IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 6, JUNE 2009
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Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory Jesús Navarro-Moreno, María Dolores Estudillo-Martínez, Rosa María Fernández-Alcalá, and Juan Carlos Ruiz-Molina
Abstract—In this paper, the problem of estimating an improper complex-valued random signal in colored noise with an additive white part is addressed. We tackle the problem from a mathematical perspective and emphasize the advantages of this rigorous treatment. The formulation considered is very general in the sense that it permits us to estimate any functional of the signal of interest. Finally, the superiority of the widely linear estimation with respect to the conventional estimation techniques is both theoretically and experimentally illustrated. Index Terms—Estimation in colored noise, Hilbert space theory, improper complex-valued random signals.
I. INTRODUCTION OMPLEX-VALUED random signals occur in fields as diverse as optics, quantum mechanics, electromagnetics, and communications [1]. They also appear in the description of narrowband signals. Indeed, the appropriate definition of instantaneous phase or amplitude of such signals requires the introduction of the so-called analytical signal, which is necessarily complex [2]. An important question when dealing with complex-valued random signals is related to its potential treatment by means of the real formalism. While it is true that complex-valued random signals can be written in terms of real-valued signals, the point is that data from some physical systems should be analyzed as complex-valued signals because the data represent motion on the complex plane. Examples of the latter are tidal analysis in oceanography and two-component observations in meteorology [3]. On the other hand, with a real-valued representation, much of the beauty and simplicity of the description would be lost. For instance, the aforementioned analytical signal is an adequate representation of the signal since it allows to show off directly the amplitude, phase, and carrier information. The complex-valued random signals can then be either proper or improper. Signals that have a vanishing complementary covariance are called proper. However, this assumption is not al-
C
Manuscript received March 03, 2006; revised January 08, 2009. Current version published May 20, 2009. This work was supported in part by the Plan Nacional de I+D+I, Ministerio de Educación y Ciencia, Spain, which is financed jointly by the FEDER, under Project MTM2007-66791. The authors are with the Department of Statistics and Operations Research, University of Jaén, 23071 Jaén, Spain (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Communicated by A. Høst-Madsen, Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109/TIT.2009.2018329
ways justified. The improper nature of some signals makes it necessary to take into account this complementary covariance in order to describe these signals’ second-order behavior completely [1], [2], [4], [5]. For example, binary phase-shift keying (BPSK) and Gaussian minimum-shift keying (GMSK) modulators produce improper communication signals [6]. The particular nature of the signal determines the type of treatment that one has to follow in its study. So, widely linear (WL) processing is the appropriate treatment for studying improper complexvalued random signals in contrast with conventional or strictly linear (SL) processing, which is adequate for proper signals (see, e.g., [4], [5], and [7]). One problem in communication engineering where the treatment of improper signals using WL processing has shown to be beneficial in improving mean square performance has been the linear least mean squares (LLMS) estimation problem [5]. Specifically, the authors consider the following formulation to represent the observation process: (1) is a proper where is the random signal of interest and zero-mean white Gaussian noise uncorrelated with . The solution obtained is based on the improper version of the Karhunen–Loève expansion [5, Th. 1] and a general result comparing the performance of WL with SL processing is also presented. They really prove that for the estimation problem in additive white Gaussian noise the performance gain, measured by mean square error (MSE), can be as large as two. The interesting results provided in [5] are derived following a heuristic procedure, i.e., a nonformal treatment of white noise processes is considered. However, the standard theory of stochastic processes provides a more formal treatment for LLMS estimation problems, which has become usual in statistical communications and signal processing (see, e.g., [8] and [9]). The mathematical approximation to the estimation problem involves the use of stochastic integration and the consideration of the Hilbert space theory. In fact, this theory provides a natural framework for solving the LLMS estimation problem since the optimum estimator is the orthogonal projection of the signal of interest onto the Hilbert space spanned by the observation process [7]–[10]. Moreover, a rigorous treatment of the problem has two remarkable advantages: potential generalizations of the problem are easier to handle and many of the results become trivial properties in this formal setting. As an illustration of the latter, the lack of optimality of the conventional estimator in
