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Target Tracking Using Monopulse MIMO Radar

With Distributed Antennas?

Sandeep Gogineni, Student Member, IEEE and Arye Nehorai∗, Fellow, IEEE

Department of Electrical and Systems Engineering

Washington University in St. Louis

One Brookings Drive, St. Louis, MO 63130, USA

Email: {sgogineni, nehorai}@ese.wustl.edu

Phone: 314-935-7520 Fax: 314-935-7500

Abstract—We propose a Multiple Input Multiple Output

(MIMO) radar system with distributed antennas that employs

monopulse processing at the receivers. We also propose an

algorithm to track a moving target using this system. This

algorithm is simple and practical to implement. It efficiently

combines the information present in the local estimates of

the receivers. Since most modern tracking radars already use

monopulse processing at the receiver, the proposed system does

not need much additional hardware. Using numerical simula-

tions, we demonstrate the advantages of the proposed system

over conventional single antenna monopulse radar. We also show

that the proposed algorithm keeps track of rapidly maneuvering

airborne and ground targets.

I. INTRODUCTION

A radar transmitter sends an electro-magnetic signal which

bounces off the surface of the target and travels in space

towards the receiver. When the electro-magnetic signal reflects

from the surface of the target, it undergoes an attenuation

which depends on the radar cross section (RCS) of the target.

This RCS varies with the angle of view of the target. We

can exploit these angle dependent fluctuations by employing

widely separated antennas [1]–[4]. When viewing the target

from different angles simultaneously, the angles which result

in a low RCS value are compensated by the others which

have a higher RCS, thereby leading to an overall improve-

ment in the performance of the radar system. This is the

motivation for using Multiple Input Multiple Output (MIMO)

radar with distributed antennas. Apart from distributed antenna

configuration, MIMO radar has also been suggested for use

in a colocated antenna configuration [5], [6]. Such a system

exploits the flexibility of transmitting different waveforms

from different elements of the array. In this paper, we will

only be dealing with MIMO radar in the context of distributed

antennas.

Most of the tracking radars have separate range tracking

systems that keep track of the range (distance) of the target and

sends only signals coming from the desired range gate to the

angle tracking system [7]. The range tracker has an estimate

?This work was supported by the Department of Defense under the Air

Force Office of Scientific Research MURI Grant FA9550-05-1-0443 and ONR

Grant N000140810849.

∗Corresponding author

of the time intervals when the target returns are expected.

The focus of this paper however is on the angle tracking

system, the most common form of which is the monopulse

radar [8]–[11]. In monopulse, we project the radar beams

slightly to either side of the radar axis in both the angular

dimensions (azimuth and elevation) [8], [9]. The beams are

generated simultaneously. Hence, monopulse radar is immune

to pulse-to-pulse target fluctuations. We compare the received

signals in each of these beams to keep track of the angular

position of the target. To perform this comparison, the system

computes a ratio which is a function of the signals received

through these beams. This ratio is called Monopulse Ratio.

Most modern radars use monopulse processors. There are

two types of monopulse tracking radars in use; amplitude-

comparison and phase-comparison. In amplitude-comparison

monopulse, the beams originate from the same phase center

whereas the beams in a phase-comparison monopulse system

are parallel to each other and originate from slightly shifted

phase centers. Essentially, the signals received from both

the beams have the same phase in amplitude-comparison

monopulse and they differ only in the amplitude. However, for

phase-comparison monopulse systems, the exact opposite is

true. In this paper, whenever we refer to monopulse, we mean

amplitude-comparison monopulse. Later in this paper (also

[12]), we propose a radar system that combines the advantages

of monopulse and distributed MIMO radar. It provides the

spatial diversity offered by MIMO radar with widely separated

antennas and is also immune to highly fluctuating target

returns just like any monopulse tracking radar.

The rest of this paper is organized as follows. In section

II, we propose a monopulse MIMO radar system and describe

its structure in detail. In section III, we describe the signal

model of our proposed system. In section IV, we propose a

tracking algorithm for this monopulse MIMO radar system.

