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Fourier Transforms - convolutions


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

Useful Related Links
  1. The convolution theorem and its applications ...Clear and comprehensive notes
  2. Efficient Calculation fourier transform via convolution .. Detailed Notes...
  3. Wikipedia Convolution .. Clear and detailed notes
  4. The Scientist and Engineer's Guide to Digital Signal Processing Chapter 6..
        A really clear explanation of the subject.

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