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Prediction on widely factorizable signals
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 96 (2012)
Abstract
This article extends the solutions to the prediction problem for factorizable real random signals to the class of improper complexvalued random signals. For that, the concept of widely factorizable signals is introduced and several real examples of signals having a widely factorizable correlation function are presented. A widely linear processing is considered in the design of both linear and nonlinear prediction algorithms which are computationally feasible from the practical standpoint. These algorithms are valid for stationary as well as nonstationary signals and they can be applied from only the knowledge of the secondorder properties of the augmented vectors involved, it not being necessary to know if the signal satisfies a statespace model.
1 Introduction
In recent research, estimation theory becomes very relevant within the complex random signals field. In fact, although traditionally the treatment of this type of problem has consisted in mere extensions of the vectorial realvalued estimation algorithms to the complex plane, a complex formalism has a special value in the description of some physical systems in such diverse fields as communications, oceanography, meteorology and optics among others (see, for example, [1] and the references therein).
The classic processing with such signals, called strictly linear (SL), assumes that the signals involved are proper.^{a} However, this assertion is not always justified. The improper nature of some signals requires that the pseudocorrelation function must be taken into account in describing and characterizing their second order properties completely [2–4].
With this motivation, [2] introduces a new methodology called widely linear (WL) whose more notable characteristic is that it utilizes not only the observed signal but also its conjugate to obtain estimators with better behavior than the conventional ones in the sense of reducing the mean square estimation error. Moreover, this kind of processing has become very usual in the last decade for designing linear and nonlinear estimation algorithms from a discretetime [1, 5–9] as well as a continuoustime 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.
On the other hand, knowledge of secondorder statistics is a key assumption in solving estimation problems (see, for example, [12]). In practice this knowledge is available because secondorder 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 secondorder 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.
In the real field, an alternative estimation methodology based on correlation information has been developed to solve very general linear and nonlinear estimation problems for the class of factorizable signals (see, for example, [14, 15]). This type of signal is characterized by having a factorizable kernel which is a very general condition valid for stationary and nonstationary signals. This formulation does not require postulating a dynamical model for the signal and thus, it is useful whenever the physical mechanism generating the signal of interest is not known or if it impossible to determine (for instance, if it does not satisfy a statespace model).
In this article, our objective is to extend this last methodology to the complex plane. For that, we introduce a new class of signals, called widely factorizable, whose correlation function of the augmented vector formed by the signal and its conjugate is a factorizable kernel. Then, a WL processing is employed in the development of both linear and nonlinear prediction algorithms for widely factorizable signals observed in the presence of noise. With this aim, in Section 2 widely factorizable signals are defined and also the basic notation and concepts about complexvalued signals are summarized. Next, the linear augmented complex prediction problem is addressed in Section 3 where a recursive algorithm is provided for computing the optimal WL prediction estimate as well as its associated error. This section also includes three numerical examples which show the enhancement of the proposed WL predictor in relation to the SL solution. Finally, by following a similar reasoning to the extended Kalman filter (EKF), the nonlinear augmented complex prediction problem is solved in Section 4 where a numerical example is also developed in order to compare the proposed WL algorithm with two conventional nonlinear augmented complex techniques, the WL EKF and the WL unscented Kalman filter (UKF) suggested in [1]. Furthermore, these solutions are also compared with the SL EKF and SL UKF.
2 Preliminaries
This section covers some of the more basic notions associated with the complexvalued random signals field and introduces a new class of signals, called widely factorizable. Moreover, the notation and hypotheses held throughout the article are also established in this section.
First of all, note that all vectors will be denoted by bold small letters and bold capital letters will be used for matrices. Also, row k of any matrix A(·) will be denoted by a_{[k]}(·). Furthermore, 0_{ p }and 0_{ p × q }represent the pvector and the p × qmatrix, respectively, whose elements are all zeros.
Unless indicated to the contrary, throughout this article we consider an improper complexvalued signal {s(t_{ i } ), t_{ i } ∈ T}, T = {t_{1}, t_{2}, ...}, with zeromean and correlation function r_{ s } (t_{ i }, t_{ j } ) = E[s(t_{ i } )s* (t_{ j } )], where the superscript '*' represents the complex conjugate.
