Utilization of a Unitary Transform for Efficient Computation in the ...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006 175 

Utilization of a Unitary Transform for Efficient 

Computation in the Matrix Pencil Method to 

Find the Direction of Arrival 

Nuri Yilmazer, Jinhwan Koh, and Tapan K. Sarkar, Fellow, IEEE 

Abstract—In this study, we use the matrix pencil (MP) method 

to compute the direction of arrival (DOA) of the signals using a 

very efficient computational procedure in which the complexity 

of the computation can be reduced significantly by using a unitary 

matrix transformation. This method applies the technique directly 

to the data without forming a covariance matrix. Simulation 

results show that the variance of the estimate approaches to 

the Cramer–Rao lower bound. Using real computations through 

the unitary transformation for the MP method leads to a very efficient 

computational methodology for real time implementation on 

a digital signal processor chip. A unitary transform can convert 

the complex matrix to a real matrix along with their eigenvectors 

and thereby reducing the computational cost at least by a factor of 

four without sacrificing accuracy. This reduction in the number of 

computations is achieved by using a transformation, which maps 

centro-hermitian matrices to real matrices. This transformation is 

based on Lee’s work on centro-hermitian matrices. 

Index Terms—Direction of arrival (DOA), matrix pencil (MP), 

unitary transformation. 

I. INTRODUCTION 

THE problem of estimating the direction of arrival (DOA) 

of the various sources impinging on a phased array 

has received considerable attention, in many fields including 

radar, sonar, radio astronomy, and mobile communications. 

There are many algorithms that exist to find the DOA and 

research is going on to increase their resolution as well as to 

reduce their computational complexity [1]. Capon’s minimum 

variance technique attempts to overcome the poor resolution 

problems associated with the delay-and-sum method [1], [2]. 

More advanced approaches are so-called super resolution 

techniques that are based on the eigen-structure of the input 

covariance matrix including multiple signal classification 

(MUSIC), Root-MUSIC and estimation of signal parameters 

via rotational invariance techniques (ESPRIT) provides the 

high resolution DOA estimation. Music algorithm proposed 

by Schmidt [3] returns the pseudo-spectrum at all frequency 

samples. Root-MUSIC [4] returns the estimated discrete frequency 

spectrum, along with the corresponding signal power 

estimates. Root-MUSIC is one of the most useful approaches 

Manuscript received March 28, 2005; revised July 20, 2005. 

N. Yilmazer and T. K. Sarkar are with the Department of Electrical Engineering 

and Computer Science, Syracuse University, Syracuse, NY 13244-1240 

USA (e-mail: nyilmaze@syr.edu; tksarkar@syr.edu). 

J. Koh is with the Department of Electronics and Electrical Engineering, 

Kyungpook National University, Taegu, Korea (e-mail: jikoh@ee.knu.ac.kr). 

Digital Object Identifier 10.1109/TAP.2005.861567 

for frequency estimation of signals made up of a sum of exponentials 

embedded in white Gaussian noise. The performance 

analysis of these methods can be found in [16]. 

The conventional signal processing algorithms using the covariance 

matrix work on the premise that the signals impinging 

on the array are not fully correlated or coherent. Under uncorrelated 

conditions, the source covariance matrix satisfies the full 

rank condition, which is the basis of the eigen-decomposition 

[1]. Many techniques involve modification of the covariance 

matrix through a preprocessing scheme called spatial smoothing 

[5]. Sarkar and Hua [6], [7] utilized the matrix pencil (MP) to 

get the DOA of the signals in a coherent multipath environment. 

In the MP method, based on the spatial samples of the data, the 

analysis is done on a snapshot-by-snapshot basis, and therefore 

nonstationary environments can be handled easily. Unlike the 

conventional covariance matrix techniques, the MP method can 

find DOA easily in the presence of multipath coherent signal 

without performing additional processing of spatial smoothing. 

On the other hand, some efforts have been done to reduce 

the computational complexity of the calculations. Huang and 

Yeh [8] have developed a unitary transform, which can convert 

a complex matrix to a real matrix along with their eigenvectors. 

Their simple transformation reduces the processing time 

by dealing with only real valued computations. The processing 

time could be reduced almost 4 times, since the complex multiplication 

cost 4 times more than that of real multiplications. 

More work has been done by Haardt and Nossek [9], [12], [13] 

and they applied the method to ESPRIT to successfully reduced 

the computational burden. 

