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Switched-Element DirectionFinding

WEI WU

CHARLIE C. COOPER, Member, IEEE

NATHAN A. GOODMAN, Senior Member, IEEEUniversity of Arizona

We introduce and evaluate the concept of switched-element

direction finding. In a switched-element system, simultaneous

data collection occurs on only a subset of the array’s N elements

at any given time. The data collection interval is divided into

subintervals, and at each subinterval a multiplexer selects

a different subset of M <N elements to connect to the data

acquisition channels. Since the switched-element system only

collects data on M channels at a time, hardware costs can be

reduced at the expense of a time-accuracy tradeoff compared to

a full-channel system.

We propose two algorithms. The first is a modification of

the root-MVDR (minimum variance distortionless-response)

algorithm that is appropriate for linear arrays and noncoherent

sources. When sources are coherent, a second algorithm based

on a compositely formed full-size covariance matrix is proposed.

Although performance depends strongly on the number of

sources, good performance can be obtained even when the

number of sources is greater than M .

Manuscript received August 16, 2007; revised January 21, 2008;released for publication August 12, 2008.

IEEE Log No. T-AES/45/3/933970.

Refereeing of this contribution was handled by M. Rangaswamy.

Authors’ current addresses: W. Wu, Aware, Inc., Bedford, MA; C.C. Cooper and N. A. Goodman, Dept. of Electrical and ComputerEngineering, University of Arizona, 1230 E. Speedway Blvd.,Tucson, AZ 85721, E-mail: ([email protected]).

0018-9251/09/$26.00 c° 2009 IEEE

I. INTRODUCTION

The typical implementation of a passive RFdirection finding (DF) system involves a dedicatedsignal acquisition channel (receiver, analog-to-digitalconverter,: : :) for each element of the antennaarray. During the data collection interval, thisstraightforward implementation simultaneouslycaptures and stores data from all N antennas inthe array. When the data collection interval ends,the DF processor analyzes the data and providesestimates of the directions of arrival (DOAs) usinga particular algorithm. There are numerous effectivetechniques for passive DF including maximumlikelihood estimation (ML) [1], the minimum variancedistortionless-response algorithm (MVDR) [2, 3],mini-norm [4, 5], multiple signal classification(MUSIC) [6—9], and estimation of signal parametersvia rotational invariance techniques (ESPRIT)[10, 11]. One common technique employed by theseDF algorithms is to form an N £N sample covariancematrix from the collection of data snapshots. Thiscovariance matrix then forms the basis for furtherprocessing. Subarray techniques such as spatialsmoothing [12]—[15] also begin with raw datacollected simultaneously over the entire array.In this paper, we propose an alternative system

implementation that simultaneously collects data froma subset of M <N antennas [16]. While the totalnumber of antenna elements remains the same, thesystem possesses only M data acquisition channels asshown in Fig. 1. The data collection interval is dividedinto subintervals, and at each subinterval a multiplexerselects a different subset of M elements to connectto the acquisition channels. The DF processor storesthe snapshots collected by each M-element subarrayover its corresponding subinterval, and combines themto provide the required DOA estimates. In this paper,we assume the system switches through every uniquecombination of M elements during the data collectioninterval. In general, however, the antenna switchingis not required to be this comprehensive. We call theproposed implementation a switched-element system.

Fig. 1. Block diagram of switched-element DF system.

The primary advantage of the switched-elementsystem is reduced receiver hardware, which impliesreduced cost, volume, and weight. Another advantageis the possibility for adaptive switching in responseto data collected in prior subintervals. The presenceof the switch could also simplify channel calibration

