By Arndt J

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**Additional info for Algorithms for Programmers**

**Example text**

C − 1 and y = 0, 1, 2, . . , R − 1) be a 2-dimensional array of data7 . 77) k=0 h=0 For a m-dimensional array ax (x = (x1 , x2 , x3 , . . , xm ), xi ∈ 0, 1, 2, . . , Si ) the m-dimensional Fourier transform ck (k = (k1 , k2 , k3 , . . , km ), ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = e± 2 π i/n , n = S1 S2 . . k where S = (S1 − 1, S2 − 1, . . 79) x=0 The inverse transform is again the one with the minus in the exponent of z. 80) x=0 which shows that the 2-dimensional FT can be accomplished by using 1-dimensional FTs to transform first the rows and then the columns8 .

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.

N-1] input,result { n := 2**ldn // length of a[] is a power of 2 for ldm:=ldn to 1 step -1 { m := 2**ldm mh := m/2 m4 := m/4 for r:=0 to n-m step m { for j:=0 to mh-1 { u := a[r+j] v := a[r+j+mh] a[r+j] := u + v a[r+j+mh] := u - v } for j:=1 to m4-1 { k := mh - j u := a[r+mh+j] v := a[r+mh+k] c := cos(j*PI/mh) CHAPTER 3. cc] TBD: higher radix FHT 54 CHAPTER 3. 3 55 Complex FT by HT The relations between the HT and the FT can be read off directly from their definitions and their symmetry relations.