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Fourier Series to Fourier Transform
For periodic signals, we can represent them as linear
combinations of harmonically related complex exponentials
To extend this to non-periodic signals, we need to consider
aperiodic signals as periodic signals with infinite period.
As the period becomes infinite, the corresponding frequency
components form a continuum and the Fourier series sum becomes an
integral.
Instead of looking at the coefficients a harmonically related
Fourier series, well now look at the Fourier transform which is a
complex valued function in the frequency domain.
Fourier Transforms are used in
X-ray diffraction
Electron microscopy (and diffraction)
NMR spectroscopy
IR spectroscopy
Fluorescence spectroscopy
Image processing
etc. etc. etc. etc.
Substituting 3.1.2 to 3.1.1
A plot of the square of the modulus of the Fourier transform (
vs ) is called the power spectrum.
It gives the amount the frequency contributes to the
waveform.
Example 3.1.3: Decaying Exponential
Consider the (non-periodic) signal
Then the Fourier transform is:
Example 3.1.4.
Fourier transform of .
However the delta function is,
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