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 45, NO. 3 JULY 2009 1209

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by switching a single antenna to all M receiver chainssimultaneously, which would reveal the relative gainand phase responses of the receiver channels inthe presence of any incident (even uncooperative)signal. Finally, in some cases the switched-elementapproach may also require reduced computationand data storage. The compromise, however, is atime-accuracy tradeoff due to less data being gatheredcompared with a full-array system. Therefore, abalance must be found to satisfy requirements for bothhardware cost and accuracy. Given this flexibility, aswitched-element system may be desirable in somepractical cases.In this paper, we propose and evaluate two

switched-element DF algorithms. First, theroot-MVDR technique [1] is extended to becompatible with switched-element direction findingof noncoherent sources. The switched-elementroot-MVDR algorithm is flexible in that it uses theM £M sample covariance matrices of all subarraysdirectly; however, this flexibility also makes itunsuitable for use with coherent sources. For coherentsources, we assemble the full N £N covariance matrixfrom components of the individual M £M covariancematrices. This composite covariance matrix (CCM)can be used with traditional spatial smoothing andMUSIC to resolve multiple coherent signal sources.The paper is organized as follows. Section II

defines our narrowband signal model andproblem statement. Section III presents the twoswitched-element DF algorithms mentioned above.Simulation results are presented and discussed inSection IV, and Section V contains our conclusions.

II. PROBLEM FORMULATION

A. Signal Model

In the narrowband signal model for arrayprocessing, far-field sources arriving at the antennaarray produce a set of phase shifts across the antennaelements. The relative phase shift for an antennaelement located at coordinates (xn,yn) is

'n =¡kxxn¡ kyyn =¡k(μ) ¢ rn (1)

where k is the wavenumber vector with componentsrelated to the source DOA by kx = k0 sinμ and ky =k0 cosμ, rn is the antenna element position vector,and k0 = 2¼f0=c is the wavenumber in radians/meterat the source’s center frequency f0 (c is the wave’spropagation speed). Defining the array manifoldvector for an N-element array as

a(μ) = [exp(¡jk(μ) ¢ r1) exp(¡jk(μ) ¢ r2)¢ ¢ ¢exp(¡jk(μ) ¢ rN)]T (2)

where (¢)T is the transpose operator, the signalreceived by the array due to a single source is

s1(·)a(μ) where s1(·) is the signal amplitude attemporal sample index ·. Adding receiver noise andallowing for D sources, the signal model for the datasnapshot at time index · is [1, 17]

z· =As·+w· (3)where

A= [a(μ1) a(μ2) ¢ ¢ ¢a(μD)] (4)

s· = [s1(·) s2(·) ¢ ¢ ¢sD(·)]T (5)

and w· is additive spatially and temporally whitecomplex Gaussian noise (AWGN) with unit power.We assume that the signals are zero-mean complexGaussian random processes; therefore, each sd(·) (thedth source’s signal amplitude at sample index ·) isa complex zero-mean Gaussian random variable. Weassume that snapshots collected at different timesare independent and identically distributed. Betweensignals, however, we consider both the correlatedand uncorrelated cases. The signal covariance matrix,S= E[s·s

H· ], therefore, is a Hermitian matrix whose

dth diagonal entry is the average power Pd of thedth source. Given the normalization of noise powerto one, the signal-to-noise ratio (SNR) of the dthsource at the output of a given antenna element isSNRd = Pd=1 = Pd. This is the SNR definition used forall results presented in this paper. For the uncorrelatedsignal case, the signal covariance matrix is a diagonalmatrix. When the signal amplitude coefficients arecorrelated, the signal covariance matrix has non-zerovalues off the main diagonal. When signal amplitudecoefficients are perfectly correlated, such as whensi(·) = ®sj(·) for some constant ® and i 6= j, we saythe signals are coherent.Algorithms such as MVDR and MUSIC typically

require data from all N sensors throughout the datacollection process for all snapshots. In this paper,however, we focus on the premise that only Mchannels of data can be collected at any given time.Consequently, we define zq· as the length-M datasnapshot formed by extracting M elements fromthe full-array data vector where the superscript qdenotes the qth subarray of the switching operation.In other words, a different subset of M elements isextracted for each value of q. We also define Kq to bethe number of (still independent) snapshots collectedin the qth switching configuration and Q to be thenumber of unique configurations.The problem can now be stated as follows. We

assume that only M channels of data can be collectedsimultaneously, but that a multiplexer allows differentsubsets of M antennas to be passed to the dataacquisition hardware at different times. Our goal isto develop and analyze algorithms for estimating theDOAs of incoming sources. Although we know thehardware cost can be reduced, the switched-elementstructure causes unique processing issues. First, while

