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Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2011, Article ID 980349, 11 pagesdoi:10.1155/2011/980349

Research Article

Two-Dimensional Frequencies EstimationUsing Two-Stage Separated Virtual SteeringVector-Based Algorithm

Ding Liu and Junli Liang

School of Automation & Information Engineering, Xi’an University of Technology,Xi’an 710048, China

Correspondence should be addressed to Junli Liang, heery [email protected]

Received 10 May 2010; Accepted 25 January 2011

Academic Editor: Laurence Mailaender

Copyright © 2011 D. Liu and J. Liang. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In this paper, we develop a novel two-stage separated virtual steering vector- (SVSV-) based algorithm without associationoperation to estimate 2D frequencies. The key points of this algorithm are (i) in the first stage, this paper rearranges themeasurement data as virtual rectangular array data matrix and obtains the propagator from the data matrix using least-squaresoperator. In addition, the virtual steering vector can be separated into two parts using the introduced electric angle thatcombines 2D frequencies (to avoid incorrect association especially when multiple 2D frequencies have the same frequency atsome dimension), and thus the electric angle and the first part of separated steering vector can be estimated using the derivedrank-reduction propagator method; (ii) in the second stage, this paper estimates the second part of separated steering vector usinganother least-squares operator and obtains 2D frequencies from the recovered steering vector. The resultant SVSV algorithm doesnot require spectral search or pairing parameters or singular value decomposition (SVD) of data matrix. Simulation results arepresented to validate the performance of the proposed method.

1. Introduction

Estimation of two-dimensional (2D) frequencies is a keyproblem in many areas such as wireless communications,joint frequency and wave-number estimation in arrayprocessing, synthetic aperture radar imaging, and nuclearmagnetic resonance imaging [1–12]. For example, in thefurnace temperature control system of silicon single crystalgrowth [13], the temperature measurement is affected bythe Argon inflation, Crucible turn and rise, Crystal rotationand ascent, and so forth. To suppress these 2D harmonicinterferences, it is necessary to estimate their frequenciesaccurately, as shown in Figure 1. Numerous methods havebeen developed to estimate 2D frequencies. The maximum-likelihood (ML) method [3–5], despite its theoretical opti-mality, has extremely demanding computational complexity.Some high-resolution techniques, such as autoregressive

method [6] and maximum entropy method [7], have limitedapplications due to spectral peak search in 2D plane. Kay andNekovei proposed a computationally efficient algorithm in[8], but it is applied to only the single 2D sinusoid case. Boththe algebraically coupled matrix pencils (ACMPs) algorithm[9] and 2D ESPRITmethod [10] combine twomatrix pencilsto obtain correct association. The matrix enhancementand matrix pencil (MEMP) method [11] estimates 1Dfrequencies along each dimension and matches them bymaximizing a certain criterion. Unfortunately, the MEMPmethod does not always provide the correct association. Toimprove its matching performance, the modified MEMP(MMEMP)method is proposed by Chen et al. [12]. Recently,J. Liu, and X. Liu proposed an eigenvector-based approach,and Haardt et al. developed a higher-order singular valuedecomposition- (SVD-)based algorithm [14, 15], but both ofthem require SVD. Besides, they [9–15] suffer performance

2 EURASIP Journal on Advances in Signal Processing

Figure 1: TDR-150 CZ crystal furnace.

degradation especially when multiple 2D frequencies havethe same frequency at some dimension [16, 17].

In this paper, we develop a novel two-stage separatedvirtual steering vector- (SVSV-) based algorithm withoutassociation operation to estimate 2D frequencies. Firstly, thispaper rearranges themeasurement data as virtual rectangulararray data and obtains the propagator using least-squaresoperator. Secondly, this paper introduces a novel electricangle, which is the combination of 2D frequencies to avoidincorrect match especially when multiple 2D frequencieshave the same frequency at some dimension. Thirdly, the vir-tual steering vector can be separated into two parts using theintroduced electric angle, the first one of which is the matrixmerely containing the introduced electric angle; whereas thesecond one is the vector containing 1D frequency. Fourthly,the electric angle and the first part of separated steeringvector can be estimated using the derived rank-reductionpropagator method rather than the conventional ones [18,19], whereas the second part of separated steering vectorcan be obtained using another least-squares operator. Finally,the 2D frequencies can be obtained from the recoveredsteering vector. The resultant SVSV algorithm does notrequire spectral search or pairing parameters or singularvalue decomposition(SVD) of data matrix.

