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Fast Fourier Transform Key Papers in Computer Science Seminar 2005 Dima Batenkov Weizmann Institute of Science � A fast algorithm for computing the Discrete Fourier Transform 1 � (Re)discovered by Cooley & Tukey in 1965 and widely adopted thereafter � Has a long and fascinating history Fast Fourier Transform - Overview »Fast Fourier Transform - J. W. Cooley and J. W. Tukey. An algorithm for the machine Overview calculation of complex Fourier series. Mathematics of Computation, 19:297�301, 1965 p.2/33 1 � (Re)discovered by Cooley & Tukey in 1965 and widely adopted thereafter � Has a long and fascinating history Fast Fourier Transform - Overview »Fast Fourier Transform - J. W. Cooley and J. W. Tukey. An algorithm for the machine Overview calculation of complex Fourier series. Mathematics of Computation, 19:297�301, 1965 � A fast algorithm for computing the Discrete Fourier Transform p.2/33 � Has a long and fascinating history Fast Fourier Transform - Overview »Fast Fourier Transform - J. W. Cooley and J. W. Tukey. An algorithm for the machine Overview calculation of complex Fourier series. Mathematics of Computation, 19:297�301, 1965 � A fast algorithm for computing the Discrete Fourier Transform 1 � (Re)discovered by Cooley & Tukey in 1965 and widely adopted thereafter p.2/33 Fast Fourier Transform - Overview »Fast Fourier Transform - J. W. Cooley and J. W. Tukey. An algorithm for the machine Overview calculation of complex Fourier series. Mathematics of Computation, 19:297�301, 1965 � A fast algorithm for computing the Discrete Fourier Transform 1 � (Re)discovered by Cooley & Tukey in 1965 and widely adopted thereafter � Has a long and fascinating history p.2/33 » Fast Fourier Transform - Overview Fourier Analysis » Fourier Series » Continuous Fourier Transform » Discrete Fourier Transform » Useful properties 6 » Applications Fourier Analysis p.3/33 Fourier Series »Fast Fourier Transform - � Expresses a (real) periodic function x(t) as a sum of Overview trigonometric series (�L < t < L) Fourier Analysis » Fourier Series » Continuous Fourier Transform ¥ » Discrete Fourier Transform 1 pn pn »Useful properties 6 x(t) = a0 + � (an cos t + bn sin t) »Applications 2 L L n=1 � Coef�cients can be computed by Z L 1 pn an = x(t) cos t dt L �L L Z L 1 pn bn = x(t) sin t dt L �L L p.4/33 Fourier Series »Fast Fourier Transform - � Generalized to complex-valued functions as Overview Fourier Analysis ¥ » Fourier Series i pn t » Continuous Fourier Transform x(t) = � cne L » Discrete Fourier Transform » Useful properties 6 n=�¥ »Applications Z L 1 pn �i t cn = x(t)e L dt 2L �L p.4/33 Fourier Series »Fast Fourier Transform - � Generalized to complex-valued functions as Overview Fourier Analysis ¥ » Fourier Series i pn t » Continuous Fourier Transform x(t) = � cne L » Discrete Fourier Transform » Useful properties 6 n=�¥ »Applications Z L 1 pn �i t cn = x(t)e L dt 2L �L � Studied by D.Bernoulli and L.Euler � Used by Fourier to solve the heat equation p.4/33 Fourier Series »Fast Fourier Transform - � Generalized to complex-valued functions as Overview Fourier Analysis ¥ » Fourier Series i pn t » Continuous Fourier Transform x(t) = � cne L » Discrete Fourier Transform » Useful properties 6 n=�¥ »Applications Z L 1 pn �i t cn = x(t)e L dt 2L �L � Studied by D.Bernoulli and L.Euler � Used by Fourier to solve the heat equation � Converges for almost all �nice� functions (piecewise smooth, 2 L etc.) p.4/33