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Unformatted text preview: Lecture 12 Discrete and Fast Fourier Transforms 12.1 Introduction The goal of the chapter is to study the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT). In the course of the chapter we will see several similarities between Fourier series and wavelets, namely Orthonormal bases make it simple to calculate coefficients, Algebraic relations allow for fast transform, and Complete bases allow for arbitrarily precise approximations. There is, however, a very important difference between Fourier series and wavelets, namely Wavelets have compact support, Fourier series do not. 12.2 The Discrete Fourier Transform (DFT) 12.2.1 Definition and Inversion Let ~ e , . . . ,~ e N 1 denote the usual standard basis for C n . A vector ~ f = ( f , . . . , f N 1 ) C N may then be written as ~ f = f ~ e + + f N 1 ~ e n 1 . 77 78 LECTURE 12. DISCRETE AND FAST FOURIER TRANSFORMS Important example: Assume the array ~ f is a sample from a function f : R C , that is, we use the sample points x := 0 , . . . , x ` := ` (2 /N ) , . . . , x N 1 := ( N 1) (2 /N ) with values ~ f = ( f ( x ) , . . . , f ( x ` ) , . . . , f ( x N 1 )). The Discrete Fourier Transform expresses such an array ~ f with linear combinations of arrays of the type ~ w k := ( e ikx ` ) N 1 ` =0 = ( 1 , e ik 2 /N , . . . , e ik` 2 /N , . . . , e ik ( N 1) 2 /N ) = ( ~ w k ) ` = ( e i 2 /N ) k` =: k` N . Definition For each positive integer N , we define an inner product on C N by h ~ z, ~ w i N = 1 N N 1 X m =0 z m w m . Lemma For each positive integer N , the set { ~ w k  k { , . . . , N 1 }} is orthonormal with respect to the inner product h , i N . In fact { ~w , . . . , ~w N 1 } is an orthonormal basis for C N . Sketch of Proof For all k, ` Z so that ` = k + JN for some J , we have h ~w k , ~w ` i N = 1 N N 1 X m =0 ( ~w k ) m ( ~w ` ) m = 1 N N 1 X m =0 e ikm 2 /N e i`m 2 /N = 1 N N 1 X m =0 e ikm 2 /N e i ( k + JN ) m 2 /N = 1 N N 1 X m =0 1 = 1 . For the remaining k, ` Z we use the geometric series to see that h ~ w k , ~ w ` i N = 1 N N 1 X m =0 ( ~ w k ) m ( ~ w ` ) m = 1 N N 1 X m =0 e i [ k ` ] m 2 /N = 1 N N 1 X m =0 ( e i [ k ` ] 2 /N ) m = 1 N 1 ( e i [ k ` ] m 2 /N ) N 1 e i [ k ` ] 2 /N = 1 N 1 ( e 2 i ) [ k ` ] 1 e i [ k ` ] 2 /N = 1 N 1 1 1 e i [ k ` ] 2 /N = 0 . 12.3. DISCRETE FOURIER TRANSFORM 79 12.2.2 The Fourier matrix Definition For each positive integer N , define the Fourier matrix N F by N F k,` = ( ~ w ` ) k = e ik` 2 /N = k` N . Example: If N = 1, then N = 1 = 1, and N F = 1. If N = 2, then N = 2 = 1, and 2 F = ( 2 ) ( 1 2 ) ( 2 ) 1 ( 1 2 ) 1 = 1 1 1 1 (the Haar matrix) If N = 4, then N = 4 = e i 2 / 4 = i , and 4 F = 1 ( i 1 ) ( i 2 ) ( i 3 ) 1 ( i 1 ) 1 ( i 2 ) 1 ( i 3 ) 1 1 ( i 1 ) 2 ( i 2 ) 2 ( i 3 ) 2 1 ( i 1 ) 3 ( i 2 ) 3...
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 Spring '08
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