Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory

A general solution to the continuous-time estimation problem under widely linear processing

“…Note that the formulation of the estimation problem treated in [29] is much more restrictive than that studied in the present paper.…”

“…The key question is whether the conditions of Theorem 3 are fulfilled. An example of the latter is the classical problem of estimating an improper complex-valued random signal in colored noise with an additive white part addressed in [29]. Specifically, the observation process considered is…”

“…The WL estimation problem under a continuous-time formulation was initially dealt with in [27,28] and [29]. More precisely, the particular problem of estimating a complex signal in additive complex white noise is solved in [27] or [28] through an improper version of the Karhunen-Loève expansion.…”

“…A general result comparing the performance of WL and SL processing is also presented in which it is shown that the performance gain, measured by MS error, can be as large as 2. Finally, [29] provides an extension of the previous problem to the case in which the additive noise is made up of the sum of a colored component plus a white one. The handicaps of both solutions are: i) they are limited to MS continuous signals, ii) the signals must be defined on finite intervals, iii) the model for the observation process involves additive noise (white noise in the case of [27] and [28]), and iv) they are only devoted to solving a smoothing problem.…”

“…In this paper, we address a more general estimation problem than those solved in [27][28][29]. For that, we consider the general formulation of the estimation problem given in [15], and we solve it by using WL processing.…”