# New PDF release: Algorithms for Discrete Fourier Transform and Convolution 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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Extra resources for Algorithms for Discrete Fourier Transform and Convolution

Example text

The unit group U of F[xJI/{x), consisting of all polynomials g{x) in F[xJI/{x) having multiplicative inverse is U = {h{x) E F[xJI/{x) I (h{x),/{x)) = I}. 9. CRT for Polynomial Rings Identifying F with the constant polynomials in F[xl/ /(x), we have Fe F[xJI/(x). (15) If p(x) is an irreducible polynomial of degree n, then K = F[xl/p(x). (16) is a field extension of F which can also be viewed as a vector space of dimension n over F. Suppose F = Zip, (17) then K is a finite field of order pR. We state without proof the next result.

For any g{x) E FIx], denote by g{x) mod [(x), (5) the remainder of the division of g{x) by [(x). Then g{x) mod [(x) E F[xl/ [(x). 8. The Ring F[xJ/ f(x) Define multiplication in F[xl/ f(x) by (g(x) h(x)) mod f(x), g(x), h(x) E F[xJ/ f(x). (7) Direct computation shows that the vector space F[xl/ f(x) becomes an algebra over F with the multiplication (7). Two polynomials g(x) and h(x) over F are said to be congruent mod f(x), and we write g(x) == h(x) mod f(x), if g(x) mod f(x) (8) h(x) mod f(x).

It strides through ~ with stride of length M. Example 1. Take N 0 0 P(4,2) = 1 0 Example 2. Take N = 4. 0 1 0 0 [~ P(S,4) = S. 1 0 0 0 0 0 0 0 Then ~l ' 0 0 1 0 0 0 0 0 Then 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 and P(4,2)F 0 1 0 0 0 0 0 0 Xo X4 Xl P(S,4)~ = X5 X2 X6 X3 X7 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 [~J , Chapter 2. Tensor Product and Stride Permutation Example 3. Take N P(6,3) = 6. Then 1 0 0 0 0 1 0 0 0 0 0 0 = 45 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 , and P(6,3)~ = Suppose now that N = RS.