# Digital Signal Processing. Applications to Communications by Salvatore D. Morgera

By Salvatore D. Morgera

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4: Complexity of Cyclic Convolution of Some Factor Lengths Over a Field of Constants GF{Z). Length ΛΓ No. 1. 5: Complexity of Cyclic Convolution Algorithm Over a Field of Constants GF(4). Length N 15 63 255 4095 No. of Mults. 1. Just a fac­ tor of two increase in field size is seen to substantially decrease (by approximately 30%) the multiplicative complexity; however, computa­ tion of the linear forms associated with the matrices A, B, and C whose elements are now over the enlarged field is somewhat more complex.

The index set S = {0,1,3}. Let a denote the primitive element of GF(2S) satisfying a 3 + a + 1 = 0. The only divisor, i, of n in this case is 3, excluding t = 1, since p = 2; thus, Θ = (pn - l)/(pl - 1) = 1. For Θ = 1, the primitive polynomial is P\{u) = (u — a)(u — a 2 )(u — α 4 ) = υ? + u + 1. Since σ\ = 3, we now refer to the polynomial multiplication algorithm for degree σ\ — 1 = 2 in Appendix B. Using the linear forms in the /fc's, we proceed in the following manner to complete the first σ\ = 3 elements of each row of A\.

The vector ψ — x * y is periodic with period N and ψ = (Nf modp)ip. The algorithm for length pn — 1 is given by V> = C ( A x # B y ) , where AK = [A^, Aj2, · · ·] x, with N'-l (Ai)kj = Σ (Ai)kj+eN, itS. A similar relation holds for the term By. We have AQ = (Nf mod P)AQ. For any nonzero zeS, it is possible from P 3 to find a Θ = {pn - 1)/{ρσθ - 1) such that i is a multiple of Θ. 3, we also know that the rows of AQ are the MLRS's generated by PQ{U). ,n-l. 1. CYCLIC CONVOLUTION (Ai)kj = 49 tr(ßkafl).