The signal {s(t_{ i } ), t_{ i } ∈ T} is said to be factorizable if there exist two lvectors α(t_{ i } ) and β(t_{ i } ) such that its correlation function r_{ s } (t_{ i }, t_{ j } ) can be expressed in the form
where the superscript 'T' denotes the transpose. Note that this type of signal is very general and includes both stationary and nonstationary signals. However, a new class of signals is possible by imposing the condition of factorizable kernel on the correlation function R_{ s }(t_{ i }, t_{ j } ) = E[s(t_{ i } )s^{H}(t_{ j } )], with the superscript 'H' denoting the conjugate transpose, of the augmented vector s(t_{ i } ) = [s(t_{ i } ), s* (t_{ i } )] ^{T}. This type of signal, called widely factorizable, is introduced next.
Definition 2.1 A signal {s(t_{ i } ), t_{ i } ∈ T} is said to be widely factorizable if and only if there exist two 2 × mmatrices A(t_{ i } ) and B(t_{ i } ) such that the correlation function R_{ s }(t_{ i }, t_{ j } ) of the augmented vector s(t_{ i } ) can be expressed in the form
where the superscript 'H' denotes the conjugate transpose.
Note that condition (2) implies (1), however condition (1) does not assure that the correlation function of the augmented vector satisfies (2). As a consequence, we have that all widely factorizable signals are also factorizable but the converse does not hold.
Some illustrative examples of widely factorizable signals are the following:
i) The rotation of a real factorizable zeromean signal x(t_{ i } ) by an independent random phase θ, s(t_{ i } ) = e^{θ j}x(t_{ i } ), with $\text{j}\phantom{\rule{0.3em}{0ex}}\text{=}\phantom{\rule{0.3em}{0ex}}\sqrt{1}$. Assume
with a(t_{ i } ) and b(t_{ i } ) two real lvectors. Thus,
where φ_{ θ } (·) is the characteristic function of θ.
ii) The seismic ground acceleration can be represented by a uniformly modulated nonstationary process given by s(t_{ i } ) = d(t_{ i } )x(t_{ i } ), where x(t_{ i } ) is a stationary process with zero mean and known secondorder statistics, and d(t_{ i } ) is the time modulating function. Both x(t_{ i } ) and d(t_{ i } ) can be either real or complexvalued [16]. Thus, if d(t_{ i } ) is complex and x(t_{ i } ) is real and factorizable with correlation function given by (3) then, s(t_{ i } ) is widely factorizable being
Common choices are $d\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{e}^{\text{j}2\mathbf{\pi}{\mathbf{\omega}}_{0}{t}_{i}}$ with ω_{0} a constant frequency and x(t_{ i } ) the OrnsteinUhlenbeck process.
On the other hand, if both d(t_{ i } ) and x(t_{ i } ) are complex and x(t_{ i } ) is also widely factorizable with
then, s(t_{ i } ) is widely factorizable with
iii) In electromagnetic theory, the timevarying position of the electric field vector can be represented as the following signal $s\left({t}_{i}\right)=A{e}^{\mathrm{j}{w}_{0}{t}_{i}}+B{e}^{\mathrm{j}{w}_{0}{t}_{i}}$ where the expressions for the random coefficients A and B can be found in [[17], p. 7]. Hence, s(t_{ i } ) is widely factorizable with
where $e\left({t}_{i}\right)={e}^{\text{j}{\mathbf{\omega}}_{0}{t}_{i}},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{r}_{A}=E\left[A{A}^{*}\right],\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{r}_{AB}=E\left[A{B}^{*}\right],\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{\gamma}}_{A}=E\left[AA\right],\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{\gamma}}_{B}=E\left[BB\right]$, and γ_{ AB }= E[AB].
