Digital Signal Processing 20 (2010) 1526–1534

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Digital Signal Processing

www.elsevier.com/locate/dsp

Computationally efficient 2D beamspace matrix pencil method fordirection of arrival estimation ✩

Muhammad Faisal Khan, Muhammad Tufail ∗

Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Nilore, Islamabad, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Available online 31 March 2010

Keywords:Matrix pencil2D matrix pencil methodDirection of arrival estimationBeamspaceDFTReduced dimensionComputational complexity

In this paper, we propose a new 2-dimensional beamspace matrix pencil (2D BMP) methodfor direction of arrival (DOA) estimation of plane wave signals using a uniform rectangulararray (URA). Based on some a priori information about DOA, the proposed method trans-forms the complex signal subspace in 2D matrix pencil (2D MP) method [Y. Hua, Estimatingtwo-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Trans. SignalProcess. 40 (9) (1992) 2267–2280] into a real and reduced dimensional beamspace usingthe discrete Fourier transform (DFT) matrix transformation. Consequently, the computa-tional complexity is reduced (several times) in comparison with 2D MP method. Computersimulations are provided to show that 2D BMP method gives comparable performance interms of average mean square error of the estimated DOA with lesser floating point oper-ations as compared to the existing (MP) methods.

© 2010 Elsevier Inc. All rights reserved.

1. Introduction

Many methods and techniques have been developed in the last few decades to estimate unknown parameters of planewave signals using sensor array [1,2]. Super resolution techniques like MUSIC [3] and ESPRIT [4] are widely used to estimatethe unknown parameters by exploiting the eigenstructure of the covariance matrix. Another method for estimating the un-known parameters of sinusoids in noise is MP method [5–8]. It is a high resolution DOA estimation method that analyzesthe (array) data on snapshot-by-snapshot basis; consequently, a non-stationary environment can be handled easily. Further-more, in comparison with covariance based methods, the MP method can find DOA in the presence of coherent signalswithout requiring additional spatial smoothing [8].

In the context of a real-time implementation, it is highly desirable to develop methods that not only give good (DOA)estimation results but also require significantly reduced computational burden. The most computationally expensive step inMP method is to find the signal subspace. This is done by applying a singular value decomposition (SVD) on a complexdata matrix. In 1D unitary matrix pencil (1D UMP) method [9], a unitary matrix transformation [10] is efficiently utilized toconvert the complex data matrix in 1D MP method into a real matrix. As a result, the computational burden is reduced toabout 1/4 to that of 1D MP method. For 2D scenarios, this method is extended in [11] to develop 2D UMP method.

The computational burden can be further reduced if some a priori information is available about the DOA, as in radarapplications. In this case, a reduced-dimension processing of data matrix is possible as in beamspace MUSIC, ESPRIT andCB-DOA methods [12–14]. Likewise, the beamspace technique is successfully applied by the authors in 1D BMP method [15]to significantly reduce the computational complexity of 1D MP method.

✩ This work was sponsored by Higher Education Commission, Government of Pakistan.

* Corresponding author.E-mail addresses: faisalmk72@yahoo.com (M.F. Khan), m_tufail@pieas.edu.pk (M. Tufail).

1051-2004/$ – see front matter © 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2010.03.016

M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534 1527

In this paper, we extend the existing 1D BMP method into 2D and propose a new 2D BMP method which transformsthe complex data matrix in 2D MP method [16] into a real and reduced dimensional matrix using selected rows of the DFTmatrix. The rows which are selected to estimate DOA in a sector of interest are based upon a priori information. Dependingupon the number of rows selected, the computational burden is decreased several times than 2D UMP method. If there isno a priori information available then BMP method can be applied via parallel processing with overlapped sectors [13].

The rest of the paper is organized as follows. In Sections 2 and 3, signal model and basic theory of 2D MP and UMPmethods are briefly discussed. In Section 4, 2D BMP method is developed and its computational complexity is analyzed. InSection 5, simulation results are presented to compare the performance of proposed method with existing 2D MP and UMPmethods.

2. Problem formulation

Consider a 2D URA of sensors in space. The distances between array elements are �x and �y along x and y directions,respectively, and the corresponding number of sensors are M and N . If I narrow band signals with wavelength λ, andazimuth and elevation angles (φi , θi ) arrive at the input of this array then the noiseless signal at the sensor located atCartesian coordinates (m,n) can be expressed as

z(m,n) =I∑

i=1

αi xmi yn

i (1)

where αi is the ith signal (complex) amplitude and the poles xi and yi are given by

xi = exp

(j2π

λ�xui

)and yi = exp

(j2π

λ�yvi

)(2)

where ui = cosφi sin θi and vi = sin φi sin θi are the direction cosines of DOA along x-axis and y-axis, respectively. Now, ifμi = 2π

λ�xui and νi = 2π

λ�yvi are spatial frequencies in x and y directions, respectively, then we have

xi = exp( jμi) and yi = exp( jνi). (3)

The main focus in this paper will be on the estimation of direction cosines ui and vi .