This paper introduces an efficient computational approach 

that reduces the burden of complex calculations in the matrix 

pencil (MP) method. The MP method can find the DOA in the 

presence of multipath coherent signals without performing additional 

processing like spatial smoothing as required in some of 

the conventional covariance matrix based techniques. By taking 

advantage of the centro-hermitian property of a matrix, a unitary 

transformation can be applied which converts a complex 

matrix to a real matrix along with their eigenvectors, which significantly 

reduces the processing time. 

The rest of the paper is organized as follows. In Section II, 

the unitary transform and the related theorems are given. In Section 

III, the MP method is explained in detail and the utilization 

of unitary transform is illustrated. The new algorithm is summarized 

in Section IV. The computer simulation is provided in 

Section V, followed by the conclusions. 

0018-926X/$20.00 © 2006 IEEE

176 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006 

II. THE UNITARY TRANSFORM 

A square matrix, , is called unitary, if it satisfies 

. The superscript denotes the complex conjugate 

transpose of a matrix. Hence , where is the identity 

matrix. For the matrix to be unitary, the columns of the matrix 

must be orthonormal. This implies . 

A matrix A, where 

, is called centro-hermitian 

[9]–[11], if it satisfies 

is called the exchange matrix and defined as 

. 

. . . . . 

. 

. 

. 

. 

. 

(1) 

. Here, is a square 

matrix, and is conjugate of . 

Theorem 1: If a vector 

, where is an odd number, is centro-hermitian, that is 

, then the following matrix : 

. 

. 

. 

. 

. (6) 

. 

Therefore, . is centro-hermitian. 

Theorem 3: If the matrix is centro-hermitian, then 

is a real matrix. Here the matrix is unitary, whose 

columns are conjugate symmetric and has a sparse structure 

[8], [14]. For even, we have 

Here, and are matrices that have the dimension of and 

. When is odd, we have 

(7) 

(8) 

. 

. 

. 

. 

. .. 

. 

. 

is also centro hermitian. 

Proof: Since 

, then 

for 

. Therefore 

(2) 

Here and are matrices that have the dimension of , 

and 0 is a vector whose elements are 0. 

Proof: Using , the conjugate of is 

Since 

(9) 

(10) 

(11) 

. 

. 

. 

. 

. .. 

. 

. 

(3) 

Therefore, 

is a real matrix. 

. 

. 

. .. . . 

(4) 

Hence, 

is a also centro-hermitian matrix. 

Theorem 2: The following matrix 

. 

is centro hermitian for any matrix [9]. 

Proof: Using 

. 

. , and 

. 

. ,wehave 

III. THE MP METHOD 

The narrowband sources located in the far field of a uniformly 

spaced array consisting of isotropic omnidirectional point sensors 

radiating in free space is considered. This results in a uniform 

linear array (ULA). The focus here is to use the unitary 

transform to convert the complex matrices used in the MP formulation 

to real matrices and use these matrices to estimate the 

DOA of multiple signals simultaneously impinging on the ULA. 

The vector is the set of voltages measured at the feed point 

of the antenna elements of the ULA. Therefore, can be 

modeled by a sum of complex exponentials, i.e. 

(12) 

. 

. 

. (5) 

where 

observed voltages at a specific instance ; 

noise associated with the observation; 

actual noise free signal; 

;

YILMAZER et al.: UTILIZATION OF A UNITARY TRANSFORM FOR EFFICIENT COMPUTATION 177 

residues or complex amplitudes at ; 

angular frequencies ; 

damping factors. 

Therefore, one can write the sampled signal as 

One can also write 

where 

(19) 

(20) 

(13) 

(14) 

. 

. 

. 

. 

. .. 

. 

. 

where, , for . 

In this study, it has been assumed that the damping factor 

is equal to zero. The objective is to find the best estimates 

for , and from . Let us consider the matrix , 

which is obtained directly from . is a Hankel matrix, 

and each column of is a windowed part of the original data 

vector, 

. It is assumed that, there 

are data samples 

. 

. 

. 

. 

. .. 

. 

. 

Now, let us consider the MP 

(21) 

(22) 

(23) 

(24) 

. 

. 

. 

. 

. .. 

. 

. 

(15) 

The parameter is called the pencil parameter. is chosen between 

and for efficient noise filtering [7]. The variance 

of the estimated values of and will be minimal, if the 

values of are chosen in this range. 

The matrix is obtained from (14) by writing it in the following 

matrix form, shown in (16) at the bottom of the page. 

From the matrix , we can define two submatrices, say 

. 

. 

. .. . 