iv) A signal widely used in many areas of signal processing is a linear frequency modulation or chirp with random phase. The chirp process can be expressed as $s\left({t}_{i}\right)={e}^{\text{j}\mathbf{\pi}\left(2\mathit{\alpha}{t}_{i}+\mathit{\beta}{t}_{i}^{2}+2\mathbf{\theta}\right)}$ where θ is the random phase, α determines the starting instantaneous frequency of the chirp and β is the chirp rate. Then, the chirp process is widely factorizable with
with $c\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{e}^{\text{j}\mathbf{\pi}\left(2\mathit{\alpha}{t}_{i}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\mathit{\beta}{t}_{i}^{2}\right)}$ and φ_{ θ } (·) the characteristic function of θ.
v) An application of the complex OrnsteinUhlenbeck process is the description of the motion of the instantaneous axis of the Earth's rotation [18]. This motion has an 1 year period and if it is removed, there remains the socalled Chandler Wobble, which has a period of about 435 days (14 months). Kolmogorov proposed the complex stochastic process $s\left({t}_{i}\right)=x\left({t}_{i}\right)+\text{j}y\left({t}_{i}\right)=\mathbf{\sigma}{e}^{\text{j}2\mathbf{\pi}{t}_{i}}+\mathbf{\xi}\left({t}_{i}\right)$ to describe the Chandler Wobble, i.e., the motion of the pole, where x(t_{ i } ) and y(t_{ i } ) are the coordinates of the deviation of the instantaneous pole from the North Pole. In that model the first term is a periodical component, and the second term ξ(t_{ i } ) is a complex OrnsteinUhlenbeck process. It is not difficult to check that the signal s(t_{ i } ) is proper and then,
where λ > 0 is the drift parameter and ω ∈ ℝ is the period.
On the other hand, we provide a simple example of a factorizable signal which is not widely factorizable: a zeromean complexvalued signal {s(t_{ i } ), t_{ i } = i/100, i = 1, ..., 100} with ${r}_{s}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}{t}_{j}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}2{e}^{\left{t}_{i}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{t}_{j}\right}$ and ${\rho}_{s}\left({t}_{i},{t}_{j}\right)=\text{j}0.5{e}^{{t}_{i}{t}_{j}}.$
In the following, we also assume that the signal of interest s(t_{ i } ) is widely factorizable in the sense of Definition 2.1.
Finally, R_{ sy }(t_{ i }, t_{ j } ) = E[s(t_{ i } )y^{H}(t_{ j } )] denotes the crosscorrelation function between any two augmented signals s(t_{ i } ) and y(t_{ i } ), and r_{ s y }(t_{ i }, t_{ j } ) = E[s(t_{ i } )y^{H}(t_{ j } )] represents the crosscorrelation function between s(t_{ i } ) and the augmented vector y(t_{ i } ).
3 Linear augmented complex prediction
Assume that the signal s(t_{ i } ) established in Section 2 is observed through the following linear equation:
where g(t_{ i } ) is a deterministic complexvalued function and v(t_{ i } ) is a doubly white noise^{b} correlated with s(t_{ i } ). Moreover, the crosscorrelation function and the augmented signal s(t_{ i } ) and the augmented noise v(t_{ i } ) is of the form
where C(t_{ i } ), D(t_{ i } ), E(t_{ i } ), and F(t_{ i } ) are matrices of dimensions 2 × l, 2 × l, 2 × l^{'}, and 2 × l^{'}, respectively.
We consider the problem of obtaining the optimal (in the sense of minimizing the WL mean square error) estimator^{c} of the signal s(t_{ k } ) as a function of the information given by the observations {y(t_{1}), ..., y(t_{ n } ), y*(t_{1}), ..., y*(t_{ n } )}, for t_{ k } ≥ t_{ n } . It is known that such an estimator can be expressed as a linear function of the set of augmented observations {y(t_{1}), ..., y(t_{ n } )} as follows [2]
where the 2D vector h(t_{ k }, t_{ j }, t_{ n } ), called the impulse response function, satisfies the equation
where $\mathbf{R}\left({t}_{i},{t}_{j}\right)=\mathbf{G}\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}{\mathbf{R}}_{s}\left({t}_{i},{t}_{j}\right){\mathbf{G}}^{\text{H}}\left({t}_{j}\right)+\mathbf{G}\left({t}_{i}\right){\mathbf{R}}_{\mathbf{s}\mathbf{v}}\left({t}_{i},{t}_{j}\right)+{\mathbf{R}}_{\mathbf{s}\mathbf{v}}^{\text{H}}\left({t}_{j},{t}_{i}\right){\mathbf{G}}^{\text{H}}\left({t}_{j}\right)$, with G(t_{ i } ) denoting the 2 × 2diagonal matrix G(t_{ i } ) = diag(g(t_{ i } ), g*(t_{ i } )), and E[v(t_{ i } )v^{H}(t_{ i } )] = Σ.
