Scale-invariant nonlinear digital filters

Many signal processing applications involve procedures with simple, known dependences on positive rescalings of the input data; examples include correlation and spectral analysis, quadratic time-frequency distributions, and coherence analysis. Often, system performance can be improved with pre- and/or post-processing procedures, and one of the advantages of linear procedures (e.g., smoothing and sharpening filters) is their scale-invariance (x_k→y_k implies λx_k→λy_k). There are, however, important cases where linear processing is inadequate, motivating interest in nonlinear digital filters. This paper considers the general problem of designing nonlinear filters that exhibit the following scaling behavior: x_k→y_k implies λx_k→λ^νy_k for some ν>0, with particular emphasis on the case v=1. Results are presented for two general design approaches. The first is the top-down design of these filters, in which a relatively weak structural constraint is imposed (e.g., membership in the nonlinear FIR class), and a complete characterization is sought for all filters satisfying the scaling criterion for some fixed ν. The second approach is the bottom-up design of filters satisfying specified scaling behavior by interconnecting simpler filter structures with known scaling behavior. Examples are presented to illustrate both the simplicity and the utility of these design approaches