On the redundancy of lossy source coding with abstract alphabets

The redundancy problem of lossy source coding with abstract source and repr oduction alphabets is considered. For coding at a fixed rate level, it is s hown that for any fixed rate R > 0 and any memoryless abstract alphabet sou rce P satisfying some mild conditions, there exists a sequence {Cn}(n=1)(in finity) of block codes at the rate R such that the distortion redundancy of C-n (defined as the difference between the performance of C-n and the dist ortion rate function d(P, R) of P] is upper-bounded by \partial derivative d(P, R)/partial derivative R\ ln n/2n + o (In n/n), For coding at a fixed d istortion level, it is demonstrated that for any d > 0 and any memoryless a bstract alphabet source P satisfying some mild conditions, there exists a s equence {C-n}(n=1)(infinity) of block codes at the fixed distortion d such that the rate redundancy of C-n, (defined as the difference between the per formance of C-n and the rate distortion function R(P, d) of P) is upper-bou nded by (7 ln n)/(6n) + o(ln n/n). These results strengthen the traditional Berger's abstract alphabet source coding theorem, and extend the positive redundancy results of Zhang, Yang, and Wei on lossy source coding with fini te alphabets and the redundancy result of Wyner on block coding of memoryle ss Gaussian sources.