(25) 

Here is the identity matrix. One can show that the rank 

of will be , provided that [7]. 

However, if , , the row of 

is zero, then the rank of this matrix is . Therefore, the 

parameters can be found as the generalized eigenvalues of 

the matrix pair . So, the solution to the problem can be 

reduced to an ordinary eigenvalue problem, and will be the 

eigenvalues of 

(26) 

where is the Moore–Penrose pseudo inverse of , which is 

defined as 

(27) 

. 

. 

. .. . 

(17) 

The DOA is obtained from , where 

. 

For the noisy data, the singular value decomposition (SVD) 

is useful to reduce some of the noise effect [15]. The matrix 

can be written as 

(18) 

(28) 

. 

. 

. .. . . 

(16)


Here, and are unitary matrices whose columns are the 

eigenvectors of , and , respectively. is the singular 

values of , which are located on the main diagonals in the descending 

order 

. If the data is noiseless, 

the first singular values are nonzero, the rest is zero, where 

(29) 

(30) 

If the data is noisy, needs to be estimated, and how to do it 

is illustrated in [7]. The ratio of each of the singular value to the 

largest one determines the value of . We choose 

, where is the number of accurate significant decimal 

digits of the data vector . One can rewrite the (25) as 

with 

Since, 

Therefore 

(39) 

(40) 

(41) 

(42) 

where , and , with 

. 

The matrices , and are defined as follows: 

. 

. 

. 

. .. . 

. 

(31) 

. .. . . 

(32) 

For data contaminated with noise, it is better to take the 

SVD of so that some of the noise effect can be reduced. 

It can be shown that the singular vectors of 

correspond to the signal space [9]. One can then solve for 

as the generalized eigenvalue of the matrix 

and , where, is the 

largest singular vectors of . is estimated number of the 

signals. That is, . Here, , 

and are orthogonal matrices. is a diagonal matrix 

with its element . , such that 

the corresponding singular values 

satisfy 

.Or 

is the 

eigenvalue of . 

Since we do not know how many frequency components exist 

in the signal, the number of the estimated frequencies should 

be determined using some criteria. Typically the singular values 

beyond are set equal to zero [7]. 

. 

. 

. 

. .. . . 

(33) 

IV. SUMMARY OF THE ALGORITHM 

Basically the algorithm can be summarized as follows: 

From (31) and (33), one can write 

(34) 

(35) 

Note that , and is real from the theorems 

(1, 2, 3). Therefore 

(36) 

It can be shown that , and 

, , and therefore, (36) can 

be represented by 

(37) 

• Step 1: From the data vector , 

find the MP from (2); 

• Step 2: Compute the real data matrix 

, using theorem (3); 

Or, 

from theorem (2); 

• Step 3: Evaluate and 

; 

• Step 4: Perform a SVD of and 

calculate , which is the 

largest singular vectors of ; 

• Step 5: Calculate the generalized 

eigenvalues 

of 

and ; 

• Step 6: Calculate . 

. 

. 

Hence, (36) converts to 

(38) 

In this algorithm all computations are made using real numbers 

and therefore, no variable is complex including the eigenvalues 

and the eigenvectors in this procedure.


TABLE I 

SUMMARY OF SIGNAL FEATURES INCIDENT ON THE ANTENNA ARRAY 

One should state that premultiplying and postmultiplying 

by and require only additions and scaling. For 

example, in the case of sensor elements, and for a given 

pencil parameter , the computations done for the unitary 

transformation, 

, requires around 

real additions. This is negligible as compared to 

the computations done for the computation involved in the 

eigendecomposition [8], [9]. 

Eigenstructure based methods for estimating DOA of the 

sources impinging on a ULA requires complex calculations 

in computing the eigenvectors and the eigenvalues. The MP 

method, in addition, requires the computation of a SVD of the 

complex-valued data. It should be stated that eigendecomposition 

with complex-valued data matrix is quite computation 

intensive. The eigen-decomposition process consists of a large 

portion of the whole computational load. To reduce the computational 

complexity during eigen-decomposition, application 

of a unitary transformation is proposed for DOA estimation 

by using a real-valued SVD. Computing the eigen-components 

of the unitary transformed data matrix requires only real 

computations. Real-valued eigendecomposition has a reduced 

computational complexity than the complex one approximately 

by a factor of four. 

The unitary MP (UMP) method is thus a completely realvalued 

algorithm as it requires only real-valued computations. 

