Coding and Complexity by Giuseppe Longo (eds.)

By Giuseppe Longo (eds.)

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9. An Error Bound for Specific Convolutional Encoders Consider a specific code (xl , x 2 ) containing two codewords of length N for use on a given DMC. Let y = [y1, Y2, ... YN] denote the received N-tuple. For a maximum likelihood decoder (MLD), the decoding region Yt for x 1 is the set of all y such that P(y lxd ~ P(y lx2 ) [except that those y for which P(y lxd = P(y lxv can be assigned arbitrarily either to Y1 or toY 2, the decoding region for x 2 ~ Now consider the decoding error probability given that :X2 is transmitted which we shall denote~ 12 .

The symbol A denotes the "empty string" which is the path to the root node of the tree. Fori= A , we have x = A and L c(x) = 0. Step 0: Place [A, 01 into the initially empty stack. Step 1: Extend the top entry [i, Lc(x) 1 in the stack by forming [i * 0, Lc(x) + Lc(xo) 1 and then deleting [i ,Lc(x)1 from the stack. Step 2: Place the two newly-formed entries into the stack so that the stack remains ordered with entries with greater metric higher in the stack. in the stack is a path through the Step 3: If the tqp entry [i, ~f(x)1 entire tree, stop and choose i10 ,L + T) = i.

E. WH (8 1o,co)) = oo . co)) < oo but result in WH (6 10,co)) = oo . Conversely if an FCE is non-catastrophic, no channel-decoder pair can ever result in WH (e[O,co)) < oo but WH (6[0,co)) = oo. By a "realistic" channel-decoder pair we mean a pair such that, regardless of w~at i 10 ,co) is encoded, the channel can behave so as to cause the decoder to decide i10 , co) = 0 . This rules out, for instance, the "noisless" BSC with 0 crossover probability and other "perfect" channels, and also rules out decoders that never estimate 0 but rules out no combination of a "real" channel and a "reasonable" decoder.

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