Matrix pencil method for simultaneously estimating azimuth

Matrix pencil method for simultaneously estimating azimuth and elevation angles of arrival along with the frequency ofthe incoming signals Nuri Yilmazer∗, Raul Fernandez-Recio, Tapan K..

Matrix pencil method for simultaneously estimating azimuth and elevation angles of arrival along with the frequency of

the incoming signals Nuri Yilmazer∗, Raul Fernandez-Recio, Tapan K Sarkar

Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244-1240, USA

Available online 17 July 2006

Abstract

In this paper we describe a method for simultaneously estimating the direction of arrival (DOA) of the signal along with itsunknown frequency In a typical DOA estimation problem it is often assumed that all the signals are arriving at the antenna array atthe same frequency which is assumed to be known The antenna elements in the array are then placed half wavelength apart at thefrequency of operation However, in practice seldom all the signals arrive at the antenna array at a single pre-specified frequency,but at different frequencies The question then is what to do when there are signals at multiple frequencies, which are unknown Thispaper presents an extension of the matrix pencil method to simultaneously estimate the DOA along with the operating frequency ofeach of the signals This novel approach involves approximating the voltages that are induced in a three-dimensional antenna array,

by a sum of complex exponentials by jointly estimating the direction of arrival (both azimuth and elevation angles) along withthe carrier frequencies of multiple far-field sources impinging on the array by using the three-dimensional matrix pencil method.The matrix pencil method is a direct data domain method for approximating a function by a sum of complex exponentials inthe presence of noise The variances of the estimates computed by the matrix pencil method are quite close to the Cramer–Raobound Finally, we illustrate how to carry out the broadband DOA estimation procedure using realistic antenna elements located

in a conformal array Some numerical examples are presented to illustrate the applicability of this methodology in the presence ofnoise It is shown that the variance decreases as the SNR increases The Cramer–Rao bound for the estimators are also provided toillustrate the accuracy and the computational efficiency of this new methodology

©2006 Elsevier Inc All rights reserved

Keywords: Matrix pencil method; 3-D matrix pencil method; Simultaneous azimuth; Elevation angles and frequency estimation;

Wavelength estimation; Direction of arrival estimation; Angle of arrival estimation; DOA estimation; Estimation of frequency;

Broadband DOA estimation; Conformal array

1 Introduction

In contemporary literature, most of the efforts have primarily been directed for estimating the two dimensional

spa-tial frequencies (namely, the azimuth angle φ and the elevation angle θ ) of plane waves that are arriving at an antenna

* Corresponding author.

E-mail address: nyilmaze@syr.edu (N Yilmazer).

1051-2004/$ – see front matter © 2006 Elsevier Inc All rights reserved.

doi:10.1016/j.dsp.2006.05.009

array It is generally assumed that all the signals have the same frequency of operation even though they are arriving

from different directions and that the antenna elements in a linear array that are uniformly spaced at 0.5λ, where λ

is the wavelength at the frequency of operation We now describe a methodology for simultaneously estimating thefrequency of operation and the DOA of the signals using a three-dimensional antenna array The voltages induced inthe antenna elements of the three-dimensional antenna arrays are used to estimate the frequency of operation and theDOA of the signal simultaneously using the matrix pencil method

The problem of estimating the DOA of multiple far field sources can be found in many application areas such as inradar, sonar, and radio communication and is being studied extensively by many researchers [1–4] The matrix pencilmethod, a high-resolution DOA estimation algorithm, has been applied to find the azimuth, and elevation angles ofthe narrow band sources, which are in the far field of the receiver antennas In some cases the carrier frequency ofthe multiple signals may not be known, in that case, the carrier and the DOA angles, azimuth, and elevation must beestimated simultaneously In the one-dimensional (1-D) uniform linear array (ULA) [1,7], it has been assumed that thefrequency of the signals is known As an extension to 1-D and 2-D matrix pencil method, 3-D matrix pencil method

is presented for estimating three-dimensional (3-D) frequencies is explained in Section 2

Many efforts have been devoted for estimating the two-dimensional (2-D) frequencies for a 2-D data set Theapplications cover a wide range of fields from synthetic aperture radar imaging or frequency, wavenumber estimation

in array processing, nuclear magnetic resonance imaging, radar, array signal processing, radar surveillance systems,and space-division-multiple-access (SDMA) for mobile communication At the same time, high resolution direction

of arrival (DOA) estimation techniques are very important in systems such as radar or sonar applications too

