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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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