SPECTRAL ESTIMATION AND INVERSE INITIAL BOUNDARY VALUE ...

Inverse Problems and Imaging

doi:10.3934/ipi.2010.4.1

Volume 4, No. 1, 2010, 1–9

SPECTRAL ESTIMATION AND INVERSE INITIAL BOUNDARY VALUE PROBLEMS

Sergei Avdonin Department of Mathematics and Statistics, University of Alaska Fairbanks, AK 99775-6660, USA

Fritz Gesztesy and Konstantin A. Makarov Department of Mathematics, University of Missouri Columbia, MO 65211, USA

(Communicated by Matti Lassas) Abstract. We extend the classical spectral estimation problem to the infinitedimensional case and propose a new approach to this problem using the Boundary Control (BC) method. Several applications to inverse problems for partial differential equations are provided.

1. Introduction. The classical spectral estimation problem consists of the recovery of the coefficients an , λn , n = 1, . . . , N , N ∈ N, of a signal (1.1)

s(t) =

N X

an eλn t ,

t ≥ 0,

n=1

from the given observations s(j), j = 0, . . . , 2N − 1. Here the coefficients an , λn , n = 1, . . . , N , may be arbitrary complex numbers. This problem is important for signal processing, with applications in the areas of wireless communications, antenna array design, bio-medical imaging, high-speed circuit analysis, and other areas (see, e.g., [19, 27]). There exist many methods for solving the spectral estimation problem: The method of Prony and its numerous modifications [18, 24]; the matrix pencil method developed by Hua and Sarkar [20, 19, 27]; iterative maximum likelihood methods (cf., e.g., [25]); MUSIC (Multiple Signal Classification) [28], and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithms [26], and others. Badeau, Bertrand, and Gaël [9] developed a generalized ESPRIT algorithm for the estimation of parameters of a signal modeled by the Polynomial Amplitude Complex Exponentials model. In [4, 5] a new approach to this important problem was proposed: Using the signal s(t), a convolution operator regarded as an input-output map of some linear discrete-time dynamical system was constructed. Knowing the input-output map then makes it possible to recover some parameters of the system. While the system realized from an input-output map is not unique, it was proved that the coefficients an and λn , n = 1, . . . , N , of s(t) can be determined exactly from the spectral data of any such system. 2000 Mathematics Subject Classification. Primary: 93B05, 93B15; Secondary: 35P10, 35R30. Key words and phrases. Spectral estimation, inverse problems, boundary control method. 1

c

2010 American Institute of Mathematical Sciences

2

Sergei Avdonin, Fritz Gesztesy and Konstantin A. Makarov

This approach is based on a “non-self-adjoint” version of the Boundary Control (BC) method [1]. The BC method has been recently developed for solving boundary spectral and dynamical inverse problems for partial differential equations (see, e.g., [2, 10, 14, 15]). Proposed originally for solving the boundary inverse problem for the multi-dimensional wave equation, it was successfully extended to the heat, Schrödinger, and other fundamental equations of mathematical physics (see, e.g., [3, 7, 8, 11, 12, 13, 21]). The BC method reveals that the two central problems of inverse and control theory of distributed parameter systems are intimately connected: The first problem consists of the recovery of unknown coefficients, the second problem entails the controllability of the corresponding initial boundary value problem. Roughly speaking, the BC method gives the realization of Kalman’s idea that the controllable (or observable) part of a system can be identified. Using the Boundary Control method it was shown in [5] that the coefficients {λn }, n = 1, . . . , N , can be obtained as in the matrix pencil method by solving the generalized eigenvalue problem for the following matrices A and B: (1.2)

b Af = λBf,

Am,n = s(m + n − 1), Bm,n = s(m + n − 2), m, n = 1, . . . , N, bn , n = 1, . . . , N . In addition, this method yields exact using the relation, λn = ln λ formulas for the computation of the amplitudes an , n = 1, . . . , N , in terms of the eigenvectors of the above eigenvalue problem and the data s(j), j = 0, . . . , 2N − 1. We also note that N may be unknown and can be found as part of the procedure. In the present paper we generalize the classical spectral estimation problem to the infinite-dimensional case. More precisely, we solve the following problem: Problem 1. Recover the coefficients an , λn , n ∈ N, of a signal (1.3)