The rest of this paper is organized as follows. The signalmodel is introduced in Section 2. The two-stage separated

virtual steering vector-based algorithm is developed inSection 3. Simulation results are presented in Section 4.Conclusions are drawn in Section 5.

Throughout this paper, we will vectorize anm×nmatrixA, for example, vec(A), which is the mn × 1 column vectorby stacking the row transpose of the matrix A on top of

one another. For example, for the 2 × 2 matrix A =[a bc d

],

vec(A) = [a b c d]T .

2. Signal Model

Consider K superimposed 2D complex-valued sinusoidalsignals

x(m1,m2) =K∑

k=1ake

jφk e j2π[m1 fk1+m2 fk2] + v(m1,m2), (1)

where m1 = 0, 1, . . . ,M1 − 1, m2 = 0, 1, . . . ,M2 − 1. K isthe number of sinusoidal signals. ak, φk, and { fk1, fk2} arethe amplitude, phase, and 2D frequencies of the kth signal,respectively. v(m1,m2) is the 2D additive white Gaussiannoise (AWGN).

In this paper, we assume that multiple 2D frequenciesmay have the same frequency at some dimension, that is,fi1 = f j1 or fi2 = f j2, where i, j ∈ {1, 2, . . . ,K}, butfi1 − fi2 are different from each other, i ∈ {1, 2, . . . ,K}.The objective of this paper is to jointly estimate the 2Dfrequencies { fk1, fk2}, k = 1, 2, . . . ,K , for K 2D sinusoidalsignals, given the 2D data x(m1,m2).

3. Proposed Algorithm

3.1. The First Stage of the Proposed SVSV Algorithm. Asshown in Figure 2, the original M1 × M2-dimensionalmeasurement data can be partitioned into

∏2i=1(Mi − L)

snapshots of a virtual rectangular-shaped array with(L+ 1)2 sensors. When the reference sensor (the origin pointof rectangular-shaped array) lies on the coordinates(0, 0), the single snapshot data of virtual rectangular-shaped array consists of x(0, 0), x(0, 1), . . . , x(0,L), . . . ,x(L, 0), x(L, 1), . . . , x(L,L), which can be represented in avector form as

r(0) = vec

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x(0, 0) x(0, 1) x(0, 2) · · · x(0,L− 1) x(0,L)x(1, 0) x(1, 1) x(1, 2) · · · x(1,L− 1) x(1,L)x(2, 0) x(2, 1) x(2, 2) · · · x(2,L− 1) x(2,L)

......

... · · · ... ...x(L− 1, 0) x(L− 1, 1) x(L− 1, 2) · · · x(L− 1,L− 1) x(L− 1,L)x(L, 0) x(L, 1) x(L, 2) · · · x(L,L− 1) x(L,L)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (2)

EURASIP Journal on Advances in Signal Processing 3

Note that r(0) has the following form:

r(0) = As(0) + n(0), (3)

where

A = [a(γ1,φ1) · · · a(γk,φk

) · · · a(γK ,φK)],

γk = e j2π fk1 ,

φk = e j2π fk2 ,

s(0) =[a1e

jφ1 · · · ake jφk · · · aKe jφK]T

,

n(0) = vec

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

v(0, 0) v(0, 1) v(0, 2) · · · v(0,L− 1) v(0,L)v(1, 0) v(1, 1) v(1, 2) · · · v(1,L− 1) v(1,L)v(2, 0) v(2, 1) v(2, 2) · · · v(2,L− 1) v(2,L)

......

... · · · ... ...v(L− 1, 0) v(L− 1, 1) v(L− 1, 2) · · · v(L− 1,L− 1) v(L− 1,L)v(L, 0) v(L, 1) v(L, 2) · · · v(L,L− 1) v(L,L)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(4)

and, the virtual steering vector a(γk,φk),

a(γk,φk

)

= vec

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 φk φ2k · · · φL−1k φLkγk γkφk γkφ

2k · · · γkφL−1k γkφLk

γ2k γ2kφk γ

2kφ

2k · · · γ2kφL−1k γ2kφLk

......

