Efficient Beamspace Eigen-Based Direction of Arrival Estimation schemes

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Purdue University
Purdue e-Pubs
ECE Technical Reports Electrical and Computer Engineering
5-1-1994
Efficient Beamspace Eigen-Based Direction of
Arrival Estimation schemes
Gregory M. Kautz
Purdue University School of Electrical Engineering
Michael D. Zoltowski
Purdue University School of Electrical Engineering
Follow this and additional works at:http://docs.lib.purdue.edu/ecetr
Kautz, Gregory M. and Zoltowski, Michael D., "Efficient Beamspace Eigen-Based Direction of Arrival Estimation schemes" (1994).
ECE Technical Reports.Paper 184.
http://docs.lib.purdue.edu/ecetr/184
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TR-EE 94-15
MAY 1994
Efficient Beamspace Eigen-Based
Direction of Arrival Estimation schemes1
Gregory M. Kautz and Michael D. Zoltowski
School of Electrical Engineering
1285 Electrical Engineering Building
Purdue University
West Lafayette, IN 47907-1285
'This work was supported by the National Science Foundation under grant no. MIP-
9320890 and by AFOSR under contract no. F49620-92-J-0198 in conjunction with Wright
Laboratories.

TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
.
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
.
2 DEVELOPMENT OF BEAMSPACE ROOT-MUSIC . . . . . . . . 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
DFT Based Beamspace Root-MUSIC With Reduced Degree Polynomia.1 8
Previous Bea.mforming Methods t.o Achieve Reduced Degree Polynomial 19
Real Covariance Matrix Processing in Beamspace . . . . . . . . . . . 22
VirtualTapering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.1 The Cosine Window . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.2 The Hanning and Haillining Windows . . . . . . . . . . . . . . 31
Constructioil of Interference Cancellation Matrix Beamformer . . . . 34
Computer Experiments/Simulations . . . . . . . . . . . . . . . . . . . 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary 51
.
3 PERFORMANCE ANALYSIS OF BEAMSPACE ROOT-MUSIC
EMPLOYING CONJUGATE SYMMETRIC BEAMFORMERS . 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Performance Analysis of Real Co~rarianceB eamspace hlUSIC . . . . . 54
3.2.1 Stat.i stics of the Signal Sl~bspaceE igenvalues and Eigenvectors 55
3.2.2 hilean and Variance of the Spect.r al h4USIC Angle Estimate . . 57
3.3 Observations of the Theosetical Variance Equations . . . . . . . . . . 61
3.4 Validation of the Theoretical Expressions . . . . . . . . . . . . . . . . 64
3.5 Merit of Employing Tapered Bealnformers . . . . . . . . . . . . . . . 72
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
.
4 BEAMSPACE DOA ESTIMATION FEATURING MU'LTIRATE
EIGENVECTOR PROCESSING . . . . . . . . . . . . . . . . . . . . 79
Page
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Development of DOA Estimators Featuring Multirate Eigenvector Pro-
cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Multirate Noise Eigenvector Processing . . . . . . . . . . . . .
4.2.2 Incorporation of Filter Deconvolution . . . . . . . . . . . . . .
4.2.3 Root-MUSIC Incorporating Multirate Eigenvector Processing
4.2.4 TLS-ESPRIT Incorporating Mult irate Eigenvector Processing
4.3 Theoretical Performance Analysis . . . . . . . . . . . . . . . . . . . .
4.3.1 Performance Analysis of Root-MUSIC Formulation . . . . . .
4.3.2 Performance Analysis of ESPRIT Formulation . . . . . . . . .
4.4 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
5 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . 123
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
APPENDICES
Appendix A: Simple Expression for Q ~ .' . . . . . . . . . . . . . . . . . 129
Appendix B: Simple Expression for DSFT of Coefficient Vector for Com-
mon Roots Polynomial . . . . . . . . . . . . . . . . . . . . . 133
Appendix C: On the Distribution of the Real Part of the Beamspace Sam-
ple Covariance Matrix in the Case of Uncorrelated Gaussian
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Appendix D: Derivation of the Asymptotic Distribution of the Signal Sub-
space Eigenvalues/Eigenvectors . . . . . . . . . . . . . . . . 137
Appendix E: Asymptotic Variance of ESPRIT Formulation . . . . . . . . 143
LIST OF FIGURES
Figure Page
2.1 (a) Nb= 8 unweighted spatial DFT beams for N=32 element ULA; beams
encompass the null-to-null subband -10/N < u < SIN. (~b)R oots of
beamspace Root-hlUSIC polynomia.1 in no source case. . . . . . . . . . . 18
2.2 (a) Six equi-spaced beams forined via virtual Hamming tapering. (b)
Orthogonalized set of beams derived from t,hose in (a). . . . . . . . . . . 33
2.3 (a) "Adapted" beams derived from those in Figure 2.1 (a) via interference
cancellation matrix beamformiilg method. Ea.ch beam exhibits a null in
each of the two interference locations. (11) Seven beams derived from those
in (a); each beam is the sum of a.n a.dja.cent pair of beams. . . . . . . . . 39
2.4 Beamspace Root-MUSIC vs. Element-space Root-MUSIC for beam set of
Figure 1. (a) Bea.mspace Root-R'IUSIC. (11) Element-spa.ce Root-MUSIC 46
2.5 Empirical probability of resolution versus SNR curves: Beamspace Root-
MUSIC versus beamspace spectral AllTSIC with two equi-powered and
uncorrelated in-band sources locatcrl at - 1 /(2N) and 1/(2N). Nb = 4
out of the eight beains displayed in Figure 2.l(a). . . . . . . . . . . . . . 47
2.6 Beamspa.ce Root-MUSIC versus 11ea.mspace spectral MUSIC. Same sce-