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estimating an improper signal can be seen clearly by observing the basic properties of the Hilbert spaces. In this paper, we address a generalization of the linear estimation problem of improper complex-valued random signals treated in [5] from a formal perspective. The extensions consist first in assuming that the additive noise is nonwhite and contains an uncorrelated white component, and second, in estimating a functional of the signal process. Indeed, many time-varying phenomena of interest can be modeled in this way (see, e.g., [8] or [11]). Specifically, we focus our attention on the class of functionals that are elements of the Hilbert space spanned by the signal process and its conjugate. Cambanis [10] studied a more general estimation problem from a similar perspective but using SL processing. Thus, the estimator that we propose outperforms that of Cambanis under the formulation here considered. Furthermore, such an estimator can be approximated through a recursive suboptimal estimator. This advantage justifies why we consider a less general formulation of the problem. This paper is organized as follows. The basic notation and mathematical concepts are introduced in Section II. Sections III and IV present the main results of the paper. Based on eigenvalues and eigenfunctions, we first derive in Section III an expression for the (LLMS) WL estimator of a functional of the signal which is corrupted by a colored noise with an additive white part (noise floor) as well as its MSE. Then, we give a closed form of this estimator by means of a WL operation1 on the realizations of the observation process. Section IV provides a suboptimum estimate that approaches the optimum one and can be efficiently calculated through a recursive algorithm. Finally, an example illustrates the superior performance of WL processing with respect to SL processing, on one hand, and the implementation of the recursive algorithm suggested, on the other hand. II. PRELIMINARIES Let denote the space of complex-valued square intewith inner product given by grable functions on
(4) (5) with defined by
and where
is the indicator function in if if
.
Let be an improper complex-valued continuous zero-mean random signal with autocorrelation function . The autocorrelation function of the augmented signal will be denoted by H , where is the conjugate transpose, and its eigenand , respecvalues and associated eigenfunctions2 by have a particular structure given tively. The eigenfunctions by , with , and are orthonormals in the following sense [5], [12]: H
(6)
with representing the real part. The Hilbert space spanned by the signal process and by the and , respecaugmented signal will be denoted3 by tively. The same notation will be used for other processes, i.e., denotes the autocorrelation of a process , denotes the , and autocorrelation of the augmented process and are their associated Hilbert spaces, respectively. The main mathematical shortcoming in the formulation given by (1) lies in the fact that the white noise is not a second-order process [8, p. 289]. A standard mathematical interpretation of this (see, for example, [8] and [9]) that avoids the use of generalized random processes is that the observed signal is defined by (7)
and the set of continuous complex-valued functions on this inverval. be a complex-valued Wiener process Let . We denote the complexwith valued Wiener integral of by , which satisfies the following properties [8, p. 299]:
where is an improper complex-valued zero-mean mean square continuous signal and is the complex-valued Wiener process uncorrelated with the signal and the integral is a complex-valued mean square Riemann integral. The process takes and is mean square continuous on this invalues in terval. Hence, complex-valued mean square Riemann–Stieltjes , with , exist [9, integrals of the form , then [9, Ch. 2] p. 123]. Furthermore, if
(2) (8)
(3) 1A
WL operation on a stochastic signal is a linear operation that depends on both the signal and its conjugate [5].
2Two Karhunen–Loève expansions for improper complex-valued signals are possible. The expansion considered here is the one given in [12] and it has real random variables. Another is the representation proposed in [5] and it contains complex improper random variables. 3Note that ( ) = spf ( ) ( ); 2 [0 ]g consists of all the finite linear combinations of both ( ) and ( ) and their quadratic mean limits. ( ) is i = a Hilbert space under the usual operations and the inner product h [ ]; 2 ( ) [13].
Hs
E xy x; y H s
st
s t ;s t t s t
;T
Hs x; y s
NAVARRO-MORENO et al.: ESTIMATION OF IMPROPER COMPLEX-VALUED RANDOM SIGNALS IN COLORED NOISE
where the first integral on the right-hand side member is a complex-valued mean square Riemann integral and the second one is a complex-valued Wiener integral.
