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Extra resources for Algorithms for Programmers

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7 (complex FT by HT, version 1) Pseudo code for the complex Fourier transform that uses the Hartley transform, is must be -1 or +1: fft_by_fht1(a[],b[],n,is) CHAPTER 3. n-1] input,result (imaginary part) { fht_fft_conversion(a[], b[], n, is) fht(a[], n) fht(b[], n) } Note that the real and imaginary parts of the FT are computed independently by this procedure. 8 for the backward transform. The complex squarings are then combined with the pre- and postprocessing steps, thereby interleaving the most nonlocal memory accesses with several arithmetic operations.

In Z/mZ things are not that simple since primitive roots of unity do not exist for arbitrary n, they exist for some maximal order R only. Roots of unity of an order different from R are available only for the divisors di of R: rR/di is a di -th root of unity because (rR/di )di = rR = 1. 1) The operations needed in FFTs are addition, subtraction and multiplication. Division is not needed, except for division by n for the final normalization after transform and backtransform. Division by n is multiplication by the inverse of n.

Cf. ) Cf. cc] The underlying idea can be derived by closely looking at the convolution of real sequences by the radix-2 FHT. g. in the MFA-based convolution for real input. ’ We want to do FFTs in Z/mZ (the ring of integers modulo some integer m) instead of C, the (field of the) complex numbers. These FFTs are called numbertheoretic transforms (NTTs), mod m FFTs or (if m is a prime) prime modulus transforms. There is a restriction for the choice of m: For a length n FFT we need a primitive n-th root of unity.

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