Fundamental Theory and Techniques for Nonlinear Signal Processing

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In telecommunications, video-technology, and multimedia applications, a real challenge is posed by some genuinely nonlinear problems for the solution of which, unlike in the linear case, a final unified theory is not yet available. Such nonlinear problems include cancellation of large nonlinear distortion in very-large-scale electronic circuits and systems (VLSI), nonlinear sharpening of contrast in noisy images, modeling and identification of nonlinear signal and image sources, and content-based nonlinear compression and decompression of signals and images. The goal of the current project is to develop a rigorous and yet practical framework for the design/synthesis of nonlinear analog, digital, and mixed mode filters for the solution of the above types of nonlinear problems. The approach followed in this project is based on an Generalized Fock (GF) space framework developed by the PI (R.J.P. deFigueiredo), T.A. Dwyer, and L.V. Zyla. A GF space is a reproducing kernel Hilbert space of discrete- or continuous- parameter Volterra functional series with a problem-dependent weighted inner product. In the context of a given application-specific problem, the optimal nonlinear filter structure is obtained by an orthogonal projection of the desired filter into the subspace spanned by the representers of interpolating, smoothing, and other design constraint functionals in an appropriate GF space. A number of analysis, synthesis, and implementation issues constitute the focus of the investigation.