two "top hat" functions, A and B. If we move the lower function along
one point at a time and take the area under the product of the two functions,
we have the convolution of the two.
This applet is part of a "virtual lab" that was written
to support an Instumentation course. It can be used to demonstrate the convolution of two functions ('top hats'
and 'sinc squared' functions).
Convolution theorem also relates
to the Fourier transform of the two functions. The theorem states:"The
Fourier transform of the product of two functions is the convolution of their
individual Fourier transforms". It follows that "The Fourier
transform of the convolution of two functions is the product of their individual
Fourier transforms". This can be seen quite intuitively if we consider
the two functions A and B. It is well known result that the Fourier transform
of a "top hat" is a sinc function. The product of two top hats together
obviously results in another top hat function. Thus by the reckoning above,
the convolution of two sinc functions must also form a sinc function, this is
far less than obvious!
This applet can be used to show the convolution of two 'top hat'
or 'sinc squared' functions.
The type of function to convolute
can be selected via the choice box in the upper left corner of the applet. In
the case of sinc functions, the period of the upper function can be altered
with the slider. The other controls can be used to run the animation.
This applet gives a clear and quick
example of the convolution theorm. Try altering the peroid of the sinc functions
and observe the function that is formed.
Applet produced and written by Douglas Robertson, from an intital
idea by Dr Iain McKenzie.