MIMO Radar Signal Processing || Generalized MIMO Radar Ambiguity Functions

3GENERALIZED MIMO RADARAMBIGUITY FUNCTIONS1

GEOFFREY SAN ANTONIO AND DANIEL R. FUHRMANN

Department of Electrical and System Engineering, Washington University, St. Louis, Missouri

FRANK C. ROBEY

MIT Lincoln Laboratory, Lexington, Massachusetts

3.1 INTRODUCTION

Multiple-input multiple-output (MIMO) systems have gained popularity and attractedattention for their ability to enhance all areas of system performance. MIMO ideas arenot new, in fact their origin can be traced to the control systems literature. Mehra [1]discussed, the idea of optimally selecting multiple system inputs to enhanceparameter estimation. The early 1990s saw an emergence of MIMO ideas into thefield of communication systems [2]. More recently, one will find the ideas ofMIMO appearing in sensor and radar systems.

A MIMO radar system [3,4] consists of transmit and receive sensors; the transmitsensors have the ability to transmit arbitrary and independent waveforms. In manyways a MIMO radar is similar to a MIMO communication system. Although themission of a radar system is quite different, among the many possible uses of aradar system, tracking and detecting targets, estimating target model parameters,and creating images of targets are some of the most common. Various authorshave all shown how these system tasks can be enhanced by using MIMO radar.

1This work is sponsored in part under Air Force Contract FA8721-05-C-0002. Opinions, interpretations,recommendations, and conclusions are those of the authors and are not necessarily endorsed by theUnited States government.

MIMO Radar Signal Processing, edited by Jian Li and Petre StoicaCopyright # 2009 John Wiley & Sons, Inc.

123

Fishler et al. [4] have shown how detection performance might be improved usingMIMO techniques. In the issue of how to make use of the extra degrees offreedom (DOF) offered by MIMO has also been addressed [5–9]. Designing transmitbeampatterns is significantly enhanced by using MIMO ideas. Also, adaptive tech-niques such as MVDR and the GLRT can be enhanced by MIMO [3,10]. Earlywork [11–13] emphasized some basic performance gains that might be achievable.

As mentioned above, two of the primary functions of a radar are to detect targetsand estimate parameters of a model used to describe those targets. Early radars coulddistinguish one unambiguous parameter, range. Continuous-wave (CW) radars couldidentify only range rate. Pulse–Doppler radar can simultaneously identify range andrange rate. Radar arrays allow for estimation of angular parameters. The work in thischapter shows how MIMO radar can enable the unambiguous observation ofadditional target parameters.

Modern radar systems are designed to be highly accurate for their intendedpurpose. Designers and engineers need to know the level of resolution to expectfrom a particular system configuration. Some of the tools used to characterize per-formance are statistical parameter estimation bounds and ambiguity functions.Typically, parameter estimation bounds such as the Cramer–Rao (CR) bounddepend on the ambiguity function. The CR bound is a local bound; it depends onthe shape of the ambiguity function in the local region surrounding the parameterestimate. The classic ambiguity function was introduced by Woodward and is usedto characterize the local and global resolution properties of time delay and Dopplerfor narrowband waveforms. Other authors have extended Woodward’s ideas tolarger classes of waveforms and whole radar systems.

The purpose of this chapter is to extend the ideas of ambiguity analysis to MIMOradar systems. An ambiguity function is developed that expresses the new degrees offreedom offered by MIMO radar while reducing to Woodward’s ambiguity functionfor a simple-single sensor narrowband system. The functions presented herein are thenecessary tools for effectively evaluating whole sets or classes of waveforms to beused in a MIMO radar system. Section 3.2 presents an overview of previous workdone on ambiguity functions, providing a background for establishing the MIMOambiguity function. In Section 3.3 the signal model used to describe a MIMOradar is presented. Section 3.4 introduces a parametric model that characterizestargets using a six-parameter vector. The MIMO ambiguity function is developedin Section 3.5. Section 3.6 presents some examples of MIMO ambiguity functionsand highlights some new signal design problems.

3.2 BACKGROUND

MIMO radar is in many ways a generalization of traditional radar technology. As apreliminary step to defining the MIMO radar ambiguity function, we need to establishwhat previous work has been done in the area of ambiguity function analysis. Theresults given later in this chapter reveal that the MIMO ambiguity function is ageneralization of most of the previous contributions.