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Proof: By using the Karhunen–Loève expansion of the augmented process , we obtain [12]
III. SIGNAL ESTIMATION IN COLORED NOISE The estimation problem that we wish to solve corresponds to a more general formulation than (7) in the sense that the noise is nonwhite but containing a white component [8], [11]. Specifically, the observation process that we consider is the following: (9) where and are the processes defined in the previous section and the colored noise component is a complex-valued continuous zero-mean stochastic process uncorrelated with . The aim is to estimate a complex-valued process belonging to the , based on Hilbert space spanned by the augmented signal . As particular cases, the the vector of observations process can be a functional of or a WL operation on . The generality of this formulation allows us to estimate a variety of magnitudes related with the signal without modifying the structure of the estimator. For example, we can estimate the angular acceleration or the angular displacement of a body from its angular velocity because mean square linear operations of the . signal, such as derivatives or integrals, are elements of The following result provides an expression for the optimum WL estimator and its associated MSE. Theorem 1: Denote and let and the eigenvalues and the corresponding eigenfunctions of autocorrelation function of the augmented process Then, the (LLMS) WL estimator of is
be , the .
(15) are defined in (12) and where the real random variables . From (15), we have that the family of random variables is an orthonormal basis of the Hilbert space gener. Hence, we have ated by and ,
(16)
(17) On the other hand, taking (2) and (3) into account, it follows that
where the last equality is a consequence of the orthonormality in the sense given by (6). of Also, from (4), (5), and (13), we have (18)
(10) where (11)
Now, let be the Hilbert space spanned by the variables and an orthonormal basis of the in . Then, from orthogonal complement of (18), we obtain
and
(19) (12) is given by (13) and . where Hence, from (9), (16), (17), and (19), we have
(13) (20) are real random variables. Moreover, the MSE associated with (10) is given by
As a consequence of (20), if we denote the Hilbert space gen, , and erated by the random variables by , it is verified that (14)
(21)
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On the other hand, it follows that
Theorem 2: The estimator (10) can be expressed in the next closed form (22)
(24)
where the equality is a direct result of (8). the Hilbert space spanned Hence, if we denote by , from (21) and (22), we have by the variables
where both integrals are complex-valued mean square Riemann–Stieltjes integrals and
Let and be the projections of onto the and , respectively. Next, spaces they are calculated. For that, note that since is uncorrelated and with both and , then . Thus
Proof: First, we prove that for all . Applying the inequality of Cauchy–Schwarz, we get (25) where
(26) where
is given by (11). Similarly, we get By examining (23), it is not difficult to identify the function as the error associated with the problem of estimating on the basis of the vector of observations . Moreover, let be the projection of the process onto the Hilbert . Then, following the proof of Theorem 1, we obtain space
(27) where the last equality is obtained from the uncorrelation between and . Now, by the uniqueness of the projection theorem of Hilbert a.s. and then spaces [9, p. 116], we have (10) holds. Moreover, the associated MSE is
Remark 1: In the particular case that , we have that and MSE is
and . Moreover, the
(23)
Thus, from (14) and (27), we have
and hence
for all . By integrating (25), we obtain
where we have taken into account that is continuous and . This last inequality follows and from (26), and the fact that [cf., with (15)]. . Therefore, Similarly, we can check that the expression
which is the solution given in [5]. From a practical point of view, it would be interesting to derive a closed-form expression for (10). This form of the optimal estimator is obtained by means of a WL operation on the realizations of the observation process and is presented in the following result.
is well defined and exists for all , where the integrals are complex-valued mean square Riemann–Stieltjes integrals. Finally, from (22), we obtain that the estimator (10) can be expressed in the closed form (24).
NAVARRO-MORENO et al.: ESTIMATION OF IMPROPER COMPLEX-VALUED RANDOM SIGNALS IN COLORED NOISE
Remark 2: Under the more restrictive hypotheses used in [5] and stated in Remark 1, it follows that
Remark 3: Now, we can see how the basic properties of the Hilbert spaces easily allow us to show the lack of optimality of the conventional estimator in order to estimate an improper signal. Although this checking can be performed for the general problem of estimating , we illustrate this fact for the hypotheses used in Remark 1. By using the well-known projection theorem of the Hilbert , spaces, the proper estimator is the projection of onto the Hilbert space spanned by , whereas the improper estimator is the projection of onto , the Hilbert space generated , then leads trivially to a by and . Since smaller error variance than . Remark 4: The estimators (10) and (24) depend on the eigenvalues and the corresponding eigenfunctions of , the autocorrelation function of the augmented process . Except for some special cases, finding the eigenvalues and the eigenfunctions of a linear integral equation with arbitrary kernel is a very difficult task and even sometimes impossible. However, we can avoid the calculation of the true eigenvalues and eigenfunctions by means of the Rayleigh–Ritz method, which is a procedure for numerically solving operator equations involving only elementary calculus and simple linear algebra [14]. An illustration of its application to obtain an approximate series expansion of a stochastic process can be consulted in [15].