s(t) =

∞ X

an eλn t ,

t ∈ (0, T ),

n=1

from the given data s(t), t ∈ (0, T ). Here T > 0 is a fixed number and we suppose that the series (1.3) converges in L2 ((0, T ); dt). The solution of Problem 1 will be presented in Section 2. The main idea of this paper (as well as of [4, 5]) exploits a fundamental result of the BC method — the possibility to extract the inverse spectral data from the dynamical data provided the system is spectrally controllable [11, 3, 7, 8] (see also the closely related papers [22, 23]). In paper [4], the BC method was applied to the spectral estimation problem PN λn t for a signal s(t) = , where the amplitudes an (t), n = 1, . . . , N , n=1 an (t)e are polynomials in t. This result can also be extended to the infinite-dimensional situation, and that will be the subject of a forthcoming paper. In the present paper we restrict ourselves to the case of a signal with constant amplitudes an ∈ C, n ∈ N. This problem is important in signal processing and has also interesting applications to the inverse theory for certain partial differential equations. Some of these applications are discussed in Section 3. 2. The spectral estimation problem. We consider the following differential equation in a separable complex Hilbert space H, (2.4)

x(t) ˙ = Ax(t) + bf (t), t ∈ (0, T ),


x(0) = 0, Volume 4, No. 1 (2010), 1–9

Spectral Estimation and Inverse Problems

3

and suppose that the operator A has eigenvectors {ϕn }n∈N , which form a Riesz basis (a basis equivalent to an orthonormal one) in H with corresponding eigenvalues {λn }n∈N ⊂ C, b ∈ H, f ∈ L2 ((0, T ); dt) for some fixed T > 0. Let {ψn }n∈N and {λn }n∈N ⊂ C be eigenvectors and eigenvalues of the adjoint operator A∗ , and βn = (b, ψn )H , n ∈ N, where (·, ·)H denotes the scalar product in H. We suppose that βn 6= 0 for all n ∈ N. Looking for a solution of (2.4) in the form of a series (2.5)

x(t) =

∞ X

xn (t)ϕn ,

t ∈ (0, T ),

n=1

and using that (ϕn , ψk )H = δn,k , one obtains (2.6)

x˙ n (t) = λn xn (t) + βn f (t), t ∈ (0, T ),

xn (0) = 0, n ∈ N.

Therefore, (2.7)

xn (t) = βn

Z

t

eλn (t−τ ) f (τ ) dτ,

t ∈ (0, T ), n ∈ N.

0

We suppose that the dynamical system (2.4) is spectrally controllable. More precisely, we need the following property: For any n ∈ N, there exists a control fn ∈ H01 ((0, T )) such that xfn (T ) = ϕn . The spectral can be expressed in controllability terms of minimality of the exponential family eλn t n∈N in L2 ((0, T ); dt). A family {en }n∈N in the Hilbert space H is called minimal if any en does not belong to the closure of the linear span of all other elements {ek }k∈N\{n} . We set (2.8)

f˙(t) = h(t), h ∈ L2 ((0, T ); dt), where f˙(t) := df (t)/dt,

t ∈ (0, T ).

Integrating by parts in (2.7), and taking into account that f (0) = f (T ) = 0, one infers ( RT βn eλn (T −t) h(t) dt, λn 6= 0, (2.9) xn (T ) = λn R 0T n ∈ N, βn 0 (T − t) h(t) dt, λn = 0, Z T (2.10) h(t) dt = 0. 0

These equations imply that the system (2.4) is spectrally controllable if and only if for any k ∈ N, there exists hk ∈ L2 ((0, T ); dt) such that ( RT βn eλn (T −t) hk (t) dt, λn 6= 0, (2.11) δn,k = λn R 0T n ∈ N, βn 0 (T − t) hk (t) dt, λn = 0, Z T (2.12) hk (t) dt = 0. 0