3. A brief review of 2D MP and 2D UMP methods

In 2D MP method [16], a data enhancement matrix De is constructed as

De =

⎡⎢⎢⎢⎣

D0 D1 · · · DM−K

D1 D2 · · · DM−K+1

......

. . ....

D K−1 D K · · · DM−1

⎤⎥⎥⎥⎦ (4)

where

Dm =

⎡⎢⎢⎢⎣

z(m,0) z(m,1) · · · z(m, N − L)

z(m,1) z(m,2) · · · z(m, N − L + 1)

......

. . ....

z(m, L − 1) z(m, L) · · · z(m, N − 1)

⎤⎥⎥⎥⎦ . (5)

Here, K and L are pencil parameters and the size of De is K L × (M − K + 1)(N − L + 1). Applying an SVD on De , we get

De = UΣ V H = UsΣs V Hs + UnΣn V H

n (6)

where Us containing I principle left singular vectors spans the signal subspace and Un containing the remaining singularvectors spans the noise subspace. Now, we define the matrices Ux1, Ux2, U y1 and U y2 as

Ux1/Ux2 = Us with last/first L rows deleted, (7)

U y1/U y2 = P Us with last/first K rows deleted (8)

where P is a shuffling matrix defined as

1528 M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534

P =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

s(1)

s(1 + L)...

s(1 + (k − 1)L)

s(2)

s(2 + L)...

s(2 + (k − 1)L)......

s(L)

s(L + L)...

s(L + (k − 1)L)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (9)

In the above equation, s(i) is a row vector of size K L with 1 at the ith position and zero elsewhere. Now, it can be shownthat ui ’s can be obtained from the eigenvalues of U †

x1Ux2 and vi ’s can be obtained from the eigenvalues of U †y1U y2. Here,

† denotes the Moore–Penrose matrix inverse. A pairing algorithm is subsequently employed to properly group ui ’s and vi ’s.In 2D UMP method [11], the matrix De is converted into a real matrix using a unitary matrix transformation, which

reduces the computational complexity to about 1/4 of 2D MP method.

4. 2D beamspace matrix pencil method

We can write the data matrix De as

De = EL AE R (10)

where

EL = XL�Y L, (11)

E TR = X T

R �Y TR (12)

and A = diag(α1,α2, . . . ,αI ). In the above equations, � denotes the Khatri–Rao product [2] and the matrices XL , Y L , XR

and Y R are given by

XL =

⎡⎢⎢⎢⎣

1 1 · · · 1x1 x2 · · · xI

......

...

xK−11 xK−1

2 · · · xK−1I

⎤⎥⎥⎥⎦ , (13)

Y L =

⎡⎢⎢⎢⎣

1 1 · · · 1y1 y2 · · · yI

......

...

yL−11 yL−1

2 · · · yL−1I

⎤⎥⎥⎥⎦ , (14)

XR =

⎡⎢⎢⎢⎢⎣

1 x1 · · · xM−K1

1 x2 · · · xM−K2

......

...

1 xI · · · xM−KI

⎤⎥⎥⎥⎥⎦ , (15)

Y R =

⎡⎢⎢⎢⎢⎣

1 y1 · · · yN−L1

1 y2 · · · yN−L2

......

...N−L

⎤⎥⎥⎥⎥⎦ . (16)

1 yI · · · yI

M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534 1529

The matrices XL and Y L can be further decomposed as

XL = X ′L Xo, (17)

Y L = Y ′L Yo (18)

where

X ′L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x− K−1

21 x

− K−12

2 · · · x− K−1

2I

x− K−3

21 x

− K−32

2 · · · x− K−3

2I

......

...

xK−3

21 x

K−32

2 · · · xK−3

2I

xK−1

21 x

K−32

2 · · · xK−1

2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (19)

Y ′L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y− L−1

21 y

− L−12

2 · · · y− L−1

2I

y− L−3

21 y

− L−32

2 · · · y− L−3

2I

......