... · · · ... ...γL−1k γ

L−1k φk γ

L−1k φ

2k · · · γL−1k φL−1k γL−1k φLk

γLk γLkφk γ

Lkφ

2k · · · γLkφL−1k γLkφLk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(5)

When the reference sensor “moves” from coordinates (0, 0)to (0,M2 − L− 1), from (1, 0) to (1,M2 − L− 1), from (2, 0)to (2,M2−L−1), . . ., from (M1−L−1, 0) to (M1−L−1,M2−L−1) with other sensors of virtual rectangular-shaped array,∏2

i=1Mi− l virtual snapshots are available and can be writtenin the following form:

r(t) = As(t) + n(t), t = 0, 1, . . . ,2∏

i=1(Mi − L)− 1. (6)

Let q = t mod (M2 − L), where “mod” stands for theremainder operator and q is the remainder. Thus, p = (t−q)/

(M2 − L) is the corresponding quotient. Therefore, s(t) hasthe following form:

s(t) =[a1e

jφ1e j2π( f11 p+ f12q) · · · ake jφk e j2π( fk1 p+ fk2q)

· · · × aKe jφK e j2π( fK1 p+ fK2q)]T

,

t = 0, 1, . . . ,2∏

i=1(Mi − L)− 1.

(7)

Therefore, r(t), t = 0, 1, . . . ,∏2i=1(Mi − L) − 1, can becombined to form the following data matrix:

Ru = [r(0) · · · r((M1 − L)× (M2 − L)− 1)]= A[s(0) · · · s((M1 − L)× (M2 − L)− 1)]+ [n(0) · · ·n((M1 − L)× (M2 − L)− 1)].

(8)

Considering that spatial smoothing can improve theestimation accuracy [20], we use R = [Ru JR∗u ] as thevirtual array data, where J is an antidiagonal matrix in whichall nonzero elements are equal to one.

In the conventional case, multiple 2D frequencies havethe same frequency at some dimension, that is, fi1 = f j1 orfi2 = f j2, where i, j ∈ {1, 2, . . . ,K}. Some existing methodsestimate 2D frequencies along each dimension and associatethe two sets of separately estimated one-dimensional (1D)frequencies into a set of 2D frequencies. Therefore, incorrectmatch is often encountered in this case, which results insevere performance degradation.

To overcome this difficulty, we define

zk = φkγk= e

j2π fk2

e j2π fk1= e j2π( fk2− fk1), k = 1, . . . ,K. (9)


(0, 0) (0,L) (0,M2 − L− 1) (0,M2 − 1)

(L, 0) (L,L) (L,M2 − L− 1) (L,M2 − 1)

(M1 − L− 1, 0) (M1 − L− 1,L) (M1 − L− 1,M2 − L− 1) (M1 − L− 1,M2 − 1)

(M1 − 1, 0) (M1 − 1,L) (M1 − 1,M2 − L− 1) (M1 − 1,M2 − 1)

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

···

···

···

···

···

···

···

···

Figure 2: The formation of virtual array data from x(m1,m2).

Since |zk| = 1, zk can be represented in another form as[21]

zk = e jθk , θk ∈ [−π,π], k = 1, . . . ,K. (10)

Although e jθk = e j2π( fk2− fk1), θk ∈ [−π,π] may not equal2π( fk2− fk1) due to 2π fk1, 2π fk2 ∈ [−π,π] and thus 2π( fk2−fk1) may exceed [−π,π], that is, 2π( fk2 − fk1) cannot beestimated from θk directly, but e j2π( fk2− fk1) can be estimatedfrom e jθk by estimating θk in [−π,π].

Based on (9)–(10), φk can be represented as γkzk.Therefore, the steering vector a(γk,φk) in (5) can be writtenin another form as follows:

a(γk, zk

)

= vec

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 γkzk γ2kz2k · · · γL−1k zL−1k γLk zLk

γk γ2kzk γ

3kz

2k · · · γLkzL−1k γL+1k zLk

γ2k γ3kzk γ

4kz

2k · · · γL+1k zL−1k γL+2k zLk

......

