interference cancellation and signal direction finding with low complexity

8/8/2019 Interference Cancellation and Signal Direction Finding With Low Complexity

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Interference Cancellation and

Signal Direction Finding with

Low Complexity

LANCE SCHMIEDER

DON MELLON

MOHAMMAD SAQUIB, Senior Member, IEEE

The University of Texas at Dallas

We propose a novel beamforming algorithm for a

three-element system that suppresses an interference signal while

still being able to measure a targets interferometer phases.

Unlike most direction-of-arrival (DOA) estimation algorithms, our

algorithm does not use a grid search. Instead the estimates result

from a closed-form solution, a great advantage in time-sensitive

applications. The derivation of the algorithm is presented, and its

statistical performance is examined with simulations. Additionally,

our numerical results demonstrate that our algorithm is capable

of achieving more reliable DOA estimates than those found with

the well-known multiple signal classification (MUSIC) algorithm.

Finally, a radar signal processing example is presented.

Manuscript received April 25, 2008; revised November 17, 2008;

released for publication February 27, 2009.

IEEE Log No. T-AES/46/3/937959.

Refereeing of this contribution was handled by R. Adve.

Authors addresses: L. Schmieder, Mustang Technology Group,

400 W. Bethany Dr., Suite 100, Allen, TX 75013-3714, E-mail:

([email protected]); D. Mellon, 136 County

Road 1998, Yantis, TX 75497; M. Saquib, Dept. of Electrical

Engineering, The University of Texas at Dallas, 2601 N. Floyd Rd.,

Richardson, TX 75083-0688.

0018-9251/10/$26.00 c 2010 IEEE

I. INTRODUCTION

A conventional technique of processing temporalsensor array measurements for signal estimation,interference suppression, or source direction andspectrum estimation is beamforming [13]. It has beenexploited in numerous applications (e.g., radar, sonar,wireless communications, speech processing, medicalimaging, radioastronomy).

The beamforming algorithm presented in thispaper is motivated by analyzing a low-cost radarsystem that provides wide spatial coverage and veryrapid target detection as well as tracking. Designingtowards these goals, a reasonable and mostly genericreceiver would employ a three-antenna receiver.Because the minimum number of sensing elementsneeded to determine two-dimensional (2D) angles isthree, the system cost has been mostly minimized.Furthermore, our antennas should emit very widebeams to eliminate the need for scanning the beamsthrough space and consequently improving the speedof our detections. We now consider the problem of

using our low-cost system to detect and estimate thedirection of arrival (DOA) of a desired signal in thepresence of a dominant interfering signal.

Considering existing algorithms, two approachesare generally taken to estimate desired signal DOAs inthe presence of interference. The first is an applicationof beamforming, and the second is a collectionof subspace-based and parametric methods. Thetraditional beamforming approach is to spatiallyfilter the interference and then make DOA estimatesbased on received power as a function of the beamslocation. However, using amplitude variation as ameans for DOA estimation would be a poor choice

because of our wide beams; also as previously stated,scanning is undesirable. These reasons lead us to seekalternative methods.

Subspace-based and parametric methods estimatethe DOA of every incident signal (interfering anddesired) by using mathematical properties to estimatethe steering vector of each signal. These methodsinclude such algorithms as Capon [4], multiple signalclassification (MUSIC) [5], maximum-likelihood(ML) estimation [6], propagation methods (PM) [7],and estimation of signal parameters via rotationalinvariance techniques (ESPRIT) [8]. Unfortunately,Capon, MUSIC, ML, and PM estimation require agrid search to estimate DOA. A grid search can betime consuming and is thus undesirable given ourstated priority of designing a low-cost rapid-responsesystem. (The root-MUSIC algorithm does not requirea grid search, but it can only used with a uniformlinear array and hence is only be able to estimateangles along a single dimension.) ESPRIT is capableof yielding estimates from closed-form equations,but with a three-element array, it can only be usedto estimate the DOA of a lone signal. If a powerful

1052 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 3 JULY 2010


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jamming signal is present, ESPRIT would not beable to estimate the DOA of any additional desirablesignal. Thus, it is not usable in the problem we aretrying to solve [9].

Despite the stated disadvantages, MUSIC is theexisting algorithm that seems best suited for oursystem. Since it is usable with our simple hardwarestructure, MUSIC is more statistically efficient thanCapon and offers a much reduced computational

burden as compared with ML estimation [6]. Becauseof its popularity in literature, an analytical andnumerical comparison of MUSIC and our algorithmwill be given later.

