By Arndt J.

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Spr] It is assumed that the procedure fft() does no normalization. 0/n and multiply as divisions are much slower than multiplications. 6) ax ay x+y≡τ (n) The corresponding procedure should be obvious. cc] CHAPTER 2. e. x + y = n + τ ) and those where simply x + y = τ holds. These are (following the notation in [18]) denoted by h(1) and h(0) respectively. 7) where h(0) = ax bτ −x x≤τ h(1) = ax bn+τ −x x>τ There is a simple way to seperate h(0) and h(1) as the left and right half of a length-2 n sequence.

CONVOLUTIONS 45 1. Apply a FFT on each column. 2. R − 1 the index of the row, C the length of each row (or, equivalently the total number columns) 3. Apply a FFT on each column (of the transposed matrix). 21) For the acyclic (or linear) convolution of sequences one can use the cyclic convolution of the zero padded sequences zx := {x0 , x1 , . . , nn−1 , 0, 0, . . e. x with n zeros appended). ω) B {ω} B is the cc. of C {ω2 } C and therefore every B {} B-term is the cc. of the C {} C-term in the same line.

It is of course possible to lower this ‘worst case factor’ to 6 by using highly composite L slightly greater than 2 n. 4 Fractional Fourier transform by ZT The z-transform with z = eα 2 π i/n and α = 1 is called the fractional Fourier transform (FRFT). g. the computation of the DFT for data sets that have only few nonzero elements and the detection of frequencies that are not integer multiples of the lowest frequency of the DFT. A thorough discussion can be found in [35]. cc] 8 cf. 1 Definition of the HT The Hartley transform (HT) is defined like the Fourier transform with ‘cos + sin’ instead of ‘cos +i · sin’.

### Algorithms for programmers. Ideas and source code by Arndt J.

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