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Conventional Fourier series analysis enables one to extract information from a periodic function of time. In practical situations, however, one often deals with finite time functions, where conventional Fourier series techniques are inadequate. Finite functions result from a singular or finite set disturbances. The transient response of a circuit is an example of such a disturbance. The theory incorporated in finite Fourier filters enables one to deal with the type of time functions mentioned above. This technical note attempts to summarize, in a logical development, the basic theory of finite Fourier analysis, and the synthesis which incorporates the theory in the finite Fourier filters.