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41) Finite State Channels 42 This expression is somewhat messy to analyze, but if we consider the QN £N, that yields the function (i) that yields ~ N s0 and the we can make some statements easily. First, Eo,N (e I aN' So) is convex " in Q and there- fore we get the following parametric equations in Q. (2. N (e. R= (2. 43) aN,so) dQ g By evaluating this partial derivative at find that at this point R• ~N =0 1 we and that the exponent given by the maximum in (2. 42) is zero. As Q increases, R decreases and the exponent increases.

34) This completes the proof since {3. 32) and {3. 34) imply (3. 24). Lower Bound to R0 (g) 63 The above theorem is due to Haskell {Trans. I. , Sept. 1969, p. 525). In many cases, particularly when the U and V al- F (w) , but F (w), phabets are infinite, it is difficult to find for any given since F(w) W, still serves as an upper bound on R 0 (Q), and is convex U in W, this upper bound is easy to work with. We now go on to find a corresponding lower bound to R0(g). (u) on the input space, h(u).

2. 65) as before and its largest eigen- J. As before, and eigenvector 'l1 we obtain (2. 66) Since there are J N different output sequences ~, the sum over ~just introduces a multiplicative factor of J N. (Q)- tn J. (2. 67) Source Coding with a Distortion Measure Many of the sources encountered in communication theory produce waveforms, pictures, or a sequence of analogue variables as their output. These can not be recreated exactly at the destination and often much of the detail in the source output is irrelevant.