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.

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

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