The Art of DSP by Bore Chris

Author:Bore, Chris
Language: eng
Format: epub
Tags: DSP
Publisher: BORES Signal Processing
Published: 2014-09-17T16:00:00+00:00

If the signal can be made by adding frequency components, then how can we calculate what each of those frequency components is? The Fourier Transform is just a way to calculate the frequency components H(w) given the time domain signal h(t). The derivation and proof of the Fourier Transform is quoted in many sources, so here we can just state it:

where the sum is now over the set of times t[n].

Note that I have again made an implicit assumption: that the frequency spectrum H(w) can be made by adding signal values at a finite set of times t[n]. This restricts the frequency spectra you can make.

What if we are given the frequency components H(w) and want to know the signal h(t)? Well we already know that because it is the equation we started from:

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