In the current situation, we have a 3-D data set or, as we will see later, a data set placed in a 3-D space such as two2-D data sets (e.g., planes) placed orthogonal to each other

In Section 2, the formulation of the problem is introduced and the signal model is given In Section 3 the trix pencil method is described for the 3-D case In Section 4, the pairing method for the 3-D poles is introduced.Sample numerical results through computer simulations are provided in Section 5 Finally, Section 6 is the conclu-sions

frequencies for each 3-D sinusoid and group them appropriately

The estimation of the complex amplitudes of the signals can be reduced to the solution of a least square problem

group them to identify each sinusoid correctly For the DOA estimation problem they will be elevation, and azimuthangles and the carrier frequencies of each of the far-field sources The original data, from which we can extractthe frequencies from, can be a data cube or two planes of data as will be explained later on, situated along threedimensions These three dimensions can be the three spatial coordinates or data from two spatial coordinates atdifferent instants of time Graphically, we can represent the two-dimensional data as a plane and three-dimensionaldata as a cube A cube, in this context, is a set of planes situated along one direction, as it is shown in Fig 1a As wehave mentioned, the original data can also be along two planes situated along two orthogonal planes and this situation

is illustrated graphically in Fig 1b

We can rewrite Eq (1) as follows:

Fig 1 Illustration of the three-dimensional data possibilities for the 3-D matrix pencil method.

where{α i ; i = 1, , I} are the complex amplitudes of each of the 3-D sinusoids and {(x i , y i , z i ) ; i = 1, , I} are

the 3-D poles that we want to extract from the data The frequencies can be obtained uniquely from the 3-D poles, i.e.,

us assume that there are I far field sources for the 3-D antenna elements This problem is equivalent to estimating I

undamped signals with different frequencies, elevation and azimuth angles for the DOA problem The induced voltage

in each of the antenna elements in the three-dimensional grid due to I 3-D sinusoids is represented by (1) The goal is

to estimate correctly each of the three frequencies for each 3-D sinusoid and group them adequately We can rewrite

λi x cos φ i sin θ i a+2π

λi y sin φ i sin θ i b+2π

λi z cos θ i c

Once the poles are found along each plane, the corresponding elevation, azimuth angles and the wavelength of the

signals θ , φ, and λ are obtained for each source as

3 3-D matrix pencil theory

The 3-D data matrix can be enhanced by using the partition-and-stacking process [5–7] The enhanced-matrixmatrix pencil technique for estimating the three-dimensional frequencies is constructed for handling the estimation of

the three-dimensional frequency The column vectors along the x-direction is enhanced by a pencil parameter L and

using a similar procedure as

The enhanced data matrix Deis now used to obtain the three-dimensional poles The matrix D y,zcan be decomposedas

4 Eigenstructure of the data matrixDe

value decomposition (SVD) [1,5] must be first applied to remove the effects of noise before applying the matrix pencil

technique The matrix Decan be written as

com-ponents of the signal and noise component, respectively Here U and V are unitary matrices whose columns are the

nonzero, the rest is zero, where

σ i >0 for i = 0, 1, , I,

If the data is noisy, I needs to be estimated, and how to do it is illustrated in [1] The ratio of each of the singular value to the largest one determines the value of I After computing the SVD, the data is split along the signal subspace and the noise subspace The signal subspace has dimension I (the number of signals) that corresponds to the main eigenvalues of Λ and the noise subspace that is related to the rest of eigenvalues [8] They are represented by

Next we define a permutation matrix that is used for the extraction of the poles [5] The permutation (shuffling)matrix in this case will be:

s(L) s(L + L)

and the other shuffling matrix is defined as

4.4 Pole paring for 3-D matrix pencil method

get the correct pairs, so that it will correspond to the true azimuth and elevation angles for the specific wavelength

of the impinging far field sources So there is a need for establishing a pairing algorithm By exploiting the property

maximizing the criterion below

where⊗ is the Kronecker product, and x L , y L , and z L are defined as follows: x L = [1, x, , x K−1]T, y L = [1, y, ,

y L−1]T, z L = [1, z, , z R−1]T Here{u p ; p = 1, 2, , I} are the I principles eigenvectors of the matrix De

Pole pairing algorithm for three-dimensional matrix pencil method

i= 1

End k

End j

Remove the paired poles from the set (corresponding to the maximum)

Increment i

End

λi x cos φ i sin θ i a+2π

λi y sin φ i sin θ i b+2π

λi z cos θ i c

Table 1

Summary of the signal features incident on the antenna array

Fig 2 The variance −10 log 10( var(φ1)), 3-D MP and the CRB are plotted against the SNR.

