Progress In Electromagnetics Research, Vol. 128, 195–214, 2012

CONSTRAINED TRILINEAR DECOMPOSITION WITHAPPLICATION TO ARRAY SIGNAL PROCESSING

X. Liu1, *, T. Jiang2, L. X. Yang1, and H. B. Zhu1

1College of Telecommunications and information Engineering, NanjingUniversity of Posts & Telecommunications, Jiangsu Key Laboratory ofWireless Communication, Nanjing, China2Nanjing Panda Technology Co., Ltd., Nanjing, China

Abstract—This paper links the constrained trilinear tensor modelinto array signal processing. The structure properties of basebandsignal, such as the Constant-Modulus (CM) and Finite Alphabet(FA) structures which are already known in the receiving array,are exploited in trilinear decomposition. Two novel algorithms forconstrained trilinear decomposition are proposed and applied to arraysignal processing. The distinguishing features of the proposed modeland algorithms compared to the traditional trilinear signal processingmethods are: (i) the proposed model has a better performance andlower computation complexity. (ii) it can still work well even ifdegeneracy of factors are involved in the data model, which is notvalid in traditional algorithms. Simulation results are presented toillustrate the application of the constrained trilinear decomposition toarray signal processing and evaluate the performance of the proposedalgorithms in DOAs estimation.

1. INTRODUCTION

In electromagnetics and signal processing area, antenna arrays havebeen widely applied to locate various types of signals and improvesystem performance [1]. The main work of array signal processing isthe estimation of parameters and signals by utilizing temporal andspatial information from the samples of receive antenna arrays [2–5]. Many effective methods, such as MUSIC, ESPRIT, PCAand HOS, have been developed to deal with direction finding,

Received 14 March 2012, Accepted 15 May 2012, Scheduled 29 May 2012* Corresponding author: Xu Liu (liuxu@njupt.edu.cn).

196 Liu et al.

polarization estimation, and signal detection [6–8]. Recently, trilineardecomposition has been exploited to be an effective method inarray signal processing. Some trilinear models, such as PARAllelFACtor (PARAFAC) and PARAllel profiles with LINear Dependencies(PARALIND), have been applied to parameter estimation, signaldetection, etc. [9–11], due to their powerful unique propertieswhich are not shared by vector or matrix models [12]. In arraysignal processing aspect, trilinear decomposition can be classifiedas a high-dimension generalization of ESPRIT algorithm and jointapproximate diagonalization (JAD) method [13]. Sidiropoulos etal. firstly introduced trilinear analysis to multiple invariance sensorarray processing (MI-SAP) [9]. They applied PARAFAC model toestimate the azimuth and elevation angles from different sources ina uniform square array. Beamforming [14, 15], polarization sensitivearray processing [16–18] and MIMO radar location [19, 20] have alsobeen linked to trilinear analysis. The common characteristic of tensormodeling approach in these applications is that baseband signals andarray response vectors, which are always involved in data model, aretreated as pure ‘double-precision’ or ‘complex’ data during trilineardecomposition. However, this kind of simplication implies that somesuperior structure properties of these data, such as constant-modulus(for phase/frequency modulated signal) and Vandermonde structure(for uniform array response), are ignored, which may cause potentialperformance loss during signal processing.

Trilinear alternating least square (TALS) algorithm is oftenused to accomplish trilinear decomposition in PARAFAC-based signalprocessing [21, 22]. Although TALS has some superior properties, suchas simplicity formulation and monotone convergence, it still suffersfrom some inherent problems, for example, low convergence speed, infollowing cases:a) High correlated factors involved in trilinear model. This condition

is often defined as “presence of degeneracy” and appears whencoherent signals are considered [23–25].

b) Initialized by random numbers. Sometimes, random initializationmay cause local minimum or low convergent speed in TALS.