We describe the various steps involved in tracking the location

of the target. In section V, we use numerical simulations

to demonstrate the improvement in performance offered by

this proposed system over conventional Single Input Single

Output (SISO) monopulse systems. We also show that the

proposed algorithm keeps track of an airborne target even

when it maneuvers quickly and changes directions. Further, we

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demonstrate that the proposed radar system efficiently keeps

track of a ground target that changes directions at sharp angles.

Finally, in section VI, we conclude this paper.

II. SYSTEM DESCRIPTION

In this section, we give a brief description of our proposed

system. Fig. 1 gives the basic structure of our monopulse

MIMO radar system. The system has M transmit antennas

and N receive antennas. The different transmitters illuminate

the target from multiple angles and the reflected signals from

the surface of the target are captured by widely separated

receivers. All the receivers are connected to a fusion center

which can be a separate block by itself or one of the receivers

can function as the fusion center. Before initializing the

tracking process, the fusion center makes the boresight axes

of all the receivers point towards the same point in space.

The fusion center has knowledge of the exact locations of all

the transmit and receive antennas and hence it can direct the

receivers to align their respective axes accordingly.

Fig. 1. Our proposed monopulse MIMO radar system.

Each of the receivers generates two overlapping receive

beams on either side of the boresight axis (see Fig. 2). In this

paper, we assume that the target moves only in the azimuth

plane scanned by these beams. However, we can easily extend

this to the other angular dimension (elevation) without loss of

generality by adding the extra beams. We compare the signals

arriving through the two beams at each of the receivers in

order to update the estimate of the angular position of the

target and subsequently update the location of the boresight

axis. All the receivers send their new local angular estimates

to the fusion center. The fusion center makes use of all the

information sent to it and makes a final global decision on the

location where the target could be present. It instructs all the

receivers to align their boresight axes towards this estimated

target location.

Target

Y

Fig. 2. Monopulse MIMO radar receivers.

III. SIGNAL MODEL

A. Transmitted Waveforms

Let ? si(t),i = 1,...,M, denote the complex baseband

modulation, the bandpass signal emanating from the ithtrans-

mit antenna is given as

si(t) = Re?? si(t)ej2πfct?,

are narrowband waveforms with pulse duration T seconds

and are repeated once every TRseconds. We do not impose

any further constraints on these waveforms. Especially, note

that we do not need orthogonality between the different

transmitted waveforms unlike conventional MIMO radar with

widely separated antennas. As we shall see later in the paper,

the reason for this is that we do not need a mechanism to

separate these waveforms at the receivers. We process the sum

of the signals coming from different transmitters collectively

without separating them. This is another advantage of the

proposed system because the assumption that the waveforms

remain orthogonal for different delays and doppler shifts is

unrealistic. In section V (numerical results), we considered

rectangular pulses.

waveform transmitted from the ithantenna. Therefore, after

(1)

where fcdenotes the carrier frequency. We assume that ? si(t)

B. Target and Received Signals

We assume a far-field point target with its RCS varying with

the angle of view. Let aik(t) denote the complex attenuation

factor due to the distance of travel and the target RCS for the

signal transmitted from the ithtransmitter and reaching the

kthreceiver and τikis the corresponding time delay. Note that

for a colocated MIMO system, aik(t) for different transmitter-

receiver pairs will be the same because all the antennas will

be viewing the target from closely spaced angles. Different

models have been proposed in literature to model the time

varying fluctuations in these attenuations aik(t) [13]–[15].

Some of these models incorporate pulse-to-pulse fluctuations,

scan-to-scan fluctuations, etc. These correspond to fast moving

and slow moving targets respectively. In our numerical simu-

lations, we consider a rapidly fluctuating scenario where these

attenuations keep varying from one pulse instant to another

because of the motion of the target. We assume aik(t) to be

constant over the duration of one pulse. These attenuations

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aik(t) are not known at the receivers. Under the narrowband

assumption for the complex envelopes of the transmitted

waveforms, and further assuming the target velocity to be

much smaller than the speed of propagation of the wave in

the medium, the bandpass received waveform is

yk(t) =

M

?

i=1

Re

?

aik(t)? si(t − τik)ej2π(fc(t−τik)+fDik(t−τik))?

where fDik is the Doppler shift along the path from the ith

transmitter to the kthreceiver,

fDik=fc

c

where− →

uRk denote the target velocity vector, unit

vector from the ithtransmitter to the target and the unit vector

from the target to the kthreceiver, respectively; ?,? is the inner

product operator, and c is the speed of propagation of the wave

in the medium. Equation (2) is valid only when the target is

moving with constant velocity.