...

yL−3

21 y

L−32

2 · · · yL−3

2I

yL−1

21 y

L−32

2 · · · yL−1

2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (20)

Xo = diag(x

K−12

1 , xK−1

22 , . . . , x

K−12

I

), (21)

Yo = diag(

yL−1

21 , y

L−12

2 , . . . , yL−1

2I

). (22)

Using (17) and (18) in (11) and simplifying, we get

EL = X ′L Xo�Y ′

L Yo

= (X ′

L�Y ′L

)XoYo (23)

so that De in (10) becomes

De = (X ′

L�Y ′L

)XoYo AE R . (24)

If W K and W L are DFT matrices of dimensions K × K and L × L, respectively, then the kth row (w Hk ) of W H

K and the lthrow (w H

l ) of W HL can be written as

w Hk = e j( K−1

2 )k 2πK · [1, e− jk 2π

K , . . . , e− j(K−1)k 2πK

], (25)

w Hl = e j( L−1

2 )l 2πL · [1, e− jl 2π

L , . . . , e− j(L−1)l 2πL

]. (26)

The row vectors w HK and w H

l represent DFT beams steered at spatial frequencies μ = k 2πK and ν = l 2π

L , respectively. Now,if B K = W H

K X ′L and BL = W H

L Y ′L are beamspace manifold matrices of dimensions K × I and L × I , respectively then it can

be easily seen that the ith columns (b(μi) and b(νi), respectively) of B K and BL are given by

b(μi) = [b0(μi),b1(μi), . . . ,bK−1(μi)

]T, (27)

b(νi) = [b0(νi),b1(νi), . . . ,bL−1(νi)

]T(28)

where

bk(μi) = sin[ K2 (μi − k 2π

K )]sin[ 1

2 (μi − k 2πK )] (0 � k � K − 1), (29)

bl(νi) = sin[ L2 (νi − l 2π

L )]sin[ 1

2 (νi − l 2πL )] (0 � l � L − 1). (30)

Furthermore, the relation between two consecutive beams bk and bk+1 can be shown to be [13]

1530 M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534

tan

(μi

2

){cos

(kπ

K

)bk(μi) + cos

((k + 1)

π

K

)bk+1(μi)

}

= sin

(kπ

K

)bk(μi) + sin

((k + 1)

π

K

)bk+1(μi). (31)

From (31), we get K equations for 0 � k � K − 1, which can be written in compact form as

tan

(μi

2

)Γ1b(μi) = Γ2b(μi) (32)

where

Γ1 =

⎡⎢⎢⎢⎢⎢⎢⎣

1 cos(πK ) 0 · · · 0

0 cos(πK ) cos(2 π

K ) · · · 0

0 0 cos(2 πK ) · · · 0

......

.... . .

...

(−1)K 0 0 · · · cos[(K − 1)πK ]

⎤⎥⎥⎥⎥⎥⎥⎦

, (33)

Γ2 =

⎡⎢⎢⎢⎢⎢⎢⎣

0 sin(πK ) 0 · · · 0

0 sin(πK ) sin(2 π

K ) · · · 0

0 0 sin(2 πK ) · · · 0

......

.... . .

...

0 0 0 · · · sin[(K − 1)πK ]

⎤⎥⎥⎥⎥⎥⎥⎦

. (34)

If there are I sources then (32) becomes

Γ1 B K Ωμ = Γ2 B K (35)

where

Ωμ = diag

[tan

(μ1

2

), tan

(μ2

2

), . . . , tan

(μI

2

)]. (36)

A relation similar to (31) exists between the beams bl and bl+1. Using this relation, it can be easily shown that

Γ3 BLΩν = Γ4 BL (37)

where Γ3 and Γ4 are similar to Γ1 and Γ2, respectively with K replaced by L and Ων is similar to Ωμ with μ replacedby ν . Now, if we define

W = W K ⊗ W L, (38)

B = B K �BL (39)

where ⊗ denotes the Kronecker product then by using the properties of Kronecker and Khatri–Rao products we can writeY = W H De as

Y = (W H

K ⊗ W HL

)(X ′

L�Y ′L

)XoYo AE R

= (W H

K X ′L�W H

L Y ′L

)XoYo AE R

= B XoYo AE R . (40)

Next, if we define Γμ1 = Γ1 ⊗ I L and Γμ2 = Γ2 ⊗ I L , where I L is L × L identity matrix, then we have

Γμ1 B = (Γ1 ⊗ I L)(B K �BL)

= (Γ1 B K )�BL (41)

and

Γμ2 B = (Γ2 ⊗ I L)(B K �BL)

= (Γ2 B K )�BL . (42)

