)
Assuming F( p )) is the Fourier of f(x) then
Quick proof
10)
The Fourier Transform of a unit Triangle FunctionΛ (1 unit high and 2 units wide) is easily obtained as the convolution of two unit Top Hat (rectangle) Functions Π each 1 unit wide and one unit high which results from the product of the Transforms of the functions...
11)
Considering a Triangle Function above. This is single entity. To produce a periodic expansion it is necessary to perform a convolution operation with a Dirac comb as follows
In accordance with the convolution theorem the Fourier Transform resulting from the convolution of the two functions f(x) * g(x) is the product of the respective Fourier Transforms i.e F ( p ) G ( p )
Y ( p )= F( p ). G ( p)
Note: If the continuous function is continuous at p = n / T1 the product of a continuous function and an impulse function has the property that:
Therefore
The equation for the Fourier Series expansion for a periodic function fs(x) of Period T has been developed fourier Transforms Intro..
The functions fs(x) can be replaced by f(x) setting T = T1 resulting in.
It is clear that the coefficients derived by use of the Fourier Integral and those by the conventional Fourier Series are the same when the function is periodic.
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