Focused detection via multiresolution analysis

In many applications, there are strong discrepancies between the signal models assumed in the design phase and the actual signals encountered in the field. These discrepancies penalize significantly the performance of the matched filter that is fine tuned to the preassumed conditions. We propose a geometric framework that designs, via wavelet multiresolution-based techniques, a receiver whose performance is to a large degree insensitive to these mismatches. We say that the receiver is a focused detector. The approach defines a signal set S that identifies the class of diverse conditions that are expected to arise. We illustrate the method in the context of multipath problems. The matched filter, which is a simple receiver, assumes that S is a singleton. When this is not the case, the matched filter experiences strong degradation. On the other hand, the optimal receiver for the signal set S is practically infeasible since it requires a multidimensional nonlinear optimization. The paper designs the focused receiver as a good compromise between these two extremes. We replace the signal set S by a linear subspace G-the representation subspace-that minimizes a measure of similarity with S. We choose G to be a multiresolution subspace. This choice resolves to satisfaction several issues. The subspace design is reduced to the design of a single shiftable scaling function, the similarity between S and G can be computed explicitly, and the focused receiver that computes the energy of the orthogonal projection on G is implemented by a bank of correlators matched to scaled/delayed versions of the reshaped scaling function followed by an energy detector. We assume that the transmitted signal is a sample of a random process. The signal set S becomes an ensemble of linear spaces. We introduce the modified deflection as the appropriate similarity measure. The paper details our algorithm, describes how to compute the modified deflection, and illustrates the performance results that can be obtained.