,

(2)

??− →

v ,− →

uRk? − ?− →

v ,− →

uTi??,

(3)

v ,− →

uTi,− →

C. Beamforming

The receive beams are generated using Capon beamformers

[16]. Capon beamformer is the minimum variance distortion-

less spatial filter. Each receiver generates two beams located

at the same phase center using two linear arrays. Each array

has L elements, each separated by a uniform distance of

where λ =

Under the given antenna spacing, the steering vector of the

beamformers becomes

?

where [·]Tdenotes the transpose. Let θkbe the angle between

λ

2,

c

fcis the wavelength corresponding to the carrier.

d(θ,f) =1,e−jπfλ

ccosθ,...,e−j(L−1)πfλ

ccosθ?T

,

(4)

Fig. 3.Spatial beamformer at the receiver.

the approaching plane wave and the two linear arrays at the kth

receiver (see Fig. 3). The received signals are first demodulated

before passing through the two beamformers. Define the

outputs of the two beamformers as yl

superscripts l and r correspond to the left and the right beams,

k(t) and yr

k(t), where the

respectively. Also, let wl

[wr

the beamformers. Similarly, el

and er

vectors of these two spatial filters. Defining

k=

?wl

k(t) =

kL(t)]T

k1,...,wl

kL

?T

and wr

k=

k1,...,wr

kL]Tdenote the corresponding weight vectors of

?el

k1(t),...,el

are the additive noise

kL(t)?T

k(t) = [er

k1(t),...,er

xk(t) ?

M

?

i=1

aik(t)? si(t − τik)ej2π(fc(−τik)+fDik(t−τik)), (5)

and sampling the outputs of the spatial filters, we get

xk[n]?wl

yr

=

k

We assume that the additive noise vectors at the two ar-

rays of sensors have zero mean and covariance matrices

Rl

the beams by minimizing

subjecttothe constraints

??wr

[17]

?Rl

d(θl

?Rr

d(θr

where θl

directed. Hence, boresight axis of the receiver is located at

an angle θb

2

. In practice, the covariance matrices Rl

and Rr

are approximated using the sample covariance matrices?

We evaluate the sum and the difference of the absolute

values of the complex outputs at the two beamformers and

send these measurements to the monopulse processor for the

decision making about the angular location of the target.

?

yd

= abs

yl

yl

k[n]

k[n]

=

k

?Hd(θk,fc+ fDik) +?wl

k

?Hel

k[n], (6)

k[n]. (7)

xk[n]?wr

?Hd(θk,fc+ fDik) +?wr

k

?Her

kand Rr

k, respectively. The Capon beamformer creates

?wl

k

?Hd(θr

k

?HRl

kwl

?Hd(θl

kand

?wr

k

?HRr

kwr

and

k

??wl

k,fc) = 1

?

kk,fc) = 1

?

, respectively. The solution to this

optimization problem gives the weights of the beamformers

wl

k

=

k

?−1d(θl

?−1d(θr

k,fc)

?−1d(θl

?−1d(θr

k,fc)H?Rl

k

k,fc)H?Rr

k

k,fc)

,

(8)

wr

k

=

k,fc)

k

k,fc)

,

(9)

k, θr

kare the angles at which both the beams are

k=

θl

k+θr

k

k

kare not known at the receiver apriori. Therefore, they

Rl

k

and?

Rr

k.

ys

k[n]=abs

yl

k[n]

?

+ abs

?

yr

k[n]

?

,

(10)

k[n]

?

k[n]

?

− abs

?

yr

k[n]

?

.

(11)

IV. TRACKING ALGORITHM

A. Initialization

The fusion center has the information about the exact

locations of all the receivers. It will initialize the tracking

algorithm by making sure that the boresight axes of all the

receivers intersect at the same point in space.

B. Monopulse Processing: Local Angular Estimates

After obtaining the measurements from the sum and the

difference channels, each of the receivers computes the

monopulse ratio

Mk[n] =yd

k[n]

ys

k[n].

