The Design of Discrete-Time Systems With Prescribed Probability Distribution and Power Spectrum Poles

This paper addresses the problem of designing nonlinear discrete-time dynamical systems for prospective use in low-complexity random signal generators. Drawing upon ergodic systems theory, we derive a novel method for designing nonlinear systems with zero-input responses that simultaneously satisfy two user-specified statistical metrics as specifications. These metrics are a piecewise-constant probability density function and a power spectral density expressed as a rational function with an arbitrary number of prescribed non-repeating poles. This new method configures a memoryless nonlinearity (i.e., a map) with a low-complexity hat-like structure as a recursive system with adjustable parameters, thereby yielding a simplified and explicit relationship between the system’s parameters and the statistical metrics of its response. This yields a matrix design approach that, unlike existing methods, affords the freedom to prescribe all of the poles as a set of non-repeating complex values without resorting to a numerical search over the parameter space for a candidate solution. Simulations are presented of elementary systems constructed using the novel method, and the link between the maps’ structure and the observed statistical behavior is expounded. More sophisticated systems with richer statistical metrics are constructed as examples that demonstrate the method’s versatility. We also demonstrate the independent adjustment of the center frequency and bandwidth of power spectrum modes during system design, which affords greater freedom than existing methods to select the power spectral density. We anticipate that the proposed method will find application in the construction of efficient random signal generators for Monte Carlo simulation in domains such as mobile communications, radar engineering, data encryption and optimization.