Several methods have been proposed to improve the performance ofTALS algorithm, some of which are compression [26], line search [27],enhanced line search [28, 29], and optimized initialization [30]. Influorescence spectroscopy and flow injection analysis, Bro et al. havepointed out that the performance of trilinear decomposition can beimproved when the multi-way data have some special structure [31]. Aswe have mentioned, baseband signals have some structural propertieswhere the receiving end is always aware of. This paper exploits

Progress In Electromagnetics Research, Vol. 128, 2012 197

structural properties of baseband signal in PARAFAC analysis. Aconstrained PARAFAC model is introduced as a mathematical toolfor modeling array signals. Two novel algorithms are proposedfor constrained PARAFAC decomposition and then applied toparameter estimation in array signal processing. The distinguishingimprovements of the proposed model and algorithms compared to thealready existing tensor-based signal processing methods are:

i). The proposed trilinear model exploits the structure of signalsto improve the performance of parameter estimation in arraysignal processing applications. Simulations will give an examplethat constrained PARAFAC decomposition has better accuracy inparameters estimation than traditional methods.

ii). As an iteration method, the proposed trilinear decomposition algo-rithm has better convergence performance and lower computationcomplexity than TALS-based trilinear signal processing method.

iii). The proposed method still has good performance when somefactors of model are “degeneracy”, which is not valid in thetraditional TALS-based method.

The rest of this paper is outlined as follows. Section 2 definesthe constrained PARAFAC model. In Section 3, two iterativealgorithms are proposed for constrained PARAFAC decomposition.Some related works, such as convergence property, complexity analysis,and initialization methods, are also discussed in this section. InSection 4, we apply constrained trilinear model to array signalprocessing. Simulations are presented to evaluate the given algorithmsin parameter estimation. In the last section, we summarize theconclusion.

Some notation conventions will be used in this paper.diag ([a, b . . .]) denotes the diagonal matrix with diagonal scalar en-tries a, b, . . . while blockdiag ([A, B . . .]) denotes the block diagonalmatrix with diagonal matrix entries A, B, . . .. (·)T and (·)† stand fortranspose and pseudo-inverse, respectively. || · ||2F stands for Frobe-nius norm. vec (·) stacks the columns of its matrix argument in avector. unvec (·) is the inverse operation of vec (·), unvec(c, I, J) =[c(1 : J), c(J +1 : 2J), . . . , c((I − 1)J : IJ)]. ⊗ is Kronecker product.¯ denotes the Khatri-Rao product, which is a column-wise Kroneckerproduct. Define A = [a1, . . . , aR] ∈ CI×R, B = [b1, . . . , bR] ∈ CJ×R.The Khatri-Rao product of A and B is:

A¯B = [a1 ⊗ b1, . . . ,aR ⊗ bR]

198 Liu et al.

2. CONSTRAINED PARAFAC MODEL

In this section, constrained PARAFAC model is formulated as the basicmodeling tool in array signal processing. Firstly, we give the basicform of PARAFAC model. Define a three-order tensor X ∈ CI×J×K

with elements of xi,j,k. X can be represented as a sum of tensorproducts [30].

X = a1 b1 c1 + . . . + aR bR cR =R∑

r=1

ar br cr (1)

where ar ∈ CI×1, br ∈ CJ×1, cr ∈ CK×1, r = 1, . . . , R and denotesthe outer product of tensors. Figure 1 gives a illustration of (1).

R is defined as the rank of three-order tensor X. Assume thatai,r, bj,r, ck,r are elements of ar, br, cr. As the scalar form of (1),xi,j,k can be formulated as a sum of triple products.

xi,j,k = ai,1bj,1ck,1+, . . . ,+ai,Rbj,Rck,R =R∑

r=1

ai,rbj,rck,r (2)

Equations (1) and (2) are variably known as PARAFAC model,trilinear decomposition or canonical decomposition [32]. Define threematrices A = [a1, . . . , aR], B = [b1, . . . , bR], C = [c1, . . . , cR].A, B, C are defined as “mode matrix” of a given PARAFAC model.

When PARAFAC analysis is adopted in array signal processing,the mode matrices represent meaningful physical interpretation andusually have structure properties. Especially, baseband signal matrixis usually one of the mode matrices, for example, in applicationsof [9, 10, 13]. Modulated signals usually have special structures whichare already known in the receiving end. Many works have exploitedthese structure properties to bilinear decomposition (matrix analysis)for blind signal separation and parameter estimation [33, 34]. However,as we have mentioned, baseband signal is only considered as simple‘data’, not a ‘signal’ in traditional trilinear signal processing. The

= + ... +

a1

b1

c1

aR

bR

cR

X

Figure 1. PARAFAC model.

