131x0b2ec1

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Unformatted text preview: ch this Recommended Standard is based. F2 TRANSFORMATION Referring to figure F-1, it can be seen that information symbols I entering and check symbols C emanating from the Berlekamp R-S encoder are interpreted as [z0, z1, ... , z7] where the components zi are coefficients of li, respectively: z0l0 + z1l1 + ... + z7l7 Information symbols I ' entering and check symbols C ' emanating from the conventional R-S encoder are interpreted as [u7, u6, ... , u0] where the components uj are coefficients of α j, respectively: u7α7 + u6α6 + ... + u0 A pre- and post-transformation is required when employing a conventional R-S encoder. CCSDS 131.0-B-2 Page F-1 August 2011 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING I C BERLEKAMP R-S ENCODER C T -1 T I' CONVENTIONAL R-S ENCODER C' Figure F-1: Transformational Equivalence Conventional and Berlekamp types of (255,k) Reed-Solomon encoders are assumed to have the same self-reciprocal generator polynomial whose coefficients appear in 4.3.3 and 4.3.4. The representation of symbols associated with the conventional encoder is the polynomials in ‘α’ appearing in table F-1. Corresponding to each polynomial in ‘α’ is the representation in the dual basis of symbols associated with the Berlekamp type encoder. Given αi = u7α7 + u6α6 + ... + u0 where 0 ≤ i < 255 (and α* denotes the zero polynomial, u7, u6, ... = 0, 0, ...), the corresponding element is z = z0l0 + z1l1 + ... + z7l7 where [z0, z1, ..., z7] = [u7, u6, ..., u0] Tα l and T = 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 Row 1, row 2, ... , and row 8 in Tα l are representations in the dual basis of α7 (10 ... 0), α6 (010 ... 0), ... , and α0 (00 ... 01), respectively. CCSDS 131.0-B-2 Page F-2 August 2011 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING The inverse of Tα l is T -1 1 0 0 1 1 0 1 1 = 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0...
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This document was uploaded on 03/06/2014.

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