Waveform-Agile Sensingfor Range and DoA Estimation

in MIMO RadarsBhavana B. Manjunath†, Jun Jason Zhang†, Antonia Papandreou-Suppappola†, and Darryl Morrell‡

†SenSIP Center, Department of Electrical Engineering, Arizona State University, Tempe, AZ‡Department of Engineering, Arizona State University, Polytechnic Campus, Mesa, AZ

E-mail: Bhavana.Manjunath@asu.edu, Jun.Zhang.EE@asu.edu, papandreou@asu.edu, morrell@asu.edu

Abstract— We propose an agile sensing algorithm to optimallyselect the transmission waveform of a multiple-input, multiple-output (MIMO) radar system in order to improve target lo-calization. Specifically, we first derive the Cramer-Rao lowerbound (CRLB) for the joint estimation of the antenna reflectioncoefficients and the range and direction-of-arrival of a stationarytarget using MIMO radar with colocated antennas. The resultingCRLB, that is a function of the transmitted waveform, isthen compared to the estimation performance of a maximum-likelihood estimator. We configure waveform parameters to min-imize the trace of the predicted error covariance by assuming thatthe covariance of the observation noise is approximated by theCRLB for high signal-to-noise ratios. In particular, we optimallyselect the duration and phase function parameters of generalizedfrequency-modulated chirps to minimize the estimation mean-squared error under constraints of fixed transmission energyand constant time-bandwidth product.

I. INTRODUCTION

MIMO radar systems are emerging as a promising newtechnology as they have the potential to enhance target de-tection and identification performance [1]. In addition to otherbenefits [2]–[5], MIMO radar systems provide the flexibilityof transmitting a completely different waveform on each of itscolocated antennas. This flexibility can lead to agile sensingand waveform diversity and thus improve target detection andestimation performance by optimally designing the transmis-sion waveforms.

Waveform selection techniques have been developed inorder to maximize desirable performance metrics. In [3],beamforming was used to improve the output signal-to-noiseratio (SNR) and increase detection and estimation performancefor MIMO radars with colocated antennas. Optimal waveformdesign for MIMO radars was achieved using an informationtheoretic approach in [6]. In [7], a procedure was developedfor designing the waveform that maximized the signal-to-interference plus-noise ratio for target detection. In [8], theCRLB was used to design a random transmission waveformto achieve a desired beam pattern.

This work was supported by the Department of Defense MURI GrantNo. AFOSR FA9550-05-1-0443.

In our work, we combine the use of MIMO radar withcolocated antennas with waveform agile sensing by optimallyselecting a different waveform to transmit on each antenna. Inparticular, we incorporate the waveform-dependent CRLB forthe joint estimation of range and direction-of-arrival (DoA) ofa stationary target, and we make use of generalized frequency-modulated (GFM) waveforms with time-varying phase func-tions. For NT transmission antennas, our aim is to optimallyselect Φ = [φT

1 φT

2 . . . φT

NT]T , where the waveform

parameter vector φT

i = [λi bi ξi(t/tr)] consists of theduration λi, frequency-modulation (FM) bi and phase functionξi(t/tr) of the GFM waveform transmitted by the ith antenna.Here, tr is a normalization time constant, and T denotesvector transpose. In order to localize a stationary target, thetransmission antenna waveform parameters are selected suchthat the trace of the predicted estimation error covarianceis minimized. The error covariance is obtained under theassumption that the covariance of the observation noise canbe closely approximated by the CRLB when the SNR is high.Thus, minimizing the waveform-dependent CRLB correspondsto minimizing the estimation mean-squared error (MSE). Us-ing a maximum likelihood estimator, we provide simulationresults to demonstrate that the estimate covariance is in closeagreement to the CRLB, thus validating the use of the CRLBas the optimization performance criterion.

The paper is organized as follows. We discuss the structureof the waveform used in the configuration process in SectionII. The received waveform for the MIMO radar systemis provided in Section III, together with the computationof the CRLB for the joint estimation of the antennatransmission coefficients and the range and DoA of thetarget. The optimization criteria for configuring linear FM(LFM) waveforms and GFM waveforms with varying phasefunctions are considered in Section IV, and simulation resultsare provided in Section V.