From (2) and (7), it is easy to check that r_{ s y }(t_{ k }, t_{ j } ) and R(t_{ i }, t_{ j } ) can be written as follows:
where ψ_{[1]}(t_{ k } ) is the first row of the 2 × qmatrix $\mathbf{\Psi}\left({t}_{i}\right)=\left[\mathbf{A}\left({t}_{i}\right),\mathbf{C}\left({t}_{i}\right),{0}_{2\times {l}^{\mathbf{\prime}}}\right]$, π_{[1]}(t_{ k } ) is the first row of the 2×qmatrix Π(t_{ i } ) = [B(t_{ i } ), 0_{2×l}, E(t_{ i } )], Φ(t_{ i } ) = [G(t_{ i } )A(t_{ i } ), G(t_{ i } ) C(t_{ i } ), F(t_{ i } )], and Γ(t_{ i } ) = [G(t_{ i } ) B(t_{ i } ), D(t_{ i } ), G(t_{ i } ) E(t_{ i } )] are also matrices of dimensions 2 × q, with q = m + l + l^{'} .
Although the problem is completely determined from the computation of the impulse response function by solving Equation (9), our aim here is to provide a recursive algorithm for its computation. Next, the recursive formulas for computing the estimator (8) and its associated error p(t_{ k } t_{ n } ) = E[s(t_{ k } )  ŝ(t_{ k } t_{ n } )^{2}] are devised.
Theorem 3.1 The optimal WL estimate ŝ(t_{ k } t_{ n } ) defined in (8) can be recursively computed as follows:
where the qvector ϵ (t_{ n } ) is recursively computed from the expression
with the q × 2matrix J(t_{ n }, t_{ n } ) given by the equation
with the 2 × 2matrix Ω(t_{ n } ) = Σ + [Γ(t_{ n } )  Φ(t_{ n } ) Q (t_{n1})] Φ ^{H} (t_{ n } ) and the q × qmatrix Q(t_{ n } ) satisfying the recursive equation
Moreover, the associated error is given by the expression
Proof. From (10) and (11), Equation (9) can be rewritten as
Then, if we introduce a function J(t_{ j }, t_{ n } ) satisfying the equation
we obtain that
and then, substituting (18) in (8), and defining the function
the Equation (12) for the optimal estimator is devised.
Now, subtracting the Equation (17) for t_{ n } and t_{n 1}and taking (11) into account, we can write
Thus, from (17), we have the relation
As a consequence, subtracting the Equation (19) for t_{ n } and t_{n 1}and using (20) in the resulting equation, the recursive expression (13) is obtained.
Next, we proceed to derive expression (14) for J(t_{ n }, t_{ n } ). By taking t_{ j } = t_{ n } in (17) and using (11), we have
where we have introduced the q × qmatrix
Moreover, if we subtract Q(t_{n 1}) from Q(t_{ n } ), and use (20) and (22) in the resulting expression, the recursive Equation (15) for Q(t_{ n } ) is derived. Finally, using (15) in (21), it is easy to check that J(t_{ n }, t_{ n } ) satisfies the expression (14).
Finally, in order to derive expression (16) for the error p(t_{ k }t_{ n } ) associated with the above estimate, we remark that, from the orthogonal projection lemma, this function can be expressed as
Then, substituting (12) in the above equation and using (17) and (19), we check that
As a consequence, from (22), (16) is obtained.