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,
where ,
, and
The associated MSE of (28) is
Observe that the expression (28) of the optimum estimator of is made up by the sum of two terms where the first is the same as the one obtained using the conventional estimator. This case has been studied in [4] under a discrete-time formulation of the problem (Example B. Circular observation). , , the In the particular case of conventional and the WL estimators coincide. This situation has also been analyzed in [4] for discrete-time systems (Example A. Jointly circular case). Moreover, as an example of this particular case, we can mention the situation where it is assumed that is proper, , and . The proper estimation problem under these assumptions has been examined in [5] with . IV. A RECURSIVE ALGORITHM FOR SUBOPTIMUM ESTIMATION A computationally more amenable form of the estimate is desired. The special structure of the observation process (9) allows us to develop a recursive algorithm in order to approach the optimum estimate. To this end, we consider a truncated version of the optimum estimate (10) denoted by (29)
A. Study of the Proper Case Now, we endeavour to solve the proper signal estimation problem by using WL processing. In our study, we consider that is proper. Note that this does not imply that or are necessarily proper (our case is really more general). We denote , the eigenvalues and the eigenfunctions associated with , by and , respectively. Subsequently, it is not has eigenvalues and associated difficult to prove that eigenfunctions . Hence, is generated with by the family of random variables . Following a similar reasoning to that used in Theorem 1, we is find that the (LLMS) WL estimator of
The main characteristic of the suboptimum estimate (29) is that it can be computed through a stochastic differential equation such as that shown in the next result. In this theorem, and with the object of emphasizing the dependence on of the vectors and matrices involved, we denote their dimensions by will denote a vector of dimension subindexes. Therefore, and an matrix. Theorem 3:
where chastic differential equation
and
H
with (28)
the vector of
zeros,
,
obeys the sto-
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and
obtained from the differential equation
90% of the amount of the total variance of is accounted for. The total variance of is calculated in the following way:
H
Proof: Consider the matrix H and the vector
Thus, the algorithm is especially suitable for those cases in which the optimum estimate (10) can be adequately approximated by means of a finite-series representation with a short number of terms (e.g., for processes with smooth autocorrelation function [17]). Remark 7: The estimator (28) for the proper case can also be approached by using a similar algorithm. Its form is the following:
Then, we have that H H
(30)
On the other hand, from (29) and (30), it follows that
where
, obeys the stochastic
, and differential equation H
(31)
We define H
with the differential equation
and
obtained from
Thus, by applying the Ito differentiation rule [8, Ch. 7] and (31), the theorem is proven. Remark 5: In practice, the observations are taken at discretetime instants. In this case, a discretization of Theorem 3 is required. To this end, we follow the procedure suggested in [16]. Specifically, given a realization of a sequence of observations , , we have that
where
with the initialization
and H H
with
.
Remark 6: Notice that the efficiency of the algorithm in Theorem 3 depends heavily on the number of summands in (29). A rule of thumb that helps to select a minimum number of terms in the expansion without an unnecessary excess of computation can be the following: select in such a way that at least
V. NUMERICAL EXAMPLE We consider a generalization of the communication example is transmitted studied in [5]. Specifically, a real waveform over a channel that rotates it by some random phase and adds complex-valued white Gaussian noise. Thus, the signal . In contrast to [5], we asof interest is sume the presence of a colored noise component of the form , with and and are real processes with zero-mean and where where both have the same covariance function characterized by and the corresponding eigenfunctions the eigenvalues . Moreover, we assume that and are independent and identically distributed random variables and that , , , and are independent of each other. These types of processes appear in [18, p. 357]. We seek to estimate the mean square integral of the process , i.e., . Hence, both the inclusion of a colored noise component and the estimation of a linear operation of the process justify the generality of the estimator (24) with respect to the one given in [5]. We also aim to compare the performance of WL processing in relation to SL processing regarding several choices of the random processes and variables involved. A general expression for the performance gain attained by the WL estimator with respect to the SL one was derived in [4]. Notice that this expression depends on the second-order characteristics of the signals
NAVARRO-MORENO et al.: ESTIMATION OF IMPROPER COMPLEX-VALUED RANDOM SIGNALS IN COLORED NOISE
Fig. 1. MSEs for SL processing (dashed line) and WL processing (solid line) where x (t) is the standard Wiener process and and are distributed as a standard normal.