Progress In Electromagnetics Research, Vol. 128, 2012 199

traditional algorithms usually do not hold and utilize these priorstructures.

In this study, structure properties are exploited in PARAFACanalysis. Assume that mode matrix C stands for the signal matrixin (2). The scalar form of the constrained PARAFAC model can beformulated as follow:

xi,j,k =R∑

r=1

ai,rbj,rck,r, ck,r ∈ Ω (3)

where Ω denotes a constrained data set. In this paper, Ω stands for FAor CM constraint. By exploiting the structure constraints, model (3) ismore appropriate to characterize array signals than traditional trilineardata model. Furthermore, it is shown that the proposed fittingalgorithm, which will be given in the next section, can improve theperformance in array signal processing applications.

3. ALGORITHMS

In this section, we propose two algorithms, named Trilinear AlternatingLeast Square with Projection (TALSP) and Trilinear AlternatingLeast Square with Successive Interference Cancellation (TALSSIC),for constrained PARAFAC decomposition. TALSP algorithm utilizes‘projection’ method to reconstruct structure of baseband signals, whichcan be viewed as a high order extension of FA-based projectionmethod [33]. It has simple formulation and low computationcomplexity. TALSSIC algorithm exploits ‘successive interferencecancellation’ to reconstruct structure in column wise and is guaranteedto be strictly monotonically convergent during iteration. With a goodinitialization, TALSSIC algorithm can converge rapidly to the MLestimation.

Before giving the algorithms, we rearrange 3-D tensor to 2-Dmatrices to simplify algorithm analysis. Consider the formulation ofX in the presence of noise

X = X + E (4)

where E is Gauss noise triple matrix. Slice X along three directionsand rearrange the elements of X. X can be formulated into three ‘slice’matrices X, Y, Z [20]

XIJ×K = (A¯B)CT + EX

YJK×I = (B¯C)AT + EY

ZKI×J = (C¯A)BT + EZ

(5)

200 Liu et al.

where [X](i−1)J+j,k = [Y](j−1)K+k,i = [Z](k−1)I+i,j = xi,j,k. EX,Y,Z arethree Gauss noise slice matrices. The cost function of matrix variablesA, B and C is defined as:

f(A,B,C;X) =∣∣∣∣∣∣X− (A¯B)CT

∣∣∣∣∣∣2

F(6)

3.1. TALSP Algorithm

TALSP is a kind of alternating iteration algorithm. Let (·)(k) standfor the value obtained in the k-th iteration. There are two steps in thek-th update procedure of matrix C.

First step: given estimation A(k−1), B(k−1), C(k) is the leastsquare solution of the minimization of f(A(k−1), B(k−1), C; X) withrespect to unconstrained continuous C

C(k)=arg minC

∣∣∣∣∣∣X−

(A(k−1)¯B(k−1)

)CT

∣∣∣∣∣∣2

F=

((A(k−1)¯B(k−1)

)†X

)T

(7)

Second step: project elements of C(k), ck,r, k = 1, . . . , K, r =1, . . . , R, to the constraint set Ω.

ck,r = projΩ(ck,r) (8)

where projΩ(•) is the projection operator. According to differentconstraints, projΩ(•) denotes different operations. Taking CMconstraint for example, the projection operator is

projCM(c) = c/||c||22 (9)

Then the k-th estimation of C, denoted as C(k) with elements ck,r, isobtained. The k-th least square update of A is obtained by minimizingthe cost function, keeping B(k−1) and C(k) fixed.

A(k) =arg minA

∣∣∣∣∣∣Y−

(B(k−1)¯C(k)

)AT

∣∣∣∣∣∣2

F=

((B(k−1)¯C(k)

)†Y

)T

(10)

Similarly, a better estimation of B is obtained as

B(k) =arg minB

∣∣∣∣∣∣Z−

(C(k)¯A(k)

)BT

∣∣∣∣∣∣2

F=

((C(k)¯A(k)

)†Z)T

(11)

Continue this process until A, B and C converge. TALSP algorithmis presented in Table 1.