The rest of the paper is organized as follows.In Section I, first, we give a full description ofour algorithm, starting with the system model andcontinuing with a tabular list of algorithm steps.Next, we proceed with a detailed description on ourmethodology for interference cancellation, targetdetection, and phase angle estimation. Afterwards, weanalytically identify the spatial scenarios of a jammerand a target in which the proposed technique will

reliably estimate a targets DOA. A similar analysisof MUSIC is given that suggests the two algorithmswill exhibit the same spatial dependency. Next, inSection III the statistical performance of the algorithmis examined through a collection of simulations. InSection IV, we present a brief radar signal processingexample to demonstrate how our proposed algorithmmight be used in practice. Finally, Section V containsthe conclusions of this work.

II. SYSTEM MODEL

Three antennas in an arbitrary geometry make upour receiver structure. The received signal at the ithelement at time n, is denoted by xi(n) and is formedfrom the coherent addition of the target signal ti(n),the jammer signal ui(n), and the noise vi(n). Therefore,

xi(n) = ti(n) + ui(n) + vi(n), i = 1,2, 3: (1)

Assuming point sources and equal gains for the threereceivers, the target and interfering signal at eachsensor will be phased replicas [10]. We also assumenarrowband signals, which means that the relativephases of the received signals will be constant acrossthe entire band. The target signals are modeled as

t2(n) = t1(n)ej, t3(n) = t1(n)e

j (2)

wheret1(n) = (n)e

j(n) (3)

and the interfering signals are

u2(n) = u1(n)ej, u3(n) = u1(n)e

j (4)

whereu1(n) = (n)e

j(n): (5)

TABLE I

Algorithm Overview

Step 1: Find beamforming weights that minimize the jammers

power.

Step 2: Apply threshold detection to the beamformer outputs

of each range-Doppler bin of interest.

Step 2a: If a target is detected, record its range and Dopplerand proceed to Step 3.

Step 2b: If no target is detected, start over with the nextcoherent processing interval (CPI).

Step 3: Estimate relative phase information for each detectedtarget.

Step 4: Calculate DOAs from the phase information.

The variables (n) and (n) respectively denotethe amplitude and the time-varying phase of the targetat antenna 1, while and denote the relative phaseangles at antenna 2 and 3. In a similar manner, theparameters (n), (n), , and denote the amplitude,time-varying phase, and electrical phase angles of the

jamming signal. The noise, vi(n) is a white zero-meancomplex random variable with variance 2 and is

uncorrelated with vm(n) for i 6= m. All Greek lettervariables represent real numbers.

We now give an overview of our algorithm whichdoes not fit either of the paradigms introduced above,i.e., we do not scan a narrow beam nor do we use aparametric method to estimate the steering vectorsof all present source signals. Throughout the rest ofthis paper, we refer to the desired signal as a targetsignal because this approach has been motivatedfrom the signal processing needs of a radar system.We have also chosen to use a noise jammer for theinterference source because of the ease at which onecan be simulated, but application need not be limited

to this case. The algorithm steps are enumerated inTable I.

Before proceeding to the algorithm details,we make some additional comments. In regardsto Step 3, in the weighted sum, the jammerspower is negligible, but the targets directionalphase information is distorted. For this reasonthe phase information has to be recovered in anovel manner. The recovery is accomplished byarranging those sums in a phasor diagram andexploiting the problems natural symmetry. At Step4, the DOAs of multiple targets can be estimated,sequentially, if those targets are resolvable in rangeor Doppler. Our approach is philosophically similarto [11] the Applebaum and Wasiewicz algorithmfor jammer suppression and target DOA estimationin monopulse systems. Like [11], instead of usingbeamforming in the traditional fashion, to maximizethe signal-to-interference-plus-noise ratio (SINR),beamforming is used to null a jamming signal.Nulling the jammer enables a reduced-complexitymathematical technique for estimating targetsignal parameters. Unlike [11], we employ phase

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interferometry and require one less receiver channel.Adapting a beam based solely on informationabout an interference signal is a common attributeamong processing techniques that attempt to reducecomputational complexity [12].