(12)

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If the Mk[n] is positive, it is highly likely for the target to

be present on the left side of the boresight axis. Similarly, a

negative Mk[n] indicates the opposite. Therefore, receiver k

will adjust its boresight axis appropriately using the following

equation

θb(new)

k

= θb

k+ δ{Mk[n]},

(13)

where δ is a positive valued design parameter. A larger δ will

enable tracking faster moving targets but will also lead to

higher steady state errors. However, a smaller δ will increase

the convergence time but the steady state errors will be less.

Each of the receivers updates its angular estimates and the axis

using the above mentioned processing.

C. Fusion Center: Global Location Estimate

The primary function of the fusion center is to combine

these decentralized estimates and arrive at a global estimate.

We have solved a similar problem for localizing acoustic

sources using Cramer-Rao bound [18]. Here, we present a

simpler method to combine the decentralized estimates. After

obtaining new angular estimates, each of the receivers sends

these new updates to the fusion center. Along with the angular

estimates, the receivers also send the instantaneous energy of

the received signal in the sum channel during that instant.

Ek[n] = (ys

k[n])2.

(14)

Fig. 4.

of three receivers.

Polygon formed by the points of intersection of the boresight axes

The fusion center forms a polygon of

connecting the points of intersection of the updated boresight

axes of each of the N receivers (see Fig. 4). The fusion

center will decide upon a point inside this polygon to be the

global estimate of the target location. Define?pxij[n],pyij[n]?

intersection of the boresight axes coming out from the ith

receiver and the jthreceiver. A linear combination of these

vertices is chosen as the estimate of the target location

N(N−1)

2

sides by

to be the cartesian coordinates of the vertex formed by the

(?

px[n], ? py[n]) =

N

?

i=1

N

?

j=i+1

αij[n]?pxij[n],pyij[n]?.

(15)

We choose the weights αij[n] to be proportional to the sum

of instantaneous energies received from the corresponding

receivers and?N

αij[n] =

?N

Finally, the fusion center sends this new estimate to all

the receivers and guides them to align their axes towards this

particular location before the next iteration.

i=1

?N

?N

j=i+1αij[n] = 1. Therefore,

Ei[n] + Ej[n]

?

i?=1

j?=i?+1

Ei?[n] + Ej?[n]

?.

(16)

V. NUMERICAL RESULTS

A. Simulated Scenario

In this section, we demonstrate the advantage of the

proposed monopulse MIMO tracking system under realistic

scenarios. We simulated such a scenario to demonstrate the

advantages of this system. First, we describe the locations of

the transmitters, receivers, and target on a cartesian coordinate

system. The simulated system has two transmitters that are

located on the y-axis at distances of 20km and 40km from

the origin, respectively. There are three receivers located on

the x-axis at the origin, 20km and 40km from the origin,

respectively. The receiver at the origin also serves as the fusion

center for this setup. The target is initially present at the

coordinate (30,35).

We chose the carrier frequency fc= 1GHz. We used com-

plex rectangular pulses each with a constant value1+√−1

bandwidth 100MHz for the transmitted baseband waveforms.

Therefore, the pulse duration T = 10−8s. The pulse repetition

interval TR = 4ms. We further had two samples per pulse

duration (Nyquist rate). We ran the simulation for 2s. Hence,

we had 500 pulses from each transmitter. The target is airborne

and moving with a constant velocity of (0.25,0.25)km/s.

There are six complex numbers {a11,a12,a13,a21,a22,a23}

describing the attenuation experienced by the signals. It is

important to realistically model these attenuations. They were

independently generated from one pulse to another using zero

mean complex normal random variables with their variances

chosen from the set {0.15,0.3,0.45,0.6,0.75,0.9}. The aik

corresponding to the antenna pair that are the closest to the

target got the higher values and vice versa. We assumed

the additive noise at every element of the receiver array is

uncorrelated zero mean complex Gaussian distributed with

variance σ2. The received powers are different at different

receivers because the attenuations aik do not have the same

variances. Therefore, we evaluate the overall signal to noise

ratio (SNR) by computing the average. For a noise variance of

σ2= 0.1, SNR=12.3dB. We further assumed the noise to be

stationary. The noise variance was estimated from a training

data set of 50 samples. We assumed that the target returns

were not present in the training samples that were used. We

independently generated the noise from one time sample to

another. The two beams at each receiver were generated using

L = 10 element linear arrays and they were made to point

5 degrees on either side of the boresight axis. The −3dB

√2

and

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beamwidth of these beams is approximately 12 degrees. We

chose the parameter δ = 0.25 degrees in our algorithm.

