Widely Linear Estimation Algorithms for Second-Order Stationary Signals

Complex-Valued Gaussian Sum Filter for Nonlinear Filtering of Non-Gaussian/Non-Circular Noise

“…(13)) and S (l) k is defined in Eq. (16), and; (ii) Term p(F (l) |Z k−1 ) which reflects the prior knowledge regarding the weights before the new measurement z k becomes available.…”

“…Intuitively speaking, the aforementioned process addresses this drawback by selecting several points around the predicted mean, running a separate C/GSF recursion (Eqs. (8)- (16)) in parallel (per step) for each, and collapsing them using multiple-model merging algorithm to select one point. (ii) The C/GSF uses a Gaussian sum approximation for the measurement update step (i.e., going from prediction to estimation).…”

“…For an improper signal, the processing model should depend on both the signal itself and its complex conjugate. Such a methodology is referred to as widely-linear processing [14]- [16] and has been utilized to develop several estimation [17], filtering [18], and detection [19] algorithms.…”

Prediction on widely factorizable signals

“…Moreover, this kind of processing has become very usual in the last decade for designing linear and nonlinear estimation algorithms from a discrete-time [1,[5][6][7][8][9] as well as a continuous-time perspective of the problem [10]. Specifically, focussing our attention on the discrete case, the recent books of Mandic and Goh [1] and Adali and Haykin [11] about WL adaptive systems can be considered as two reference texts in this area which provide a unified treatment of linear and nonlinear complexvalued adaptive filters.…”

“…In practice this knowledge is available because second-order statistics of the problem have been measured experimentally or they are understandable enough from the physical mechanism [13]. In this framework, WL estimation algorithms for computing all types (filtering, prediction, and smoothing) of estimates have been devised in [8] for second-order stationary (SOS) signals, i.e., those signals with constant mean function and both correlation and pseudocorrelation functions only dependent on the difference of time instants. Although the WL estimation problem considered is very general, its applicability is limited to SOS signals.…”