1210 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 45, NO. 3 JULY 2009


most DF algorithms are based on a single N £Nsample covariance matrix, the switched conceptis only capable of producing Q different M £Msample covariance matrices with different underlyingsubarray structures. Therefore, we must either modifyexisting algorithms to accept multiple mismatchedM £M covariance matrices or we must constructthe full size covariance matrix by piecing togetherdata gathered from multiple subarrays. Second, theswitched-element system collects less data than thefull-array system over the same amount of time. Thisresults in a compromise whereby performance of theswitched-element system is reduced in comparison toa full-array system when the two systems have equaldata collection intervals. Full-array performance canusually be achieved, however, if the switched-elementsystem is allowed to collect data over a longer timeinterval. Hence, we also quantify the time-accuracytradeoff inherent in the switched-element concept.

B. Cramer-Rao Bounds

Since the noise and signal samples are Gaussian,the observed data (for the full array) are also Gaussianwith covariance matrix

Kz = E[z·zH· ] =Ks(μ)+ IN =ASA

H + IN (6)

where μ is a vector of unknown source DOAs,IN is the N £N identity matrix, and (¢)H is theconjugate-transpose operator. Therefore, the pdf ofthe ·th data snapshot is

p(z·;μ) =1

j¼Kzjexp(¡zH·K¡1z z·): (7)

Given the independence of successive samples ofsignal and noise, the pdf for a full collection of Ksnapshots is

p(z1,z2, : : : ,zK ;μ) =KY·=1

1j¼Kzj

exp(¡zH·K¡1z z·): (8)

According to [1], the Fisher information matrix (FIM)for this scenario is

Jμμ = 2KRe½ [SAHK¡1z M]¯ [SAHK¡1z M]T+[SAHK¡1z AS]¯ [MHK¡1z M]

T

¾(9)

where ¯ is the Hadamard matrix product, Ref¢gdenotes the real part of a complex number, and M is

M=·@a(μ1)@μ1

@a(μ2)@μ2

¢ ¢ ¢ @a(μN)@μN

¸: (10)

The Cramer-Rao (CR) bound is defined by C(μ) =E[(μ̂¡μ)(μ̂¡μ)T]¸ J¡1μμ where the inequality is usedto indicate that C(μ)¡ J¡1μμ is a nonnegative definitematrix [1].A switched-element system collects data using Q

sequential subarray configurations. First, the systemconnects the receiver inputs to the outputs of the M

antenna elements in the first subarray configuration.The system stays in this configuration long enoughto collect K1 independent data snapshots (in thesame manner as the full-array system above collectsK independent snapshots, but the switched systemcollects data from M elements instead of N). Whenthe K1 snapshots are collected, the system switchesthe receiver inputs to a new set of M antenna outputsand collects K2 snapshots in this configuration. Whenthis is done, the system switches again and collects K3snapshots from a third unique subarray configuration.This process continues until data collection from theQth subarray configuration is complete.All snapshots are collected sequentially in time

such that signal amplitude and noise fluctuationsare independent, even for snapshots collected by thesame subarray. However, the spatial covariance matrixvaries depending on the structure of a particularsubarray. Define

Kqz = E[zq·(z

q·)H] =AqSA

Hq + IM (11)

as the covariance matrix for the qth arrayconfiguration where

Aq = [aq(μ1) aq(μ2) ¢ ¢ ¢aq(μD)] (12)

is the array manifold matrix valid for the qth subarray.Noting that the qth configuration is formed using asubset of M elements from the full array, we see thatthe elements of aq(μ) are equal to the elements of thefull-array manifold vector a(μ) that correspond to theelements in the subarray. For example, suppose M = 3and the first, third, and fourth antenna elements ofthe full array are selected for the qth subarray, thenaq(μ) is formed by taking the first, third, and fourthentries of a(μ). Furthermore, the 3£ 3 covariancematrix Kqz will have entries equal to the correspondingentries of the full-array covariance matrix Kz. Inthis case, the (2,1) entry of Kqz is equal to the(3,1) entry of Kz because both values describe thecorrelation between the same two antenna elements.This relationship will be exploited in Section IIIBwhere we use switched-element data to approximatea full-array covariance matrix for use with spatialsmoothing.The joint pdf of the snapshot vectors for the

switched-element system is

p(z11,z12, : : : ,z

QKQ;μ)

