Communication Sciences and Engineering Xvii. Statistical Communication Theory A. Kernels Realizable Exactly with a Finite Number of Linear Systems and Multipliers

This group is interested in various aspects of statistical communication theory with particular emphasis on nonlinear systems and noise. A special field of investigation is that of two-state modulation. Current research problems are: (1) Application of generalized superposition theory to nonlinear filtering, (2) Studies in optimum quantization, (3) Filter performance, (4) Efficient techniques for the synthesis of nonlinear systems, (5) Analysis of nonlinear systems, (6) Coupled oscillators, (7) Analysis of commutator machines with random inputs by Volterra functionals, (8) Study of nonlinear systems through Kolmogorov partial differential equations, (9) Two-state circuitry problems, (10) Noise in two-state systems, and (11) Magnetic tape noise. 1. A theory of generalized superposition which represents an application of linear algebra to the treatment of nonlinear systems was presented by A. V. Oppenheim in 1964.1,2 The proposed research is directed toward applying this theory to nonlinear filtering. In particular, the theory suggests an approach to signal design and filtering in time-variant and multipath communications channels. Also, this research will be concerned with a study leading to a generalization of the matched filter, the filter being in general a nonlinear homomorphic system whose class is specified by the manner in which signal and noise are combined in the channel. 2. In his research on optimum quantization, J. D. Bruce 3 ' 4 has derived an exact expression for the quantization error as a functiorf of the parameters that define the quantizer, the error-weighting function, and the amplitude probability density of the quantizer-input signal. An algorithm that permits the determination of the specific values of the quantizer parameters that define the optimum quantizer has been developed. This algorithm, which is based on a modified form of dynamic programming, is valid for both convex and nonconvex error-weighting functions. Furthermore, this error expression and this algorithm have been extended to the case for which the quantizer-input signal is a message signal contaminated by a noise signal. (The contamination is not required to be additive and the noise signal is not required to be independent of the message signal.) Recent progress in this research is described in this quarterly report. 3. V. R. Algazi 5 has reported on a study of the message characteristics that lead to good or poor separation from noise by linear and nonlinear filters. His work leads to the determination of lower bounds on the filtering error for various classes of messages. This research is a new …