Apart from finding the singular values and vectors, the rest of the 

calculations are also real computations as opposed to the ones 

done in the conventional MP method. A big portion of the computational 

load is occupied by the multiplication operations, so 

transforming the data can save a noticeable amount of computations 

and the processing time is reduced greatly. 

V. SIMULATION RESULTS 

In this section, the computer simulation results are given to illustrate 

the performance of the UMP method. The noisy signal 

model is formulated from (14). is treated as a zero mean 

Gaussian white noise with variance . Uniformly spaced arrays 

(ULA) of omnidirectional isotropic point sensors are considered 

in this study. The distance between any two elements 

of the ULA is half a wavelength. is the voltage induced 

at each of the antenna elements, for 

.Itis 

assumed that there are 6 antenna elements and two signals are 

impinging on the array with amplitudes 

. The 

phases, magnitudes and DOA of the two impinging signals are 

tabulated in Table I. 

Since the estimated direction of arrival is treated as a random 

variable, the stability/accuracy of the results need to be expressed 

in terms of its statistical properties, which in this case 

Fig. 1. Variance 010 log (var( ^ )) UMP, MP, and the CRLB are plotted 

against the SNR. 

are the estimated values such as the mean, variance, and so 

on of the estimate. The Cramer–Rao lower bound (CRLB) 

measures the goodness of an estimator. CRLB is the limit that 

the variance of the estimates contaminated by white Gaussian 

noise cannot be any smaller than this bound. The simulation 

results show that the variance of the estimators approaches to 

the CRLB. The bound is found by using the Fisher information 

matrix ( ), whose diagonal elements are the corresponding 

CRLB of that element. Fisher is defined by 

(43) 

(44) 

where is the estimate of . Here, is probability density 

function conditioned to an unknown vector parameter . The diagonal 

elements of the inverse of the Fisher define the bound 

on that estimated value as shown in [15]. In this study, a comparison 

in performance is made between the MP and the UMP 

method. We compute the CRLB of the variance for the estimate 

of the DOA. The inverse of the sample invariance of the estimate 

of (DOA) and the inverse variance of the MP method 

using complex data, inverse variance of the UMP method and 

the corresponding CRLB versus signal-to-noise ratio (SNR) of 

the input voltages is plotted in Fig. 1. Different values of SNR 

comprise the -axis and the inverse of the variance of the estimated 

arrival angle in logarithmic domain, 

is shown along the -axis. 

To obtain the sample variance of the estimate of , 300 

independent trials have been performed. Noise for each 

run is independent of each other. In the summary of algorithm 

section, the Step 6 implies that all the eigenvalues of 

are real. One should 

recall that the eigenvalues of a real matrix can either be real 

or they occur in complex conjugate pairs. In the case of very 

noisy environment, the eigenvalues of this matrix may fail to be 

real and they occur in complex conjugate pairs, which is also


For higher values of the SNR, the bias of the estimator decreases, 

as expected. 

Fig. 2. 

RMSE of the DOA versus the number of antenna elements. 

VI. CONCLUSION 

In this paper, it has been shown that the computational complexity 

can be reduced for the MP method for DOA estimation. 

A unitary transformation applied to the MP method has been 

successfully formulated and utilized to convert the complex data 

matrix to a real matrix, hence reducing the computational complexity 

significantly. It is seen that when the SNR of the data is 

greater than 10 dB, then both the MP and the new UMP method 

can be used to model a given data set by a sum of complex exponentials 

and the UMP can be implemented on a DSP chip using 

only real arithmetic. The surprising part is that the real computations 

come at a cost, particularly for low values of SNR where 

the variance of the estimates due to the UMP method is larger 

than that for the MP method. 

ACKNOWLEDGMENT 

Grateful acknowledgment is made to the reviewers for suggesting 

ways to improve the readability of the paper. 

Fig. 3. 

Bias of the estimator versus SNR. 

pointed out in [9]. This causes some degradation for low SNR 

case. That explains the performance difference between the MP 

and the UMP methods for the low SNR case. After 8 dB SNR, 

the eigenvalues are all real. Since the results for are similar 

to we are giving the results for only . The optimum value 

for the pencil length, , is chosen to be 3 for efficient noise 

filtering which is explained in [7]. 

The root mean square error (RMSE) of the DOA versus the 

number of antenna elements for the UMP is shown in Fig. 2. As 

the number of antenna elements increase, the RMSE decreases 

as expected. This simulation is based on the value of an SNR of 

20 dB. 

The bias for the estimate of the DOA has also been studied. 