nario as that in Figure 2.5 except SNR = OdB. (a) Root scatter plot
obtained from five independent rllns. (13) Corresponding beamspace spec-
tra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Performance of beamspace Root-hlUSIC as a functioll of position of two
uncorrelated half-Rayleigh sources (SNR = 6dB) within the subband
-10/N < IL < 8/N. ATb = S beams generated with weigl~l~inagp plied
to the set shown in Figure 2.l(a). Statistics for the "left" source com-
puted from 250 runs. (a) Sample bias. (11) Sample standard deviation. . 49
Figure Page
2.8 Beamspace Root-MUSIC scatter plots associated with the use of the rect-
angular and Hamming-weighted beam sets of Figures 2.1 (a) and (b). 'Two
6dB uncorrelated sources were located a.t 0 a.nd 1/N while an 18dB out-
of-band source was positioned a.t -17/AT. The roots in 250 trial runs .were
computed assuming only 2 signals present. (a,) no (rectangular) tapering,
(b) Hamming taper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1 Real vs. complex processing: standard devia.tion a,s f~~nctioonf phase
difference between two correlated sig11a.l~. . . . . . . . . . . . . . . . . . 68
3.2 Real vs. complex processing: bias a.s function of phase difference between
two correlated signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Real vs. complex processing: standarc1 clcviat,ion as function of sna.pshots
for two uncorrelated signals. . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Real vs. complex processing: bia.s a.s function of sna,pshots for two uncor-
related signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Real vs. complex processing: standa,rd deviation as function of SNIt for
two uncorrelatecl signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Real vs. complex processing: bias a.s fuuct.ion of SNR for t,wo uncorrela.ted
signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 Real vs. complex processing: probability of resolution a.s function of SNR
for two uncorrelatecl signals. . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.8 Candida.te beamforming a.rchitectures for sector-based processing. . . . . 76
3.9 Theoretical bias of left signal estiina,te for various bea.mforming prepro-
cessors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.10 Theoretical std. clev. of left signal (.stinrat(> fot \.arious bearnforming
preprocessors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Condition number versus number of spa.tia.1 DFT beams, ATb, for an N =
128 sensor army. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure Page
4.2 Spectrum of a transformed noise eigenvector derived from the decomposi-
tion of the ideal heamspace covariance associated with an N =: 128 sensor
ULA operated on by a Nb = S cli~r~ensiosnp atial DFT beamformer cen-
tered in space at 25/N. Two in-band signals were located at 10.6' and
11.5' and one out-of-band source was located at sin O = 69/N. . . . . . . 85
4.3 Angular responses of Nb = 8 successive DFT beanforming vectors. Beam-
forming sector centered at at sin0 = 25/N. . . . . . . . . . . . . . . . . 86
4.4 hlUSIC null spectrum after modulation to baseband and filter response
associated with the L = 12s lengtll Hamming-weighted LPF. . . . . . . . 91
4.5 MUSIC null spectrum a.fter Halnining-winclo~vI msed ba.ndpass filtering. . 92
4.6 MUSIC null spectrum after cleci~r~a.tio1n1 f~a~ct or of 12S/S = 16. . . . . . 93
4.7 Roots using tra.nsformed-modulated-filterecl-decimatedn oise eigenvectors
for both the Z and (deconvol\:c:cl) Z1 transformations. Quiescent root
locations coinputed with the use of t11e ideal heamspace covariance. True
signal locations: 10.6" and 11.5". . . . . . . . . . . . . . . . . . . . . . . 101
4.8 Spatial responses of an L = 270 lcngth eq~~i-rippflielt er and interpolated
Nb = 6 beam set derived from an N = 90 sensor array. The spatial
foldover frequency for the suh-lna~mi a1 cl(.(-imat ior1 architecture is located
at sin 0 = (1/4)(8/N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.9 Decimated filter/bea.mformer spectra a.ssocia.tetl with the filter/beam set
of Figure 4.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.10 MUSIC spectra associated with ideal white-noise heamspact: covariance
with the use of the Z and Z1 tra.nsformations derived froin the: bea.mform-
ing/filter architecture of Figure 4.8. . . . . . . . . . . . . . . . . . . . . . 1 04
4.11 Experiment 1: Empirical and tlleorctical left signal sta~ldadi deviation
versus spatial position of a 10~113h, alf-Rayleigh spaced signal set. Central
position of signal set varied from mid-l,a11c1 to 6/N. The Nb = 8 spatial
DFT beams were formed on an !I=r 128 sensor UT,A. . . . . . . . . . . . 117
4.12 Experiment 2: Left signa.1 stanclal-tl cleviation versus source SNR for the
two in-ba.nd signal, :I:=, S, beam esample scenario depicted in Figure 4.3. 118
Figure Page
4.13 Experiment 3: Location bias versus source SNR, for an Arb = 6 beam pre-
processor (un-weighted DFT or Ta.ylor-weighted beamformers) operating
on an N = 36 sensor ULA. The mea,n angle estimate for a signal located
at lo was computed over 600 trials. . . . . . . . . . . . . . . . . . . . . . 119
4.14 Experiment 4: TLS-ESPRIT left signal standarc1 deviation versus source
SNR for the two signal example of Figure 4.12. . . . . . . . . . . . . . . 120
4.15 Experiment 4: Quiescent locations of the TLS-ESPRIT eigenvalues asso-
ciated with the decomposition of the ideal beamspace covariance input. . 121