Using relations between the solvability of the moment problem and properties of the corresponding exponential families (cf., [6, Sec.III.3]), we can formulate the following statement: Lemma 2.1. The spectral of the system (2.4) is equivalent to the controllability minimality of the family eλn t n∈N ∪ {1} in L2 ((0, T ); dt) in the case λn 6= 0 for all n ∈ N (or of the family eλn t n∈N ∪ {t} in the case λn = 0 for some n ∈ N) and to the fact that βn 6= 0 for all n ∈ N. Inverse Problems and Imaging

Volume 4, No. 1 (2010), 1–9

4


Let c ∈ H and let y(t) = (x(t), c)H be the output of the dynamical system (2.4), and γn = (ϕn , c)H , n ∈ N. From (2.5) and (2.7) one gets Z tX Z t ∞ (2.13) y(t) = βn γn eλn (t−τ ) f (τ ) dτ = r(t − τ )f (τ ) dτ, t ∈ (0, T ), 0 n=1

0

where

(2.14)

r(t) =

∞ X

βn γn eλn t ,

t ∈ (0, T ).

n=1

Thus, the function r(·) has the form of the series depicted in (1.3). Next, we demonstrate how to find the exponents λn and inner products βn , γn , n ∈ N, given the function r(·). Introducing the system adjoint to (2.4), (2.13), (2.15)

w(t) ˙ = A∗ w(t) + cg(t),

z(t) = (w(t), b)H ,

one verifies that (2.16)

w(t) =

∞ X

wn (t)ψn ,

wn (t) = γ n

n=1

and (2.17)

z(t) =

Z

t ∈ (0, T ),

t

eλn (t−τ ) g(τ ) dτ,

0

t ∈ (0, T ), n ∈ N, Z

t

r(t − τ )g(τ ) dτ,

t ∈ (0, T ).

0

Next, we introduce the connecting operator C T acting in the space L2 ((0, T ); dt) through its bilinear form by (2.18) C T f, g L2 ((0,T );dt) = (x(T ), w(T ))H , where (·, ·)L2 ((0,T );dt) denotes the scalar product in L2 ((0, T ); dt). The right-hand side of (2.18) can be written as (see (2.7), (2.16)) ! Z T Z T ∞ X (2.19) xn (T )wn (T ) = r(2T − t − τ )f (t) dt g(τ ) dτ. 0

n=1

0

Therefore,

T

(2.20)

(C f )(τ ) =

Z

T

r(2T − t − τ )f (t) dt,

0 ≤ τ ≤ T.

0

By the definition of fn , (2.21)

x˙ fn (T ) = Axfn (T ) + bfn (T ) = Axfn (T ) = λn xfn (T ),

n ∈ N.

T

The definition of the operator C and equations (2.21) imply that for any g ∈ L2 ((0, T ); dt) one has ˙ C T f˙n , g L2 ((0,T );dt) = xfn (T ), wg (T ) H = x˙ fn (T ), wg (T ) H = Axfn (T ), wg (T ) H = λn xfn (T ), wg (T ) H (2.22) = λn C T fn , g L2 ((0,T );dt) , n ∈ N. Using (2.20), one gets the following integral eigenvalue equation for finding λn and fn : Z T (2.23) r(2T − t − τ ) f˙n (t) − λn r(2T − t − τ ) fn (t) dt = 0, 0


Volume 4, No. 1 (2010), 1–9


5

and finally, (2.24)

Z

T

[r(2T ˙ − t − τ ) − λn r(2T − t − τ )] fn (t) dt = 0,

0≤τ ≤T.

0

A priori, we do not suppose that r˙ ∈ L2 ((0, T ); dt), and therefore, in general, the RT integral 0 r(2T ˙ −t−τ )fn (t) dt should be understood as the action of the functional −1 r˙ ∈ H (0, T ) on fn ∈ H01 (0, T ). For algorithmic purposes it is convenient to rewrite the equations (2.23), (2.24) in the form Z T (2.25) [r(2T − t − τ ) − λn R(2T − t − τ )] hn (t) dt = 0, 0 ≤ τ ≤ T , 0