A. Interference Cancellation

If a weighted sum of the received signals is

formed, it is possible to choose non-zero, equalmagnitude weights that completely cancel, or nullthe jammer signals. The importance of the weightsbeing non-zero is obvious because we still desireto detect the target. The importance of the weightsbeing equal in magnitude is discussed later in thepaper. The time indices n representing sample timeshave been dropped below. For convenience, weintroduce the paired received signal vector xi,m andthe corresponding weight vector wi,m for antenna i

and m:1

xi,m = [xi,xm]>, i = 1,2,3 and i 6= m (6)

wi,m =1p

2[1, ej! i,m ]>: (7)

To clarify our nomenclature, each entry in a weightvector is referred to as a weight. Thus, in (7)the first weight is 1=

p2, and the second weight is

ej!i,m =p

2. As mentioned above, these two weights arereadily seen to have equal magnitudes.

For an example of cancellation, we take thesummation of antennas 1 and 2, where r1,2 is theoutput of the beamformer,

r1,2 = w>1,2x1,2 =

1

p2x1 +

1

p2x2e

j!1,2 : (8)

Notice that if !1,2 = , then u1 + u2ej!1,2 = 0, andthe jammer has been completely canceled. Simplifyingwe get

r1,2 =1p

2(t1 + v1) +

1p2

(t2 + v2)ej!1,2 : (9)

Effectively, ej!1,2 has advanced the phase of the signalat antenna 2 so that the jammer signals add together180 out of phase. Analogous weighted sums canbe formed between antennas 1 and 3 and betweenantennas 2 and 3.

Estimating the Beamforming Weights: Manytechniques exist for estimating the weight vectors,and we implement eigenvector weighting because itwas used with much success in [11]. The samplesused in estimating the weights are assumed tocontain no target signal of significant strength. If theinterference is from a noise jammer, these samples

1In this work, the notation X> is used as the transpose of the matrixX, and X denotes the hermitian of the matrix X.

can usually be gathered from range-Doppler binscorresponding to the furthest ranges or the largestDoppler frequencies. If the interference changesits DOA as it changes range or Doppler, a slidingwindow is more appropriate for sample selection[13, 14].

Below, training samples at the ith antenna arespecified by si, where

si

= ui

+ vi, i = 1, 2,3 (10)

and pairing the samples from antennas i and m, whereagain i 6= m, we define a 21 vector

si,m = [si,sm]>: (11)

For one weight pair, a 22 cross-correlationmatrix is formed for the received signals of twoantennas. The cross-correlation matrix for si,m isdenoted by

Ri,m = Efsi,msi,mg (12)and the above ensemble average is estimated with theaverage of N

tnoisy snapshots by the calculation

Ri,m =1

Nt

NtXk=1

ski,mski,m: (13)

It can be shown that the complex conjugate of theeigenvector corresponding to the minimum eigenvalueof Ri,m is the desired set of beamforming weights forantennas i and m [15]. The conjugation is a result ofdefinitions in (7) and (9). If the jammer remains at aconstant angular location, as the limit of Nt goes toinfinity, the eigenvector weights will converge to aset of optimum weights that completely eliminate the

jammer signal after processing, i.e.,

w>i,mRi,mwi,m =

2: (14)

However, we note that with finite Nt, theeigenvector weights will have some amplitude erroras well as phase error. A simple modification is tomeasure the relative phase between the elementsof the eigenvector conjugate and then reset themagnitude of each element. This modification isadvantageous because of our system model, and theunderlying assumption of receiver calibration. Becausethe jammer powers for any particular snapshotare equal in magnitude at each antenna, the only

quantities that need to be estimated are the phases!1,2, !1,3, and !2,3.

For example, if eigenvalue decomposition resultsin a vector as wi,m = [a,b]

>, then !i,m is calculated as

!i,m = 6a 6b (15)and the weighting vector that we use becomes

wi,m =1p

2[1,ej!i,m ]>: (16)



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For our three-antenna receiver, the processof averaging jammer phase angles or performingeigenvalue decomposition (EVD) only needs to becarried out twice because any third weight vector canbe calculated from the first two. From (4) and (9), weknow that the optimal !1,2 = , and the optimal!1,3 = . It can also be shown that, in order tocancel the jammer from the beamformer output ofantennas 2 and 3,

!2,3 = + : (17)By combining the equations for each of the weightphase angles we arrive at the relationship

!2,3 = !1,3 !1,2 + : (18)

B. Signal Detection

After the training snapshots are collected andthe weights have been estimated, they are used tosearch for a target signal within the jammer-dominatedreceived signal. Multiplying the weighting vectors

with their respective received signal vectors resultsin the residue power detection variable

p =1

2

3Xm=2

jw>1,mx1,mj2 =1

2

3Xm=2

jr1,mj2: (19)

This particular definition of p was chosen inanticipation of using the L-shaped array described in alater section.