Remark 1 When {s(t_{ i } ), t_{ i } ∈ T} is a factorizable realvalued signal with correlation function of the form (1), and the observations of the signal verify the complexvalued linear Equation (6) with
where c(t_{ i } ) and e(t_{ i } ) are vectors of dimensions l and l', respectively, and D(t_{ i } ) and F(t_{ i } ) are matrices of respective dimensions 2 ×l and 2 ×l^{'}, we obtain that Algorithm 3.1 holds, replacing the involved 2 × qmatrices Ψ(t_{ i } ), Π(t_{ i } ) by the vectors ${\mathit{\psi}}^{T}\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left[{\mathit{\alpha}}^{T}\left({t}_{i}\right),\phantom{\rule{0.3em}{0ex}}{\mathbf{c}}^{T}\left({t}_{i}\right),\phantom{\rule{0.3em}{0ex}}{0}_{{l}^{\mathbf{\prime}}}^{T}\right]$ and ${\mathbf{\pi}}^{T}\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[{\mathit{\beta}}^{T}\left({t}_{i}\right),\phantom{\rule{0.3em}{0ex}}{0}_{l}^{T},\phantom{\rule{0.3em}{0ex}}{\mathbf{e}}^{T}\left({t}_{i}\right)\right]$, and taking the matrices Φ(t_{ i } ) = [g(t_{ i } ) α ^{T} (t_{ i } ), g(t_{ i } )c ^{T} (t_{ i } ), F(t_{ i } )] and Γ(t_{ i } ) = [g(t_{ i } )β^{T}(t_{ i } ), D(t_{ i } ), g(t_{ i } )e ^{T} (t_{ i } )], with g(t_{ i } ) = [g(t_{ i } ), g*(t_{ i } )]^{T} .
Remark 2 The efficiency of Algorithm 3.1 is closely related to the dimensions l, l^{'}, and m of the matrices involved in the factorizations (2) and (7). Indeed, the computational complexity of this algorithm is of order q, with q = m + l + l^{'}, and thus, it involves a further complication in implementation and an increased computational burden as q grows. Since the factorization of the covariance is not unique then, the key question is in choosing the one which minimizes the dimension q. There exist simple cases, as illustrated previously, where the factorization is easily obtained. Nevertheless, in those more complex cases where this factorization is not trivial, one can use several methods available in the literature to get a factorization with minimum dimension (see, e.g., [19]).
3.1 Numerical examples
The advantages of the proposed Algorithm with respect to the SL solution are illustrated here through three numerical examples. The first one involves real correlation matrices and analyzes the effectiveness of the WL processing with respect to the SL one in terms of the impropriety degree of the observations. In the second example, complex correlation matrices are considered and the resulting WL and SL estimation errors are graphically compared. Finally, the third example shows a real application to seismic signal processing.
3.1.1 Example
Let {x(t_{ i } ), t_{1}≤ t_{ i } ≤ t_{100}}, with t_{ i } = i/ 100, i = 1, ..., 100, be an OrnsteinUhlenbeck process with correlation function
which is transmitted over a channel that rotates it by a standard normal phase θ and adds a doubly white Gaussian noise v(t_{ i } ) correlated with the signal with
Thus, the signal of interest is s(t_{ i } ) = e^{θ j}x(t_{ i } ) and the observations y(t_{ i } ) are of the form (6) with g(t_{ i } ) = 1. Moreover, we assume that θ is independent of x(t_{ i } ) and v(t_{ i } ).
Note that the correlation matrix of the augmented signal s(t_{ i } ) can be expressed in the form (2), where A and B are as in (4) with l = 1, $a\left({t}_{i}\right)={e}^{{t}_{i}}$, $b\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{e}^{{t}_{i}}$, and φ_{ θ } (2) = φ_{ θ } (2) = e^{2}. Moreover, the crosscorrelation matrix between s(t_{ i } ) and v(t_{ i } ) is of the form (7) with D(t_{ i } ) = F(t_{ i } ) = [t_{ i } /25, t_{ i } /25]^{T} and $\mathbf{C}\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathbf{E}\left({t}_{i}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left[{e}^{1/2}{t}_{i}^{2},\phantom{\rule{0.3em}{0ex}}{e}^{1/2}{t}_{i}^{2}\right]}^{\text{T}}$.