II. WAVEFORM STRUCTURE

We choose the transmission waveform from each of theco-located antennas of a MIMO radar system to be a GFM

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waveform that is given by [9]

s(t) = a(t) ej2π b ξ(t/tr) . (1)

Here, a(t) is the amplitude envelope function and b is the FMrate of the GFM. By varying the phase function ξ(t/tr), wecan obtain different time-frequency waveform signatures thatuniquely characterize the GFM. The time-frequency signaturecorresponds to the waveform instantaneous frequency obtainedusing the time-derivative of the phase function. For example,when ξ(t/tr) = (1/2)(t/tr)

2, the GFM has a linear instanta-neous frequency given by d/dt(ξ(t/tr)) = t/t2r, and it sim-plifies to an LFM chirp, a waveform commonly used in radarapplications. When ξ(t/tr) = ln (t/tr), the instantaneousfrequency is hyperbolic and the corresponding waveform isthe hyperbolic FM (HFM) chirp; this is a waveform similarin time-frequency structure to the waveforms used by bats forecho-location [9]. Note that GFM waveforms with nonlinearphase function were shown in [10] to provide more accurateestimates of the range and Doppler parameters of a target thanLFM waveforms.

The properties of the amplitude envelope function a(t) in(1) can greatly affect the choice of the waveform and hencethe target parameter estimation performance. If the envelopefunction is Gaussian, the characteristic time-frequencysignatures of different GFMs can be affected in differentways due to the low energy content at the tails of the Gaussianfunction. Although the ideal choice for the envelope would bea rectangular function, this function is not differentiable andthus the corresponding waveform-dependent CRLB cannot beobtained in closed form. As an alternative, we choose to usea sigmoid envelope function. This function is differentiableand thus can lead to a closed form CRLB expression. Inaddition, it closely approximates a rectangular window in thetime domain with an out-of-band frequency roll-off that ismuch higher than that of a rectangular window (as shown inFig. 1). The sigmoid function is defined as

a(t) = α

[1

1 + e−qt−

1

1 + e−q(t−λ)]

], (2)

where the parameter α is chosen such that s(t) in (1) has unitenergy, λ is the duration of the waveform, and q is a designparameter that controls the roll-off rate of the window in thefrequency domain.

III. RECEIVER MODEL AND CRLB

A. Received Signal Model

We consider a MIMO radar system with NT colocatedtransmitter antennas and NR receiver antennas. The trans-mitted signal si(t), i = 1, . . . , NT , of the ith antenna issampled every Ts seconds to obtain si[n] = si(nTs), n =1, . . . , N . The resulting transmitted signal matrix is given byS = [s1 s2 · · · sN ] where sn = [s1[n] s2[n] . . . sNT

[n]]T .The time-delayed (by τ ) signal due to a single target is given

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Am

plitu

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(a) Sigmoid window in the time-domain.

0 5 10 15 20−1200

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Frequency (MHz)

Nor

mal

ized

mag

nitu

de (

dB)

Sigmoid windowRectangular window

(b) Sigmoid window in the frequency-domain.

Fig. 1. Amplitude envelope functions of the GFM waveform in both thetime and frequency domains.

by si(τ)[n] = si(nTs − τ); the delayed sampled signal matrixis thus given by S(τ) = [s1(τ) s2(τ) · · · sNT

(τ)].

The signal received by the NR receivers is given by theNR × N matrix

R = β a(θ)vT (θ)S(τ) + W, (3)

where the steering vectors v(θ) and a(θ) of the transmitter andreceiver antennas, respectively, depend on the DoA θ, β is thereflection coefficient of the target, and W is an NR×N noisematrix whose elements are snapshots of zero-mean additiveGaussian noise. Vectorizing W, we obtain w = vec[W] =[wT

1 wT2 . . . w

T

N ]T , where wn = [w1[n] w2[n] . . . wNT[n]]T

and wi[n] is the nth sample of the Gaussian noise addedto the waveform transmitted by the ith antenna. The noisecovariance matrix can be expressed as Cw = E{ww

H} =CT ⊗CS , where CT and CS are the corresponding temporaland spatial noise covariance matrices, E{·} denotes statisticalexpectation, H denotes complex conjugate transpose, and ⊗is the Kronecker product. Following the same notation, wevectorize R such that r = [rT1 r

T2 . . . r

T

N ]T where rn =[r1[n] r2[n] . . . rNT

[n]]T is the nth snapshot of the receivedsignal. Note that (3) can also be extended to multiple targets.

