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Fourier Transforms - Theorems and Functions

 Heaviside Function Gaussian Function Exponential Decay Function Cosine Sine Convolutions Derivative function Triangle function Fourier series from Fourier Transforms

 Introduction 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. A typical fourier transform pair based on an x value and frequency ( p ) is as shown below . Note: Other reference sources use signs in the exponents in the inverse and Fourier transforms opposite to the ones I have selected.    As long as the signs in the two related domains are opposite and consistent they are OK. Dirichlet conditons 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 ¦ -> 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 Basic Theorems 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. Useful Functions 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 . This function has the value unity at x = 0 and is zero whenever x = nπ. 3) The Gaussian function ..... The Gaussian function 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) The exponential decay. 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) The Dirac "delta-function" δ(x) . 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" 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 = 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) 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 Quick proof 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: 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 coeffients 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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