GENERALIZED MIMO RADAR AMBIGUITY FUNCTIONS124

The earliest radar systems were designed to make simple measurements. Thesemeasurements included estimating time delay/range or velocity by using continuouswave (CW) radar. As systems became more complex and precise, the inevitableissues of accuracy and resolution arose. Early researchers introduced a functioncalled the ambiguity function that captures some of the inherent resolution propertiesof a radar system. The ambiguity function was first introduced by Ville [14];however, it is generally identified with Woodward because of his pioneering work[15,16]. Woodward was interested in characterizing how well one could identifythe target parameters of time delay (range) and Doppler (range rate) based on thetransmission of a known waveform s(t). He established his ambiguity functionby first noting that a good waveform is one that could be used to distinguishbetween radar returns with different target parameters. He defined a total mean-squared error (MSE) metric between a known waveform s(t) and a frequencyshifted and time-delayed version:

C(Dt, Dfv) ¼ð

s(t)� s(t � Dt)e�j2pDfvt�� 2dt (3:1)

When the square is expanded, the only remaining term depending on the parametersis a term that is the inner product between the original waveform s(t) and thetime-delayed/frequency-shifted version. Woodward called this the radar ambiguityfunction:

x(Dt, Dfv) ¼ð

s(t)s�(t � Dt)e�j2pDfvtdt

��2

(3:2)

A magnitude-squared operation is usually introduced to the inner product term. Inorder to minimize the function (3.1), the ambiguity function should be large for(Dt ¼ 0, Dfv ¼ 0) and small for (Dt=0, Dfv=0). An ideal ambiguity functionis one that resembles a thumbtack in the (Dt, Dfv) plane. A good description of theproperties of this function can be found in Ref. 17. Among its more well-knownproperties is the fact that there is an inherent ambiguity or duality between resolutionin time and resolution in frequency. For a given time–bandwidth product, targetscannot be resolved perfectly in time and frequency simultaneously. In fact, this iswhy (3.2) is referred to as an ambiguity function. In conjunction with this conceptis the idea that ambiguity or energy (volume under the surface) can be movedaround in the (Dt, Dfv) plane but not removed. An intuitive comparison to make isto think of the ambiguity function as analogous to a probability density function.Just as the trace of a covariance matrix bounds the spread of the density function,the time–bandwidth product bounds the spread of the ambiguity function. Itshould be noted that alternative derivations of the ambiguity function exist and areequally valid. In particular, Van Trees [18] provided a derivation of the ambiguityfunction that starts from a statistical view of the received radar signal. In

3.2 BACKGROUND 125

Section 3.4 a statistical description of the received data in a MIMO radar isdeveloped. Using this, a definition of ambiguity is proposed.

It was recognized that Woodward’s ambiguity function needed modification tohandle larger bandwidth signals, long duration signals, and targets with high velocity.Other authors and researchers have undertaken this task by considering waveformswith larger bandwidths and targets described by higher-order motion parameters.The hierarchy of generalizations can be confusing since many authors simply calltheir functions generalized ambiguity functions. Correct modeling of these types ofscenarios has been carried out [19–21]. Those authors use parametric models thatmore accurately reflect the actual physical phenomena involved with movingtargets and reflecting signals. As a traveling wavefield reflects off a moving target,the field either expands or compresses in time as a result of the movement of thetarget. When a narrowband waveform is transmitted, this compressive effect isignored for the waveform’s complex envelope and considered only for the carrier.The condition that must be met for this compressive effect to be ignored is basedon the time–bandwidth product TB, target velocity v, and the propagation speed oftraveling waves c:

2vBT

c� 1 (3:3)

The ambiguity function derived for conditions that violate (3.3) is

x(t, fv) ¼ffiffiffigp ð

s(t)s� g(t � t)ð Þe�j2pfvtdt

��2

(3:4)

The term g ¼ 1þ fv=fc specifically accounts the stretching/compressing in timeof the reflected signal. There is a scalar term in front of the integral to accountfor a change in amplitude of the reflected signal as it is stretched. The amplitudescaling is necessary for the conservation of energy when the waveform isstretched in time. Several authors have suggested similar wideband ambiguity func-tions [22–26].