... · · · ... ...γL−1k γ

Lk zk γ

L+1k z

2k · · · γ2L−2k zL−1k γ2L−1k zLk

γLk γL+1k zk γ

L+2k z

2k · · · γ2L−1k zL−1k γ2Lk zLk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(11)

Introduce the following partition of R:

R =⎡⎣R1R2

⎤⎦, (12)

where R1 and R2 are the first K rows and the rest of matrix R,respectively.

Define the propagator

P =(R1RH1

)−1R1RH2 , (13)

using least-squares operator, which satisfies with P =minP‖R2 − PHR1‖2. Based on the conventional propagatormethod [18, 19, 22], (γk, zk) satisfies

⎡⎣ P−I(L+1)2−K

⎤⎦H

a(γk, zk

) = 0, k = 1, 2, . . . ,K , (14)

or

g(γk, zk

) = aH(γk, zk)⎡⎣ P−I(L+1)2−K

⎤⎦⎡⎣ P−I(L+1)2−K

⎤⎦H

× a(γk, zk) = 0, k = 1, 2, . . . ,K.

(15)

Based on (11), the steering vector a(γk, zk) of virtual arraycan be represented in separated two parts as

a(γk, zk

) = a1(zk)a2(γk), (16)

where (2L + 1)-dimensional vector

a2(γk) =

[1 γk γ2k γ

3k · · · γ2L−1k γ2Lk

]T, (17)

and in, (L + 1)2 × (2L + 1)-dimensional matrix a1(zk),

a1(zk) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C 0(L+1)×L

0(L+1)×1C 0(L+1)×(L−1)

0(L+1)×2C 0(L+1)×(L−2)...

0(L+1)×(L−1)C 0(L+1)×1

0(L+1)×LC

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (18)


in which

C = diag{1, zk, z2k , . . . , z

Lk

}. (19)

Inserting (16) into (15), we have

g(γk, zk

)

= aH(γk, zk)⎡⎣ P−I(L+1)2−K

⎤⎦⎡⎣ P−I(L+1)2−K

⎤⎦H

a(γk, zk

)

= aH2(γk)aH1 (zk)

⎡⎣ P−I(L+1)2−K

⎤⎦⎡⎣ P−I(L+1)2−K

⎤⎦H

× a1(zk)a2(γk)

= aH2(γk)D(zk)a2

(γk)

= 0(20)

in the ideal case [18, 19], where

D(z) = aH1 (z)⎡⎣ P−I(L+1)2−K

⎤⎦⎡⎣ P−I(L+1)2−K

⎤⎦H

a1(z). (21)

Note that if 2L+1 ≤ (L+1)2−K thenD(z) is, in general,full rank because the column rank of

[P

−I(L+1)2−K]is (L+1)2−

K . Equation (20) may hold true if and only if matrix D(z)drops rank (i.e.,2L), or equivalently, when the polynomial ofz = zk (see (21) for details),

det{D(z)} = 0, (22)

which is the rank-reduction propagator method (RRPM)[23]. From the roots zk = e jθk of (22) [20], it is easy tofind θk, k = 1, 2, . . . ,K . Equations (15) and (20) imply that,substituting the estimated ẑk into a1(zk) in (20), then we findthe minima of the following function:

γ̂k = minγ

aH2(γ)D(ẑk)a2

(γ), (23)

the minima of which indicates the estimation γ̂k. Infact, Equation (20) implies that a2(γk) is just the uniqueeigenvector corresponding to the smallest eigenvalue (zero)of D(ẑk) in the ideal case. In other words,

⎡⎣ P−I(L+1)2−K

⎤⎦H

a1(zk)a2(γk) = 0. (24)

3.2. The Second Stage of the Proposed SVSV Algorithm. Define

D2 =[D21 D22

]=⎡⎣ P−I(L+1)2−K

⎤⎦H

a1(ẑk), (25)

where D21 and D22 are the first columns and last 2L columnof D2, respectively.

