Algorithms for Discrete Fourier Transform and Convolution - download pdf or read online

By R. Tolimieri, Myoung An, Chao Lu (auth.), C. S. Burrus (eds.)

ISBN-10: 1475738544

ISBN-13: 9781475738544

ISBN-10: 1475738560

ISBN-13: 9781475738568

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Additional resources for Algorithms for Discrete Fourier Transform and Convolution

Example text

Formed from the array Y has components given by Y,. = Ynm, r = n + mN, 0 ~ while the vector ~ formed from y Z, = Ynm, 8 = m + nM, t m < M, 0 ~ n < N, (13) has components given by 0 ~ m < M, 0 ~ n < N. (n + mN} = m + nM, 0 ~ m < M, 0 ~ n < N, (15) and we have Y,. ) , 0 ~ r < MN. 2. Tensor Product Example 1. Taking M = 2 and N = 3, we have "" = (0 2 4 1 3 5 ), and %0 %2 11. = %4 %1 %3 %5 To form 11. from ~ we 'stride' through In general, to form ~ ~ with length two. ®! from! ® ~ we first initialize at boao, the O-th component of !

As !! runs over all vectors of size Land ll. runs over all vectors of size M, the tensor products 11 ® ll. span the space p LM (see problems 4 and 5). To prove that the action of two matrix expressions on p LM are equal, we only need to prove that they are equal on the 11 ® ll. with!! E P Land ll. E pM. Theorem 2. If A and Care L x L matrices and B and Dare M x M matrices, then (A®B)(C® D) = AC®BD. Proof (22) Take vectors!! and ll. of sizes Land M respectively. By (20), (A ® B)(C ® D)(!! 2. , proving (22), in light of the proceeding discussion.

Introduction to Abstract Algebra 33 7. Give the table for the ring-isomorphism ~ of the CRT corresponding to factorizations n = 4 X 5 and n = 2 X 7. 8. Suppose n = nl n2 ... , nr are relatively prime in pairs. Show that 1 ~ Ie ~ r. 9. Continuing the notation and using the result of problem 8, define integers el, e2, ... , er satisfying e1c == 0 mod n" 1 ~ 1e,1 < r, Ie =/: I. These integers are uniquely determined by these conditions and form the system of idempotents corresponding to the factorization given in Problem 8.

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Algorithms for Discrete Fourier Transform and Convolution by R. Tolimieri, Myoung An, Chao Lu (auth.), C. S. Burrus (eds.)


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