More recently, connections have been made between the wideband ambiguityfunction and the continuous-time wavelet transform [27–30]. While wideband wave-forms have been shown to enhance resolution, the correct processing of such signalscan be computationally intensive. Fast Fourier transform (FFT) can be used withnarrowband signals to efficiently compute Woodward’s ambiguity function.Equation (3.2) is basically a convolution integral. In contrast, the wavelet transformis the tool associated with processing wideband signals. The wavelet transformdescribes signals that have been shifted in time and time-dilated. Techniques for com-puting wavelet transforms over a continuum of time shifts and time dilations havebeen described [31].

This description of ambiguity functions was based on radar systems operating witha single aperture. Some researchers have also defined ambiguity functions for radarsystems with multiple apertures. The first to do so was Urkowitz [32], who formulatedan ambiguity function that is a function of azimuth, elevation, range, and Doppler. In


his system, the waveform transmitted at each aperture is the same except for a timedelay or phase shift. His focus is on waveforms that are narrowband. As a result,he shows that his ambiguity function can be factored into separable space and timecomponents. It will be shown later that for certain situations, the MIMO ambiguityfunction will also be spacetime-separable. Spacetime separability has been discussedfurther in the literature see [33,34].

Formulations of the ambiguity function have also appeared for radars configuredin a bistatic configuration [35]. The term bistatic has generally been used in theliterature to refer to a radar system with a large separating baseline between transmitand receive apertures. In this configuration the resolution of target parameters such asrange and Doppler becomes highly dependent on the location of the target relative tothe transmit and receive apertures. We will use the term bistatic to refer to any systemconfiguration with separate transmit and receive sensors.

Some authors have proposed ambiguity functions that incorporate the estimationof nuisance parameters. For example, a particular problem might include the jointestimation of delay and Doppler along with the background noise power, althoughthis is not consistent with Woodward’s original concept of ambiguity. Rendas andMoura [36] propose an ambiguity function based on the Kullback–Leiblerdivergence. Their approach to defining the ambiguity function is based on thegeneral problem of parameter estimation in curved exponential families. In theirpaper they are able to show how their ambiguity function reduces toWoodward’s function under special signal model assumptions. In a relatedproblem, Dogandzic and Nehorai [37] derive the CR bounds for estimatingrange, velocity, and angle with an active array. They relate their solution to theambiguity function proposed by Rendas and Moura [36], whose definition of ambi-guity that we propose for MIMO radar is somewhat similar to ours. It should berecognized that we are not defining a notion of ambiguity that is vastly differentfrom those of other authors. Rather, the intent of this chapter is examine the roleof the different MIMO radar system components in the ambiguity function. Amajor difference between our definition of ambiguity and that of Rendas andMoura [36] is that our function is not normalized and does not strictly varybetween 0 and 1. This normalization step is somewhat arbitrary and results in aloss of information. Traditionally, the ambiguity function has been defined as a nor-malized function. Most analysis focuses on the shape of the ambiguity function;however, the overall levels can provide further information, especially withregard to the CR bounds.

3.3 MIMO SIGNAL MODEL

In this section we introduce a general signal model for MIMO radar to clarify theinteraction between the transmitted signals, the target, and the noise. In the followingsection we discuss a particular parametric model to be used in MIMO radar.

A MIMO radar system consists of NT transmit sensors and NR receive sensors. Aseries of independent signals is transmitted from each transmit sensor in a coherent

3.3 MIMO SIGNAL MODEL 127

fashion. The propagation of a signal from a transmit sensor to a receive sensor con-sists of propagation through a channel with three components: a forward-propagatingchannel to the target, a reflecting/scattering target, and a reverse channel to thereceive sensor. Both the forward and reverse channels will be jointly parameterizedby a parametric model with parameter u.

The target will be considered point-like in nature. This means that the physicaldimensions of the target are small enough that it appears to the radar as a single-point target with little or no extent. We do allow, however, for the target to possiblybe composed of many smaller scattering centers. For each pair of transmit and receivesensors, the target response or scattering function is approximated as a realization of arandom process. For the ith transmit sensor and jth receive sensor this scatteringfunction will be denoted as aj,i. The random process is such that over the timeperiod of the transmit signal duration, the realization of the random processis constant.