From (25), we can see that[D21 D22

]a2(γk)

=[D21 D22

] [1γk γ2k γ

3k · · · γ2L−1k γ2Lk

]T

= D21 +D22[γk γ

2k γ

3k · · · γ2L−1k γ2Lk

]T

= 0

(26)

or

D22[γk γ

2k γ

3k · · · γ2L−1k γ2Lk

]T = −D21, (27)

which implies that [γk γ2k γ3k · · · γ2L−1k γ2Lk ]

Tcan be

obtained using another least-square operation

[γk γ

2k γ

3k · · · γ2L−1k γ2Lk

]T = −(DH22D22

)−1DH22D21. (28)

3.3. Description of the Proposed SVSV Algorithm. Based onthe above analyses, the proposed SVSV algorithm for 2Dfrequency estimation can be described as follows:

Step 1. Construct data matrix R using (2)–(8).

Step 2. The partition of R yields R1 and R2 using (12), from

which the propagator P = (R1RH1 )−1R1RH2 is obtained.

Step 3. Obtain zk from the roots of polynomial D(z) in (22).

Step 4. a2(γ̂k) is easily constructed from the least-squaresolution of (28).

Step 5. Obtain γ̂k from a2(γ̂k) according to the followingequation:

γk = 12L2L∑

i=1

a2(γ̂k)[i + 1]

a2(γ̂k)[i]

, (29)

where a2(γ̂k)[i] denotes the ith element of a2(γ̂k).

Step 6. The estimation { f̂k1, f̂k2} can be given as follows:

fk1 = 12π∠γ̂k,

fk2 = 12π∠(γ̂kẑk

).

(30)

3.4. Discussion. In this section, we discuss the proposedalgorithm in three ways, that is, computational complexity,selection of L, and capacity for estimating 2D frequencieswhen multiple 2D frequencies have the same frequency atsome dimension.


Computational Complexity. Regarding the computationalcomplexity, we consider the major part, namely, multipli-cations and SVD. The eigenvector method addressed [14]requires O{M31M32}, and the MMEMP algorithm developedin [12] needs O{L2M21M22}. However, the proposed SVSValgorithm without SVD requires O{M1M2L}, being less thanthose of the eigenvector method and MMEMP algorithm.

About Selection of L. the original M1 × M2 measurementdata can be partitioned into

∏2i=1(Mi − L) snapshots of a

virtual rectangular-shaped array with (L + 1)2 sensors. In thearray signal processing field, the increase of sensors (largearray aperture) can improve the direction-of-arrival (DOA)estimation accuracy under the given snapshots, and so doesthe increase of snapshots under the given sensors [16–18].However, for the 2D frequencies estimation problem, therise of virtual sensors is contradictory to the increase of thevirtual snapshots, givenM1 ×M2 measurement data (see therelationship between (L+ 1)2 and

∏2i=1(Mi − L). In this case,

more virtual sensors (i.e., (L + 1)2 or larger array aperture)means that less snapshots (

∏2i=1(Mi−L) ) are available, which

causes low estimation accuracy of R or P = (R1RH1 )−1R1RH2 .On the other hand, more virtual snapshots (

∏2i=1(Mi − L))

means that less virtual sensors are available, which causessmall array aperture. Therefore, the selection of L should beevaluated from available virtual snapshots and sensors. Fromthe second experiment in Section 4, we can see that both toolarge L and small L cause low estimation accuracy, which isin agreement with the above analyses.

Capacity for Estimating 2D Frequencies When Multiple 2DFrequencies Have the Same Frequency at some Dimension. Inthe conventional case, multiple 2D frequencies have the samefrequency at some dimension, that is, fi1 = f j1 or fi2 = f j2,where i, j ∈ {1, 2, . . . ,K}. Although fi1 = f j1 or fi2 = f j2,fi1 − fi2 is different from each other, i ∈ {1, 2, . . . ,K}. zk aredetermined from the roots of polynomial D(z) in (22) andthe related a2(γk) (i.e., the unique eigenvector correspondingto the smallest eigenvalue (zero) of D(ẑk)) is obtainedusing least-squares operator. Therefore, the proposed SVSValgorithm can implement efficiently especially whenmultiple2D frequencies have the same frequency at some dimensionand avoid pairing parameters. However, when fi1 − fi2 areequals to each other, i ∈ {1, 2, . . . ,K}, polynomial D(z) in(22) has several equivalent roots, z1 = z2 = · · · = zK . Inthis case, a2(γk), k ∈ {1, 2, . . . ,K}, is no longer the uniqueeigenvector, but one of several eigenvectors corresponding tothe smallest eigenvalue (zero) of D(ẑk). Therefore, unequalfi1 − fi2, i ∈ {1, 2, . . . ,K}, are required to ensure uniqueestimation for the proposed algorithm.