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B. Joint CRLB Derivation

We want to estimate the location of a target by estimatingthe reflection coefficient, range, and DoA of the target.We denote the parameter vector to be jointly estimated asΨ = [βR βI τ θ]T , where we denote β = βR +jβI . Then, theprobability density function p(r|Ψ) of the received data giventhe unknown parameter vector is a joint Gaussian densitywith mean vector µ(Ψ) = β a(θ)vT (θ)S(τ) and covariancematrix Cw. The ijth element of the Fisher information matrixI(Ψ) for estimating Ψ is given by [11]

[I(Ψ)]ij = 2 Re

{∂µH(Ψ)

∂ΨiC−1

w

∂µ(Ψ)

∂Ψj

},

where i, j = 1, 2, 3, 4, Re{·} denotes the real part and Ψi isthe ith element of the parameter vector Ψ = [βR βI τ θ]T .The joint CRLB is thus given by CRLBΨ = (I(Ψ))−1.

We define A(θ) = a(θ)vT (θ), and the diagonalelements of the Fisher information matrix are expressedas [I(Ψ)]11 = IβR

, [I(Ψ)]22 = IβI, [I(Ψ)]33 = Iτ , and

[I(Ψ)]44 = Iθ , then we can obtain these elements in closedform as

Iθ = |β2| 2Re{

Tr(S

H(τ)∂AH (θ)∂θ C

−1S

∂A(θ)∂θ S(τ)C−1

T

)}

IβR= IβI

= 2Re{

Tr(S

H(τ)AH (θ)C−1S A(θ)S(τ)C−1

T

)}

Iτ = |β2| 2Re{

Tr(

∂SH(τ)∂τ A

H(θ)C−1S A(θ)∂S(τ)

∂τ C−1T

)}.

(4)

The other Fisher information matrix elements can besimilarly derived in closed form. Note that we obtain directDoA measurements and the range information is obtainedfrom c τ/2, where c is the speed of light in the air. Thus,the parameters of interest directly depend on the CRLB onestimating Ψ as the CRLB is obtained as inverse of theFisher information matrix. As a result, if the error covarianceof the estimate is in close agreement with the CRLB, thenthe CRLB can be used as the performance metric in thewaveform selection algorithm.

IV. WAVEFORM SELECTION

Under high SNR conditions, minimizing the parameterestimation MSE corresponds to minimizing the CRLB (whichwas shown to depend on the transmitted waveforms). As aresult, selecting the optimal waveform for each antenna inorder to minimize the MSE corresponds to searching over alibrary of waveforms and their parameters such that the traceof the error covariance matrix (corresponding to the trace ofthe CRLB) is minimized. The library of waveforms is formedby varying the phase function ξ(t/tr), FM rate b, and durationλ of the GFM waveform in (1) and (2).

In this paper, we consider three types of GFM waveforms:LFM waveforms with phase function ξ(t/tr) = (t/tr)

2 andlinear instantaneous frequency, HFM waveforms with phase

function ξ(t/tr) = ln (t/tr) and hyperbolic instantaneousfrequency, and exponential frequency-modulated (EFM) wave-forms with phase function ξ(t/tr) = et/tr and exponentialinstantaneous frequency. In the first simulation case, we con-sider a library consisting of only LFM waveforms with varyingFM rates and durations. In the second simulation case, weconsider a library consisting of all three GFM waveforms, thusallowing the phase functions as well as the durations and FMrates to vary. Once the optimal waveform for each antennais chosen, the optimal waveforms are transmitted and thereceived waveform is processed using a maximum likelihoodestimator to jointly estimate the delay and DoA of the target.

For a fair comparison between the different possible selec-tion waveforms, we fix the time-bandwidth of each waveformin the library as well as the transmitted energy of all theantenna waveforms. Fixing the total transmitted energy isequivalent to fixing the total transmission time of all the wave-forms given by λ =

∑NT

i=1 λi, where λi is the duration of thewaveform transmitted by the ith antenna. We define ∆fi to bethe frequency sweep of the ith transmit waveform (that is, thedifference between the maximum and minimum frequencies inthe waveform). Then, we choose L combinations of λi valuesthat fix λi∆fi to a constant value for varying ∆fi. The valueof L depends on how fine we want the grid resolution of thesearch in the selection algorithm to be.