Fig. 2. MSEs for SL processing (dashed line) and WL processing (solid line) where x (t) is the standard Wiener process and and are distributed as a bilateral exponential.
implicated and hence, the performance gain in our application and the function will vary according to the distribution of . has eigenvalues We first notice that and with associated eigenfunctions and , , it follows that the choice of the respectively. Since probabilistic distributions of must assure that . in the proper case is Consequently, the MSE of
(32) and the error (14) associated with (24) is given by
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Fig. 3. MSEs for SL processing (dashed line) and WL processing (solid line) where x (t) is the standard Ornstein–Uhlenbeck process and and are distributed as a standard normal.
Fig. 4. MSEs for SL processing (dashed line) and WL processing (solid line) where x (t) is the standard Ornstein–Uhlenbeck process and and are distributed as a bilateral exponential.
(33) representing the imaginary part. A second measure with for comparing the performance is the following ratio: (34) which is closely related to the performance measure considered in [5]. , we consider two well-known As models for the process stochastic processes: the standard Wiener process on the inand the standard Ornstein–Uhlenbeck process on terval . The eigenvalues and the eigenfunctions of these processes can be found in [11]. The Ornstein–Uhlenbeck process and its integral are appropriate models to describe the respective velocity and position of an object (see, for instance,
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Fig. 5. Performance measure I in (34) when x (t) is the standard Wiener process and and are distributed as a standard normal (dashed line) and as a bilateral exponential (solid line).
Fig. 6. Performance measure I in (34) when x (t) is the standard Ornstein–Uhlenbeck process and and are distributed as a standard normal (dashed line) and as a bilateral exponential (solid line).
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 6, JUNE 2009
Fig. 7. Optimum estimations (solid line) of y (t) and their corresponding suboptimum estimations with n = 10 (dashed line) and n = 20 (doted line), where x (t) is the standard Wiener process and and are distributed as a standard normal.
Finally, we illustrate the implementation of the recursive alis the stangorithm given in Remark 5. We assume that is a Gaussian process, and and dard Wiener process, are distributed as a standard normal. In order to choose an adequate , we have taken into account that the total variance of is , and thus, and represent 96.4% and 98.2% of the total variance of , respectively. Fig. 7 shows the optimum estimations of together with the corresponding suboptimum estimations with and and illustrates the convergence of the suboptimum estimations toward the optimum ones when the length of the series is increased. Notice that different simulations with different noise levels and different stochastic processes and random variables have been made in this study. The simulations presented here are representative. REFERENCES
[19]). This process also provides a model of the transistor gain. The probabilistic distributions considered for and are the standard normal and the bilateral exponential. Unless otherwise in all the simulations. indicated, we assume is the standard The functions (32) and (33) when distributed as a Wiener process are shown in Fig. 1 ( and standard normal) and Fig. 2 ( and distributed as a bilateral exponential). Moreover, the same functions are represented is the standard Ornstein–Uhlenbeck process in when distributed as a standard normal) and Fig. 4 Fig. 3 ( and distributed as a bilateral exponential). From these ( and functions, it is evident that an improper treatment improves the performance of the estimator. Figs. 5 and 6 show the performance measure (34) for difand for the processes and probabilistic disferent values of tributions mentioned above. As noted in [5], the advantage of becomes larger and disappears WL processing diminishes as . On the other hand, normal distribucompletely for tion assures a better performance of WL processing in relation to the proper one in both stochastic processes considered.
[1] P. J. Schreier and L. L. Scharf, “Second-order analysis of improper complex random vectors and processes,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 714–725, Mar. 2003. [2] B. Picinbono and P. Bondon, “Second-order statistics of complex signals,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 411–420, Feb. 1997. [3] P. Rubin-Delanchy and A. T. Walden, “Kinematics of complex-valued time series,” IEEE Trans. Signal Process., vol. 56, no. 9, pp. 4189–4198, Sep. 2008. [4] B. Picinbono and P. Chevalier, “Widely linear estimation with complex data,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 2030–2033, Aug. 1995. [5] P. J. Schreier, L. L. Scharf, and C. T. Mullis, “Detection and estimation of improper complex random signals,” IEEE, Trans. Inf. Theory, vol. 51, no. 1, pp. 306–312, Jan. 2005. [6] P. J. Schreier, L. L. Scharf, and A. Hanssen, “A generelized likelihood ratio test for impropriety of complex signals,” IEEE Signal Process. Lett., vol. 13, no. 7, pp. 433–436, Jul. 2006. [7] T. A. H. S. Kailath and B. Hassibi, Linear Estimation. Englewood Cliffs, NJ: Prentice-Hall, 2000. [8] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. New York: Springer-Verlag, 1994. [9] P. A. Ruymgaart and T. T. Soong, Mathematics of Kalman-Bucy Filtering, 2nd ed. Berlin, Germany: Springer-Verlag, 1988. [10] S. Cambanis, “A general approach to linear mean-square estimation problems,” IEEE Trans. Inf. Theory, vol. IT-19, no. 1, pp. 110–114, Jan. 1973.