Given this discussion of the target and parametric channel, the received signal atthe jth receive sensor due to the ith transmit waveform can be expressed as

r j,i(t) ¼ si(t, u, j)a j,i þ nj(t) (3:5)

The term nj(t) is an additive noise process independent of the target scattering func-tion aj,i. The term si(t, u, j) represents the ith transmitted signal modified according tothe parametric channel model with parameter u for the jth receive sensor. Thereceived signal will most likely be sampled; therefore, we can use matrix vectornotation to represent the received signal as an N � 1 vector:

r j,i ¼ si(u, j)a j,i þ nj (3:6)

Now, because there are NT transmit signals, the received signal at the jth receivesensor is the linear combination of all such signals as in Eq. (3.5):

rj ¼XNT

i¼1

r j,i

¼XNT

i¼1

si(u, j)a j,i þ nj

¼ S(u)aj þ nj (3:7)

The term S(u) is a N � NT matrix whose columns are the N � 1 vectors si(u, j):

S(u) ¼ [s1(u, j), s2(u, j), . . . , sNT (u, j)]


There are a total of NR receive sensors; the data from each can be composed into asingle vector of size NNR � 1:

r ¼ rT1 , rT

2 , . . . , rTNR

h iT

¼

S(u, 1) 0 . . . 0

0 S(u, 2) . . . 0

..

. ... . .

. ...

0 0 0 S(u, NR)

266664

377775

a1

a2

..

.

aNR

266664

377775þ

n1

n2

..

.

nNR

266664

377775

¼ S(u)aþ n (3:8)

Equation (3.8) is the signal model used throughout the rest of this chapter to describethe received data due to a set of transmit waveforms, a point-like target with responsevector a, and a parametric channel model with parameter u. The rest of this sectiondescribes the probability density function (pdf) for the data. A common pdf used inradar signal processing is the complex Gaussian distribution. Various authors haveshown how both the noise and target response are well modeled by thisdistribution (e.g., see Ref. 17). The NTNR � 1 target response vector will have the dis-tribution

a � CN (0, s2sS) (3:9)

This denotes a complex Gaussian random vector with zero mean and covariancematrix s2

sS. In a later section the characteristics of the target covariance matrixplay an important role in the ambiguity function. We shall encounter three types oftarget covariance matrices: (1) if the covariance matrix is rank 1, s2

sS ¼ s2s vvH ,

the target is called coherent; (2) if the covariance matrix is a multiple of the identitymatrix, s2

sS ¼ s2s I, the target is called noncoherent; and (3) in cases where the target

covariance matrix has rank .1 but is not a multiple of the identity matrix, the target iscalled partially coherent. The matrix s2

sS is positive semidefinite and therefore hasthe eigenvalue/eigenvector decomposition

s2sS ¼ s2

s VLVH

Also, if the rank of S is Ns � NTNR, then the covariance matrix can be factored as theproduct of two matrices:

s2sS ¼ s2

s QQH

3.3 MIMO SIGNAL MODEL 129

The matrix Q has dimensions NT NR � Ns. It is given by Q ¼ VsL1=2s . The matrix Vs

is the matrix of the first Ns columns of V. The matrix Ls is a square Ns � Ns matrixgiven by the positive definite component of the matrix L.

The noise vector will also be complex Gaussian, with the distribution

n � CN (0, s2nINNR ) (3:10)

The subscript notation on the identity matrix INNR signifies an identity matrix ofdimension NNR � NNR. The received data vector has the distribution

r � CN 0, s2s S(u)SSH(u)þ s2

nINNR

� �(3:11)

Later we will have occasion to use the data loglikelihood function. This is simply thenatural logarithm of the data pdf p(rju) viewed as a function of the parameter u:

L(u; r) ¼ log p(rju)

¼ �NNR ln(p)� ln det s2nINNR þ s2

s S(u)SSH(u)� �

� tr [s2nINNR þ s2

s S(u)SSH(u)]�1rrH� �

(3:12)

Using two matrix identities, we can simplify this expression. The first is a matrixdeterminant identity:

det ABþ Ið Þ ¼ det BAþ Ið Þ (3:13)

Using this identity, the determinant in the loglikelihood function can be written as

det s2nINNR þ s2

s S(u)SSH(u)� �

¼ (s2n)NNR det INs þ

s2s

s2n

QHSH(u)S(u)Q� �

The second identity is the matrix inversion lemma. If A ¼ Bþ CDCH , then

A�1 ¼ B�1 � B�1C D�1 þ CHB�1C� ��1

CHB�1 (3:14)