If no target signal is present, the equation for pcan be simplified as

p =1

2

3

Xm=2

1

p2(v1 + vme

j!1,m )2

=1

2

3Xm=2

1

2(v1 + vme

j!1,m )(v1 + vmej!1,m )2

:

(20)

Next, by taking the expectation of both sides, we findthat

Efpg = 14

3Xm=2

Efv21 + v2mg =1

4

3Xm=2

22 = 2: (21)

Therefore, if a target signal is not present inthe received signals and the weight vectors arenormalized, then the average residue power will beequal to the average noise power in the channels. Ifa target signal is present, then p will be larger andcan be used for detection purposes without havingto adjust constant false alarm rate (CFAR) settings.How much larger p becomes depends on the originaltarget signal power and on the spatial angle separationbetween the target and jammer signals. Weighted sumsin which the jammer has been canceled can be madefrom groups or from individual range-Doppler cells

Fig. 1. Vector diagram showing derivation of angle of r1,2(i.e., 6r1,2) in (24).

over any map region of interest. Using (19) an entirecleaned range-Doppler map can be created fromvalues of p. An example on target detection is givenin Section IV.

Depending on antenna geometry or systemrequirements, alternate definitions for a detectionvariable might be useful. One example would be touse p = 13 (

jr1,2

j2 +

jr1,3

j2 +

jr2,3

j2). Another possible

detection criteria would be to use an m-of-n thresholddetection on the three beamformer outputs, e.g.,2-of-3 jri,mj2 should exceed the threshold to declarea detection.

C. Phase Angle Estimation

The DOA can be calculated from the difference intarget signal phase at the three antennas. Referring toprevious equations, we need to estimate and . Byusing (2), (3), and (9) and by assuming the jammersignal is canceled and the noise negligible, we get aset of three equations:

r1,2 =p

2ej(1 + ej(+!1,2))

r1,3 =p

2ej(1 + ej(+!1,3))

r2,3 =p

2ej(ej + ej(+!2,3)):

(22)

In the above set of equations we have fourunknowns , , , and . We can rewrite theseequations in phasor form using r1,2 as an example:

jr

1,2j6r

1,2=

p2[6

ej +6

ej(++!1,2)]: (23)

Here we see the importance of equal magnitudeweights as mentioned in Section IIA. Becausethe components of the weight vector are of equalmagnitude, r1,2 bisects the angle subtended bythe target phasors shown in Fig. 1. The bisectionallows us to write an equation that only involves theunknown phases:

26r1,2 = + ( + + !1,2): (24)



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From (24) we define variable '1 as

'1 = 26r1,2 !1,2 = + 2: (25)Note that the quantity '1 is known, whereas thequantities on the right-most side of the above equationare unknowns. The final two weighted sum equationsfor r1,3 and r2,3 can also be reduced to relationshipsinvolving only the phase angles. These operationsrespectively yield

'2 = + 2 (26)

'3 = + + 2: (27)

Finally, using (25), (26), and (27), we obtain and as

= '3 '2, = '3 '1:Once values for and are found, it isstraightforward to obtain estimates for and ifdesired.

D. Effects from the Angular Separation of Signals

The DOA estimates will be reliable in a noisyenvironment if the phase terms 6ri,m are primarilydetermined by the target signal and not by noise.Because the average noise is unchanged by thebeamformers, it is informative to look at howthe beamformers affect target signal strength.Therefore if we again neglect noise, jr1,2j is the gainexperienced by the target signal in the beamformerfrom antennas 1 and 2. Starting with (22) and usingEulers identity and trigonometric identities, we canwrite

jr1,2j = p2 cos + !1,2

2

: (28)

Substituting !1,2 = , we can rewrite this as

jr1,2j =p

2

sin

2

: (29)

Similar to (29), we obtain the magnitudes of theoutputs of the other two beamformers as

jr1,3j =p

2

sin

2

(30)

jr2,3

j=p

2 sin + +

2 : (31)

Inspection of the above equations show that asthe jammers phase angle approaches the targets phase angle, the magnitude of r1,2 goes to zero.As this happens, the phase variable '1 becomesdominated by noise and the variance of estimationgrows. At the limit when = , cannot be estimatedat all. Similarly, as the jammers phase angleapproaches the targets phase angle, the varianceof the estimation grows. When the magnitude of

r2,3 becomes small, the variances of both target phaseestimations grow.

E. Analytical Comparison with MUSIC

Above we stated that MUSIC is the existingalgorithm that seems most suitable for our target DOAestimation scenario, for this reason, it is importantand interesting to compare their performances.

