Source Coding with Causal Side Information at the Decoder

2 ⌋ }. The i-th reconstruction is of the form X̂i(T, Y ), i.e., it is allowed to depend on the side information, but only causally. The motivation for studying this problem comes from source coding systems that are constrained to operate with no, or with limited, delay. Sequential source codes with a limited-delay encoder and a sequential, zero-delay, decoder with access to side information comprise a subfamily of the schemes that our notion of causal decoder side information allows, since our formulation does not limit the encoding delay. In particular, our formulation encompasses “real-time Wyner Ziv” coding schemes, as recently introduced in [8]. Consequently, our negative (converse) results immediately apply to all such source coding scenarios. On the positive side, as we show, in some cases, schemes from the more limited (and realistic) family of “real-time” codes can achieve optimum performance within the larger family that we allow. Another motivation for our work is to better understand the duality between channel coding and source coding problems. If the Wyner-Ziv problem [13] is the source coding analogue of the Gelfand-Pinsker setting of channel coding with non-causal state information at the transmitter [1], then the problem we consider here is the source coding analogue of channel coding with causal state information at the transmitter, as considered by Shannon in [6]. Our work characterizes the relationship between the non-causal and causal versions of the source coding problem and highlights some similarities to the relationship between the non-causal and causal channel coding problems. Our work can also be viewed a complement to the recent study in [9], which examined various causal source coding scenarios involving side information. Specifically, the effect of the introduction of side information into the causal source coding framework of Neuhoff and Gilbert [4] was studied. As was pointed out, Neuhoff and Gilbert’s notion of causal source codes is not well-matched for situations involving side information at the decoder which, consequently, were not considered in [9].