Statistical Signal Processing in Engineering

16Line Spectrum Analysis

The estimation of the frequencies of multiple sinusoidal signals is remarkably important and common in engineering, namely for the Fourier analysis of observations or experiments that model the observations as random set of values whose statistical properties depend on the angular frequencies. Since the problem itself reduces to the estimation of the (angular) frequencies as a generalization of Section 9.2 and Section 9.3, one might debate the rationale of dedicating a whole chapter to this. The problem itself needs care in handling the assumptions behind every experiment and these were deliberately omitted in the early chapters. Crucial problems addressed here are (1) resolution (estimating two sinusoids with similar frequencies), (2) finding alternative methods to solve non‐linear optimizations, and (3) evaluating accuracy wrt the CRB. All these matters triggered several frantic research efforts around the ’80s in this scientific area. A strong boost was given by array processing (Chapter 19) as the estimation of the spatial properties of wavefields from multiple radiating sources in space is equivalent to the estimation of (spatial) angular frequencies, and resolution therein pertains to the possibility to spatially distinguish the emitters. Since MLE is not trivial, and closed form optimizations are very cumbersome, the scientific community investigated a set of so called high‐resolution (or super‐resolution¹) methods based on some ingenious algebraic ...