For the LFM only library, the L combinations consist ofthe same type of waveform. When the library includes LFM,HFM and EFM waveforms with varying durations and FMrates, we again form L combinations of possible durations foreach combination of NT possible phase functions (includingthe combination of using the same phase function on differentantennas). The waveform selection algorithm then choosesthe combination of phase functions and durations that resultsin the minimum CRLB trace.

V. SIMULATION RESULTS

We tested the waveform selection algorithm for the jointestimation of the delay and DoA of a single target with 15km range and 30◦ DoA. The MIMO radar in the simulationsconsisted of NT = 3 colocated antennas, and the carrier fre-quency was fc = 10 GHz. The allowable waveform durationranged between 10 µs to 70 µs, and λ = 90 µs. The maximumallowable frequency sweep was constrained to 12 MHz.

When the waveform library consisted only of LFM wave-forms, the duration grid size was chosen to be 10 µs, and thetrace of the CRLB was computed for each of the combinations.Fig. 2 shows that the trace of the CRLB did change fordifferent combinations of LFM waveform durations. When weincluded LFM, HFM and EFM waveforms with different dura-tions in the library, many more combinations were computed.In particular, as shown in Fig. 3, the trace of the CRLB isshown for different phase and duration combinations.

The estimation of DoA strongly depends on the energy ofthe transmitted signal whereas the delay estimation is affectedby the type of signal transmitted. As a result, when we consider

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0 10 20 30 40 50 6010

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Tra

ce o

f CR

LB

Fig. 2. Trace of CRLB for different combinations of LFM waveformdurarations.

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log

10 T

race

of

CR

LB

Fig. 3. Trace of CRLB for different combinations of durations and phasefunctions.

the CRLB of the delay for different waveforms, the differencesin magnitude are more pronounced. This is demonstrated inFig. 4 when the GFM waveform library is used. The CRLBfor the delay estimate for the best and worst combinations isshown to vary by a factor of about 400 in magnitude.

In Fig. 5, we demonstrate that the CRLB can be used asa performance metric for the maximum likelihood estimator(MLE). Specifically, we show that the variance of the delayand DoA MLE estimates are in close agreement with theirCRLB.

The results of the waveform-agile estimation algorithmare demonstrated in Fig. 6. Specifically, we compared thethe CRLB and MSE performance of the waveform selectionalgorithm with the CRLB and MSE performance of fixed LFM

020

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log

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RL

Bτ

Fig. 4. CRLB for delay for different combinations of durations and phasefunctions.

2 4 6 8 10

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Var

ianc

e of

est

imat

or

Delay (CRLB)Delay (Simulation)DOA (CRLB)DOA (Simulation)

Fig. 5. Simulated variance of estimates using maximum likelihood estimationand computed CRLB.

waveforms. The LFM waveforms were chosen to have shortdurations (for good delay estimates) and wide bandwidths (forgood Doppler estimates). They also satisfied the energy andtime-bandwidth product constraints. As we can see in Fig. 6,the CRLB of the delay estimation using waveform selectionoutperformed the CRLB of the fixed LFM waveform. Also, theMSE performance when transmitting the selected waveformsoutperformed the performance of transmitting the fixed LFMwaveforms.

VI. CONCLUSION

We developed a waveform-agile algorithm for locating atarget using a MIMO radar system with colocated antennas.In particular, we designed different time-varying signaturesfor multiple transmit antennas in order to minimize theestimation error covariance of the parameters of a stationary

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Del

ay C

RLB

Fixed waveform(CRLB)Fixed waveform (MLE)Waveform selection (MLE)Waveform selection (CRLB)

Fig. 6. Waveform selection to minimize CRLB, resulting in minimizing MSEperformance.

target. Using high SNR assumptions, the estimation errorcovariance is approximated by the waveform-dependentCRLB for the joint estimation of range and direction-of-arrival (DoA) of the target. Simulation results demonstratedthe effectiveness of the waveform-agile localization algorithm.

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