Additionally, MUSIC has been shown for multiplesource cases to be asymptotically efficient when theSNR of all sources tends towards infinity [16]. In [5],the estimates for azimuth and elevation are given asthe value of azimuth (AZ) and elevation (EL) thatmaximizes the quantity in (32), where a is the steeringvector and EN is a matrix defined as having columnsmade up of eigenvectors in the noise subspace:

Pmu(AZ, EL) =1

a(AZ, EL)ENENa(AZ, EL)

:

(32)

We now analyze (32) for our target and jammerscenario, rewriting (2) and (4) in steering vector formas

t = t1[1, ej,ej]>, u = u1[1, e

j ,ej ]>: (33)

Matrix EN has only one column, and, by the proofprovided in [5], will be orthogonal to t and u. Anequation for EN can then be found by taking the crossproduct of the vectors. We derive this cross product(u t) as

EN = C[ejej ejej ,ej ej,ej ej]>

(34)

where C is a constant. The ability of a grid searchto yield two distinct peaks and thus the DOAestimations for the jammer and the target depends onthe magnitude of the entries of the vector EN. For afinite number of snapshots N and finite SNRs, smallentry magnitudes will manifest themselves in blurredpeaks in the grid search, which means less reliableestimates. After applying mathematical identities,those entry magnitudes are

jEN(1,1)j = 2C

sin

+ + 2

jEN(2,1)j = 2C

sin

2

jEN(3,1)j = 2Csin

2

:

(35)

Remarkably, other than an arbitrary scaling factor,the magnitude of the EN entries are equivalent to ouralgorithms beamformer gains that are listed in (29),(30), and (31). Whenever an entry of EN is zero, thegrid search will not yield unique peaks. We see that



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this occurs by inspecting (32). If an entry of EN iszero, any particular inner product with that vector canbe achieved by an infinite number of steering vectors.Therefore, even at MUSICs asymptotic limit, if = , an EL estimate cannot be made; if = , an AZestimate cannot be made; and if + + = 0,neither an AZ nor an EL estimate can be made.

From the above analysis, one would expect theproposed algorithm and MUSIC to behave similarly

for large numbers of snapshots. Unfortunately, itis generally very difficult to analytically trace theperformance of a DOA estimation algorithm formulti-source signal cases [10, 16, 17]. In order tocompare the performance of our proposed algorithmand MUSIC as a function of snapshots, we haveinstead conducted a simulation which is described inSection III.

It is difficult to directly compare the computationalcomplexity of our algorithm and MUSIC becauseof the various engineering decisions that must bemade before applying either. Consequently, fromour algorithm we omit from this comparison the

computations required to estimate the beamformingweights, and from MUSIC, we omit the computationsrequired to estimate the 3 3 cross-correlationmatrix, the subsequent EVD, and the selectionof the correct Pmu peak. In other words, for ouralgorithm we consider here only the computationsrequired in calculating the beamformer outputs andthe phase angle estimation. For MUSIC we consideronly the grid search of (32). In the descriptionthat follows the multiplication of two complexnumbers involves four real multiplications, andthe addition of two complex numbers two realadditions. Divisions are counted as multiplications,

and subtractions are counted as additions. Weassume that measuring the phase angle of acomplex number can be done inexpensivelythrough the use of a look-up table and that thecomplexity involved in multiplying a number by 2is negligible.

Averaging our phase angle estimates fromN jammer-plus-target-plus-noise snapshots, ouralgorithm requires 24N+ 1 real multiplications,12N1 real additions. For MUSIC, each grid pointrequires 16 real multiplications and 4 real additions.We now compare the complexity of the phase angleestimation in our algorithm with the complexity ofa MUSIC grid search in an example. If we use atypical N value of 5, our algorithm requires 121real multiplications and 59 real additions. For theMUSIC grid search we use points with 1 of precisionfrom 50 to 50 AZ and 50 to 50 EL. Thisexample grid has 10,201 points, resulting in a total of163,216 real multiplications and 40,804 real additions.Therefore, even though we limited the search span tobe much smaller than that of a full hemisphere andused a coarse step size of 1, MUSICs search in this

Fig. 2. Array geometry used in simulations.

case is more complex than our phase angle estimation

method by a factor of 1,348.

F. Application to an L-Shaped Array

Although our algorithm can be applied to generalarray shapes, to further analyze its properties we nowconsider a highly utilized array shape, the L-shapedarray. The L-shaped array is popular because it can beused to estimate DOA angles in two dimensions. Also,AZ and EL angles can be defined such that spatialangle only depends on a single electrical phase angle.Here we designate the relative signal phases at sensor1 and 2 to measure AZ, and the relative signal phases

at sensor 1 and 3 to measure EL, i.e.,

AZ = arcsin

, EL = arcsin

(36)

where the distance from antennas 1 to 2 and from 1to 3 is a half-wavelength. Thus, antenna 1 is at thecorner of the L-shape and the point 0 AZ and 0 ELcorresponds to the boresight of the receiver array. Thearray structure is depicted in Fig. 2.