Using this identity, we can express the data covariance matrix as

s2nINNR þ s2

s S(u)SSH(u)� �1 ¼ 1

s2n

� �INNR �

1s2

n

� �s2

s

s2n

� �S(u)Q

� INs þs2

s

s2n

� �QHSH(u)S(u)Q

��1

QHSH(u)


With these two simplifications the loglikelihood function may be rewritten as

L(u; r) ¼ �NNR ln (ps2n)� ln det INs þ

s2s

s2n

QHSH(u)S(u)Q� �

� 1s 2

n

� �tr(rrH)

þ s2s

s4n

� �tr INs þ

s2s

s2n

� �QHSH(u)S(u)Q

��1

QHSH(u)rrHS(u)Q

!(3:15)

3.4 MIMO PARAMETRIC CHANNEL MODEL

In this section we introduce a parametric model that describes how the transmitsignals appear at the receive sensors. The parts of this model include signal trans-mission, signal propagation, signal reflection, and signal reception.

3.4.1 Transmit Signal Model

Currently, proposed MIMO radars consist of coherent networks of transmit andreceive sensors. These sensors could be distributed apertures or elements of asingle phased array. In this development, we assume that all sensors have an isotropicradiation pattern and that no mutual coupling between sensors occurs. A MIMO radarhas NT transmit sensors and NR receive sensors. The ith transmit sensors spatiallocation will have Cartesian coordinates given by the column vector xi,T and the jthreceiver will have Cartesian coordinates xj,R. These coordinates are referenced toa predefined origin shared by both the sensors and the target. The 3 � NT and3 � NR matrices XT and XR denote the collection of all transmit and receive sensorlocations:

XT ¼ [x1,T ; x2,T ; . . . ; xNT ,T ]

XR ¼ [x1,R; x2,R; . . . ; xNR,R]

Each transmit sensor has the ability to transmit an independent waveform. The actualsignal transmitted from the ith sensors is

gi(t) ¼ 2Re{si(t)}

All sensors operate at the same carrier frequency fc that is referenced to the samephase angle on transmit and receive for all sensors. In this sense the waveforms arephase-coherent relative to the carrier. The bandwidth B and time duration Tare considered to be constant for all waveforms si(t). Moreover, it is assumedthat the signal bandwidth satisfies the condition B/2 , fc. Under this assumption,

3.4 MIMO PARAMETRIC CHANNEL MODEL 131

the common complex envelope notation can be applied to the form of the trans-mitted signal

gi(t) ¼ 2Re{si(t)ej2pfct} (3:16)

where si(t) is the complex envelope of the ith waveform.

3.4.2 Channel and Target Models

On transmission, the radar signals as given by the model in Eq. (3.16) propagatethrough free space, reflect off objects, and return to the radar receivers. Proper statisti-cal modeling of the received signals requires a thorough understanding of the propa-gation channel characteristics and the target reflection process. For each transmit/receive sensor pair there exists a forward transmit channel to the target and areverse receive channel from the target. These channels are modeled as losslesstime delay and phase shift channels. All others losses, such as R4 attenuation, willbe assumed to be due to the target reflection process.

Several authors have considered various models for target reflection in MIMOradar systems; the two extremes are noncoherent scattering and coherent scattering.The coherence of the scattering is reflected in the covariance matrix of the targetresponse. Proper model selection is based on a careful comparison of targetcomplexity, signal bandwidth, signal duration, target motion, and sensor arrayconfiguration. The target model applied in this chapter is a point-like target movingwith constant velocity.

Point targets will be described by a parameter vector u consisting of a positionvector component p and velocity vector component v. The position vector will bedefined in the same coordinate system as the transmit and receive sensor arrays.The velocity vector will also be defined in the array coordinate system. Dependingon the array configuration, each target could be identified by up to six unambiguousparameters, three for position and three for velocity. We will now show how theseparameters appear in the model. In general, if the signal h(t) is transmitted fromsensor i, reflects off a target, and is received by sensor j, the response can bedescribed by

b(u, i, j)h t � d(t, u, xi,T , x j,R)� �

(3:17)

The function d(t, u, xi,T , x j,R) is a time-varying function that depends on the targetparameters and sensor locations. Using a Taylor series expansion, we find

d(t, u, xi,T , x j,R) � ti, j(p)� f (u)fc

(t � ti, j(p)) (3:18)

