مرکز منطقه ای اطلاع رساني علوم و فناوري - Kraft inequality for decoding with respect to a fidelity criterion

Abstract :

Let $d_1(\\alpha , \\beta )$ be a distortion measure that is row balanced if the source alphabet is a finite set and that has the form $d(\\alpha - \\beta )$ if the source alphabet is the real line. For $N$ -tuples, let the distortion measure $d_N$ be the single-letter measure. Consider a variable-length code in which there are $D$ code symbols and for which the length of the codeword for $\\alpha$ is $n(\\alpha )$ . Codewords $w(u)$ for $N$ -tuples $u$ are formed by concatenating codewords for the individual letters. Let $E(N) = \\sup d_N (u,v)$ where the supremum is over all pairs $(u,v)$ for which $w(u) = w(v)$ . Call the code $\\varepsilon$ -decodable if $\\lim E(N) = \\varepsilon$ . If $0 = d_1(\\alpha ,\\beta ) < d_1(\\alpha ,\\beta )$ for $\\alpha \\neq \\beta$ , and if the code is uniquely decipherable, then $\\varepsilon = 0$ . For a discrete source it is shown that $\\sum D^{-n(i) -h(\\varepsilon )} \\leq 1$ , where $h(0) = 0$ and $h(\\varepsilon ) > 0$ if $vare\\psio\\ln > 0$ . For a continuous source for which values from the interval $[-c,c]$ are encoded, $\\int_{-c}^c D^{-n(z)-h_1 (\\varepsilon )} dz \\leq 1$ , where $h_1 (\\varepsilon )$ is a known function. These inequalities are used to obtain lower bounds on the mean length of any $\\varepsilon$ -decodable code. In many cases, these lower bounds coincide with the rate-distortion function $R(\\varepsilon )$ associated with the same distortion measure.