MIMO Radar Direction Finding Performance UsingSwerling ModelTuomas Aittomaki and Visa Koivunen

Signal Processing Laboratory, SMARAD CoEHelsinki University of Technology

P.O. Box 3000, FI-02015 HUT, FinlandEmail: taittoma@wooster.hut.fi, visa@wooster.hut.fi

Abstract—A MIMO radar uses multiple waveforms and otherforms of diversity to improve the performance. A statisticalMIMO radar that takes advantage of spatial diversity has beenproposed earlier. This radar concept aims at mitigating theperformance loss caused by fluctuations of the radio cross-sectionof the target.In this paper, we examine the direction finding performance

of the statistical MIMO radar using the Swerling scatteringmodel, which covers both fast and slow fluctuation of the targetRCS. The direction finding performance has been studied before,but so far, the different cases of the Swerling model have notbeen considered. We derive the Cramer–Rao bound or a similarperformance measure for each case. Numerical results are alsoprovided.Index Terms—MIMO systems, radar, array signal processing

I. INTRODUCTIONMultiple-input multiple-output radar is a concept in which

multiple waveforms and other diversity methods are used inthe radar to improve the performance. The motivation behindthe MIMO radar is the rapid progress that has been made inthe field of MIMO communication systems. However, due todifferent nature of radar compared to communication systems,entirely different methods are needed to exploit diversity inMIMO radars.Various methods to have been proposed that aim at improv-

ing the performance of radar by taking advantage of multiplewaveforms or other types of diversity. Transmit beamformingusing the waveform diversity has been studied, for example,in [1]–[3]. Another approach proposed in [4] is the statisticalMIMO radar which uses angular diversity in addition tomultiple waveforms.The idea of the statistical MIMO radar is to reduce the

impact of fluctuations of the target radar cross-section (RCS).These fluctuations cause fading of the received signal whichdegrades the performance of the radar. It has been suggestedthat by introducing angular diversity so that the target isseen from several angles simultaneously, the performancedegradation caused by scintillations could be mitigated[4], [5].This can be achieved by employing several transmitters orreceivers placed in separate locations.Although scattering is a deterministic phenomenon, the

rapid fluctuations that result even from small changes of the

This work was supported by the Finnish Defence Forces Technical ResearchCentre and Academy of Finland, Center of Excellence program

angle cause the fluctuations to appear random. In addition, thenumber of individual scatterers in a typical target is so largethat a deterministic model is unfeasible. Therefore, statisticalmethods can be used to model and analyze the impact of thefluctuations. In the radar community, the Swerling models arecommonly used[6].The direction finding performance of the statistical MIMO

radar has been studied in [4] and [5], but these papers useda simple scattering model. In this paper, we investigate thedirection finding performance of the statistical MIMO radarunder Swerling cases 1–4. The Cramer–Rao bound or a similarperformance measure is derived for each model, and numericalresults supporting the analysis are provided. It will be shownthat with the assumed signal and scattering models, the statis-tical MIMO radar does not provide improved direction findingperformance if the fluctuation of the RCS is fast. However, itis possible that the performance improves in other scenarios.Moreover, it has been shown that other performance gains canbe achieved with the MIMO radar concept, including improvedresolution[7] and parameter estimation[2].The rest of the paper is organized as follows: Section II

describes the signal model and the assumptions of the Swerlingmodels. The direction finding performance is analyzed inSection III, and the numerical simulation results are given inSection IV. Conclusions are discussed in Section V.

II. SIGNAL MODEL

We adopt a signal model similar to the model in [8]. Ina MIMO radar configuration with M transmitters and Nreceivers, the received array snapshot can be written as

r(t) =

√P

Mdiag(b)Cdiag(a∗)s(t− τ) + n(t), (1)

where diag(b) denotes a diagonal matrix with the elementsof the vector b on the diagonal. P is a power parameter thatdepends on transmit power, path losses, antenna gains etc. Weassume P to be same for all receiver elements and independentof the direction of arrival. The transmitter and receiver steeringvectors are denoted by a and b, respectively. The scatteringmatrix C models fading of the scattered signal caused by thefluctuation of the RCS. The transmitted signal s and receivernoise n are assumed to be zero-mean.