The same function can be derived using the Lorentz transformation, which specifieshow wavefields and signals are related for different inertial reference frames. This par-ticular derivation assumes that the target velocities of interest are significantly less


than the speed of light. The term ti, j(p) is simply the two-way time delay due to atarget located at p with transmit and receive sensors at x i,T and x j,R:

ti, j(p) ¼ ti(p)þ tj(p)

¼ kp� xi,Tkc

þ kp� x j,Rkc

(3:19)

where k.k denotes the usual Euclidean vector norm.The term f (u) is the frequency shift caused by a target moving with velocity vector

v and position p. The frequency shift is caused by an instantaneous change in path-length between transmitter, target, and receiver. For simplification assume the radar isfixed, then f (u) is defined as

f (u) ¼ 1l

d

dt(RT þ RR)

�(3:20)

where RT and RR are the transmit and receive pathlengths. Each derivative is theprojection of the target velocity vector onto either the transmitter or receiver line ofsight (LOS):

1l

d

dtRT ¼

1l

,p� xi,T

kp� xi,Tk, v fi(u)

1l

d

dtRR ¼

1l

,p� xi,R

kp� xi,Rk, v fj(u)

Now the frequency shifts can be explicitly written in terms of a transmit componentand receive component:

fi, j(u) ¼ fi(u)þ fj(u) (3:21)

Under certain assumptions concerning array geometry, simplifications can be made toboth the formulas for time delay and frequency shift. In a later section, a hierarchy ofassumptions is outlined that will have implications for the form of the MIMO ambi-guity function. It is also convenient to define the stretch factor

gi, j(u) ¼ 1þ fi, j(u)fc

3.4.3 Received Signal Parametric Model

Using the parametric model described above, the received signal at the jth receivesensor before demodulation to baseband can be written as

~rj(t, u) ¼XNT

i¼1

~a j,i

ffiffiffiffiffiffiffiffiffiffiffiffigi, j(u)

qgi gi, j(u)(t � ti, j(p))� �

þ ~nj(t)

3.4 MIMO PARAMETRIC CHANNEL MODEL 133

The quantities aj,i and nj(t) represent the target response and noise response beforedemodulation to baseband or another intermediate frequency. After complex demod-ulation to baseband, the received signal signal is

rj(t, u) ¼XNT

i¼1

a j,i

ffiffiffiffiffiffiffiffiffiffiffiffigi, j(u)

qsi gi,j(u)(t � ti, j(p))� �

e�j2pti, j(p)(fcþfi, j(u))e j2pfi, j(u)t

þ nj(t) (3:22)

This is same data model as described by Eq. (3.7), but now with the parametric modelexplicitly stated. This is a continuous-time version of the received signal.

3.5 MIMO AMBIGUITY FUNCTION

In this section we develop the ambiguity function for the MIMO radar systemdescribed in the previous section. We mentioned in Section 3.2 that the ambiguityfunction is a tool used to indicate both the local and global resolution properties ofsensing systems. It is most commonly associated with active sensing such as radar,and often is regarded as only a function of the transmit waveforms. However, thegeometric sensor configuration of a MIMO radar can have as much of an impact onresolution as the waveforms. Moreover, because different sensors can observedifferent scattering characteristics from a single-point target, the ambiguity functionis also a function of the target, or at least its statistical properties. It therefore iscorrect to view the ambiguity function as a function of the whole radar system:geometry, waveforms, and target. We define the generalized MIMO ambiguityfunction as the expected value of the data loglikelihood function for the receiveddata with respect to the distribution p(rju0). The parameter u0 is consideredto be the true target parameter, whereas the parameter u1 is the hypothesizedtarget parameter:

A(u0, u1) ¼ E p(rju0)[L(u1jz)] (3:23)

This definition results in an ambiguity function with properties similar to those ofother well-known ambiguity functions. In Section 3.2 we mentioned that Rendasand Moura [36] propose an ambiguity function that is defined in terms of I diver-gence. Our definition of the MIMO radar ambiguity function is somewhat similar.Both reduce to Woodward’s ambiguity function under the correct conditions, andboth define functions that are possibly asymmetric in the parameters u0 and u1.Woodward’s ambiguity function is a special case in which the ambiguity functionis symmetric in the two parameter arguments.