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[11] H. L. Van Trees, Detection, Estimation, and Modulation Theory. New York: Wiley, 1968. [12] E. J. Kelly and W. L. Root, “A representation of vector-valued random processes,” J. Math. Phys., vol. 39, pp. 211–216, 1960. [13] E. Masry, “Expansion of multivariate weakly stationary stochastic processes,” Inf. Sci., vol. 2, pp. 303–317, 1970. [14] M. Chen, Z. Chen, and G. Chen, Approximate Solutions of Operator Equations. Singapore: World Scientific, 1997. [15] J. Navarro-Moreno, J. C. Ruiz-Molina, and R. M. Fernández-Alcalá, “Approximate series representations of linear operations on secondorder stochastic processes. Application to simulation,” IEEE, Trans. Inf. Theory, vol. 52, no. 4, pp. 1789–1794, Apr. 2006. [16] A. H. Jazwinski, Stochastic Processes and Filtering Theory. Part I. San Diego, CA: Academic, 1970. [17] K. K. P. S. P. Huang and S. T. Quek, “Implementation of Karhunen-Loève expansion for simulation using a wavelet-Galerkin scheme,” Probab. Eng. Mech., no. 17, pp. 293–303, 2002. [18] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1989. [19] J. K. Lindsey, Statistical Analysis of Stochastic Processes in Time. Cambridge, U.K.: Cambridge Univ. Press, 2004.
Jesús Navarro-Moreno was born in Jaén, Spain, in 1970. He received the M.Sc. degree in mathematics and the Ph.D. degree in statistics from the University of Granada, Granada, Spain, in 1993 and 1998, respectively. In 1993, he joined the faculty at the University of Jaén, Jaén, Spain. In October 1998, he was promoted to Associate Professor in the Department of Statistics and Operations Research, University of Jaén. His current research interests involve the areas of second-order stochastic processes, estimation, and detection. Dr. Navarro-Moreno is a member of the Spanish Sociedad de Estadística e Investigación Operativa.
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María Dolores Estudillo-Martínez was born in Jaén, Spain, on February 19, 1977. She received the M.Sc. degree in sciences and statistical techniques from the University of Granada, Granada, Spain, in 2000. Currently, she is working towards the Ph.D. degree in statistics at the Department in Statistics and Operations Research, University of Jaén, Jaén, Spain. She is an Associate Lecturer at the Department in Statistics and Operations Research, University of Jaén.
Rosa María Fernández-Alcalá was born in 1973 in Jaén, Spain. She received the M.Sc. degree in mathematics from the University of Granada, Granada, Spain, in 1996 and the Ph.D. degree in statistics from the University of Jaén, Jaén, Spain, in 2002. In 1996, she joined the Department of Statistics and Operations Research, University of Jaén, where she is currently an Associate Professor. Dr. Fernández-Alcalá is a member of the Spanish Sociedad de Estadística e Investigación Operativa.
Juan Carlos Ruiz-Molina was born in 1966 in Jaén, Spain. He received the M.Sc. in mathematics and the Ph.D. degree in statistics from the University of Granada, Granada, Spain, in 1989 and 1993, respectively. In 1989, he joined the faculty at the University of Granada. From 1989 to 1993 he held various teaching and research positions in the Department of Statistics and Operations Research, University of Granada. Since 1993 he has been an Associate Professor at the Department of Statistics and Operations Research, University of Jaén, Jaén, Spain, and he is currently the Head of that department. His research interests include identification, estimation, and detection. He has published a textbook on estimation in Spanish. Dr. Ruiz-Molina is a member of the Spanish Sociedad de Estadística e Investigación Operativa.
