Fourier index
Fourier Transforms  convolutions
Introduction The notes on this page are provided to simply describe convolutions
and their application with respect to Continuous Fourier Transforms and Discrete Fourier Transforms.
The convolution process is completed in four steps as illustrated in the figure below 1 ) Obtain f_{2} (  u ). The process is illustrated by a very simple example. two simple example function f_{1} ( u ) and f_{2} ( u ) are shown below 1 ) Obtain f_{2} (u). 2) Obtain f_{2} ( x  u ). 3 ) Consider all possible cases of x from . Calculating the producti f_{1} ( u ). f_{2} ( x  u ). 4 ) Plot of the integral . This is the integral of the product of the overlapping part of the two functions. Convolution process with f_{1} ( u ) = 1 and f_{2} ( u ) = e ^{u} Convolution Process with impulse (Delta function)Convolution Process of f( x ) with a train of impulses (Delta function) Sampled discrete Signal Convolution Process An illustration is provided below of the convolution of a one four point input signal x(n) with an eight point response h(n) resulting in a 4+81 = 11 point output signal. y(n) The above output signal is effectively the sum of four seperate signal
responses x(0)*h(0) + x(1)*h(n1) + x(2)*h(n2) + x(3)*h(n3).
In mathematical terms, x(n) is convolved with h(n) to produce y(n) .
Each of the four samples in the input signal contributes a scaled and shifted version of the impulse response to the output signal. The four
versions are shown below. 
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Fourier index