Algorithms for programmers ideas and source code by Arndt J.

By Arndt J.

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Only a small subset of C++. 12 Actually CHAPTER 1. 75/pt The generated codes can be of great use when one wants to spot parts of the original code that need further optimization. Especially repeated trigonometric values and unused symmetries tend to be apparent in the unrolled code. g. multiplications, additions, load/stores) of the code it emits. Even better if those numbers are compared to the corresponding values found in the compiled assembler code. It is possible to have gcc produce the assembler code with the original source interlaced (which is a great tool with code optimization, cf.

106) v+ and v− are antisymmetric. u+ , u− , v+ and v− are eigenvectors of the FT, with eigenvalues +1, −1, −i and +i respectively. The eigenvectors are pairwise perpendicular. 107) we can, for a given sequence, find a transform that is the ‘square root’ of the FT: Simply compute u+ , u− , v+ , v− . 108) F 0 [a] is the identity, F 1 [a] is the (usual) FT, F 1/2 [a] (which is not unique) is a transform so that F 1/2 F 1/2 [a] = F [a], that is, a ‘square root’ of the FT. The eigenvectors of the Hartley Transform are u+ := a + H [a] (with eigenvalue +1) and u+ := a − H [a] (with eigenvalue −1).

The n log(n)-implementations of the HT are called fast Hartley transforms (FHT). 1 Radix 2 FHT algorithms Decimation in time (DIT) FHT For a sequence a of length n let X 1/2 a denote the sequence with elements ax cos π x/n + ax sin π x/n (this is the ‘shift operator’ for the Hartley transform). 8) H [a] H [a] = 55 CHAPTER 3. cc] The procedure hartley_shift replaces element ck of the input sequence c by ck cos(π k/n) + cn−k sin(π k/n). n-1] input,result { n := 2**ldn // length of a[] is a power of 2 revbin_permute(a[], n) for ldm:=1 to ldn { m := 2**ldm mh := m/2 m4 := m/4 for r:=0 to n-m step m { for j:=1 to m4-1 // hartley_shift(a+r+mh,mh,1/2) CHAPTER 3.

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