As stated, ambiguity functions provide both local and global properties ofparameter estimator performance. The Cramer–Rao lower bounds are the commontool used to evaluate the local performance of unbiased estimators. With the


definition of ambiguity as in Eq. (3.23), the curvature of the function about a point u0

gives the elements of the Fisher information matrix (FIM):

Jij ¼ �E p(rju0)@2lnp(rju1)

@ui1@u

j1

" #��u1¼u0

¼ � @2

@ui1@u

j1

A(u0, u1)

��u1¼u0

Using the definition of the data loglikelihood function in Eq. (3.15), we can writethe ambiguity function as follows. It can be expressed in terms of two functions, anautocorrelation function, and a cross-correlation function:

A(u0, u1) ¼ E p(zju0)[L(u1jz)]

¼ �NNR ln (ps2n)� ln det INs þ

s2s

s2n

f(u)

� �� 1

s2n

� �tr(E p(zju0)[rrH])

þ 1s2

n

� �s2

s

s2n

� �tr INs þ

s2s

s2n

� �f(u)

��1

QHSH(u)E p(zju0)[rrH]S(u)Q

!


s2s

s2n

f(u1)

� �� NNR �

s2s

s2n

tr f(u0)ð Þ

þ s2s tr INs þ

s2s

s2n

� �f(u1)

��1

f(u1)

!

þ s2s

s2s

s2n

tr INs þs2

s

s2n

� �f(u1)

��1

QHSH(u1)S(u0)QQHSH(u0)S(u1)Q

!


s2s

s2n

f(u1)

� �� NNR �

s2s

s2n

tr f(u0)ð Þ

þ s2s tr INs þ

s2s

s2n

� �f(u1)

��1

f(u1)

!

þ s2s

s2s

s2n

tr INs þs2

s

s2n

� �f(u1)

��1

c(u1, u0)cH(u1, u0)

!(3:24)

The functions f(u) and c(u1, u0) are matrix-valued functions defined as

f(u) ¼ QHSH(u)S(u)Q (3:25)

c(u1, u0) ¼ QHSH(u1)S(u0)Q (3:26)

3.5 MIMO AMBIGUITY FUNCTION 135

Equation (3.25) is an autocorrelation-type function that is the inner product of the signalmatrix S(u) and projects itself onto the weighted subspace of the target covariancematrix S. Similarly, Eq. (3.26) defines a cross-correlation-type function. It is theinner product of the signal matrix S(u) under two different parameter values and isthen projected onto the weighted subspace of the target covariance matrix S. If therank of the target covariance is .1, then these two functions will be matrix-valuedwith dimensions Ns � Ns.

In our definition of the ambiguity function there is a strong dependence on boththe target power level s2

s and the noise power level s2n because the definition is

based on the data loglikelihood function. Under some special conditions the shapeof the ambiguity function is largely unaffected by these two terms. Specifically,when the SNR defined as the ratio SNR ¼ s2

s=s2n is large, then the overall shape

of the ambiguity function approaches a limiting form. It is scaled according to theactual value of s2

s . At a certain threshold SNR, the ambiguity function as definedbegins to flatten out and the noise begins to dominate.

In the case of a single collocated transmitter and receiver, the expression in (3.24)reduces to a form that includes Woodward’s ambiguity function:

A(u1, u0) ¼ �N ln (ps2n)� ln 1þ s2

s

s2n

f(u1)

� �� N � s2

s

s2n

f(u0)

þ s2s

f(u1)1þ (s2

s=s2n)f(u1)

þ s2s

s2s

s2n

c(u1, u0)j j2

1þ (s2s=s

2n)f(u1)

¼ �N ln (ps2n)� ln 1þ s2

s

s2n

E

� �� N � s2

s

s2n

E

þ s2s

E

1þ (s2s=s

2n)Eþ s2

s

s2s

s2n

c(u1, u0)j j2

1þ (s2s=s

2n)E

/ c(u1, u0)j j2

¼ð ffiffiffiffiffiffiffiffiffiffiffi

g(u1)p

s(g(u1)(t � t(p1))

��ffiffiffiffiffiffiffiffiffiffiffig(u0)

ps�(g(u0)(t � t(p0))

e�j2pt(p1) ( fcþf (u1))e j2pt(p0) ( fcþf (u0))

e j2p( f (u1)�f (u0))tdt��2

�ð

s(t � t(p1))s�(t � t(p0))e j2pt( f (u1)�f (u0))dt

��2

(3:27)

