For a periodic input, it's countably infinite (Fourier series [1]). For aperiodic, its uncountably infinite ((continuous) Fourier transform). I don't think thi

For a periodic input, it's countably infinite (Fourier series [1]). For aperiodic, its uncountably infinite ((continuous) Fourier transform). I don't think this implies that there are a countably infinite number of smooth, periodic curves though, because the each one of the parameters is a real. I'm not a mathematician though, so I could be abusing these concepts... There are two more major Fourier transforms: the discrete Fourier transform, which maps a discrete aperiodic function to a continuous periodic (i.e. band-limited) frequency domain representation [3], and the discrete-time Fourier transform [4], which maps a discrete period (i.e. time-limited) function to a discrete period (i.e. band-limited) frequency representation. Notably, in the applications mentioned in the article, what's really being described is the short-time discrete-time Fourier transform (STFT [5]). In this most useful variation, you chop the input signal of arbitrary duration into fixed-sized (and possibly overlapping) chunks, (usually) apply a windowing function to temper down the edges of the chunks, and then apply DTFT to each chunk. The result is a sort of spectrogram -- a time-frequency representation of the input. In image processing this is done in two dimensions. The famous FFT [6] is a O(n * log(n)) algorithm for calculating DTFTs. [1] http://en.wikipedia.org/wiki/Fourier_series http://en.wikipedia.org/wiki/Fourier_series [2] http://en.wikipedia.org/wiki/Fourier_transform http://en.wikipedia.org/wiki/Fourier_transform [3] http://en.wikipedia.org/wiki/Discrete_Fourier_transform http://en.wikipedia.org/wiki/Discrete_Fourier_transform [4] http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform [5] http://en.wikipedia.org/wiki/STFT http://en.wikipedia.org/wiki/STFT [6] http://en.wikipedia.org/wiki/FFT http://en.wikipedia.org/wiki/FFT