A Necessary and Sufficient Condition for the Asymptotic Tightness of the Shannon Lower Bound
نویسنده
چکیده
The Shannon lower bound is one of the few lower bounds on the rate-distortion function that holds for a large class of sources. In this paper, it is demonstrated that its gap to the rate-distortion function vanishes as the allowed distortion tends to zero for all sources that have a finite differential entropy and whose integer parts have a finite entropy. Conversely, it is demonstrated that if the integer part of the source has an infinite entropy, then its rate-distortion function is infinite for any finite distortion. Consequently, the Shannon lower bound provides an asymptotically tight bound on the rate-distortion function if, and only if, the integer part of the source has a finite entropy.
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