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