Algorithms for Discrete Fourier Transform and Convolution by Richard Tolimieri PDF

By Richard Tolimieri

ISBN-10: 0387982612

ISBN-13: 9780387982618

This graduate-level textual content offers a language for figuring out, unifying, and imposing a wide selection of algorithms for electronic sign processing - particularly, to supply principles and strategies which could simplify or perhaps automate the duty of writing code for the latest parallel and vector machines. It therefore bridges the distance among electronic sign processing algorithms and their implementation on quite a few computing structures. The mathematical inspiration of tensor product is a ordinary subject during the ebook, considering that those formulations spotlight the knowledge movement, that is specially vital on supercomputers. as a result of their value in lots of functions, a lot of the dialogue centres on algorithms concerning the finite Fourier rework and to multiplicative FFT algorithms.

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Additional info for Algorithms for Discrete Fourier Transform and Convolution (Signal Processing and Digital Filtering)

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In general, we have the next result. 4 If N = ML and 7r is the perm,utation of ZIN defined by 7r(a bM) = b aL, 0 < a < M, 0 < b < L, then P(70= P(N,L). Consider the set of N x N permutation matrices {P(N,L) LI NI . 10) We will describe the permutation matrices in this set in terms of the unit group U(N - 1) of Z/(N - 1). The unit group U(N -1) is given by U (N - 1) = {0 < T < N - 1 : (T - 1) = 1} . , N - 21. 3 Stride Permutations Define the permutation 71-7-, of Z/N by the two rules 72-(k) kT mod (N —1), 0 < k < N —1, 71-2-(N —1) = (N —1).

6 Parallel Implementation 51 Alternatively, we can use the identity A0 = (P(2M, 2) 0 /L)(im 0 A® h)(P(2M, M) 0 IL). Consider the factor im 0 A0 IN. As above, we can implement the action by M parallel computations of A0 IN. If MN processors are available, we can use the identity im 0 A 0 = P(2MN,2M)(1114N 0 A)P(2MN,N) or the identity im 0 A 0 = (Im P(2N,2))(ImN 0 A)(1m P(2N, N)) to compute Im A0 IN as MN paxallel computations of A. In this way, we naturally control the granularity of the parallel computation and fit the computation to granularity and to the number of available processors.

Every divisor of f(x) and g(x) in F[x] divides d(x). Equivalently, d(x) is the unique monic polynomial over F, which is a common divisor of f (x) and g(x) of maximal degree. We call d(x) the greatest common divisor of f(x) and g(x) over F and write d(x) = (f (x), g(x)). By the divisibility condition above, (f (x), g(x)) =- a(x) f (x) b(s)g(x), where a(x) and b(x) are polynomials over F . 22) for some polynomials ao(x) and bo(x) over F. Arguing as in section 2, we have the following corresponding results.

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Algorithms for Discrete Fourier Transform and Convolution (Signal Processing and Digital Filtering) by Richard Tolimieri

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