B. Spatial Diversity

We first demonstrate the spatial diversity offered by

monopulse MIMO radar with widely separated antennas by

comparing this system with monopulse SISO radar. Since a

single receiver monopulse tracking radar can only track the

angular location of the target, we shall compare only the angle

errors of the SISO and MIMO monopulse radars. For SISO

radar, we assumed only the first transmitter (0,20) and the

first receiver (0,0) to be present. First, we assumed that the

initial estimate of the target location for 2x3 MIMO radar

is far from the actual location at (32,32). Hence, the initial

estimate was at a distance of 3.61km from the actual location.

The same initial estimate was also used for SISO radar and it

corresponds to an initial angular error of 4.3987 degrees. In

order to make the comparison fair, we deliberately increased

the transmit power per antenna for the SISO system to make

the overall transmit power the same. We chose the complex

noise variance σ2for this comparison to be 0.1. We plotted

the angular error as a function of the pulse index. Fig. 5 shows

that the MIMO system overcomes a poor initial estimate and

manages to track down the target much quicker than the SISO

radar. The SISO system takes 60 pulses to come within an

angular error of 1 degree. However, the 2x3 MIMO system

takes only 20 pulses to reach within the same level of angular

error. To obtain good accuracy, we plotted these curves by

averaging the results over 100 independent realizations.

0100200300400 500

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pulse index

Angle error (in degrees)

Monopulse SISO radar

Monopulse MIMO radar

Fig. 5.

a function of the pulse index for σ2= 0.1.

Comparing the angle error of SISO and MIMO monopulse radars as

Next, we assumed a good initial estimate of (29.9,34.9)

and plotted the average angular errors of both these systems

as a function of the complex noise variances. As expected,

Fig. 6 shows that the average angular error increases with

an increase in the noise variance. MIMO system significantly

outperforms the SISO system. The angular error of these

systems can further be reduced by using a smaller value of

δ. However, if the initial estimate of the target location is

poor, a smaller δ would mean that the convergence time of the

algorithm would increase. Hence, it is a trade-off between the

steady-state error and convergence rate. Note that as the noise

variance reduces, the gap between the performances of the

systems reduces because the advantage offered by the spatial

diversity becomes more relevant when there is more noise. The

performance of any monopulse system is independent of the

absolute values of the signals of interest. This is an outcome of

the fact that we use a ratio in monopulse processing instead

of the absolute values of the measured signals in both the

channels. As the noise variance increases, we get to see that

the improvement offered by the spatial diversity of the MIMO

system also increases.

00.511.52

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Complex noise variance (σ2)

Average angular error (in degrees)

Monopulse SISO radar

Monopulse MIMO radar

Fig. 6.

radars as a function of the complex noise variance σ2.

Comparing the average angle error of SISO and MIMO monopulse

The advantage of the proposed monopulse MIMO radar over

monopulse SISO radar stems from the fact that by employing

multiple antennas, we are exploiting the fluctuations in the

target RCS values with respect to the angle of view. Even if

the RCS between one transmitter-receiver pair is very small,

it is highly likely that the other transmitter-receiver pairs will

compensate for it. Also, in our proposed algorithm, the weights

are proportional to the received energies. Hence, with high

probability, a transmitter-receiver pair with high RCS value

will contribute significantly to the received energy at that

particular receiver.

Along with tracking the angular location of the target, the

exact coordinates of the target location can also be estimated

by evaluating the points of intersection of the boresight axes

coming from all the receivers. Since this processing is possible

only for monopulse systems with multiple receivers, in the

following simulations, we show the localizing abilities of 2x3

MIMO radar under different challenging scenarios.

C. Rapidly Maneuvering Airborne Target

A clever target would change its direction of travel at

high velocities to reduce the detectability and to confuse the

tracking radar. Hence, it is extremely important to track a

rapidly maneuvering airborne target. In order to check the

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