The bias is computed from 

(45) 

where denotes the expected value. The bias of the estimator 

versus SNR is shown in Fig. 3. 

REFERENCES 

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the parameters of a sum of complex exponentials,” IEEE Antennas 

Propag. Mag., vol. 37, no. 1, pp. 48–54, 1994. 

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975–977, 1991. 

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[11] L. Datta and D. M. Salvatore, “Some results on matrix symmetries and 

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Processing, vol. ASSP-34, no. 4, pp. 992–993, Aug. 1986. 

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Nuri Yilmazer was born in Silifke, Turkey. He 

received the B.S. degree from Cukurova University, 

Adana, Turkey, in 1996, and the M.Sc. degree from 

the University of Florida, Gainesville, in 2000. He 

is currently working toward the Ph.D. degree in the 

Department of Electrical Engineering at Syracuse 

University, Syracuse, New York. 

His current research interests include digital signal 

processing and adaptive antenna problems. 

Jinhwan Koh was born in Taegu, Korea. He received 

the B.S. degree in electronics from Inha University, 

Korea, and the M.S. and Ph.D. degrees in electrical 

engineering from Syracuse University, Syracuse, 

New York, in 1997 and 1999, respectively. 

He was formerly with Goldstar Electron Semiconductor 

Company, Korea. He is now a Professor 

in the Department of Electronics and Electrical 

Engineering, Kyungpook National University, 

Taegu, Korea. His current research interests include 

digital signal processing related to adaptive antenna 

problems and data restoration. 

Tapan K. Sarkar (S’69–M’76–SM’81–F’92) received 

the B.Tech. degree from the Indian Institute of 

Technology, Kharagpur, India, in 1969, the M.Sc.E. 

degree from the University of New Brunswick, 

Fredericton, NB, Canada, in 1971, and the M.S. and 

Ph.D. degrees from Syracuse University, Syracuse, 

NY, in 1975. 

From 1975 to 1976, he was with the TACO 

Division of General Instruments Corporation. He 

was with the Rochester Institute of Technology, 

Rochester, NY, from 1976 to 1985. He was a 

Research Fellow at the Gordon McKay Laboratory, Harvard University, 

Cambridge, MA, from 1977 to 1978. He is now a Professor in the Department 

of Electrical Engineering and Computer Science, Syracuse University. He has 

authored or coauthored more than 250 journal articles, numerous conference 

papers, chapters in books, and several books including his most recent, Iterative 

and Self Adaptive Finite-Elements in Electromagnetic Modeling (Boston, MA: 

Artech House, 1998), Wavelet Applications in Electromagnetics and Signal 

Processing(Boston, MA: Artech House, 2002), and Smart Antennas (New 

York: Wiley, 2003). He is on the editorial board of J. of Electromagnetic Waves 

and Applications and Microwave and Optical Technology Letters. His current 

research interests deal with numerical solutions of operator equations arising 

in electromagnetics and signal processing with application to system design. 

Dr. Sarkar is a Registered Professional Engineer in the State of New York. He 

is a Member of Sigma Xi and the International Union of Radio Science (URSI) 

Commissions A and B. He received the College of Engineering Research 

Award in 1996 and the Chancellor’s Citation for Excellence in Research from 

Syracuse University in 1998. He obtained one of the “Best Solution” awards 

at the Rome Air Development Center (RADC) Spectral Estimation Workshop 

in May 1977, and received the Best Paper Award from the National Radar 

Conference in 1997 and the IEEE TRANSACTIONS ON ELECTROMAGNETIC 

COMPATIBILITY in 1979. He received the title Docteur Honoris Causa from 

Universite Blaise Pascal, Clermont Ferrand, France, and from the Politechnic 

University of Madrid, Madrid, Spain, in 1998 and 2004, respectively. He 

received the medal of the Friend of the City of Clermont Ferrand, France, 

in 2000. He was the Chairman of the Inter-commission Working Group of 

International URSI on Time Domain Metrology from 1990 to 1996. He was 

an Associate Editor for feature articles of the IEEE Antennas and Propagation 

Society Newsletter from 1986 to 1988. He was a Distinguished Lecturer for 

the Antennas and Propagation Society from 2000 to 2003. He is currently a 

Member of the IEEE Electromagnetics Award Board and an Associate Editor 

for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is the Vice 

President of the Applied Computational Electromagnetics Society (ACES) and 

the technical Chair for the combined IEEE 2005 Wireless Conference along 

with ACES to be held in Hawaii.

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