ARMA Prediction of Widely Linear Systems by Using the Innovations Algorithm

Prediction on widely factorizable signals

“…As a measure for comparing the performance of the WL and SL fixedlead predictors we use the one defined in [6], the mean square of the difference between both errors, for…”

“…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.…”

Augmented second-order statistics of quaternion random signals

“…Also, the invariance of a Q-proper random vector under a linear or affine transformation (shown in Appendix X-B) is similar to that in the complex case (see Lemma 3 [11]). This invariance arises due to the properties in (16) and the condition of vanishing ı−covariance matrix, C qı = 0, in (13) is equivalent to the conditions…”

“…Based on the proof of Lemma 3 of [11] and the special properties of complementary covariance matrices in (16), y is also a Q−proper random vector, as shown by C yı = E{yy ıH } = AC qı A ıH = 0 C y = E{yy H } = AC q A H = 0 C yκ = E{yy κH} = AC qκ A κH = 0…”

“…Their work was followed by Picinbono [12] and Van Den Bos, who formulated a generic Gaussian distribution of both proper and improper complex processes, to show that the traditional definition of the complex Gaussian distribution (based on the covariance) is only a special case, applicable to proper processes only [13]. These foundations have been successfully used to design novel algorithms in adaptive signal processing [14], communications [15], autoregressive moving average (ARMA) modelling [16], and independent component analysis [17]. By virtue of augmented complex statistics, all these results are applicable to the generality of complex signals, both second order circular and noncircular.…”