Poster: On the Capacity-Delay-Error-Tradeoff of Source Coding Ralf Lübben Institute of Communications Technology Leibniz Universitat Hannover ¨ Markus Fidler Institute of Communications Technology Leibniz Universitat Hannover ¨ ralf.luebben@ikt.uni-hannover.de ABSTRACT In this note we present a statistical, non-equilibrium envelope model that, based on Legendre transforms, characterizes source coders and transmission channels by their capacity-delay-error-tradeo €. The model is proven to have the desirable property of additivity, that allows analyzing coders and channels separately. We present an example for Hu €man coding. More can be found in the report [2]. markus. dler@ikt.uni-hannover.de 100 80 delay d 60 40 20 0 2 1. ADDITIVE ENVELOPE MODEL We use a discrete time model and denote by arrivals A(t) the cumulative number of bits generated by a source in the interval [0, t]. The arrivals have statistical envelope E(t) with over ‚ow pro le ( ) if for all t, ¥ 0 and ¥ 0 [1] P[A(t + ) ’ A( ) > E(t) + ] ¤ ( ). We use the concave Fenchel conjugate, respectively, Legendre transform of E(t) de ned for any c ¥ 0 as LE (c) = sup{E(t) ’ ct} t ¥0 2.5 3 capacity c 3.5 ’12 ’9 ’6 ’3 error Figure 1: Capacity-delay-error-tradeo € of a source. We label successive symbols generated by a discrete source with constant symbol rate by t ˆ N. The cumulative number of bits generated by the encoded source becomes A(t) = t i=1 L(i) and A(0) = 0 by de nition. Assuming a memoryless source A(t) has MGF MA ( , t) = (ML ( ))t such that ( ) = 1 ln pi e li . i to characterize the source. LE (c) has the interpretation of a delay bound for the source at a channel with constant rate c. For the delay W it holds for all ¥ 0 that [2] P[W > (LE (c) + )/c] ¤ ( ), (1) (3) where ( ) is the error probability. We refer to (c, (LE (c) + )/c, ( )) as the capacity-delay-error-tradeo € of the source. Given the e €ective bandwidth of A(t) de ned for > 0 as ( , t) = 1/( t) ln MA ( , t) where MA ( , t) = E[e A(t) ] is the moment generating function (MGF) of A(t). We apply Cherno €â€™s theorem and the union bound to derive an envelope E(t) with Legendre transform [2] LE (c) = sup{( ( , t) + ’ c)t} ’ ln( )/ , t ¥0 As an example, assume an in nite alphabet with geometrically distributed symbols pi = p(1 ’ p)i for i ¥ 0. We let p = 1/2 to obtain a dyadic source. A respective Hu €man coder uses codewords of lengths li = i + 1. From (3) we derive for 0 < < ln 2 using the geometric sum that ( ) = 1 ln p1 ’ ln 2 1 ’ (1 ’ p)1 ’ ln 2 . (2) where > 0 and ˆ (0, 1/ ] are free parameters that can be optimized. Similarly, channels can be described by convex Legendre transforms, where the composition of a source and a channel corresponds to an addition of their transforms [2]. We compute LE (c) from (2) and the capacity-delay-errortrado € from (1). We optimize the free parameters and numerically. The entropy rate of the source is HX = 2 and the expected codeword length is l = 2. Fig. 1 depicts the capacity-delay-error-tradeo € of the Hu €man coded source. For c > l the delay bounds are nite, whereas the delay grows unbounded for c → l. Also, Fig. 1 shows the logarithmic growth of the delay for decaying . 2. CODED MEMORYLESS SOURCE 3.

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