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The fluctuations of the RCS are modeled as a randomprocess, and the probability distributions of the elements of thescattering matrix C depend on the assumed Swerling model aswell as the spatial positioning of the receiver and transmitterelements. Next, we focus on the correlation of the elementsof the scattering matrix C mainly following the discussion in[8].If the transmitter and receiver elements are closely spaced,

all the signals experience similar scattering, which can then bemodeled with a single scattering parameter c. All the elementsof C are then equal, i.e. C = c1M×N . The received signal is

rSISO(t) =

√P

McbaHs(t− τ) + n(t). (2)

Therefore, this would be a SISO system from the scatteringpoint of view.If the transmitter does beamforming, the transmitted signal

is s(t) = as(t). After matched filtering at the receiver, thesignal vector is

ySISO(t) =√

GPMcbg(τ) + n(t), (3)

where G is the SNR gain due to filtering, g(τ) is the autocor-relation function, and n is filtered noise. We will assume anideal autocorrelation function to simplify the analysis.On the other hand, if all the elements are sufficiently far

apart, scattering of each transmitter–receiver pair is indepen-dent. This means that

E[vec(C)vecH(C)] = IMN . (4)

Assuming that distributions of the complex scattering param-eters are circular, the receiver snapshot can be written as

rMIMO(t) =

√P

MCs(t− τ) + n(t). (5)

Due to the impact of scattering, this configuration is referredto as a MIMO system.Between these two systems with minimum and maximum

diversity, we have the cases when either the transmitter el-ements or the receiver elements are placed sufficiently farapart to induce independent scattering, but the elements areclosely spaced at the other end. These two cases are SIMOand MISO scattering, respectively[4]. Since direction findinggenerally employs coherent reception, only the latter one isconsidered here. In the MISO case, the scattering matrix Cequals 1NcT , where E[ccH ] = IN . The received snapshot canbe written as

rMISO(t) =

√P

MbcT s(t− τ) + n(t), (6)

where the circularity assumption was used. In this case,beamforming cannot be done at the transmitter, but the receivercan use matched filters to obtain M signals

y(i)MISO(t) =

√GP

Mbci

M∑k=1

gik(τ) + n(i)(t). (7)

Assuming ideal autocorrelation and cross-correlation functions

gik(τ) = δikδ(τ), (8)

the filtered signals can be written as

yMISO(t) =

√GP

Mc⊗ b + n(t) (9)

after stacking the output of each matched filter into one columnvector, where ⊗ denotes the Kronecker product.We consider the marginal distribution of the scattering

parameters next. The fading of the received signal depends onthe particular case of the Swerling model[6]. The probabilitydistribution functions of the target radar cross-section (RCS)σ is either

f(σ) =1

σe−σ/σ (10)

in cases 1 and 2, or

f(σ) =4σ

σ2e−2σ/σ (11)

in cases 3 and 4, where σ is the average RCS[6]. These arescaled χ2-distributions with two and four degrees of freedom,where the scaling factors are σ/2 and σ/4, respectively.Since we use the scattering parameters to model the fluctu-

ations of the RCS and the average RCS is taken into accountin the power parameter P , we assume σ = 1. Thus, we needto choose a scattering parameter C so that it is zero-mean andradially symmetric and the pdf of its magnitude squared wouldbe either (10) or (11) depending on the case. Choosing

C =√

XeiΘ, (12)

where X has either distribution and Θ is uniformly distributedbetween zero and 2π, it is easy to see that C has the requiredproperties.

