| The decision feedback equalizer (DFE) is an effective method of combating ISI in digital communication systems. The DFE uses feedback of a weighted sum of past decisions together with a feedforward filter to equalize the ISI. The DFE ideally performs better than a linear equalizer, although, in practice, the performance of the DFE is limited by the problem of error propagation. Error propagation ocurs in a DFE when hard decisions with errors are fed back into the feedback filter.
In order to determine conditions which guarantee the non-propagation of these errors we formulate the DFE as a dynamical system in the form of a matrix equation. This resembles a block formulation of the DFE but we emphasize that this formulation as a matrix equation is only as a tool with which to analyze the dynamics of the error propagation - the filtering operation and the tap-weight updates may be performed as symbol-by-symbol operations.
Using matrix inequalities, and a Neumann series expansion for the inverse matrix, we derive constraints on the size of the feedback tap-weights so that there is exponential decay of the errors over time. We use both a deterministic approach which restricts the feedback tap-weights to give exponential decay of errors in an absolute sense, and a less confining stochastic approach, which guarantees merely that the expected value of the error decays exponentially.
A constrained minimization of the MSE can be performed, using the derived constraints, to give filter tap-weights for which errors are guaranteed to "die out" and not propagate. Several authors have considered constrained MMSE DFE tap-weight calculations, where the constraint is designed to impede error propagation. The present paper provides a theoretical justification for the efficacy of such constraints in inhibiting error propagation.
We provide simulations to justify the use of the method as compared to unconstrained tap-weight computation with regard to error propagation.
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| Christopher Pladdy was born in Cardiff, Wales, U.K. in 1965. He received the B.Sc. degree in Mathematics from Cardiff University, Wales, U.K. in 1987, and the M.S. degree (Mathematics) and Ph.D. degree (Applied Mathematics) from the University of Alabama at Birmingham in 1991 and 1994 respectively. His dissertation work was in the areas of functional analysis and mathematical physics and he has published several research papers in these areas.
From 1994 to 1998 he was Assistant Professor of Mathematics at Nicholls State University in Thibodaux, LA, and taught courses across the mathematics spectrum including Differential Equations, Linear Algebra, Numerical Analysis, Applied Matrix Analysis, and the Calculus sequence.
From 1998 to 2001 he was an adjunct lecturer in mathematics at the University of Florida, while simultaneously pursuing graduate studies in electrical engineering. He graduated with an M.S. degree in Electrical Engineering from the University of Florida in 2001, with an emphasis in signal processing and digital communication, and joined Zenith Electronics R&D Center in Lincolnshire, IL, as a Research Engineer.
Here he worked primarily on equalization of terrestrial HDTV signals including adaptive equalization, channel estimation, diversity gain, decision feedback equalizers (time and frequency domain tap-weight calculation and filtering), and a predictive DFE architecture (linear equalizer and noise canceller, separately optimized; time and frequency domain tap-weight calculation and filtering). At Zenith the work was broadly focused on the goal of mobile reception of ATSC 8-VSB HDTV signals.
Since September 2005 he has been a Lead Operations Research Analyst with the MITRE Corporation in Leavenworth, KS. At MITRE he has worked on FCS network problems and will begin to work again on physical layer problems, including source separation and equalization for interference limited communications.
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