Rt Rt where R(t) = 0 r(τ )dτ and fn (t) = 0 hn (τ )dτ . Every function fn which solves the spectral control problem satisfies the equation (2.24), and any function satisfying this equation satisfies also the equality Axfn (T ) = λn xfn (T ). Therefore, the spectral controllability (i.e., the minimality of the exponential family) guarantees that equation (2.24) has an infinite sequence of λsolutions {λn , fn }n∈N . The functions {fn (t)}n∈N form a family biorthogonal to e n (T −t) n∈N in L2 ((0, T ); dt). Equation (2.24) determines the functions fn up to constant factors. To find these factors (and the coefficients an in (1.3)), we consider the equation analogous to (2.24) with the kernel r replaced by r. This equation yields the sequence {λn , gn }n∈N . The functions gn solve the control problems wgn (T ) = ψn , n ∈ N. Since (2.26) C T fn , gk L2 ((0,T );dt) = xfn (T ), wgk (T ) H = (ϕn , ψk )H = δn,k , the functions fn and gk should be normalized according to ! Z T Z T (2.27) r(2T − t − τ ) fn (t) dt gk (τ ) dτ = δn,k . 0

0

The coefficients an can now be found as the products (2.28)

an = βn γn ,

n ∈ N,

where (2.29)

γn = (ϕn , c)H = xfn (T ), c

and similarly, (2.30)

βn =

z gn (T )

=

H

Z

= y fn (T ) =

Z

T

r(T − t) fn (t) dt,

n ∈ N,

0

T

r(T − t) gn (t) dt,

n ∈ N.

0

Remark 1. It is possible to extend the spaces of sequences {βn }n∈N and {γn }n∈N to the case where one of them belongs to ℓ∞ (N). 3. Inverse problems. In this section we consider the following initial boundary value problem for the Schrödinger equation (3.31)

iut − uxx + q(x) u = 0,

(3.32)

ux (0, t) = u(L, t) = 0,

(3.33)

u(x, 0) = a(x),


(x, t) ∈ QT = (0, L) × (0, T ), t ∈ (0, T ),

x ∈ (0, L). Volume 4, No. 1 (2010), 1–9

6


Here q(·) ∈ L1 ((0, L); dx) is assumed to be real-valued (the case of a complexvalued potential q(·) can also be considered within our framework) and a(·) ∈ L2 ((0, L); dx). (Equation (3.32) represents a boundary condition and (3.33) an initial condition.) The inverse problem consists of the recovery of the unknown potential q(x) for a.e. x ∈ (0, L), from the given data (the “trace”) u(0, t), t ∈ (0, T ), of the solution of the problem (3.31)–(3.33) at x = 0. We emphasize that T > 0 may be arbitrarily small. We denote by λn and ϕn , n ∈ N, the eigenvalues and eigenfunctions of the mixed boundary value problem (3.34)

− ϕ′′ (x) + q(x)ϕ(x) = λϕ(x),

0 < x < L,

′

ϕ (0) = ϕ(L) = 0,

and let (3.35)

αn = (a, ϕn )L2 ((0,L);dx) ,

n ∈ N.

Then u(0, t) is represented as the series (3.36)

∞ X

u(0, t) =

αn ϕn (0)eiλn t ,

t ∈ (0, T ).

n=1

The series on the right-hand side of (3.36) has the form of (1.3), and our next step amounts to finding the exponents λn from the given data u(0, t), t ∈ (0, T ). It is known (see, e.g., [7, 8]) that the family eiλn t n∈N is minimal in L2 ((0, T ); dt) for any T > 0. Moreover, this family forms a Riesz basis in the closure of its linear span in L2 ((0, T ); dt) (see [8] for more details). Since 0 < inf n∈N |ϕn (0)| ≤ supn∈N |ϕn (0)| < ∞, the series in (3.36) converges in L2 ((0, T ); dt). Using the results of the previous Section 2, we can recover the spectrum {λn }n∈N of (3.34), provided αn 6= 0 for all n ∈ N. A recipe for constructing initial data satisfying these conditions is proposed in [17]. Applying the same method, one can recover the spectrum {µn }n∈N of the Neumann boundary value problem (3.37)