III. DIRECTION FINDING PERFORMANCEBased on the result in, for example, [9], the Cramer–

Rao lower bound of direction estimate of a signal with thecovariance matrix

R = σ2sbbH + σ2

nI

can be written as

CRB(θ) =1

2K

(1

‖b‖2σ4

n

σ4s

+σ2

n

σ2s

) (‖d‖2 − |d

Hb|2‖b‖2

)−1

,

(13)where b is the receiver steering vector and d its partialderivative with respect to the direction of arrival θ. The CRBwill be the basis of the performance analysis of directionfinding in this paper.The covariance matrix of the receiver snapshots depends

on the scattering model and the MIMO radar configuration.If the RCS fluctuations are fast (Swerling cases 2 and 4),the scattering amplitudes vary from pulse to pulse, and thecovariance matrix of the SISO configuration is

RfastSISO = GPMbbH + σ2

nI, (14)

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since E[|c|2] = 1 as discussed in Section II. The correspondingcovariance matrix of a MISO system is

RfastMISO =

GP

M(bbH)⊗ I + σ2

nI. (15)

We see that this covariance matrix consists of M blocks ofthe form GPMbbH + σ2

nI. In other words, the number ofindependently scattered samples has increased by a factor ofM compared to the SISO system.If the fluctuation is slow (Swerling cases 1 and 3), c and c

do not change during a scan and the covariance matrices forthe SISO and MISO cases can be written as

RslowSISO(c) = GPM |c|2bbH + σ2

nI (16)

RslowMISO(c) =

GP

M(b⊗ c)(b⊗ c)H + σ2

nI, (17)

respectively. Because the SNR depends on the number oftransmitters in the configuration, we introduce a power ratio

ρ =GP

σ2n

(18)

that can be used to compare different configurations fairly.Substituting the covariance matrices (14) and (15) into (13)

and also using (18), we obtain the bounds

CRBfastSISO(θ) =

1

2K

[1

N

1

M2ρ2+

1

Mρ

]×(

‖d(θ)‖2 − |dH(θ)b(θ)|2

N

)−1

(19)

CRBfastMISO(θ) =

1

2KM

[1

N

M2

ρ2+

M

ρ

]×(

‖d(θ)‖2 − |dH(θ)b(θ)|2

N

)−1

. (20)

The CRB is the same in cases 2 and 4 because the averagemagnitude of the scattering amplitudes equals one in bothmodels.It can be seen from (20) that with equal M and ρ, the

CRB of the MISO configuration is higher than that of theSISO configuration. This is caused by the lack of beamformingin the MISO configuration, and even though there are moresamples to be used in the estimation, the increased samplesupport is not enough to compensate for the increase of theCRB caused by lower SNR. Therefore, a SISO configurationwill likely perform better in a DOA estimation task than aMISO configuration if the fluctuation of the target RCS isfast.To evaluate the CRBs, we use following identities for a

uniform linear array[4]

‖d‖2 =1

6N(N − 1)(2N − 1)π2 cos2 θ (21)

|dHb|2 =1

4N2(N − 1)2π2 cos2 θ. (22)

0 5 10 15 2010−8

10−7

10−6

10−5

10−4

ρ (dB)

CR

B

Swerling Cases 2 & 4

M = 1MISO, M = 8MISO, M = 4MISO, M = 2SISO, M = 2SISO, M = 4SISO, M = 8

Fig. 1. Cramer–Rao bounds of DOA estimates as a function of the powerratio ρ when the Swerling case is 2 or 4. The CRB is displayed for SISO andMISO systems using a ULA with six receiver elements, M transmitters, and80 samples. It is seen that as the number of transmitters increases, the CRBof a MISO system increases whereas the CRB of a SISO system decreases.