With a half-wavelength spacing and traditionalinterferometry, relative signal phases would measurebetween . However, when using the proposedalgorithm, and will not fall within this range in

general. Therefore, before applying (36), and mustfirst be appropriately unwrapped.

III. NUMERICAL RESULTS

Now, we present a set of numerical results whichare obtained from simulations. All of the simulationswere performed with the L-shaped array describedabove. First, the algorithms statistical performancefor estimating target DOA is compared with thebaseline interferometer performance when onlya target signal is present, i.e., no jamming. Next,we revisit the issue of jammer-to-target angularseparation and demonstrate how the separation canaffect performance. Last, our algorithms performanceis compared with that of the well-known MUSICalgorithm.

A. Simulation 1

The target was placed at 0 AZ, 0 EL. Thejammer signal, simulated as a white noise signal,was placed at 22 AZ, 55 EL. The jammer-to-noise



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Fig. 3. Estimation variances for simulation 1.

ratio (JNR) was set to 40 dB, and 20 snapshots wereused to estimate the three cross-correlation matrices.After canceling weights were found, the JNR waskept at 40 dB, and a single snapshot was taken that

contained jammer-plus-target-plus-noise samples.The simulation was carried out 5000 times for targetSNR values ranging from 10 to 30 dB. The resultingDOA estimations were unbiased, and their variancesare plotted in Fig. 3. On the same set of axes, theperformance of traditional interferometry is plotted forthe case when there is no jammer. With no jammer,using traditional interferometry, the AZ estimationperforms identically to the EL estimation becausethe target is at boresight. For all the estimates, thevariance estimates decrease as the SNR increases.The EL estimation performance is affected moreadversely by the jammer cancelation procedure

because jr1,2j is smaller than jr1,3j. The cost of thejammer cancellation on the target DOA estimationscan be thought of in terms of additional SNR requiredto achieve the same performance. With the parametervalues simulated here, the AZ estimation requiredapproximately 1 additional dB, and the EL estimationrequired approximately 5 additional dB to achievethe target-only performance. For low-cost radarsystems, typical single bin detection threshold levelsrange from 10 to 15 dB above the noise floor. Hence,the plot shows that the proposed algorithm followsthe same SNR performance trends as traditionalinterferometry, regardless of whether that SNR

happens to be low or high.

B. Simulation 2

A target signal was simulated with an SNR of25 dB and was again placed at 0 AZ, 0 EL. Fivesnapshots that contained target signal were used tomake the final angle estimates. The jammers ELangle was set to 30 while its AZ angle is sweptthrough space. As was done in Simulation 1, each

Fig. 4. Average jri,mj= for simulation 2.


experiment was performed 5000 times so thataverage performances could be studied. As expected,regardless of the jammers position, each of theDOA estimations remained unbiased. However,the estimation variances show a strong dependenceon the separation angles. In Fig. 4 the averagemagnitude of the three beamformer outputs, scaledby 1=, are plotted as a function of jammer-to-targetseparation in AZ. The jr1,3j gain, which is a functionof jammer-to-target EL separation is constant forthis set of simulation experiments. Fig. 5 shows theresulting DOA estimation variances. As predicted inour analysis, the targets EL estimate becomes poor

when the AZ separation is small, and both estimatesare poor, around 30 of separation when jr2,3j becomessmall in magnitude.

C. Simulation 3

In this simulation, we compare the performanceof the proposed algorithm and MUSIC as a functionof snapshots N. The target was moved to 7 AZ,25 EL with an SNR of 20 dB. The jammer



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signal was placed at 46 AZ, 15 EL with a JNR of50 dB. Twenty snapshots were utilized to estimatethe jammer covariance matrices and the cancelingweights. The targets DOA was then estimated using

target-plus-jammer-plus-noise snapshots, while thenumber of these snapshots was varied from 1 to 15.For the cases when the number of snapshots wasgreater than 1, we averaged the estimates for therelative phases, and , after first properly wrappingthose values into the range. Then the averagesfor and were used to calculate AZ and EL.The same weighting was used for each snapshot ofthat particular simulation iteration. The simulationwas carried out 1000 times, and the resulting DOAestimation variances are plotted in Fig. 6. Anautomatic minimization routine was used to obtainthe MUSIC estimates instead of a grid search so that

the grid step size would not affect the simulationsstatistics. Our algorithm outperforms MUSIC whena small number of snapshots containing the targetsignal are available. Furthermore, MUSIC cannot beused with only a single snapshot. As expected, theperformance of the two algorithms begins to convergeat larger numbers of snapshots.

