New PDF release: Algorithms for Programmers

By Arndt J

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1 (cyclic convolution by definition) Compute the cyclic convolution of a[] with b[] using the definition, result is returned in c[] procedure convolution(a[],b[],c[],n) { for tau:=0 to n-1 { s := 0 for x:=0 to n-1 { tx := tau - x if tx<0 then tx := tx + n s := s + a[x] * b[tx] } c[tau] := s } } This procedure uses (for length-n sequences a, b) proportional n2 operations, therefore it is slow for large values of n. The Fourier transform provides us with a more efficient way to compute convolutions that only uses proportional n log(n) operations.

Of course the same ideas also work for separate real- and imaginaryparts. n-1] input,result { fht(a[], n) for i:=1 to n/2-1 { t := n - i u := a[i] v := a[t] a[i] := 1/2 * (u+v) a[t] := 1/2 * (u-v) } } At the end of this procedure the ordering of the output data c ∈ C is a[0] a[1] a[2] = = = c0 c1 c2 a[n/2] a[n/2 + 1] a[n/2 + 2] a[n/2 + 3] ... = = = = ... 20) CHAPTER 3. cc] Vice versa: same line of thought as for complex versions. Let Trc be the operator corresponding to the postprocessing in real_complex_fft_by_fht, and Tcr correspond to the preprocessing in complex_real_fft_by_fht.

Cf. 3 Weighted Fourier transforms Let us define a new kind of transform by slightly modifying the definition of the FT (cf. 13) n−1 ck vx ax z x k := vx = 0 ∀x x=0 where z := e± 2 π i/n . The sequence c shall be called weighted (discrete) transform of the sequence a with the weight (sequence) v. Note the vx that entered: the weighted transform with vx = √1n ∀x is just the usual Fourier transform. 14) 1 n vx n−1 1 n vy n−1 n−1 ck z −x k k=0 This can be easily seen: Wv−1 [Wv [a]]y = = 1 n vx ax z x k z −y k k=0 x=0 n−1 n−1 vx k=0 x=0 1 ax z x k z −y k vy n−1 = = 1 1 ax δx,y n vx n x=0 vy ay (cf.

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Algorithms for Programmers by Arndt J

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