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Arlon Online Engineering Library - TechRef! Fourier Series Every periodic function f(t) which has the period T = 2 / 0 and has certain continuity conditions can be represented by a series plus a constant       The above holds if f(t) has a continuous derivative f'(t) for all t. It should be noted that the various sinusoids present in the series are orthogonal on the interval 0 to T and as a result the coefficients are given by       The constants an and bn are the Fourier coefficients of f(t) for the interval 0 to T and the corresponding series is called the Fourier series of f(t) over the same interval. The integrals have the same value when evaluated over any interval of length T. If a Fourier series representing a periodic function is truncated after term n = N the mean-square value F2N of the truncated series is given by the Parseval relation. This relation says that the mean-square value is the sum of the mean-square values of the Fourier components, or       and the RMS value is then defined to be the square root of this quantity or FN. Fourier Transform The Fourier transform pair, one form of which is       can be used to characterize a broad class of signal models in terms of their frequency or spectral content. Some useful transform pairs are:       Some mathematical liberties are required to obtain the second and fourth form. Other Fourier transformations are derivable from the Laplace transform by replacing s with j provided

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