4. Simulation Results

In the simulation section, we compare the SVSV algorithmwith the MMEMP algorithm [12] and the eigenvectormethod [13] to assess the estimation performance of theproposed algorithm.

0 5 10 15 20 25 3010−5

10−4

10−3

10−2

SNR (dB)

RMSE

(Hz)

SVSVEigenvector

MMEMPCRLB

Figure 3: RMSE of the estimated frequency f11 versus SNR.

0 5 10 15 20 25 3010−5

10−4

10−3

10−2

SNR (dB)

RMSE

(Hz)

SVSVEigenvector

MMEMPCRLB


In the first experiment, 20 × 20 data samples with f11 =0.20837, f12 = 0.09943, f21 = 0.15077, f22 = 0.25789, f31 =0.10231, and f32 = 0.30987 are used. The effect of signal-to-noise ratio (SNR) on the performance of the proposedSVSV algorithm is investigated. The SNR varies from 0 dBto 30 dB. We use the root mean square error (RMSE) ofthe estimated frequencies as the performance evaluation.Figures 3, 4, 5, 6, 7, and 8, respectively, show the RMSEof estimated six frequencies from 500 independent runsusing these algorithms: the proposed algorithm (L = 6),the MMEMP algorithm (K = L = 6, enhancementmatrix [12, Equation(5.7)] for all three experiments, and theeigenvector method (K1 = K2 = 6, enhancement matrix


0 5 10 15 20 25 3010−5

10−4

10−3

10−2

SNR (dB)

RMSE

(Hz)

SVSVEigenvector

MMEMPCRLB


0 5 10 15 20 25 3010−5

10−4

10−3

10−2

SNR (dB)

RMSE

(Hz)

SVSVEigenvector

MMEMPCRLB


[14]). In addition, the related CRLB on the variance of theestimated parameters is obtained from the inverse of theFisher information matrix by averaging 500 computations[23]. From these figures, we can see that the eigenvectormethod and the proposed algorithm slightly outperform theMMEMPmethod in the estimation accuracy under differentSNRs, and their RMSE approach to the CRLB.

In the second experiment, 20×20 data samples with f11 =0.20837, f12 = 0.09943, f21 = 0.15077, f22 = 0.25789, f31 =0.10231, and f32 = 0.30987 are used. The effect L on theperformance of the proposed SVSV algorithm is investigated.The SNR is set to 10 dB. Figure 9 shows the RMSE of theestimated frequencies for 500 independent runs using the

0 5 10 15 20 25 3010−5

10−4

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(Hz)

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MMEMPCRLB


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(Hz)

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proposed SVSV algorithm with different L. From it, we cansee that for the simulation data the proposed algorithm hasbetter estimation performance when L = 6.

In the third experiment, 20× 20 data samples with f11 =0.20837, f12 = 0.09943, f21 = 0.10231, f22 = 0.25789, f31 =0.10231, and f32 = 0.09943) are used. The ability of theproposed SVSV algorithm to deal with the same frequenciesat some dimension is investigated. The SNR varies from 0 dBto 30 dB. Figures 10, 11, 12, 13, 14, and 15, respectively,show the RMSE of the estimated six frequencies for 500independent runs using these algorithms: the proposedalgorithm (L = 6), the MMEMP algorithm (K = L =6, enhancement matrix[12, Equations (5–7)] for all three


3 3.5 4 4.5 5 5.5 6 6.5 710−4

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L

RMSE

(Hz)

f11f21f31

f12f22f32

Figure 9: RMSE of the estimated frequencies versus L.

