These k symbols are passed directly to the output of

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Unformatted text preview: utput of each encoder. The synchronized action of S2 reassembles the symbols at the port labeled ‘OUT’ in the same way as they entered at ‘IN’. Following this, each encoder outputs its 2E check symbols, one symbol at a time, as it is sampled in sequence by S2. If, for I=5, the original symbol stream is 1 51 5 1 5 1 12 2 k k [2E × 5 spaces] d ... d d ... d ... d ... d then the output is the same sequence with the [2E × 5 spaces] filled by the [2E× 5] check symbols as shown below: 1 5 1 1 1 2E p ... p ... p 5 ... p 2E where i dd i 12 ... d i K i i 1 2E p ... p is the R-S codeword produced by the ith encoder. If q virtual fill symbols are used in each codeword, then replace k by (k – q) in the above discussion. CCSDS 131.0-B-2 Page 4-7 August 2011 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING With this method of interleaving, the original kI consecutive information symbols that entered the encoder appear unchanged at the output of the encoder with 2EI R-S check symbols appended. 4.4.2 DUAL BASIS SYMBOL REPRESENTATION AND ORDERING FOR TRANSMISSION Each 8-bit Reed-Solomon symbol is an element of the finite field GF(256). Since GF(256) is a vector space of dimension 8 over the binary field GF(2), the actual 8-bit representation of a symbol is a function of the particular basis that is chosen. One basis for GF(256) over GF(2) is the set ( 1, α1, α2, . . ., α7). This means that any element of GF(256) has a representation of the form u7α7 + u6α6 + . . . + u1α1 + u0α0 where each ui is either a zero or a one. Another basis over GF(2) is the set ( 1, β 1, β 2, . . ., β 7) where β = α 117. To this basis there exists a so-called ‘dual basis’ (l0, l1, . . ., l7). It has the property that ⎧ 1 if i = j Tr(liβ j ) = ⎨ 0 otherwise ⎩ for each j = 0, 1, . . ., 7. The function Tr(z), called the ‘trace’, is defined by 7 Tr(z) = ∑ z2 k K=0 for each element z of GF(256). Each Reed-Solomon symbol can also be represented as z0l0 + z1l1 + . . . + z7l7 where ea...
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This document was uploaded on 03/06/2014.

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