14. 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.

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

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 14.1, shows.

Different Fourier transforms exist, depending if the original function is:

• time discrete/time continuous

• periodic/non-periodic