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Applied to functions of continuous arguments, Fourier-related transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform Fourier transform, with special cases: Fourier series When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients. Sine and cosine transforms: When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform. Hartley transform Short-time Fourier transform (or short-term Fourier transform) (STFT) Chirplet transform Fractional Fourier transform (FRFT) Hankel transform: related to the Fourier Transform of radial functions. |
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