In this example, we consider the problem of computing the fixedlead predictor ŝ(t_{k+10}t_{ k } ). 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
with τ varying within the interval [1, 2):
with p_{ τ } (t_{k+10}t_{ k } ) and ${\stackrel{\u0303}{p}}_{\tau}\left({t}_{k+10}{t}_{k}\right)$ denoting the WL and SL fixedlead prediction errors, respectively, for every value τ. The results obtained are displayed in Figure 1 which shows that, not only the WL fixedlead predictor presents a better behavior than the SL fixedlead predictor but also the difference between both errors (in the mean square sense) increases with τ , and hence, the WL technique becomes more effective.
3.1.2 Example
Let {s(t_{ i } ), t_{1}≤ t_{ i } ≤ t_{100}}, with t_{ i } = i/100, i = 1, ..., 100, be a signal of the form
where A and B are complex random variables. In this example, the signal is assumed to be widely stationary, that is E[AA] = E[BB] = E[AB*] = 0 (see [[17], p. 24]), and also we consider that E[AA*] = E[BB*] = 1 and E[AB] = 0.8. Thus, by using (5), the correlation matrix of the signal s(t_{ i } ) can be expressed in the form (2). Moreover, we consider that the observations y(t_{ i } ) are of the form (6) with g(t_{ i } ) = 1 and where the noise ν(t_{ i } ) is uncorrelated with the signal and its augmented variance matrix is
On the basis of the set of observations {y(t_{1}), y(t_{2}), ..., y(t_{100})}, we consider the problem of computing the fixedlead predictor ŝ(t_{k+10}t_{ k } ). Then, Algorithm 3.1 is used to obtain the WL fixedlead prediction error p_{ τ } (t_{k+10}t_{ k } ) which is compared with the SL fixedlead prediction error ${\overline{p}}_{\tau}\left({t}_{k+10}{t}_{k}\right)$ in Figure 2. As could be expected, this figure shows that the WL fixedlead predictor presents a better behavior than the SL fixedlead predictor. Finally, Figure 3 depicts the performance measure given by (24) with τ ∈ [1, 2) and Σ as in (23). Again, the improved precision attained with the WL fixedlead predictor with respect to the SL fixedlead predictor is observed as τ increases.
3.1.3 Example
As indicated in Section 2, uniformly modulated nonstationary processes are often used to model seismic records, especially acceleration records. The modulated nonstationary process is given by s(t_{ i } ) = d(t_{ i } )x(t_{ i } ), where x(t_{ i } ) is a stationary process with zero mean and known secondorder statistics, and d(t_{ i } ) is the time modulating function. A stochastic earthquake model commonly used for x(t_{ i } ) is the KanaiTajimi process (see, e.g., [20]). It is wellknown that the KanaiTajimi earthquake model is covariance equivalent with the subset of the ARMA(2,1) model corresponding to a unit value of the springdashpot input ratio [21]. For firm ground conditions, at moderate epicentral distance, Kanai and Tajimi have suggested specific values for the parameters in the equation of motion in continuous time whose corresponding discrete ARMA(2,1) model is [21]
with ${\sigma}_{e}^{2}=\text{39}.0\text{8}$ and Δt = t_{ i } t_{i 1}= 0.02 s. This ARMA model has a factorizable correlation function given by (3). Moreover, we have chosen the modulating function $d\left({t}_{i}\right)={e}^{\text{j}{t}_{i}}$ and the augmented error covariance matrix of the form
with τ ∈ [0, 1). Here we have studied the filtering problem. Figure 4 illustrates the enhancement of the WL filter in relation to the SL one by using the measure
Similar to the previous examples, we observe the superiority of the WL estimate as τ increases.
4 Nonlinear augmented complex prediction
Given the same conditions on the signal established in Section 2, we suppose that the observation process can be given by a nonlinear relation of the form
where z(·) is a complexvalued nonlinear function and the signal s(t_{ i } ) is uncorrelated with the noise ν(t_{ i } ). As in the linear case, the aim here is to estimate the signal s(t_{ k } ) on the basis of the observation set {y(t_{1}), ..., y(t_{ n } ), y*(t_{1}), ..., y*(t_{ n } )}, with t_{ k } ≥ t_{ n } .
