130x1g2 - CCSDS

TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEThe paths are said to have diverged at some state, and some depth j, if at depth j+1, theirinformation bit disagree. Later, paths can remerge after (K–1) consecutive identicalinformation bits. The maximum-likelihood sequence estimation problem is formally identicalto the problem of finding the shortest route through a certain graph. The Viterbi algorithmthen arises as a natural recursive solution. Consider a rate 1/n convolutional code. Letu 0 … u t–1 u t u t+1 … denote the information bits input to the encoder. At time t define theencoder state ass t = u t … u t – K + 1 (2)Given a sequence of observations y 0 , y 1 , … y L , where y i = (y i1 … y in ), every path may beassigned a ‘length’ proportional to metric –log p(y|s), where p(y|s) is the likelihood functionand s = (s 0 , …, s L ) is the state sequence associated with that path.The Viterbi algorithm solves the problem of finding the state sequence for which p(y|s) ismaximum, or equivalently of finding the path whose length –log p(y|s) is minimum. It shouldbe noted that to every possible state sequence s there corresponds a unique path through thetrellis, and vice versa. If the channel is memoryless, thenwheret=1( )− log p( y| s) = λ s , s −L∑tt1λ(s t , s t–1 ) = –log p(y t |s t ,s t–1 ) = –log p(y t |s t )is the branch ‘length’ or metric. T t (s t ,s t–1 ) denotes the transition from state s t–1 to s t associatedwith branch symbols x t = (x t1 … x tn ), which correspond to the information sequenceu t … u t–KTherefore, the state transition can be defined as T t (s t ,s t–1 ) = u t …u t–K . By s(s t ) is denoted asegment (s 0 , s 1 , …, s t ) consisting of the states up to time t of the state sequence s. In thetrellis, s(s t ) corresponds to a path segment starting at the state s 0 and terminating at state s t .For any particular time t and state s t , there will in general be several such path segments,each with some lengtht∑( s ( st)) ( si,si− 1)λ = λThe shortest such path segment is called the survivor, corresponding to the state s t , and isdenoted ŝ(s t ). For any time t>0, there are 2 m survivors in all, one for each s t .i=1CCSDS 130.1-G-2 Page 4-6 November 2012