Encoder-Decoder Design for Perfect Reconstruction: A Robust Control Perspective

Abstract

In this paper we consider the transmission of discrete data via a communication channel that is subject to (additive) noise with a known upper bound on its magnitude but otherwise completely unknown. We are interested in designing transmitter-receiver pairs that perfectly reconstruct the discrete data with a given delay under all possible realizations of channel noise. A Decision Feedback Equalizer (DFE) structure is assumed for the receiver while a linear structure is imposed on the transmitter along with the requirement that the power of the transmission is limited. Under these circumstances, we build on our previous work to provide necessary and sufficient conditions for perfect reconstruction in terms of the 1 norms of appropriate maps. An 1 iteration procedure that results in parametric linear programs is developed to optimize the design parameters for the transmitterreceiver pair. This is done for both the (standard) case when no feedback from the receiver to the transmitter is available and for the case when feedback is available. When only a delayed binary decision is fed back to the transmitter, which is a special instance of the second case, we also provide an implementation for finite time error recovery in terms of an additional 1 optimization.

On the role of feedback in the design of perfectly reconstructing equalizers

“…In [10] we discussed the problem of designing the DFE along with feedback/preprocessing and showed that the problem is non-convex. In addition, we proposed a method based on parameterizing the feedforward filter Q and using exhaustive search on the parameters (the number of parameters equals the delay K in the decision and for small decision delays this procedure is very efficient).…”

“…Furthermore, feedback from the received signal to the preprocessor was allowed in order to increase the tolerable noise bound even more; Figure 6 illustrates the setup of that work. The requirement of limited transmitted power was captured in [10] by requiring that |u(k)| remains within predefined limits at all times k, i.e., (s,n) → u ≤ 1.…”

“…We now restrict ourselves to DFE structures. In [10], in addition to the DFE structure, preprocessing was utilized to improve the maximum tolerable noise bound. Furthermore, feedback from the received signal to the preprocessor was allowed in order to increase the tolerable noise bound even more; Figure 6 illustrates the setup of that work.…”

“…The case when feedback from the receiver to the transmitter is available was briefly discussed in the DFE design scheme in [10]; here we discuss conditions to improve perfect reconstruction in a general setting and we show that a necessary condition for linear feedback and preprocessing to Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. [12][13][14]2007 ThPI22.24 improve the perfect reconstructability margin is that both the feedback and preprocessing units should be unstable systems. We would like to mention that consideration of feedback in communication systems using a combined control and information theoretic point of view has recently received considerable attention (e.g., [11], [12]).…”

“…For the noise we assume that we have no prior information other than a knowledge of an upper bound on its magnitude and (in the case of filtered noise) the coloring filter. In [10] we considered the design of transmitter-receiver pairs that perfectly reconstruct the discrete data with a given delay under all possible realizations of channel noise. A DFE structure was assumed for the receiver while a linear transmitter structure was imposed together with the requirement that the power of the transmission is limited.…”