The solution to this problem is addressed by following a similar philosophy to the EKF [22]. Specifically, following the basic idea of the EKF the nonlinear function z(s(t_{ n } ), t_{ n } ) is linearized at each time instant by a firstorder Taylor series expanded about the estimated signal ŝ(t_{ n } t_{n1})
Consequently, we can proceed to approximate the nonlinear observation Equation (25) as shown by
where
and
and thus, s(t_{ k } ) can be estimated in terms of the set of observations $\left\{\mathit{\u0233}\left({t}_{1}\right),\dots ,\mathit{\u0233}\left({t}_{n}\right),{\mathit{\u0233}}^{*}\left({t}_{1}\right),\dots ,{\mathit{\u0233}}^{*}\left({t}_{n}\right)\right\}$ from the relation (26). For that, the formulas given in Algorithm 3.1 for the WL predictor can be used. Note that in the case of the signal and observation noise being uncorrelated, Algorithm 3.1 holds with Ψ(t_{ i } ) = A(t_{ i } ), Π(t_{ i } ) = B(t_{ i } ), Φ(t_{ i } ) = G(t_{ i } )A(t_{ i } ) and Γ(t_{ i } ) = G(t_{ i } )B(t_{ i } ). Then, as in the EKF, in the resulting formulas we also use the linearized observation function (27) in place of the previous function g(t_{ n } ) and the term G(t_{ n } )A(t_{ n } )ϵ(t_{n 1}) is replaced by the vector
Next the formulas of the proposed Algorithm are summarized.
Theorem 4.1 A nonlinear augmented complex predictor of the signal s(t_{ k } ) based on the set of nonlinear observations {y(t_{1}), ..., y(t_{ n } ), y* (t_{1}), ..., y*(t_{ n } )} of the form (25) can be determined through the equation
where the mvector ϵ (t_{ n } ) is recursively computed from the expression
with
where Ω(t_{ n } ) = Σ + G(t_{ n } ) [B(t_{ n } )  A(t_{ n } )Q(t_{n 1})] A ^{H} (t_{ n } )G ^{H} (t_{ n } ), G(t_{ n } ) = diag(g(t_{ n } ), g*(t_{ n } )), with$g\left({t}_{n}\right)=\partial z\left(s,\phantom{\rule{2.77695pt}{0ex}}{t}_{n}\right)/\partial s{}_{s=\mathit{\u015d}\left({t}_{n}{t}_{n1}\right)},$and Q(t_{ n } ) satisfies the recursive equation
Remark 3 Similarly to Remark 1, if the signal {s(t_{ i } ), t_{ i } ∈ T} is a realvalued signal with factorizable kernel of the form (1) which is observed through a complexvalued nonlinear equation of the form (25), Algorithm 4.1 holds with a_{[1]}(t_{ i } ) = α^{T}(t_{ i } ), g(t_{ i } ) = [g(t_{ i } ), g*(t_{ i } )] ^{T} and replacing the 2 × qmatrices A(t_{ i } ) and B(t_{ i } ) by the vectors α (t_{ i } ) and β (t_{ i } ), respectively.
4.1 Numerical example
In this Example, the estimation of a real signal on the basis of a set of complexvalued observations is considered. Specifically, let {s(t_{ i } ), t_{1}≤ t_{ i } ≤ t_{100}} be a real Wiener process with unit variance parameter and the observation equation
where v(t_{ i } ) = e^{θ j}u(t_{ i } ), with θ a standard normal phase and u(t_{ i } ) a white Gaussian noise with unit spectral height and uncorrelated with the signal s(t_{ i } ).
In this case, the correlation function of the signal s(t_{ i } ) can be expressed in the form (1) with α(t_{ i } ) = 1 and β(t_{ i } ) = t_{ i } .
With the aim of examining the good behavior of the WL solution proposed in Algorithm 4.1, the estimation error of the WL filter is compared with the errors associated with SL and WL conventional Algorithms. Specifically, the standard EKF and UKF and the WL EKF and WL UKF proposed in [1] have been implemented. For that, we use the fact that the signal s(t_{ i } ) obeys the state equation
with w(t_{ i } ) a centered Gaussian signal with variance parameter 10^{2}.