D. Simulation 4

In the analysis of the preceding section, weshowed that the relative AZ and EL of the jammerand target incident signals affects the DOA estimationperformance of MUSIC in a similar way to how itaffects our algorithm. Although performance trendscould be identified, a full analytical descriptionof MUSICs performance cannot be found [10].As a result, it seems useful to further the MUSICcomparison with another simulation. The target iskept at 7 AZ, 25 EL with an SNR of 20 dB.The jammer signal was simulated with a constant15 EL, while its AZ was swept across space.Twenty snapshots of the jammer at 50 dB JNR were


again used to estimate the beamforming weights.In Simulation 3, 15 target-plus-jammer-plus-noisesnapshots resulted in nearly equivalent averageperformance of the two algorithms, so 15 snapshots

were used here as well. The estimation variancesare plotted in Fig. 7 after 1000 iterations. Theperformance of the two algorithms is quite similar.For the scenarios where our algorithm is relatively lessreliable, MUSIC is also. The results of Simulation 3and Simulation 4 suggest that, while being a powerfulalgorithm and useful for complex multi-signalscenarios, MUSICs computational complexity isunwarranted in the problem on which we are focused.

IV. RADAR EXAMPLE

For this example, a CW radar is simulated with

a 10 GHz center frequency. The radar samples 256times per CPI at a rate of 51.2 kHz, which results in aCPI of 5 ms. CW was chosen for convenience, but allof the below steps could be used with a pulse-Dopplersystem. Using standard equations from [18], we findthat the maximum unambiguous range-rate, or relativevelocity, is 384 m/s, and the range-rate resolution is3 m/s.

Reflections from two targets are simulated. Thefirst target is closing at rate of 121 m/s (or 121 m/s)from the direction of51 AZ and 8 EL, and thesecond target is opening at a rate of 86 m/s (or+86 m/s) from the direction of 19 AZ and

30 EL.

White Gaussian noise (WGN), uncorrelated fromantenna to antenna, is added to the received signalsuch that the noise floor will correspond to 0 dB onthe generated radar maps. With a JNR of 40 dB, the

jamming signal is simulated at a location of 46 AZand 37 EL relative to the receiver. The resultingmap from a single CPI is shown after Dopplerprocessing in Fig. 8, and we notice that withoutfurther work, only the jamming signal can be seen.The target signals we are trying to detect and track are



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Fig. 8. Resultant Doppler data from one CPI.

Fig. 9. Target detections after jammer cancellation.

completely hidden, but are plotted on the same axisfor reference. We now apply our proposed algorithm.

Step 1 The beamforming weights areestimated from a set of snapshots that contain only

jammer-plus-noise information. Let us assume thatno targets of interest are expected to reach velocitiesup to 300 m/s. Using snapshots from bins less than300 m/s and greater than 300 m/s, 56 trainingsnapshots are thus provided.

The eigenvalues that are calculated as a byproductof our weight estimation are also useful. Thedifferences in the minimum eigenvalues and maximumeigenvalues of this example are approximately 43 dB,which is consistent with the JNR defined above. Theextra 3 dB results from the possibility that, if desired,the jamming signal could be added perfectly in phaseby the beamformer, thereby doubling its power. Thisspread in eigenvalues can be used by a radar analystto determine that the receiver is, in fact, being jammedand to predict that roughly 40 dB of suppression canbe expected from applying the beamforming weights.

Step 2 After the beamforming weights are found,they are used to create a new radar map where the

Fig. 10. Beamformer output jr1,2j= as function of signal AZ.

Fig. 11. Beamformer output jr1,3j= as function of signal EL.

Fig. 12. Beamformer output jr2,3j= as function of signalAZ and EL.

jammer has been cancelled. For each range-rate binfrom 300 m/s to 300 m/s, the detection variable pis calculated as in (19), converted to dB scale, andreplotted in Fig. 9. A detection threshold of 15 dBabove the noise floor is chosen, resulting in detections



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Fig. 13. Detection variable, p, as a function of signal

AZ and EL.

in seven bins. As seen in the figure, those detectionscan be easily associated into two targets. Taking thetwo peak detections as the target relative velocities,target 1 is closing at approximately 120 m/s and target

2 is opening at 87 m/s. Both estimated values arewithin the range-rate resolution of our system.

