By Salvatore D. Morgera

ISBN-10: 0125069952

ISBN-13: 9780125069953

ISBN-10: 0323158935

ISBN-13: 9780323158930

**Read or Download Digital Signal Processing. Applications to Communications and Algebraic Coding Theories PDF**

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**Extra info for Digital Signal Processing. Applications to Communications and Algebraic Coding Theories**

**Example text**

If we let yN-j = ^5 j = 0 , 1 , . . , N — 1, and take yjy = j/o ? 17) and appending k — 1 zeroes to the indeterminates {y0, j / i , . . 18) 0 £fc+d-2 ^0 ^1 * 0 We have N = k + d — 1 and { ^ } 5 {Φί}\ specifically, tßt = ^ η t = 0 , 1 , . . , k — 1. 18) represents the manner in which Φ can be computed as a cyclic correlation. 19) CHAPTER 2. 19) are of circulant, or cyclic, form. In the first part of Chapter 3, efficient JVC algorithms of the type C(Ax · By) for computing length TV cyclic convolutions over finite fields are derived and discussed.

Let a denote the primitive element of GF(32) satisfying a2 + a + 2 = 0. The divisors t of n are 1 and 2; thus, 0 = 4 and 1, respectively. For Θ = 1 and 4, the primitive polynomials are P\{u) = u2 + u + 2 and P±{u) — u + 1, and the (monic) reciprocal polynomials are P\{u) = u2 + 2u + 2 and P±{u) = u + 1, respectively. In determining the first σ\ = 2 elements of each row of Αχ, we use the degree σ\ — 1 = 1 polynomial multiplication algorithm of Appendix B. In determining the first σ\ — 2 rows of Ci, the following values for tr(-) over GF(3 2 ) are required: 50 CHAPTER 3.

CYCLIC CONVOLUTION 31 where xz- and y ; are vectors over GF(p) and a is a primitive element of GF(pm). Using bilinearity, we obtain TO— 1 TO—1 TO—1 4 771—1 Φ = ( Σ xitt ) * ( Σ y ^ ) = Σ Σ <*i+j (* * yy)· t=o i=o »=o i=o Further, since x» and yy are over GF(p), we may use the bilinear algo rithm to compute x,· * y^·; thus, TO-l TO-l ^ = Σ Σ^Μ^^··^·)] ι=0 j=0 TO—1 TO—1 = c | Σ Σ ( W · ayy«*) i=o i=o TO—1 TO — 1 = C (A j ; χίβ ) · (B Σ 4 i=0 i=0 yy«'') = C ( A x # £ y ) . This result is useful, as it allows us to confine consideration to a smaller field of constants, GF(p).