Progress In Electromagnetics Research, Vol. 128, 2012 201

Table 1. TALSP algorithm.

Step 1: Initialize three mode matrices A(0), B(0), C(0), k = 0;

Step 2: k = k + 1

Step 3: According to (7)–(8), update C using X, A(k−1), B(k−1);

Step 4: According to (10), update A using Y, B(k−1), C(k);

Step 5: According to (11), update B using Z, C(k), A(k);

Step 6: repeat steps 2–5. Define a fitting error variable

ε = ||X−(A(k−1) ¯ B(k−1)

)C(k−1)T ||2F

−||X−(A(k) ¯ B(k)

)C(k)T ||2F

When ε becomes very small (less than10−8), algorithm is finished.

3.2. TALSSIC Algorithm

A limitation of TALSP is that the projection procedure in C-updatebreaks its monotone convergence, which means that TALSP algorithmcannot guarantee to decrease or hold the value of cost function in eachmatrix updating procedure†. TALSSIC is also a nonlinear alternatingiterative algorithm. However, it is most important that TALSSICcan be guaranteed to monotonically converge to the constrained leastsquare solution.

The main difference between TALSSIC and TALSP is the updateprocedure of mode matrix C. TALSSIC exploits successive interferencecancelation strategy to update C in column wise. Assume that A(k),B(k), C(k) stand for the k-th updates of A, B, and C, respectively.Define H(k) = A(k) ¯ B(k). a(k)

r , b(k)r , c(k)

r , h(k)r are columns of A(k),

B(k), C(k), H(k). Consider (7) in column wise during the (k − 1)-thiteration:

X =(A(k−1) ¯ B(k−1)

)C(k−1)T

+ EX = H(k−1)C(k−1)T+ EX

=h(k−1)1 c(k−1)T

1 +R∑

r=2

h(k−1)r c(k−1)T

r +EX =h(k−1)1 c(k−1)T

1 +X1+E1 (12)

where X1 =R∑

r=2h(k−1)

r c(k−1)T

r . E1 is the additive noise during the

† Although we have found that TALSP is monotonically convergent mostly, a strictlydemonstration is failed.

202 Liu et al.

update of c1. The unconstrained least square solution of c1 is

c(k)1 =arg min

c1

∣∣∣∣∣∣X−h(k−1)

1 cT1 −X1

∣∣∣∣∣∣2

F=

(h(k−1)

1

)H

∣∣∣∣∣∣h(k−1)

1

∣∣∣∣∣∣2

2

(X−X1)

T

(13)

Project the elements of c(k)1 to constraint set Ω

c(k)1 = projΩ

(c(k)1

)(14)

where c(k)1 is the estimation of c1 in the k-th iteration. Then update

c(k)1 to C(k−1) (replace c(k−1)

1 with c(k)1 ) and continue to update cr,

r = 2, . . . , R. Without loss of generality, we give the update procedureof cr. Define matrix Xr

Xr =r−1∑

i=1

h(k−1)i c(k)T

i +R∑

i=r+1

h(k−1)i c(k−1)T

i (15)

The formulation of X in the rth update is

X = h(k−1)r c(k−1)T

r + Xr + Er (16)

where Er is the additive noise during cr-update. c(k)r can be obtained

as follow

c(k)r = projΩ

(h(k−1)

r

)H

∣∣∣∣∣∣h(k−1)

r

∣∣∣∣∣∣2

r

(X−Xr)

T

(17)

C(k) is then obtained when all columns of C are updated. The updateprocedures of A and B are the same as TALSP algorithm. TALSSICalgorithm is concluded in Table 2.

