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Digital Signal Processing Handbook P28

The focus of this chapter is on inverse problems — what they are, where they manifest themselves in the realm of digital signal processing (DSP), and how they might be “solved1 .” Inverse problems deal with estimating hidden causes, such as a set of transmitted symbols ftg, given observable effects such as a set of received symbols frg and a system (H) responsible for mapping ftg into frg. Inverse problems are succinctly stated using vector-space notation and take the form of | K. Venkatesh Prasad. Inverse Problems Statistical Mechanics and Simulated Annealing. 2000 CRC Press LLC. http www.engnetbase.com . Inverse Problems Statistical Mechanics and Simulated Annealing K. Venkatesh Prasad Ford Motor Company 28.1 Background 28.2 Inverse Problems in DSP 28.3 Analogies with Statistical Mechanics Combinatorial Optimization The Metropolis Criterion Gibbs Distribution 28.4 The Simulated Annealing Procedure Defining Terms References Further Reading 28.1 Background The focus of this chapter is on inverse problems what they are where they manifest themselves in the realm of digital signal processing DSP and how they might be solved1. Inverse problems deal with estimating hidden causes such as a set of transmitted symbols t given observable effects such as a set of received symbols r and a system H responsible for mapping t into r . Inverse problems are succinctly stated using vector-space notation and take the form of estimating t RM given r Ht 28.1 where r RN and H RM N and R denotes the space of real numbers whose dimensions are specified in the superscript s . Such problems call for the inversion of H an operation which may or may not be numerically possible. We will shortly address these issues but we should note here for completeness that these problems contrast with direct problems where r is to be directly without matrix inversion estimated given H and t. 1 The quotes are used to stress that unique deterministic solutions might not exist for such problems and the observed effects might not continuously track the underlying causes. Formally speaking this is a result of such problems of being ill-posed in the sense of Hadamard 1 . What is typically sought is an optimal solution such as a minimum norm minimum energy solution. c 1999 by CRC Press LLC 28.2 Inverse Problems in DSP Inverse problems manifest themselves in a broad range of DSP applications in fields as diverse as digital astronomy electronic communications geophysics 2 medicine 3 and