# Fourier Transforms convolutions

### 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. Mathematically, a convolution of two function f1( x ) and f2( x ) is defined as the integral over all space of the product of one function at f1( u ) and another function at f1( x - u ).    (The variable x being substituted by u)     The convolution operation is illustrated below.    The * is used to indicate the convolution operation. Basic Convolution Process Note: f1 ( x ) * f2 ( x ) = f2 ( x ) * f1 ( x ) The convolution process is completed in four steps as illustrated in the figure below 1 ) Obtain f2 ( - u ). 2 ) Obtain f2 ( x - u ). 3 ) Consider all possible cases of x from .     Calculate the producti f1 ( u ). f2 ( x - u ) when they both have values 4 ) Determine the integral. The process is illustrated by a very simple example. two simple example function f1 ( u ) and f2 ( u ) are shown below 1 ) Obtain f2 (-u).        2) Obtain f2 ( x - u ). 3 ) Consider all possible cases of x from . Calculating the producti f1 ( u ). f2 ( 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 f1 ( u ) = 1 and f2 ( 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+8-1 = 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(n-1) + x(2)*h(n-2) + x(3)*h(n-3).     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.At x(0) the signal value = 1 is decomposed into an impulse δ(0) and the resulting output signal = 1.h(0). That is the contribution is the response scaled by 1 and unshifted. At x(2) the signal value = 2 is decomposed into an impulse 2.δ(n-2) and the resulting signal output signal = 2.h(n-2). That is the contribution is the response scaled by 2 and shifted 2 to the right