21. Fourier Transforms#
Fourier transforms are linear transformations of functions usually relating a physical domain of time and/or space, with a domain of frequency and/or wave-vectors.
Contents.
Fourier series. Fourier series is defined for finite-domain or periodic, time-continuous functions, or - more generally - continuous functions in the physical domain.
Fourier transform. Fourier transform is defined for infinite-domain non-periodic, time-continuous functions, or - more generally - continuous functions in the physical domain.
Relations between Fourier transforms and sampling. Fourier series, Fourier transform, discrete time Fourier transform and discrete Fourier transforms are presented, and their relations discussed. Fundamental results about evenly-spaced sampling seamlessly follows, as Shannon-Nyquist theorem, Theorem 21.1, shows.
Different Fourier transforms exist, depending if the original function is:
time discrete/time continuous
periodic/non-periodic
Transform |
Time domain |
Frequency domain |
|---|---|---|
Fourier transform |
Continuous function in infinite domain |
Continuous function in infinite domain |
Fourier series |
Continuous function in finite domain (or periodic function in infinite domain) |
Discrete function in compact range |
Discrete-time Fourier transform |
Discrete function (or continuous function sampled with Dirac’s comb) in infinite domain |
Continous periodic spectrum in infinite domain |
Discrete Fourier transform |
Discrete (or continuous, sampled with Dirac’s comb) function in finite domain (or periodic in infinite domain) |
Discrete periodic spectrum in infinite domain |
Fourier transforms can be useful in:
highlighting the frequency content of functions
solving problems: sometimes, it can be easier to transform a problem in frequency domain, solve it in frequency domain, and transform the solution back to the physical domain