On the other hand, for computing the estimation errors, Monte Carlo simulations have been performed. Figure 5 shows the results obtained with 5000 sample paths, confirming the better behavior of a WL processing in the nonlinear estimation problem. In fact, the dashed line represents the errors associated with the SL EKF and SL UKF (the differences between them are negligible) and the solid line depicts the errors associated with the WL UKF, WL EKF and the filter given in Algorithm 4.1 (again the differences between them are negligible). Obviously, the similar behavior shown here by the three WL filters has not to be repeated in other examples. As occurs in the standard estimation techniques, there is not a best nonlinear estimator either. In each application one has to pick the appropriate nonlinear estimation method. Really, in every particular case one has to choose the estimator which is found to best trade off various properties such as estimation accuracy, ease of implementation, numerical robustness, and computational burden [22]. Note that unlike UKF and EKF, Algorithm 4.1 does not require a state space model but only the knowledge of the secondorder statistics of the processes involved.
Endnotes
^{a}A zeromean complexvalued signal {s(t_{ i } ), t_{ i } ∈ T}, T = {t_{1}, t_{2}, ..., } is said to be proper if the pseudocorrelation function, ρ_{ s } (t_{ i }, t_{ j } ) = E [s(t_{ i } )s(t_{ j } )], is null for all t_{ i }, t_{ j } ∈ T. Otherwise, it is called improper. ^{b}v(t_{ i } ) is said to be a doubly white noise if E[v(t_{ i } )v* (t_{ j } )] = σ_{1}δ_{ ij } and E[v(t_{ i } )v(t_{ j } )] = σ_{2}δ_{ ij } , with σ_{2} < σ_{1} and δ_{ ij } stands for the Kronecker delta function [3]. ^{c}A simple application of the Hilbert space projection theorem shows that WL estimation outperforms SL estimation for general complexvalued signals. Specifically, denote $\overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right)\right\}$ and $\overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right),{y}^{*}\left({t}_{\text{1}}\right),\dots ,{y}^{*}\left({t}_{n}\right)\right\}$ the closed spans of the following sets {y(t_{1}), ..., y(t_{ n } )} and {y(t_{1}), ..., y(t_{ n } ), y*(t_{1}), ..., y (t_{ n } )}, respectively. Let $\tilde{s}\left({t}_{k}{t}_{n}\right)$ and ŝ(t_{ k } t_{ n } ) be the projections of s(t_{ k } ), t_{ k } ≥ t_{ n } , onto the spaces $\overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right)\right\}$ and $\overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right),{y}^{*}\left({t}_{\text{1}}\right),\dots ,{y}^{*}\left({t}_{n}\right)\right\}$ respectively. Thus, $\tilde{s}\left({t}_{k}{t}_{n}\right)$ is the SL estimate of s(t_{ k } ) and ŝ(t_{ k } t_{ n } ) is its WL estimate. Since $\overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right)\right\}\subseteq \overline{\text{sp}}\left\{y\left({t}_{\text{1}}\right),\dots ,y\left({t}_{n}\right),{y}^{*}\left({t}_{\text{1}}\right),\dots ,{y}^{*}\left({t}_{n}\right)\right\}$, then, by the projection theorem, it follows that the mean square error of ŝ(t_{ k } t_{ n } ) is smaller than that of ŝ(t_{ k } t_{ n } ).
Abbreviations
 EKF:

extended Kalman filter
 SL:

strictly linear
 SOS:

secondorder stationary
 UKF:

unscented Kalman filter
 WL:

widely linear.
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Acknowledgements
This study was supported in part by Project MTM200766791 of the Plan Nacional de I+D+I, Ministerio de Educación y Ciencia, Spain, which was financed jointly by the FEDER.
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FernándezAlcalá, R.M., NavarroMoreno, J. & RuizMolina, J.C. Prediction on widely factorizable signals. EURASIP J. Adv. Signal Process. 2012, 96 (2012). https://doi.org/10.1186/16876180201296
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Keywords
 complexvalued improper signals
 prediction problems
 widely factorizable signals
 widely linear processing.