− ϕ′′ (x) + q(x)ϕ(x) = µϕ(x), ′

0 < x < L,

′

ϕ (0) = ϕ (L) = 0,

and then find the potential q(·) from the two spectra {λn }n∈N and {µn }n∈N . Following the same ideas, one can solve similar inverse problems for the heat and wave equations. For instance, consider the heat equation (3.38)

ut = c(x) uxx ,

(x, t) ∈ QT = (0, L) × (0, T ),

(3.39)

ux (0, t) = u(L, t) = 0,

(3.40)

u(x, 0) = a(x),

t ∈ (0, T ),

x ∈ (0, L),

where c(·) ∈ C 1 ([0, L]) is assumed to be strictly positive and a(·) ∈ L2 ((0, L); dx). (Here (3.39) represents a boundary condition and (3.40) an initial condition.) The function u(0, t) is then represented as the series (3.41)

u(0, t) =

∞ X

αn ϕn (0)e−λn t ,

t ∈ (0, T ),

n=1 Inverse Problems and Imaging

Volume 4, No. 1 (2010), 1–9


7

convergent in L2 ((0, T ); dt), where αn , ϕn , n ∈ N, are defined in (3.34), (3.35). It is known (cf. [6]) that for any T > 0 the exponential family {exp(−λn t)}n∈N is minimal in L2 ((0, T ); dt). Therefore, applying the results of Section 2, one can recover the exponents {λn }n∈N from the data u(0, t), provided αn 6= 0 for all n ∈ N. Similarly, one can recover the spectrum {µn }n∈N of the boundary value problem (3.37), and finally, find the coefficient c(·). The inverse approach to this type of problem was studied in [17]. The authors in [17] proposed a method recovering c(·) from the data u(0, t), t ∈ (0, ∞). Our approach allows us to use the same data on an arbitrarily small time interval (0, T ). The analogous inverse problem for the string equation is the recovery of the density ρ(x) from the given data u(0, t), t ∈ (0, T ), where u(x, t) solves the initial boundary value problem (3.42)

ρ(x) utt = uxx ,

(x, t) ∈ QT = (0, L) × (0, T ),

(3.43)

ux (0, t) = u(L, t) = 0,

(3.44)

u(x, 0) = a(x),

t ∈ (0, T ),

ut (x, 0) = 0,

x ∈ (0, L).

Here ρ(·) ∈ C 1 ([0, L]) is assumed to be strictly positive and a(·) ∈ L2 ((0, L); dx). Similarly to (3.36) and (3.41), one can represent u(0, t) as the series u(0, t) =

∞ X

αn ϕn (0) cos

n=1

(3.45)

∞ √ √ p 1X λn t = αn ϕn (0) ei λn t + e−i λn t , 2 n=1

t ∈ (0, T ),

2

convergent in L ((0, T ); dt), with αn , ϕn , n ∈ N, defined as above. Next, we denote by Lopt the optical length of the string, Z Lp ρ(x) dx. (3.46) Lopt := 0

√ Then the exponential family e±i λn t n∈N forms a Riesz basis in the Hilbert space L2 ((0, 2Lopt ); dt); this family forms a Riesz basis in the closure of its linear span in L2 ((0, T ); dt) for T > 2Lopt and it is not minimal in L2 ((0, T ); dt) for T < 2Lopt (see [6]). Therefore, using the results of Section 2, one can recover the exponents {λn }n∈N from the data u(0, t), t ∈ (0, T ), T ≥ 2Lopt , provided αn 6= 0 for all n ∈ N. Similarly, one can recover the spectrum {µn }n∈N of the boundary value problem (3.37), and then find the coefficient ρ(·). We note that in the case a(x) = δ(x), this inverse problem for the string equation has been studied by several authors, starting with [16]. It can be solved by the BC method (see, e.g., [2]) without the spectral estimation described in Section 2, however, this approach does not work directly for other choices of a(·). Acknowledgments. The research of Sergei Avdonin was supported in part by the NSF grant ARC 0724860. This work has been started when the first author visited the University of Missouri, Columbia, as a Miller scholar in October and November of 2008. He is grateful to the Department of Mathematics for its hospitality. REFERENCES [1] S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm–Liouville operator, Control Cybernetics, 25 (1996), 429–440. Inverse Problems and Imaging