=QYq=1

KqY·=1

1j¼Kqz j exp[¡(z

q·)H(Kqz )

¡1(zq·)]: (13)

The FIM for a single subarray configuration pairtakes the same form as (9), but with the arraymanifold and covariance matrix appropriate to that

WU ET AL.: SWITCHED-ELEMENT DIRECTION FINDING 1211


subarray:

Jqμμ = 2KqRe

½[SAHq (K

qz )¡1Mq]¯ [SAHq (Kqz )¡1Mq]

T

+[SAHq (Kqz )¡1AqS]¯ [MH

q (Kqz )¡1Mq]

T

¾:

(14)

Based on the independence between snapshotscollected by different subarrays, the total FIM is thesum of the individual FIMs for each subarray [1];therefore, the CR bound for the switched-elementsystem is

Csw(μ)¸ (Jswμμ)¡1 (15)

where Jswμμ =PQ

q=1 Jqμμ.

Since the switched system’s total FIM is a sum ofindividual FIMs, and the individual FIMs are scaledby the number of snapshots for that subarray, wenote that the CR bound varies depending on how thetotal data collection time is allocated to the differentsubarrays–in other words, through the selectionof the Kq’s. It is interesting to consider whether theCR bound can be used to optimally allocate a finitenumber of snapshots across the individual subarrayconfigurations. Unfortunately, since the CR bound isa local error measure [1, 18], optimal allocation interms of the CR bound would assign all snapshots tothe subarray configuration with the widest baselinesince it will have the best angular resolution. Whilethis approach will minimize the CR bound, we mustconsider that subarrays are formed by selecting asubset of antennas from the full array. This impliesthat individual subarrays will be undersampled, andswitching between a variety of different subarrayswith different spatial configurations is essential foravoiding ambiguity-type errors. Hence, allocatingsnapshots to the individual subarray configurationsis a compromise between making global errors andreducing the variance of local errors.In Section III, two algorithms are proposed. The

first one involves the direct processing of a set ofM £M sample covariance matrices. This algorithmworks well for uncorrelated signals. When the signalamplitude coefficients are correlated, the rank of thesignal covariance matrix degenerates to a value lessthan the number of sources D. This is known to affectcovariance-based DF algorithms significantly, but incertain situations the spatial smoothing technique canrestore much of the performance. Spatial smoothing,however, requires two or more identically structuredsubarrays that differ only by a shift in position.While in some cases the subarray corresponding toa particular switch configuration could be partitionedeven further for the purpose of spatial smoothing, thesubarray structure of the different switched-elementconfigurations is not guaranteed to support thisin general. Hence, our second algorithm forms acomposite full size matrix, and by incorporating thespatial smoothing technique, it can deal with coherentsignal sources.

III. DF ALGORITHMS FOR A SWITCHED-ELEMENTARRAY

A. Switched-Element Root-MVDR Algorithm

As mentioned above, most DF algorithms arebased on the sample covariance matrix (SCM), whichfor the full array of N elements is formed by

Rz =1K

KX·=1

z·zH· (16)

where K is the total number of snapshots. The MVDRspectrum [2] is given by

P̂mvdr(μ) =1

aH(μ)R¡1z a(μ)(17)

where a(μ) is defined in (2). Peaks in P̂mvdr(μ) indicatesource DOAs.In a switched-element system, however, we cannot

form the full SCM directly. Alternatively, we can findthe SCM for each switch configuration, which for theqth configuration is

Rqz =1Kq

KqX·=1

zq·(zq·)H: (18)

To obtain a switched-element MVDR algorithm,we define a spectrum that incorporates all Qconfigurations according to

P̂SW(μ) =1PQ

q=1 aHq (μ)(R

qz)¡1aq(μ)

: (19)