The numerical values were evaluated for SISO and MISOconfigurations with six receivers. The DOA was 0◦ and thesample support K = 80. The results are shown in Fig.1. It isseen that the CRB increases if the number of transmitters inincreased in a MISO system, whereas the bound decreases fora SISO system. This follows from the lack of beamforminggain in the MISO system as discussed earlier.It was shown that in the slow fluctuation Swerling cases

1 and 3, the expressions for the covariance matrices containthe scattering amplitudes, which means that the power of thereceived signal is not constant. If the power of the signal variesbetween pulse trains, so does the CRB of the direction esti-mate. In order to find the average performance of an estimator,we assume that the estimator is a uniformly minimum-varianceunbiased estimator (UMVUE), so its variance is the same asthe CRB for any SNR.If X is UMVUE of the direction of arrival, the mean square

error equals the variance because the estimator is unbiased.The marginal density function of the estimator can be writtenas

fX(x) =

∫ ∞

−∞

fXC(x, c)dc =

∫ ∞

−∞

fX|C(x)fC(c)dc, (23)

where C is the scattering amplitude. The variance is then

Var(X) =

∫ ∞

−∞

∫ ∞

−∞

(x− θ)2fX|C(x)fC(c)dcdx

=

∫ ∞

−∞

Var(X|C)fC(c)dc

= EC [Var(X|C)] (24)

The order of integration had to be changed to get the final

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result. According to Fubini’s theorem[10], this can be done if∫R2

∣∣(x− θ)2fXC(x, c)∣∣ d(x× c) (25)

is finite, i.e. the variance exists. This derivation shows that thevariance of an UMVUE is the same as the average CRB usedin [4].In the Swerling cases with slow fluctuation, we can assume

that the scattering amplitudes are independent of θ. Althoughin reality, the RCS is a deterministic function of θ and thetarget orientation, the exact dependence is unknown and thefluctuations of the RCS are modeled as a random processwhich is independent of θ. Changes in the average RCS canbe assumed to affect the signal power parameter P . Therefore,

∂

∂θ(b⊗ c) = d⊗ c. (26)

Using identities

‖b⊗ c‖2 =N ‖c‖2 (27)‖d⊗ c‖2 = ‖d‖2 ‖c‖2 (28)

|(d⊗ c)H(b⊗ c)|2 =|dHb|2 ‖c‖4 , (29)

we obtain

CRBslowSISO(θ|c) =

1

2K

[1

N

1

M2ρ2|c|4 +1

Mρ|c|2]×(

‖d(θ)‖2 − |dH(θ)b(θ)|2

N

)−1

(30)

and

CRBslowMISO(θ|c) =

1

2KM

[1

N

M2

ρ2 ‖c‖4 +M

ρ ‖c‖2]×

(‖d(θ)‖2 − |d

H(θ)b(θ)|2N

)−1

. (31)

In order to get the variances of the UMVUEs, we need totake expectation of these two expressions. It is straightforwardto show that if X has a chi-squared distribution with n degreesof freedom, then

E[Xk

]= 2k Γ(n/2 + k)

Γ(n/2). (32)

Since the magnitude squared of each scattering amplitude hasa scaled chi-squared distribution with 2M or 4M degrees offreedom,

1

‖c‖2 =2

2(M − 1), (33)

1

‖c‖4 =4

4(M − 1)(M − 2)(34)

for Case 2 and1

‖c‖2 =4

2(2M − 1), (35)

1

‖c‖4 =16

4(2M − 1)(2M − 2)(36)

0 5 10 15 2010−6

10−5

10−4

10−3

10−2

ρ (dB)

Var

Swerling Cases 1 & 3, MISO

Case 3, M = 8Case 1, M = 8Case 3, M = 6Case 1, M = 6Case 3, M = 4Case 1, M = 4

Fig. 2. Variance of an uniformly minimum-variance unbiased DOA estimatoras a function of the power ratio ρ when the Swerling case is 1 or 3. Thevariances were evaluated for MISO systems with a ULA with six receiverelements, M transmitters, and 80 samples. The variance is not defined if aSISO system is used. It can be observed that as the number of transmittersincreases, the variance of the estimates increase and the performance degradeswhen a MISO system is used.

for Case 4. Unfortunately, the expected values of the CRBs arenot defined for SISO system or a MISO system with fewer thanthree transmitters under Swerling case 1. Numerical analysisor other performance measure should be used in these cases.Numerical values of the variance of the UMVUE estimator

are shown in Fig.2. The variance is shown in both Swerlingcases 1 and 3 for a MISO system with N = 6 receivers andK = 80 samples. The DOA, which was zero again. As incases 2 and 4 with fast fluctuation, one can see that when thenumber of transmitters is increased, the variance increases sothe DOA estimation performance degrades. Regardless of theSwerling case, the gains resulting from the spatial diversityare not enough to offset the lack of beamforming gain withthe displayed number of transmitters.