The values for jri,mj= are plotted in Figs. 1012.The plots are generated from (29), (30), and (31).Whereas the jammers relative phase angles and are held constant based on its AZ and EL definedabove, the targets relative phase angles and areswept from to . Then for viewing purposes, therelative phase angles are mapped to AZ and EL. Toadd clarity, the angular locations of the jammer, target1 and target 2 are marked on the curves (or surfacefor Fig. 12) by the terms J, T1, and T2, respectively.By design of the beamformers, the jammer is located

in the nulls of the plots. The detection beam shown inFig. 13 is obtained from averaging jr1,2j= and jr1,3j=according to (19).

Step 3 Because targets have been detected,we now wish to estimate the target signals DOA.From bins where detections were made, the complexquantities r1,2, r1,3, and r2,3 are used as inputs into(25), (26), and (36).

Step 4 For this simulation iteration, the DOA oftarget 1 is estimated at 50:4 AZ and 8:2 EL, andtarget 2 is estimated at 17:8 AZ and 30:1 EL. Bothof these estimates are close to the true values.

V. CONCLUSIONS

While in the presence of a dominant interferencesource, our proposed algorithm yields unbiasedtarget DOA estimates from a low-cost, three-elementreceiver. We also mathematically identified thespatial scenarios where those estimates will have lowvariances. Unlike most DOA estimation methods, ourestimates are found from closed-form expressions.In contrast to MUSIC, our algorithm performs

well even when the number of target-containingsnapshots available is small. This property makes itattractive for use in post-Doppler processing where itis common for a target signal to straddle only a fewrange-Doppler bins. The DOAs of multiple targetscan be estimated from one CPI as long as those targetsignals are resolvable in range or Doppler.

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Lance Schmieder received a B.S.E.E. from the University of Tennessee inKnoxville in 2005 and was recognized as a Top Collegiate Scholar for theCollege of Engineering. In 2008, he completed his M.S.E.E. at the University ofTexas at Dallas (UTD).

He was a Jonsson Fellowship recipient at UTD and a researcher in the

Wireless Communications Research Laboratory. Currently, he is an analyst withthe Mustang Technology Group in Allen, TX, where his primary duties involvesignal processing, system simulation, and algorithm development.

Mr. Schmieder is a member of the Tau Beta Pi and Eta Kappa Nu honorsocieties.

Donald Mellon received an advanced degree is in solid state physics from Iowa

State University in 1970.He is currently an independent contractor supporting radar development

programs at Mustang Technology Group in Allen, TX. Dr. Mellon has over30 years experience as an analyst and systems engineer on advanced radarand communication systems obtained primarily in Texas Instruments DefenseElectronics organization.

Mohammad Saquib (SM09) received the B.Sc. degree (1991) in electricaland electronics engineering from Bangladesh University of Engineering &Technology, Bangladesh. He received the M.S. (1995) and the Ph.D. (1998)degrees in electrical engineering from Rutgers University, New Brunswick, NJ,where he was a graduate research assistant in the Wireless Information Networks

Laboratory (WINLAB).He worked as a system analyst (19911992) at the Energy Research

Corporation, Danbury, CT. From 1998 to 1999, he was with the MIT LincolnLaboratory, Lexington, MA, as a member of the technical staff. In January 1999,he joined the Electrical and Computer Engineering Department at LouisianaState University, where he was the Donald Ceil & Elaine T. Delaune EndowedAssistant Professor. Since July 2000, he has been with the Electrical EngineeringDepartment at the University of Texas at Dallas, where presently he is anassociate professor. His research interests include various aspects of wirelessdata transmission including system modeling and performance, signal processing,and radio resource management, with emphasis on open-access techniques forspectrum sharing. His research interests also include designing signal processingtechniques for low-cost radar and medical applications.

Professor Saquib served on the editorial board of the IEEE Transactions onWireless Communications from 20042009 and that of the IEEE Communications

Letters from 20022008. He received the Best Teaching Award for excellencein teaching Electrical Engineering and Telecommunications Classes from theDean of the School of Engineering, the University of Texas at Dallas, 20022003.He was a corecipient of the best Paper Award in International Test SynthesisWorkshop (ITSW), 2007, for the paper A Robust Interconnect Mechanism forNanometer VLSI. He coauthored the paper Signal Direction Finding with LowComplexity which received the best student paper award in IEEE InternationalWaveform Diversity and Design Conference, 2009.


interference cancellation and signal direction finding with low complexity

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