Defining the first element in each switch configurationas the reference element for that subarray, themanifold vector aq(μ) is

aq(μ) = [1 e¡jk(μ)¢(rq

2¡rq

1) ¢ e¡jk(μ)¢(rqM¡rq1)]T (20)

where rqi is the position vector of the ith elementin the qth subarray. Note that Rqz is always M £Mand aq(μ) is M £ 1. By evaluating over all μ, wecan estimate the DOAs by finding the D maximain the spectrum. This is the spectral version of theswitched-element MVDR algorithm.The root-MVDR algorithm is an efficient method

to evaluate the MVDR kernel [1]. Rather thanmaximizing (17), the root-MVDR algorithm uses apolynomial root-finding approach to minimize thedenominator of (17), but for the switched-elementconcept, we apply the polynomial root-findingtechnique to find the minima of the denominator of(19). If the antenna array is linear with all elementslocated at integer multiples of a fundamental spacingd0, then the array manifold vector for the qth subarray(referenced to the first element of the subarray) is

aq(μ) = [1 e¡jkx(mq

2¡mq

1)d0 ¢ ¢ ¢e¡jkx(mqM¡mq1)d0 ]T (21)



where mqi is an integer that defines the location ofthe ith element of the qth subarray in terms of thefundamental spacing. Defining b = exp(jkxd0), thearray manifold vector becomes

aq(μ) = [1 b¡(mq

2¡mq

1) ¢ ¢ ¢b¡(mqM¡mq1)]T (22)

which consists of integer powers of the variable b.Letting Gq = (R

qz )¡1, the denominator of (19) is

L̂(μ) =1

P̂SW(μ)=

QXq=1

aHq (μ)Gqaq(μ): (23)

To see that (23) is a polynomial, consider the M = 2case. Letting

Gq =·gq11 gq12

gq21 gq22

¸(24)

we have

aHqGqaq = gq11 + g

q12b

¡(mq2¡mq

1) + gq21b(mq2¡m

q

1) + g22:

(25)Substituting (25) into (23), we obtain

L̂(μ) =QXq=1

(gq11 + gq12b

¡(mq2¡mq

1) + gq21b(mq2¡m

q

1) + gq22)

(26)

which is also a polynomial. At this point, we canuse an efficient polynomial rooting algorithm tofind the minima of (26). Note, however, that thedefinition of b above implies that roots correspondingto sinusoidal plane waves should have unit magnitude.Therefore, the roots of (26) closest to the unit circleare the ones that correspond to propagating sources,and the DOA estimates are obtained by setting thephase of the root equal to kxd0. Furthermore, sinceL̂(μ) is conjugate symmetric, each root of (26) hasa conjugate reciprocal root. Hence, we only needto examine either the roots inside the unit circle oroutside the unit circle, but not both.One advantage of the switched root-MVDR

algorithm is its compatibility with theswitched-element concept. Since the switchedroot-MVDR algorithm works directly on the M £Mcovariance matrices of the individual subarrays, weare not required to include every antenna pair in theswitching pattern. By contrast, if an individual antennaelement is to be included in the CCM (describedbelow), the switching pattern must ensure thatsometime during the data collection this antenna is“on” at the same time as each of the other antennas inthe array. Otherwise, elements of the full covariancematrix will be missing. Unfortunately, this placesa constraint on the switching pattern that may notbe consistent with best performance. For example,if input SNR is well above the threshold SNR [1]such that ambiguity-type errors are unlikely, it wouldbenefit the system to collect more snapshots with large

antenna baselines to refine the angle estimate(s) ratherthan smaller baselines that prevent ambiguity errors.The switched root-MVDR algorithm is consistent withthis desired flexibility.One disadvantage of the root-MVDR algorithm,

however, is its inability to handle coherent sources.Because the algorithm operates on multiple M £Mcovariance matrices formed from subarrays withdifferent structures, the spatial smoothing techniquecannot be applied directly. The switched-elementroot-MVDR algorithm works well for noncoherentsources, but when coherent signals are known to bepresent, we propose the CCM-based root-MUSICalgorithm described next.

B. CCM-Based Root-MUSIC with Spatial Smoothing

Spatial smoothing requires two or more identicallystructured subarrays that differ only by a shift inposition. While the switched-element system couldcertainly select different subsets of M antennas withthe same subarray structure (assuming the full arrayhas the proper structure to begin with), the datacollected by these different subsets must be collectedat different times. Therefore, the signal coefficientrealizations (the sd(·)’s) are different for eachsubarray, which leads to poor results for the switchedroot-MVDR algorithm. Although the CCM is alsoformed from data snapshots containing different signalcoefficient realizations, the CCM-based root-MUSICtechnique is less sensitive to this difficulty.For a given subset of antennas, the M £M SCM is

Rqz =

26666664¾q11 ¾q12 ¢ ¢ ¢ ¾q1M

¾q21 ¾q22...