Volume 4, No. 1 (2010), 1–9

8


[2] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Boundary control and an inverse matrix problem for the equation utt − uxx + V (x)u = 0, Mat. Sb., 182 (1991), 307–331; (Russian.) Engl. transl. in Math. USSR-Sb., 72 (1992), 287–310. [3] S. A. Avdonin, M. I. Belishev and Yu. S. Rozhkov, The BC-method in the inverse problem for the heat equation, J. Inv. Ill-Posed Probl., 5 (1997), 1–14. [4] S. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem. The case of multiple poles, Math. Contr. Sign. Syst., (2010), to appear. [5] S. Avdonin, A. Bulanova and D. Nicolsky, Boundary Control apprach to the spectral estimation problem. The case of simple poles, Sampl. Theory Signal Image Proc., 8 (2009), 225–248. [6] S. A. Avdonin and S. A. Ivanov, “Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,” Cambridge University Press, Cambridge, 1995. [7] S. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schr¨ odinger equation by the boundary control method, Inverse Problems, 18 (2002), 41–57. [8] S. Avdonin, S. Lenhart and V. Protopopescu, Determining the potential in the Schr¨ odinger equation from the Dirichlet to Neumann map by the boundary control method, J. Inverse and Ill-Posed Problems, 13 (2005), 317–330. [9] R. Badeau, D. Bertrand and R. Ga¨ el, High-Resolution spectral analysis of mixtures of complex exponentials modulated by polynomials, IEEE Transactions on Signal Processing, 54 (2006), 1341–1350. [10] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524–527; (Russian). Engl. transl. in Soviet Math. Dokl., 36 (1988), 481–484. [11] M. I. Belishev, Canonical model of a dynamical system with boundary control in inverse problem for the heat equation, St. Petersburg Math. J., 7 (1996), 869–890. [12] M. I. Belishev, Boundary Control method in reconstruction of manifolds and metrics (the BC method ), Inverse Problems, 13 (1997), R1–R45. [13] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. [14] M. I. Belishev and Ya. V. Kurylev, A nonstationary inverse problem for the multidimensional wave equation “in the large”, J. Soviet Math., 50 (1990), 1944–1951. [15] M. I. Belishev and Ya. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Commun. Partial Diff. Eq., 17 (1992), 767–804. [16] A. S. Blagoveschenskii, On a local approach to the solution of the dynamical inverse problem for an inhomogeneous string, Trudy MIAN, 115 (1971), 28–38 (Russian). [17] A. Boumenir and V. K. Tuan, An inverse problem for the heat equation, submitted preprint. [18] G. R. de Prony, Essai ´ experimental et analytique: sur les lois de la dilatabilit´ e des fluides ´ elastiques et sur celles de la force expansive de la vapeur de l’alkool, ` a diff´ erentes ´ temp´ eratures, Journal de l’ Ecole Polytechnique, 1 (1795), 24–76. [19] Y. Hua, A. B. Gershman and Q. Cheng (eds.), “High-Resolution and Robust Signal Processing,” Marcel Dekker, New York, Basel, 2004. [20] Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE transactions of acoustics, speech, and signal processing, 38 (1990), 814–824. [21] A. Katchalov, Ya. Kurylev and M. Lassas, “Inverse Boundary Spectral Problems,” Chapman Hall/CRC, Boca Raton, FL, 2001. [22] A. P. Katchalov, Ya. V. Kurylev, M. Lassas and N. Mandache, Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems, 20 (2004), 419–436. [23] Ya. V. Kurylev and M. Lassas, Gelf ’and inverse problem for a quadratic operator pencil, J. Funct. Anal., 176 (2000), 247–263. [24] S. L. Marple, “Digital Spectral Analysis with Applications,” Prentice-Hall, Englewood Cliffs, NJ, 1987. [25] V. Nagesha and S. Kay, On frequency estimation with the IQML algorithm, IEEE Trans. Signal Processing, 42 (1994), 2509–2513. Inverse Problems and Imaging

Volume 4, No. 1 (2010), 1–9


9

[26] R. Roy, A. Paulraj and T. Kailath, Multiple emitter location and signal parameter estimation, IEEE Trans. Acoust., Speech, Signal Process., 34 (1986), 1340–1342. [27] T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, “Smart Antennas,” Wiley, Hoboken, New Jersey, 2003. [28] R. O. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., 34 (1986), 276–280.

Received March 2009; revised October 2009. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]


Volume 4, No. 1 (2010), 1–9