Exact integral equation solutions and synthesis for a large class of optimum time variable linear filters

This paper presents the exact integral equation solution and synthesis for a large class of optimum time variable linear filters characterizing many physical problems. The signal random process is expressed in nonstationary Fourier series ensemble form, with certain statistical information assumed about its coefficients. The noise perturbation is represented by a damped exponential-cosine autocorrelation function, which is of major importance in fields of physics and engineering, such as radar, meteorology, and automatic control. For any finite operating period from 0 tot, the optimum time variable weighting functionh(\tau, t)is found to be of a separable form, consisting of functions of parameter\taumultiplied by functions of parametert, plus two delta function contributions at the beginning and end. Valid synthesis designs are developed for such separable weighting functions. Asymptotic synthesis techniques are formulated which cover special situations of long-time or short-time operation. The results are applied to two examples of practical interest.