3.3. Convergence

Unlike TALSP algorithm, TALSSIC algorithm is monotonicallyconvergent. Here we give a simple demonstration, which is equivalentto show that matrices update procedures of TALSSIC only decrease orhold, but never increase the value of the cost function. The updatesof A and B are standard least square procedures. Then the followinginequalities are always hold

f(A(k),B,C;X

)≤ f

(A(k−1),B,C;X

)(18)

f(A,B(k),C;X

)≤ f

(A,B(k−1),C;X

)(19)

Progress In Electromagnetics Research, Vol. 128, 2012 203

Table 2. TALSSIC algorithm.

Step 1: Initialize three mode matrices A(0), B(0), C(0), k = 0;Step 2: k = k + 1Step 3: According to (12)–(17), update C using X, A(k−1), B(k−1);Step 4: According to (10), update A using Y, B(k−1), C(k);Step 5: According to (11), update B using Z, C(k), A(k);Step 6: repeat steps 2–5. Define a fitting error variable

ε = ||X−(A(k−1) ¯ B(k−1)

)C(k−1)T ||2F

−||X−(A(k) ¯ B(k)

)C(k)T ||2F

When ε becomes very small (less than 10−8), algorithm is finished.

Therefore, if

f(A,B,C(k);X

)≤ f

(A,B,C(k−1);X

)(20)

is satisfied, TALSSIC algorithm can be demonstrated monotonicallyconvergent. Now we prove the validity of (20). Consider the followingconstrained least square problem:

c = arg minc∈Ω

∣∣∣∣∣∣X− hcT

∣∣∣∣∣∣2

F(21)

[35] has pointed out that (21) is equivalent to two procedures:

c = arg min∣∣∣∣∣∣X− hcT

∣∣∣∣∣∣2

F(22)

c = arg minc∈Ω

∣∣∣∣c− c∣∣∣∣2

F(23)

where c is the unconstrained least square solution of (21). Therefore,(17) is the constrained least square solution of (16), and the followinginequality is always satisfied

∣∣∣∣∣∣X−Xr − hrc(k)T

r

∣∣∣∣∣∣2

F

= mincr

∣∣∣∣∣∣X−Xr − hrcT

r

∣∣∣∣∣∣2

F≤

∣∣∣∣∣∣X−Xr − hrc(k−1)T

r

∣∣∣∣∣∣2

F(24)

Then it follows

f(A,B,C(k);X

)

=∣∣∣∣∣∣X−(A¯B)C(k)T

∣∣∣∣∣∣2

F=

∣∣∣∣∣∣X−HC(k)T

∣∣∣∣∣∣2

F=

∣∣∣∣∣

∣∣∣∣∣X−R∑

r=1

hrc(k)T

r

∣∣∣∣∣

∣∣∣∣∣2

F

204 Liu et al.

=∣∣∣∣∣∣X−XR−1 − hRc(k)T

R

∣∣∣∣∣∣2

F= min

c

∣∣∣∣∣∣X−XR−1 − hRcT

∣∣∣∣∣∣2

F

≤∣∣∣∣∣∣X−XR−1 − hRc(k−1)T

R

∣∣∣∣∣∣2

F=

∣∣∣∣∣∣X−XR−2 − hR−1c

(k)T

R−1

∣∣∣∣∣∣2

F

= minc

∣∣∣∣∣∣X−XR−2 − hR−1cT

∣∣∣∣∣∣2

F≤

∣∣∣∣∣∣X−XR−2 − hR−1c

(k−1)T

R−1

∣∣∣∣∣∣2

F

...

≤∣∣∣∣∣∣X−X1 − h1c

(k−1)T

1

∣∣∣∣∣∣2

F=

∣∣∣∣∣∣X−

R∑

r=1

hrc(k−1)T

r

∣∣∣∣∣∣2

F

=∣∣∣∣∣∣X−HC(k−1)T

∣∣∣∣∣∣2

F=

∣∣∣∣∣∣X− (A¯B)C(k−1)T

∣∣∣∣∣∣2

F

= f(A,B,C(k−1);X) (25)

The proof of (20) is finished. According to (18) ∼ (20), TALSSICalgorithm is demonstrated monotonically convergent.