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RMSE

(Hz)

SVSVEigenvector

MMEMPCRLB


experiments, and the eigenvector method (K1 = K2 =6, enhancement matrix [14]). The eigenvector algorithmobtains the eigenvectors being corresponding to eigenvaluese j2π f11 , e j2π f21 , and e j2π f31 , and thus has poor estimationperformance due to f21 = f31 at the first dimension. Thesimilar reason holds for the MMEMP method. From Figures13 and 15, we can see that the eigenvector algorithm andMMEMP method have low estimation accuracy for f22 andf32 due to f21 = f31. However, the proposed SVSV algorithmperforms well in distinguishing the three signals.

In the fourth experiment, 20 × 20 data samples withf11 = 0.20837, f12 = 0.09943, f21 = 0.10231, f22 =0.25789, f31 = 0.10231, and f32 = 0.09943 are used.

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MMEMPCRLB


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The SNR is set to 10 dB. Figures 16, 17, and 18 show 2Dplot of the estimated frequencies for 500 independent runsusing the proposed algorithm, the MMEMP algorithm, andthe eigenvector method, respectively. From these figures,we can see that the MMEMP and eigenvector methodsimplement poorly when multiple 2D frequencies have thesame frequency at some dimension, that is, f21 = f31.However, the proposed algorithm performs well becausef11− f12, f21− f22, and f31− f32 are different from each other.

In the fifth experiment, 20 × 20 data samples with twosignals are used, in which the first one is with 2D frequenciesf11 = 0.20837, f12 = 0.09943), and the second one is withf21 = 0.15077. The SNR is set as 10 dB. When f22 varies from


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−0.00817 to 0.09183 (i.e., the frequency gap θ2 − θ1 variesfrom−0.05 to 0.05), the RMSEs of the frequency estimationsare shown in Figure 19. From it, we can see that the proposedalgorithm is slightly sensitive to the frequency gap. When theabsolute value of the frequency gap is larger than 0.03, theproposed algorithm performs well.

5. Conclusion

A novel two-stage separated virtual steering vector-basedalgorithm is proposed for 2D frequencies estimation prob-

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10−1


0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

0.05

0.1

0.15

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f1

f 2

Figure 16: 2D plot of 500 independent estimates for three 2Dfrequencies using the proposed algorithm.

lem. The algorithm arranges the measurement data asvirtual rectangular-shaped data. Moreover, by introducingone special electric angle which is the combination of 2Dfrequencies, we derive a rank-reduction propagator methodto estimate the introduced electric angle and the firstpart of separated virtual steering vector. The second partof the separated virtual steering vector is obtained usingleast-square operator. Finally, 2D frequencies are obtainedfrom the recovered steering vector. Therefore, the resultantSVSV algorithm does not require spectral search or pairingparameters or singular value decomposition (SVD) of datamatrix.


0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

0.05

0.1

0.15

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0.25

0.3

f1

f 2

Figure 17: 2D plot of 500 independent estimates for three 2Dfrequencies using the MMEMP algorithm.

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

0.05

0.1

0.15

0.2

0.25

0.3

f1

f 2

Figure 18: 2D plot of 500 independent estimates for three 2Dfrequencies using the Eigenvector algorithm.

−0.05 0 0.0510−4

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θ2 − θ1 (Hz)

RMSE

(Hz)

f11f21

f12f22

Figure 19: RMSE of the estimated frequencies versus variedfrequency of f22.

Acknowledgments

This work was supported by the Important NationalScience and Technology Specific Project under Grant2009x02011001, by the National Natural Science Founda-tions of China under grant 60901059 and 61075044, bythe China Postdoctoral Science Foundation funded projectsunder grant 201003679 and 20100481355, by the Educa-tional Department Foundation grant 09JK629 and NaturalScience Foundation under grant 2010JQ8001 of ShaanxiProvince, and by theDiscipline Union Fund under grant 116-210905 of Xi’an University of Technology.

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1. Introduction2. Signal Model3. Proposed Algorithm3.1. The First Stage of the Proposed SVSV Algorithm.3.2. The Second Stage of the Proposed SVSV Algorithm.3.3. Description of the Proposed SVSV Algorithm.3.4. Discussion.

4. Simulation Results5. ConclusionAcknowledgmentsReferences

two-dimensionalfrequenciesestimation usingtwo … · 2017. 8. 27. · require spectral search or...

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