IV. SIMULATION RESULTS

The Cramer–Rao bounds and the variance of a UMVUEestimator were derived in the previous section for the fourdifferent Swerling cases. In this section, we show the results ofestimating the DOA using subspace-based estimation methodMUSIC. The MSE of MUSIC was estimated using Monte-Carlo method with 1000 independent trials. Estimation wasdone for both Case 2 and 4 with the same antenna config-urations as in the previous section, namely M transmitters,a ULA with six elements as a receiver, 80 samples, and theDOA equal to zero. The results are shown in Figures 3 and 4.The lines in these figures are the same CRB values as in Fig.1.Even though the MSE values are close to the CRBs, MUSICis not optimal and does not generally achieve the bound.Based on the simulation results, we can say that the Cramer–

Rao bounds reflect the direction finding performance of system

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0 5 10 15 2010−8

10−7

10−6

10−5

10−4

ρ (dB)

MS

EMSE of MUSIC, Swerling Case 2

M = 1MISO, M = 8MISO, M = 4MISO, M = 2SISO, M = 2SISO, M = 4SISO, M = 8

Fig. 3. MSE of MUSIC as a function of the power ratio ρ. Swerling model2 was used. The radars had a ULA with six elements as a receiver, M

transmitters, and 80 samples were available. Lines are the same CRB valuesas in Fig.1.

0 5 10 15 2010−8

10−7

10−6

10−5

10−4

ρ (dB)

MS

E

MSE of MUSIC, Swerling Case 4

M = 1MISO, M = 8MISO, M = 4MISO, M = 2SISO, M = 2SISO, M = 4SISO, M = 8

Fig. 4. MSE of MUSIC as a function of the power ratio ρ. Swerling model4 was used. The radars had a ULA with six elements as a receiver, M

transmitters, and 80 samples were available. Lines are the same CRB valuesas in Fig.1.

well in both models. The MSE of the actual estimates exhibitssimilar behavior as the CRB increasing in the case of a MISOsystem when the number of transmitters is increased.

V. CONCLUSIONSMIMO Radar is a new radar concept in which diversity

methods are used to improve performance. Previously, astatistical MIMO radar concept was introduced in [4]. Thegoal of the statistical MIMO radar is to use spatial diversityto reduce the impact of fluctuations of the target radar cross-section. Direction finding performance was discussed in [4]and [5]; however, these papers did not consider the Swerling

cases, which are used widely in the radar community to modelfluctuating target RCS.In this paper, we have derived the Cramer–Rao bound or the

variance of a uniformly minimum-variance unbiased estimatorfor each Swerling case. Our results indicate that with theassumed signal model, the transmit diversity used to mitigatethe impact of scintillation does not improve direction findingperformance in Swerling cases 2 and 4, i.e. the scenarios inwhich the fluctuation of the RCS is fast. The simulation resultsprovided also support this conclusion.Based on the derived variance of an UMVUE, it seems that

in the models with slow fluctuations, increasing the angulardiversity beyond certain point is not beneficial. However, theperformance measures considered were not defined for someconfigurations. The performance of the MIMO radar underSwerling models 1 and 3 will be investigated in more detailin future work. A more realistic signal model should be usedas well.Although the results of this study did not show improved

direction finding performance for the statistical MIMO radarwhen the RCS fluctuations were fast, it is not to say thatMIMO radar with angular diversity is without a merit. Onecould ask if the performance is high enough if there areother benefits – it has been shown that the spatial diversitycan be used to improve target detection[8]. MIMO radarmight also provide gains in estimation methods that are lesssensitive to SNR. There are also other methods that improvethe performance of a MIMO radar by exploiting the waveformdiversity.

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