Fig 3 The variance −10 log 10( var(θ1)), 3-D MP and the CRB are plotted against the SNR.

Fig 4 The variance −10 log 10( var(λ1)), 3-D MP and the CRB are plotted against the SNR.

Fig 5 The scatter plots of elevation and azimuth angles for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB.

(a) (b)

Fig 6 The histogram of azimuth angle for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB.

where w(a, b, c) is a zero mean Gaussian white noise with variance κ A three dimensional array of omni directional

examples illustrate the performance of the estimator in the presence of white Gaussian noise The attributes of the

estimated DOA angles and the wavelength will have a bias and a variance due to noise In the case of noisy data, theestimated values will also be random variables The stability/accuracy of the results needs to be expressed in terms

of its statistical properties, which in this case are the estimated values such as the mean, variance, and so on of theestimate These results can be obtained with Monte Carlo simulations

The Cramer–Rao bound (CRB) measures the goodness of an estimator This bound is the smallest limit for thevariance of the estimated values under noisy measurements with white Gaussian noise The bound is found fromusing the Fisher information matrix, whose diagonal elements are the corresponding CRB of that element The Fisherinformation matrix and how it relates to the CRB is explained in Appendix A The simulation results show that the

(a) (b)

Fig 7 The histogram of elevation angle for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB.

(SNR) of the incoming signals and are plotted in Figs 2–4 Different values of SNR are plotted along the x-axis

and the inverse of the variance of the estimated azimuth, elevation angles and wavelength are in logarithmic domain,

The variances of the estimated values of elevation and azimuth angle and the wavelength of the first source plottedagainst SNR are shown below The results are based on 1000 Monte Carlo simulations Since the simulation resultsfor other two signals have very similar characteristic, we only provided the first signals results

The scatter plot of the estimated elevation and azimuth angles are shown in Fig 5 for different signal-to-noise

when the SNR increases, the estimated values approach to its true values in the scatter plot

The histogram of the estimated elevation and azimuth angles and the wavelength of the sources are shown in

Carlo simulations As can be seen, for the increased SNR values, the estimated values approach to its true values inthe histogram plot

(a) (b)

Fig 8 The histogram of wavelength for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB.

The bias of the estimator has also been studied to see the efficiency of the new method For the elevation andazimuth angles and the wavelength, the bias of the estimator is computed The bias is calculated as

the estimator for azimuth and elevation angles, and the wavelength versus SNR is shown in Figs 9–11

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

6 Conclusions

In this study, a new methodology has been presented to extract the 3-D frequencies from the sampled values ofthe voltages induced in an array This 3-D estimation method can easily be applied to a generalized DOA estimation

Fig 9 Bias of the estimator for azimuth angle versus SNR.

Fig 10 Bias of the estimator for elevation angle versus SNR.

Fig 11 Bias of the estimator for wavelength versus SNR.

problem, where one can estimate not only the azimuth and elevation angles of the various signals impinging on arraybut also their wavelengths of operation It has been shown how to form the enhanced data matrix and by using this datamatrix one then pair the respective poles to obtain the corresponding angle of arrivals and the wavelengths Numericalexamples are provided to illustrate the validity of the new method The statistical performance parameters such asvariance of the estimator have also been compared with the Cramer–Rao bound to observe the accuracy of the newmethod.

Acknowledgment

Grateful acknowledgment is made to the reviewers for suggesting ways to improve the readability of the paper

Appendix A CRB for the direction of arrival estimation

s cos φ i sin θ i a x+2πfi

s sin φ i sin θ i b y+2πfi

s cos φ i sin θ i a x+2πfi

s sin φ i sin θ i b y+2πfi

s cos φ i sin θ i a x+2πfi

s sin φ i sin θ i b y+2πfi

s cos θ i c z

; a, b, c



s cos φ i sin θ i a x+2πfi

s sin φ i sin θ i b y+2πfi

s cos φ i sin θ i a x+2πfi

s sin φ i sin θ i b y+2πfi

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z

(f i cos φ i sin θ i − f j cos φ j sin θ j )a x

+ (f i sin φ i sin θ i − f j sin φ j sin θ j )b y + (f i cos θ i − f j cos θ j )c z