Exponential Decay Function
Fourier series from Fourier Transforms
The notes on this page are provided simply to identify basic Fourier transfroms and
some of the theorems and calculation rules applicable to their use.
The detailed exploration of this subject is far beyond the range of this website
and it is recommended that engineers involved in serious work using Fourier Transforms
use quality reference sources.
Not all functions have fourier transforms. The necessary conditions required for a function to be transformable are called the Dirichlet conditons. 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 ->
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 )
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 Top Hat function...Π a
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) The sinc function .....sinc (x) = sin(x) /x .
3) The Gaussian function .....
G(x) = e -x2 / a2
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) The exponential decay.
This is accepted as the positive part of the function e-xa The fourier transform is complex.
A variation of the exponential decay function with a ceofficient A.
5) The Dirac "delta-function" δ(x) .
δ(x) = 0 unless x = 0
This function disobeys the Dirichlet condition 4 as it is not bounded at x = 0 .
5) The Heaviside Function" H(x) .
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) Two symmetrical dirac Functions.
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
A.j.δ( x - a ) - A.jδ( x + a ) A.j.e2πjpa - A.j.e-2πjpa = 2A sin (2 πpa)
7) Convolutions and Convolution Therorems. Ref. Convolutions
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) The Dirac Comb IIIa( x )
This function is an infinite set of equally spaced δ-functions that is
The Fourier transfor of a Dirac comb III a ( x ) is another Dirac comb ( 1/a ) III 1 / a( p )
9) Derivative Therorems.
Assuming F( p )) is the Fourier of f(x) then
10) Triangle Function
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...
10) Fourier Series from Fourier Transforms
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 Tranforms 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:
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 coeffients derived by use of the Fourier Integral and those by the conventional Fourier Series are the same when the function is periodic.