On the redundancy of trellis lossy source coding

It is well known that trellis lossy source codes have better performance/complexity tradeoff than block codes, as shown by simulations. This makes the trellis coding technique attractive in practice. To get a better understanding of this fact, this paper studies the redundancy of trellis coding for memoryless sources and compares it with a similar result for block codes. It was known that for block codes of block length n, the nth-order distortion redundancy D-n(R) at fixed rate R greater than or equal to 0 equals - partial derivatived(p, R/partial derivativeR) ln n/2n + o(ln n/n), where partial derivatived(p, R/partial derivativeR) is the partial derivative of d(p, R), the distortion-rate function of a source p, evaluated at R and assumed to exist. Since e(nR), the number of codewords of the block code, can be used as an approximate measure of both the storage complexity C-s of the code and the computational complexity C-c per source symbol for full search encoding, the redundancy can be written as functions of the complexity measures in the form D(R, C-s) approximate to partial derivatived(p, R)/partial derivativeR R ln ln C-s/2 ln C-s and D(R, C-s) approximate to partial derivatived(p, R)/partial derivativeR R ln ln C-c/2 ln C-c In this paper, it is demonstrated that for a particular trellis lossy source code with storage complexity C-s = e(2nR) and computational complexity C-c = e(2nR) (assuming the Viterbi algorithm is used for encoding), the distortion redundancy satisfies D-n(R) less than or equal to c(p, R)/n where c(p, R) is a constant independent of n. For this code, the complexity/redundancy tradeoff can be written as D(R, C-s) approximate to 2Rc(p, R)/ln C-c + o(1/ln C-s) and D(R, C-c) approximate to 2Rc(p, R)/ln C-c + o(1/ln C-c) which shows that trellis coding improves the redundancy/complexity tradeoff over block coding by roughly a factor ln ln C-c.