Using (35) and (41) in the above equation and simplifying we get

M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534 1531

Γμ2 B = Γμ1 BΩμ. (43)

In a similar fashion, if we define Γν1 = I K ⊗ Γ3 and Γν2 = I K ⊗ Γ4 then we obtain the following relation

Γν2 B = Γν1 BΩν. (44)

It can be easily observed that all the matrices to the right of B in (40) have full row rank. This implies that the real matrix Band the complex matrix Y share the same column space. For noisy data, the largest I left singular vectors of [Re{Y }, Im{Y }]span the column space of B . If Es contains these left singular vectors then we have

Es = BT (45)

where T is a non-singular I × I matrix. Using (45) in (43) and (44) we can write

Γμ1 EsΨμ = Γμ2 Es ⇒ Ψμ = (Γμ1 Es)†Γμ2 Es, (46)

Γν1 EsΨν = Γν2 Es ⇒ Ψν = (Γν1 Es)†Γν2 Es (47)

where Ψμ = T −1ΩμT and Ψν = T −1Ων T . Due to similarity transformation, the eigenvalues of Ψμ and Ψν are the diagonalelements of Ωμ and Ων , respectively. These eigenvalues can be subsequently used to estimate the required direction cosines.

4.1. Reduced dimension 2D BMP method

The advantage of 2D BMP method is evident when one employs a subset of p rows of W HK and q rows of W H

L (wherep � K and q � L). In this case, a reduced dimensional beamspace data matrix is obtained as

Y r = [W r]H

De (48)

where the superscript r is used to denote the matrix with reduced dimensions and the matrix W r is defined as

W r = W rK ⊗ W r

L . (49)

Here, W rK contains p selected columns of W K and W r

L contains q selected columns of W L . If we use these reduced DFTmatrices then (35) and (37) are modified as

Γ r1 Br

K Ωμ = Γ r2 Br

K , (50)

Γ r3 Br

LΩν = Γ r4 Br

K . (51)

Here, BrK and Br

L are beamspace array manifold matrices of dimensions p × I and q × I , respectively, Γ r1 and Γ r

2 of di-mensions (p − 1) × p are appropriate sub-blocks of Γ1 and Γ2, respectively, and similarly Γ r

3 and Γ r4 of dimensions

(q − 1) × q are appropriate sub-blocks of Γ3 and Γ4, respectively. Now, if Ers contains the largest I left singular vectors

of [Re{Y r}, Im{Y r}] then Ψμ and Ψν can be obtained as

Ψμ = [Γ r

μ1Er

s

]†Γ r

μ2Er

s, (52)

Ψν = [Γ r

ν1Er

s

]†Γ r

ν2Er

s (53)

where Γ rμ1

= Γ r1 ⊗ Iq , Γ r

μ2= Γ r

2 ⊗ Iq , Γ rν1

= I p ⊗ Γ r3 and Γ r

ν2= I p ⊗ Γ r

4 . Finally, a procedure similar to the one given in [13]or [17] can be adopted in order to properly group the required parameters.

4.2. Computational complexity

In all MP methods, the most computationally expensive step is to estimate the signal subspace, which requires an SVDof a data matrix. In the conventional 2D MP method, the data matrix is complex of size K L × 2(M − K + 1)(N − L + 1). In2D UMP method, the data matrix is transformed into a real matrix Dr (with negligible computations) of same size. Since,one complex multiplication requires four real multiplications, therefore 2D UMP requires four times less computations ascompared to 2D MP method. In 2D BMP method, the determination of signal subspace can be divided into two sub-steps.In the first sub-step, a matrix [W r]H of dimension pq × K L is multiplied with the data matrix De . In the second sub-step,an SVD of a real matrix of size pq × 2(M − K + 1)(N − L + 1) is computed. In 2D BMP method, normally pq � K L, so thedata matrix is not only real but also has reduced dimensions. Hence, the computational complexity of 2D BMP method isseveral times less than 2D UMP method.

A detailed comparison of the computational complexity of these methods in order to determine the signal subspace isshown in Table 1 (see also [18]).

As mentioned earlier, the computational complexity required for 2D UMP method is four times less in comparisonwith 2D MP method. In order to compare the computational burden of 2D BMP and 2D UMP methods, we define thecomputations ratio ρ as

1532 M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534

Table 1Comparison of computational complexity of various MP methods*.

