THE REDUNDANCY OF SOURCE-CODING WITH A FIDELITY-CRITERION .1. KNOWN STATISTICS

The problem of redundancy of source coding with respect to a fidelity criterion is considered, For any fixed rate R > 0 and any memoryless s ource with finite source and reproduction alphabets and a common distr ibution p, the nth-order distortion redundancy D-n(R) of fixed-rate co ding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate R and t he distortion rate function d(p, R) of the source p. It is demonstrate d that for sufficiently large n, D-n(R) is equal to -(partial derivati ve/partial derivative R)d(p, R) 1n n/2n + o(1n n/n), where (partial de rivative/partial derivative)d(p, R) is the partial derivative of d(p, R) evaluated at R and assumed to exist. For any fixed distortion level d > 0 and any memoryless source p, the nth-order rate redundancy R(n) (d) of coding at fixed distortion level d (or by using d-semifaithful codes) is defined as the minimum of the difference between the expecte d rate per symbol of any d-semifaithful code of length n and the rate- distortion function R(p, d) of p evaluated at d, It is proved that for sufficiently large n, R(n)(d) is upper-bounded by In nln + o(ln n/n) and lower-bounded by In n/2n + o(ln n/n). As a by-product, the lower b ound of R(n)(d) derived in this paper gives a positive answer to a rec ent conjecture proposed by Yu and Speed.