.... . .

¾qM1 ¢ ¢ ¢ ¾qMM

37777775 (27)

where ¾qij is the sample correlation between the ithand jth antennas of the qth subarray. If the systemcollects and stores data from all combinations ofantennas, the full-dimension N £N CCM can beconstructed by substituting entries from the individualRqz in (21). For example, if the ith and jth antennasof the qth subarray are the same as the mth andnth antennas of the full array, respectively, then wecan set R̃z(m,n) =R

qz (i,j) where R̃z is the CCM.

Since the data from each subarray configuration areindependent, repeated measurement of the variances(diagonal entries) can be averaged to make full useof the collected data (however, this seems to havelittle performance benefit). The CCM has the sameform as a regular SCM; therefore, given an array withproper structure, we can apply the spatial smoothingtechnique to resolve correlated sources. For simplicity,in the next section we assume a fully populated lineararray for all simulations involving coherent sources.



An interesting characteristic of the CCM is thatit is not positive definite, which we expect from aproper covariance matrix. This characteristic occursbecause different elements of the CCM are estimatedfrom data collected at different times. Since the dataare collected at different times, the signal coefficientrealizations are different. This creates an inconsistencythat manifests itself as real, negative eigenvalues.Unfortunately, this characteristic prohibits applyingthe root-MVDR algorithm to the CCM. The MUSICalgorithm requires only the eigenvectors of thesmoothed CCM; therefore, the MUSIC algorithm isunaffected by negative eigenvalues. Once the CCM isformed, we apply the root-MUSIC algorithm in thesame way that we would for a traditionally formedSCM.

IV. PERFORMANCE AND ANALYSIS

In this section we evaluate the performance andbehavior of switched-element direction finding.The above algorithms are tested against single- andmultiple-signal cases. For multiple signals, bothcoherent and noncoherent cases are considered. Unlessotherwise noted, a linear array comprised of N = 8elements is used to estimate the DOA(s). In somesimulations, the array elements are uniformly spaced;in others they are sparsely spaced on a uniform grid.For the switched implementation, we assume thatthe system collects M = 2 channels of data at anygiven time. Therefore, there are Q =N(N ¡ 1)=2 = 28unique antenna pair combinations. We assume thatthe system switches through all 28 combinations andcollects an equal number of snapshots for each pair.In order to make a proper comparison, it is

necessary to define the relative amounts of datacollected by the full and switched implementations.Possible scenarios include collecting the same totalnumber of snapshots, the same total amount ofdata, or collecting the same number of snapshotson each subarray configuration as the full-arraysystem collects. When the switched and full-arrayimplementations collect the same total number ofsnapshots, the collection time interval is the same.Therefore, this comparison shows the performancesacrificed by collecting data on M <N antennassimultaneously.First, we compare the performance of the

switched-element system to a full, nonswitchedsystem for a single source. The source arrives fromfive degrees relative to the array normal. The systemcollects 20 snapshots for each antenna pair, resultingin 20 ¤ 28 = 560 total snapshots. The full arraysystem also collects 560 snapshots. Because theswitched-element system collects only two channelsof data rather than eight for each snapshot, the totalamount of data gathered by the switched-elementsystem is one-fourth of the data collected by the

Fig. 2. DF performance for full-array root-MUSIC,switched-element root-MVDR, and CCM root-MUSIC algorithms.

