Explicit bounds to R(D) for a binary symmetric Markov source

Abstract

A new upper hound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{u}(D)</tex> and lower hound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{\ell}(D)</tex> are developed for the rate-distortion function of a binary symmetric Markov source with respect to the frequency of error criterion. Both hounds are explicit in the sense that they do not depend on a blocklength parameter. In the interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{c} &lt; D &lt; 1/2 = D_{max}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{c}</tex> is Gray's critical value of distortion, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{u}(D)</tex> is convex downward and possesses the correct value and the correct slope at both endpoints. The new lower bound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{ell}(D)</tex> diverges from the Shannon lower bound at the same value of distortion as does the second-order Wyner-Ziv lower bound. However, it remains strictly positive for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D \leq 1/2</tex> and therefore eventually rises above all the Wyner-Ziv lower bounds as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> approaches <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> . Some generalizations suggested by the analytical and geometrical techniques employed to derive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{u}(D)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{ell}(D)</tex> are discussed.