3.4. Complexity and Initialization

In the C-update procedure of algorithms, the multiplication costof TALSP is IJR2 + IJKR while the complexity of TALSSIC is3IJR + IJR2 + IJKR. Note that TALSSIC algorithm has highercomputation complexity than TALSP algorithm. However, because ofthe similarity of these two algorithms, we can use TALSP to ‘initialize’TALSSIC to ensure convergence and decrease complexity during signalprocessing. The main procedure can be summarized as two steps:TALSP is firstly used to pre-fit the triple data (e.g., the receivingsignals of sensor array). When TALSP worsens the fit or the fittingerror variable ε is relatively small (less than 1e-4, for example) inthe (k+1)th iteration, then we use the estimations A(k), B(k) andC(k), obtained from TALSP, to initialize TALSSIC and continue tofit the data until algorithm converges. This method combines theadvantage of TALSP (low computation costs) and the benefits ofTALSSIC (monotone convergence). In the following simulations, weuse this initialization method to evaluate the performance of proposedalgorithms.

4. SIMULATION RESULTS

In this section, we apply constrained PARAFAC decomposition tomodel multiple invariance sensor array signals and accomplish jointazimuth and elevation angles estimation by using TALSSIC algorithm.

Progress In Electromagnetics Research, Vol. 128, 2012 205

Subarray 3

Subarray 4

Subarray 1 Subarray 2

Figure 2. The arrangement of 6×5 rectangular array and subarraies.

The following four simulations will illustrate the improvements of theproposed constrained model and algorithms listed in Section 1. Firstly,simulations 1 and 2 evaluate the performance of parameters estimationof the TALSSIC-based algorithm in the forms of both scatter diagramand mean squared error curve, compared with TALS-based algorithmand traditional ESPRIT algorithm. Secondly, simulation 3 depictsthe convergence performance of the proposed algorithms. Finally,simulation 4 gives the performance of TALSSIC-based and TALS-basedalgorithms when correlated factors are involved in the data model.

As discussed in Section 3, TALSP algorithm is used to initializeTALSSIC. Consider a 6 × 5 rectangular array and four subarraies,depicted in Figure 2 [9, 36].

Assume that the baseband signals are modulated as QPSK withCM structure. The received signal of multiple invariance sensor arraycan be formed as constrained PARAFAC model.

X = (Φ¯A)ST + E, S ∈ Ω (26)

Three mode matrices are the subarray response matrix A, parametersmatrix Φ, and baseband signal matrix S, formulated as

A = [a1, . . . ,aR]=

1 . . . 1

e−j 2πdλ

sin(θ1) . . . e−j 2πdλ

sin(θR)

......

...

e−j2π(I−1)d

λsin(θ1) . . . e−j

2π(I−1)dλ

sin(θR)

206 Liu et al.

Φ=

1 . . . 1e−j 2πd

λsin(θ1) . . . e−j 2πd

λsin(θR)

e−j 2πdλ

cos(θ1) sin(φ1) . . . e−j 2πdλ

cos(θR) sin(φR)

e−j 2πdλ

(sin(θ1)+cos(θ1) sin(φ1)) . . . e−j 2πdλ

(sin(θR)+cos(θR) sin(φR))

S=

s11, s12 . . . , s1R

s21, s22 . . . , s2R...

......

sK1, sK2 . . . , sKR

(27)

where θ, φ denote azimuth and elevation. λ is the wavelength ofsignals. d is the distance between adjacent antenna. R is the number ofsources. K denotes the length of data block and is assumed to be 100in the following simulations. I is the length of subarray. According toFigure 2, I is equal to 5. Parameters, including azimuth and elevation,are involved in Φ. The TALS-based algorithm [9] and traditionalESPRIT algorithm [36] are also simulated just for comparison. SNRis defined in terms of the noisy data model (26)

SNR = 10 log10

∣∣∣∣X∣∣∣∣2F

||E||2F(28)

Three sources are considered in the simulation. The azimuths andelevations are listed in Table 3.