No. Description 2D MP 2D UMP 2D BMP

1 Data matrixtransformation

NA De → Dr

negligible computationsDe → Y r pqK LR Scomplex multiplications

2 SVD 11(K L)3 + 4(K L)2 R Scomplex multiplications

11(K L)3 + 4(K L)2 R Sreal multiplications

11(pq)3 + 4(pq)2 R Sreal multiplications

3 Computations in the examplegiven in Section 5

4 × 10 935 = 43 740real multiplications

10 935real multiplications

1280 + 1296 = 2576real multiplications

* R = (M − K + 1) and S = (N − L + 1).

Fig. 1. Computational complexity comparison of 2D UMP and BMP methods.

ρ = Multiplications required for SVD in 2D BMP

Multiplications required for SVD in 2D UMP. (54)

For optimum performance it is recommended to use K = M+12 and L = N+1

2 [16]. For simplicity, we also assume thatM = N and p = q. Now, if r = K

p is the ratio of number of original beams to the number of selected beams then we have

ρ = 4r4+4r2+1115r6 , which is plotted in Fig. 1 as a function of beams ratio. From this figure, it can be easily seen that as r

increases (less beams selected) the computational complexity of 2D BMP method decreases significantly in comparison with2D UMP method.

It may be noted that the computational burden of other steps is negligible and therefore not discussed here.

5. Simulation results

The noise contaminated signal model is given by

z(m,n) = z(m,n) + w(m,n) (55)

where w(m,n) is assumed to be zero mean white Gaussian noise of variance σ 2. In the simulation, we considered twoclosely spaced signals with direction cosines (0.40,0.25) and (0.25,0.40) impinging on a URA with �x = �y = λ/2 andM = N = 5. The pencil parameters are chosen to be K = L = 3 and the complex amplitudes of signals are α1 = α2 = 1. Inorder to compare the performance of various methods, we define the mean square error (MSE) for the ith source as

MSEi = E(ui − ui)2 + E(vi − v i)

2

2(56)

where ui and v i are the estimates of ui and vi , respectively, and E is the expectation operator. In Fig. 2a and b, 10 log(MSEi)

is plotted (for 2D MP, UMP and BMP methods) versus signal to noise ratio (SNR) for sources 1 and 2, respectively, whereSNR is defined as

SNR = 10 log

(1

2

). (57)

σ

M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534 1533

(a)

(b)

Fig. 2. Performance comparison of 2D MP, UMP and BMP methods: (a) source 1, (b) source 2.

The Cramer–Rao lower bound (CRLB) for the estimates is also calculated and shown in these figures. In 2D BMP method, wechoose p = q = 2 (r = 1.5).

It is evident from Fig. 2a and b that (on average) for low SNR values 2D UMP method performs better than 2D MPmethod (as indicated in [19]) whereas the performance of 2D BMP is slightly better than both 2D MP and UMP methods.For high values of SNR, the estimation accuracy of all the methods is almost the same.

The computational requirements of 2D MP, UMP and BMP methods for this particular scenario are given in the 3rd rowof Table 1. It can be easily seen from Table 1 (and Fig. 1) that in this case (r = 1.5), the computational burden of 2D BMP isabout 1

4 to that of 2D UMP method and about 116 to that of 2D MP method.

6. Conclusion

In this paper, we have applied the DFT beamspace technique to 2D MP method in order to develop 2D BMP methodfor DOA estimation. In situations (such as radar applications), where some prior information about DOA is available, thereduced 2D BMP method gives improved performance at low SNRs as compared to 2D MP and UMP methods while at highSNRs the performance is comparable. From the computational complexity point of view, depending upon the number ofrows of the DFT matrix selected, 2D BMP method is several times less expensive than 2D MP and UMP methods.

1534 M.F. Khan, M. Tufail / Digital Signal Processing 20 (2010) 1526–1534

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Muhammad Tufail received the M.Sc. degree in Electronics from University of Peshawar, Pakistan in 1997, the M.Sc. degree in SystemsEngineering from Quaid-i-Azam University, Islamabad, Pakistan in 1999, and the Ph.D. degree in Electronic Engineering from TohokuUniversity, Sendai, Japan in 2006. Since October, 2006 he has been working as Senior Scientist in the Department of Electrical Engineering,Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan. His research interests include blind source separation,array signal processing and active noise control.

Muhammad Faisal Khan received his B.E. degree in Electronics from NED University of Engineering and Technology, Karachi, Pakistanin 1995. He completed his M.Sc. in Systems Engineering from Quaid-e-Azam University, Islamabad, Pakistan in 1997. From 1997 to 2005,he worked as a researcher in a public sector institute in Pakistan. Since July 2005, he has been pursuing a Doctor of Engineering degreeat PIEAS, Islamabad, Pakistan. His research interests include sensor array signal processing and beamforming.