One signal impinging from 5 deg.

full-array system. We use the root-MUSIC algorithmfor the full-array data. Fig. 2 shows the mean-squarederror performance and CR bounds for the full-arrayand switched-element systems. Both switched-elementalgorithms perform similarly for a single source whilethe full-array system obviously performs the best dueto collecting the most data. The performance of thefull-array system approaches the CR bound at lowerSNR than the switched system and has smaller errorfor the same input SNR.Results in [16] show that the full and switched

systems perform similarly if the total data collectedis equal rather than the number of snapshots (in thiscase, for example, the equal-data equivalence wouldrequire the switched system to collect 80 snapshotsper antenna pair). In general, the switched systemcan achieve the same single-source performanceas the full-array system with an increase in datacollection time of approximately N=M. Consideringhardware costs, the time-performance tradeoff ofa switched-element system may be an acceptablecompromise.Next, we estimate the DOAs of two noncoherent

signals arriving from 10 and 55 deg. Because thesignals are noncoherent, we can use a nonredundant[18] linear array. In Fig. 3, we notice that theswitched-element root-MVDR algorithm outperformsthe CCM root-MUSIC algorithm (Figs. 3—6 showthe mean-squared error performance of locatingthe source at 10 deg in the presence of the othersources, not an average over the different sources).This pattern of performance has been tested overvarious multiple-source (noncoherent) scenarios withthe root-MVDR algorithm consistently outperformingCCM-based root music.Another interesting aspect seen in Fig. 3 is the

floor observed for the switched-element algorithms aswell as in the CR bound. This floor is a consequenceof the fact that the number of sources is equal to



Fig. 3. DF performance for switched-element root-MVDR andCCM root-MUSIC algorithms. Two noncoherent signals arriving

from 10 and 55 deg.

Fig. 4. Performance of CCM root-MUSIC for two noncoherentsources parameterized by number of snapshots.

the number of simultaneous receiving channels. Afull-array system has the same behavior, but it israrely seen because the number of signal sourcesis rarely equal to the number of antenna elements.One explanation for this behavior can be obtainedby considering the way in which training data andSNR affect estimation of the signal-plus-noise andnoise-only subspaces of a covariance matrix. At highSNR, the signal-plus-noise subspace is defined mostlyby the array manifold vectors and powers of thearriving sources. As SNR increases, it becomes easierto accurately define and separate the signal-plus-noiseand noise-only subspaces. The amount of trainingdata will still control how accurately the internalstructure of the signal-plus-noise subspace is estimated(and, therefore, the accuracy of DOA estimates), butthe span of the subspace will be accurate. Hence,performance constantly improves with increasingSNR, especially if the number of sources is muchsmaller than the size of the covariance matrix.

Fig. 5. DF performance for switched-element root-MVDR andCCM root-MUSIC algorithms, with two perfectly correlated

signals impinging from 10 and 55 deg.

Fig. 6. DF performance for switched-element root-MVDR andCCM root-MUSIC algorithms, with three perfectly correlated

signals impinging from ¡40, 10, and 55 deg.

When the number of sources is equal to the size ofthe covariance matrix, the noise-only subspace doesnot exist. Therefore, we only need to estimate thestructure of the matrix, but this depends strongly onthe amount of training data because it is necessary toaverage out signal amplitude fluctuations. Even if thenoise power is low, accurate angle estimation requiresaccurate estimation of the array manifold vectorsin the proper proportions of power. This estimationdepends on the amount of available training data, noton SNR. Hence, when the number of sources is equalto the size of the covariance matrix, performanceeventually achieves a floor where performance isflat with increasing SNR. The MUSIC algorithm,for example, relies on finding valid array manifoldvectors that are orthogonal to the noise-only subspace.If the noise-only subspace cannot be accuratelydefined, performance of the algorithm suffers. Sincethe switched-element system of Fig. 3 consists of



two-element subarrays, the performance floor canbe observed when only two signals are present. Ifthe floor is unacceptable, the number of elements ineach switching configuration must be increased. Fig. 4demonstrates the relationship between snapshots andthe performance floor for the CCM root-MUSICalgorithm.The switched-element root-MVDR algorithm

provides good results for noncoherent signals.However, we cannot effectively apply the switchedroot-MVDR algorithm to coherent signals. Thespatial smoothing technique is essential for theestimation of coherent signals and multiple identicalsubarrays are needed before we can implement thealgorithm. In the following simulations, we comparethe switched-element root-MVDR and CCM-basedroot-MUSIC algorithms using a 16-element uniformlinear array. Spatial smoothing is applied to the CCM.Fig. 5 shows simulated results for two perfectly