4.1. Simulation 1

The performance of azimuth and elevation estimations of TALSSIC-based algorithm, TALS-based algorithm and ESPRIT algorithm areevaluated in this simulation. CM constraint is involved in TALSSIC-based algorithm. SNR is 40 dB, and 100 independent Monte Carlo runsare used to evaluate performance. Figures 3–5 depict the azimuth-elevation scatter diagrams of three algorithms. The X coordinatestands for azimuth, and Y coordinate stands for elevation. Theleft sub-figure of each figure depicts the estimated azimuth-elevationdiagrams of all 3 waves, and the right two sub-figures, titled as (a)

Table 3. Azimuths and elevations of sources.

Azimuth (θ) Elevation (φ)source 1 5 10

source 2 10 15

source 3 15 20

Progress In Electromagnetics Research, Vol. 128, 2012 207

and (b), are the zoom-in diagrams of two waves with parameters (5,10) and (10, 15). It is illustrated that the proposed TALSSIC-basedalgorithm has a better performance than TALS-based algorithm andESPRIT algorithm.

4.2. Simulation 2

This simulation evaluates the root mean squared error (RMSE)performance of the three algorithms in parameter estimation. Sincematrix Φ involves the full information of the azimuth and elevation,

4 6 8 10 12 14 168

10

12

14

16

18

20

22

azimuth

ele

vat

ion

4.5 5 5.59.5

10

10.5

azimuthele

vatio

n

9.5 10 10.514.5

15

15.5

azimuth

ele

vatio

n

(a)

(b)

Figure 3. Azimuth-elevation scatter diagram of TALSSIC-basedalgorithm.

4 6 8 10 12 14 168

10

12

14

16

18

20

22

4.5 5 5.59.5

10

10.5

9.5 10 10.514.5

15

15.5

azimuth

elev

atio

n

azimuth

elev

atio

n

azimuth

elev

atio

n

(a)

(b)

Figure 4. Azimuth-elevation scatter diagram of TALS-basedalgorithm.

208 Liu et al.

4 6 8 10 12 14 168

10

12

14

16

18

20

22

azimuth

elevation 4.5 5 5.5

9.5

10

10.5

azimuth(a)

elevation

9.5 10 10.514.5

15

15.5

azimuth(b)

elevation

Figure 5. Azimuth-elevation scatter diagram of ESPRIT algorithm.

20 22 24 26 28 30 32 34 36 38 4010

-3

10-2

10-1

10 0

SNR/dB

RM

SE

of

ma

trix

B

TALS-based algorithm

TALSSIC-based algorithm

ESPR IT algorithm

Figure 6. RMSE curve of param-eter estimation versus SNR.

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

1800

2000

Monte Carlo runs

iter

atio

n n

um

ber

TALS-based algorithmTALSS IC-based algorithm

Figure 7. Iteration number in100 Monte Carlo runs (SNR =20dB).

we calculate the RMSE between the estimated Φ and the original Φ toevaluate the RMSE performance of parameters estimation, formulatedas the Frobenius norm of (Φ− Φi)

ξ =1Q

Q∑

i=1

∣∣∣∣∣∣Φ− Φi

∣∣∣∣∣∣F

(29)

where Φi is the estimated parameter matrix in the ith simulation andQ the number of independent runs. SNR varies from 20 dB to 40 dB.100 independent Monte Carlo runs are simulated. Figure 6 plots theRMSE curve versus SNR in logarithmic coordinates. It is depicted thatthe RMSE values of the proposed TALSSIC-based algorithm are lessthan the other two algorithms under either high or low SNR conditions.

The results of simulations 1 and 2 show that TALSSIC-based

Progress In Electromagnetics Research, Vol. 128, 2012 209

20 22 24 26 28 30 32 34 36 38 400

100

200

300

400

500

600

700

SNR/dB

aver

age

iter

atio

n n

um

ber

TALS-based algorithm

TALSS IC-based algori thm

Figure 8. Average iteration number curve versus SNR.

algorithm has better accuracy of parameter estimation than TALS-based algorithm and ESPRIT algorithm, because TALSSIC-basedalgorithm not only makes use of three diversities of trilinear datamodel, but also exploits the structure of signals to improve theperformance of parameter estimation.

