# Reconstruction of a Sequence of Impulses

Cuthbert Nyack

A function f(t) consisting of a periodic set of impulses(delta functions) each of magnitude T and
separated by T can be written as:-
Its Fourier series is:-

The reconstruction of the impulses from the series is shown by the applet below:-

f(t) is shown by the magenta line. The reconstructed function is shown in red. Horizontal
axis is from -1.25T to 1.25T. Since a_{n} = 2 for all n then the series does not
converge in the normal way. At t = nT the sum increases linearly with the number of
terms summed so it is not possible to say what value f(t) converges to at these points.
However the width also decreases linearly with the number of terms summed so the area under
each is ~T. The sum therefore converges to a set of delta functions as usually defined. In
the above plot, the sum is "normalised" to avoid the result from going off scale.

This is relevant for the Fourier representation of sampled signals.

When activated the following gif image show how the applet should appear.

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COPYRIGHT © 1999 Cuthbert A. Nyack.