Topic 30: Intuitive Derivation of the Fourier Transform from Complex Fourier Series (SNADA, Chapter 6)
From James Hopgood
This Topic introduces the Fourier transform as the limiting operation
of taking a periodic signal, and sending the period to infinity, so as
to represent a non-periodic signal. The Topic considers this by looking
at a particular example, namely the periodic pulse wave, where the
spectrum has an elegant shape that is easy to analyse. The properties of
this spectrum are then considered as the period is increased. It is
noticed that the shape of the spectrum doesn't change, but the gaps
between the frequency components are filled in with more frequencies. In
the limit, the discrete-spectrum becomes continuous. This phenomena is
then investigated further using some MATLAB code that can simulate this
process for different periodic waveforms. The limiting spectrum is the
Fourier transform of the non-periodic signal.
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