4.3. Simulation 3

The convergence performance of constrained and unconstrainedalgorithms is evaluated in this simulation. Figure 7 depicts theiteration number of TALS-based algorithm and TALSSIC-basedalgorithm in 100 independent runs when SNR is 20 dB. Figure 8 depictsthe average iteration number curve of two algorithms versus SNR.Obviously, TALSSIC-based algorithm converges faster than TALS-based algorithm, which illustrates that the structure reconstructionduring matrix update procedure can improve the convergence speed.Note that the complexity of an iterative algorithm is related to itsconvergence speed and the computation cost of each iteration. Thesimulation results also imply that TALSSIC needs lower computationcomplexity than TALS in array signal processing.

4.4. Simulation 4

As mentioned in Section 1, correlated factors in mode matrix may leadto performance loss in trilinear decomposition. This simulation willevaluate the performance of algorithms when correlated factors areinvolved in subarray matrix A. Due to (27), the correlation propertyof factors in A can be characterized by the similarity of θ1, . . . , θR. Tosimplify the calculation, we change the azimuth of source 1, denoted as

210 Liu et al.

θ1, from 5 to 9.9 and keep azimuth of source 2 and source 3 constant,as θ2 = 10, θ3 = 15. The correlation degree of factors a1 and a2,denoted as ρ1,2, can be characterized by the inverse of their Edicesdistance.

ρ1,2 =1√

(a1 − a2)H (a1 − a2)

(30)

Assume that SNR is 40 dB. Table 4 presents the azimuths of threesources and the related ρ1,2.

ρ1,2 is increased when θ1 and θ2 become closer and closer. Figure 9presents the RMSE curve of parameter estimation versus ρ1,2 whileFigure 10 depicts the average iteration number curve versus ρ1,2.

The simulation results illustrate that the performances of bothparameter estimation and convergence of TALS-based algorithm aredecreased sharply as ρ1,2 increased, which implies that TALS-basedalgorithm is not suitable for data with correlated factors. However,

Table 4. Azimuths of three sources and the related ρ1,2.

θ1 θ2 θ3 ρ1,2

5 10 15 0.696 10 15 0.867 10 15 1.138 10 15 1.79 10 15 3.38

9.5 10 15 6.759.9 10 15 33.8

0 5 10 15 20 25 30 3510

-3

10-2

10-1

10 0

1,2

RM

SE

of

matr

ix

TALS-based algorithm

TALSSIC-based algorithm

ρ

Φ

Figure 9. RMSE curve ofparameter estimation versus ρ1,2.

0 5 10 15 20 25 30 350

500

1000

1500

2000

2500

1,2

aver

age

iter

atio

n n

um

ber

TALS-based algorithm

TALS S IC-based algorithm

ρ

Figure 10. Average iterationnumber curve versus ρ1,2.

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TALSSIC-based algorithm still performs very well in either low orhigh ρ1,2 conditions. In array signal processing aspect, high correlatedfactors are often caused by coherent signals in multipath transmittingand radar receiving scenarios. It can be concluded that TALSSIC-based algorithm is still valid in these applications.

5. CONCLUSION

This paper exploits structure constraint in trilinear decomposition andlinks it to array signal processing. Constrained PARAFAC model isintroduced for model analysis. Two new algorithms, named TALSPand TALSSIC, are proposed to accomplish constrained PARAFACdecomposition in signal processing. We apply the proposed model andalgorithms to azimuth and elevation estimation of multiple invariancesensor arrays. Simulations show that TALSSIC-based algorithmoutperforms traditional TALS-based algorithm and ESPRIT algorithmin terms of the performance of both parameter estimation andconvergence. Especially, the proposed algorithms can still work welleven if high correlated columns are involved in the data model, whichmay be encountered in multipath transmitting and radar receivingscenarios.

ACKNOWLEDGMENT

This work is supported by China NSF Grants (61101104,61071090, 61100195), National Science and Technology Major Project(2011ZX03005-004-03), Jiangsu “973” Project (BK2011027), a projectfunded by the Priority Academic Program Development of JiangsuHigher Education Institutions-Information and Communication Engi-neering; Nanjing University of Posts & Telecommunications Project(NY211010). The authors wish to thank the anonymous reviewers fortheir valuable suggestions on improving this paper.

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