coherent signals arriving from 10 and 55 deg. Thespatially smoothed version of CCM root-MUSICclearly performs the best, which is not surprising sincethe switched-element root-MVDR algorithm operatesdirectly on 2£ 2 covariance matrices without spatialsmoothing.Next, we simulate three perfectly correlated

signals impinging from ¡40, 10, and 55 deg. Resultsshown in Fig. 6 indicate that CCM root-MUSICcontinues to provide better performance thanswitched-element root-MVDR. In comparing Fig. 6 toFig. 5, we see that mean-squared error has increasedsignificantly due to the presence of an additionalsource; however, it is encouraging that we are ableto obtain modest performance despite the presenceof three correlated signals and only two elements perswitch configuration.

V. CONCLUSIONS

We have proposed two algorithms for directionfinding with a switched-element system. Thetime-accuracy tradeoff is clear: we cannot achieve thesame performance as a full-array system for the samecollection time since the switched system collects lessdata. The advantage of a switched-element systemis that only M <N receiving channels are needed;therefore, reduced hardware cost could outweighthe sacrifice in performance. We have demonstratedthat well-known DF algorithms such as root-MVDR,root-MUSIC, and spatial smoothing can be adaptedto work with a switched-element system. Our resultsshow that switched-element root-MVDR performswell when the signal sources are uncorrelated. Onthe other hand, if coherent signals are present, theCCM-based root-MUSIC with spatial smoothingappears to be the best choice. As in traditionaldirection finding, performance can be improved bycollecting more data snapshots and/or by increasing

the size of the antenna array. For noncoherent sources,the size of the array can be increased by using anonredundant array, which integrates very well withthe switched-element approach.

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[15] Friedlander, B., and Weiss, A. J.Direction finding using spatial smoothing withinterpolated arrays.IEEE Transactions on Aerospace and Electronic Systems,28 (Apr. 1992), 574—587.

[16] Cooper, C. C.Performance of a switched element direction finding array.Master’s Thesis, The University of Arizona, Tucson, AZ,2006.

Wei Wu received the B.S. degree from Fudan University, Shanghai, China in2005, and the M.S. degree from the University of Arizona, Tucson, in 2007, bothin electrical engineering.From 2006—2007, he was a graduate research assistant in the Laboratory

for Sensor and Array Processing in the ECE Department at the Universityof Arizona. He is currently a DSP engineer with Aware, Inc., Bedford,Massachusetts.

Charles C. Cooper (M’08) received a B.S. in psychology, a B.S. in computerscience, and an M.S. in electrical engineering from the University of Arizona,Tucson, in 2002, 2003, and 2006, respectively.At the University of Arizona, he was a graduate research assistant for the

microelectronics design group from 2003 to 2004, and an active member of theCubesat satellite program from 2001 to 2006. Also in 2006, he was a graduateresearch assistant for the Laboratory for Sensor and Array Processing. Hehas worked for Rincon Research Corp. as a digital signal processing engineersince 2004, and is currently a doctoral student in electrical engineering at theUniversity of Arizona. His research interests include application of digital signalprocessing to RF source localization and array processing.

Nathan A. Goodman (S’98–M’02–SM’07) received the B.S., M.S., and Ph.D.degrees in electrical engineering from the University of Kansas, Lawrence, in1995, 1997, and 2002, respectively.From 1996 to 1998, he was an RF Systems Engineer for Texas Instruments,

Dallas, TX. From 1998 to 2002, he was a graduate research assistant in the RadarSystems and Remote Sensing Laboratory, University of Kansas. He is currentlyan associate professor in the Department of Electrical and Computer Engineering,University of Arizona, Tucson. Within the department, he directs the Laboratoryfor Sensor and Array Processing. His research interests are in radar and arraysignal processing.Dr. Goodman was awarded the Madison A. and Lila Self Graduate Fellowship

from the University of Kansas in 1998. He was also awarded the IEEE 2001International Geoscience and Remote Sensing Symposium Interactive SessionPrize Paper Award.

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[18] Johnson, D. H., and Dudgeon, D. E.Array Signal Processing: Concepts and Techniques.Upper Saddle River, NJ: Prentice-Hall, 1993.



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