¼ð

s(t)s�(t � t)e j2ptfvdt

��2

¼ x(t, fv) (3:28)


The approximation in (3.27) is the standard narrowband assumption; the frequencyshift due to target motion does not affect the complex envelope for small time–bandwidth signals. We see that (3.24) does indeed reduce to (3.28) [the sameas (3.2)], which is solely a function of relative time shift t and Doppler shift fv.Woodward’s ambiguity function has been obtained; however, if no terms aredropped, (3.24) actually equals Woodward’s up to an additive scalar and scalarmultiplicative factor. Equations (3.27)–(3.28) have applied the fact that in this scen-ario the autocorrelation function (3.25) is equal to a scalar constant E (signal energy)for all u.

In the remainder of this section we examine Eq. (3.24) further and explore aspectsof signal design in determining radar resolution capability. All the expressions thatfollow use the coherent target model. Without loss of generality, the target covarianceis given as S ¼ 11T. Additionally, we specialize the MIMO ambiguity function to afew scenarios of interest.

3.5.1 MIMO Ambiguity Function Composition

When the cross-correlation function (3.26) is expanded by substituting the definitionof received signal rj(u), the result is a triple sum over the indices ( j, i, i ):

c(u1, u0) ¼ 1T SH(u1)S(u0)1

¼XNR

j¼1

XNT

i¼1

XNT

i¼1

ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffigi, j(u1)

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffigi,j(u0)

q

si gi, j(u1)(t � ti, j(p1))� �

s�igi, j(u0)(t � ti, j(p0))�

e�j2pti, j(p1)(fcþfi, j(u1))e j2pti, j(p0)(fcþfi, j(u0))

e j2p(fi, j(u1)�fi, j (u0))tdt (3:29)

One observation that can be made is that the exponential terms are components intransmit and receive array steering vectors. In its present form with no simplificationsbased on array geometry and target location, there is very little, if anything, that canbe done to simplify this expression. One notational simplification that can be made isto collect all the inner product terms into a single matrix. Define the NT � NT matrixR(u1, u2, j), where the (i, i) element is equal to

Ri, i(u1, u0, j) ¼ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gi, j(u1)q

si gi, j(u1)(t � ti, j(p1))� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffigi, j(u0)

qs�

igi, j (u0)(t � ti, j (p0))�

e j2p ( fi, j (u1)�f i, j (u0))tdt (3:30)


Every element of the matrix has a functional dependence on u1, u0, and j. Using thisnotation, we can rewrite the cross-correlation function as

c(u1, u0) ¼XNR

j¼1

XNT

i¼1

XNT

i¼1

Ri, i(u1, u0, j)

e�j2pti, j(p1)(fcþfi, j(u1))e j2pt i, j(p0)(fcþf i, j(u0)) (3:31)

One further simplification would be to write out a combined transmit–receive steer-ing vector. We define the following NT � 1 vector:

aTR(u, j) ¼ e j2pt1, j (p)(fcþf1, j (u)); e j2pt2, j(p)(fcþf2, j(u));�

. . . ; e j2ptNT , j(p)( fcþfNT , j(u))T

(3:32)

Using the notation in Eq. (3.32), we can express the cross-correlation function as asum over a series of quadratic forms:

c(u1, u0) ¼XNR

j¼1

aHTR(u1, j)R(u1, u0, j)aTR(u0, j) (3:33)

Some simple observations can be made about the cross-correlation function whenwritten in this form. The transmit signals play a role only in the covariancefunction. By changing the signals that are sent, one can control the covariancefunction over the parameter space u ¼ u1 � u0. In the case of transmitting arbi-trary wideband signals while observing high-velocity targets using a nonsimplearray geometry, all target parameters become coupled in the covariance function.As we show later, under the assumption of narrowband signals there is a space-time separability that falls out of the cross-correlation function. In the followingsections a few specific geometric array configurations and signal scenarios areexamined. These scenarios were chosen to highlight certain key factors at playin the cross-correlation function and thus the ambiguity function. There will bea hierarchy to the simplifications in that the progression will be from more toless complexity.

3.5.2 Cross-Correlation Function under Model Simplifications

The first simplification to be made concerns the impact of the target velocity. Aswas shown in a previous section, when high-velocity targets are observed withhigh time–bandwidth product transmit signals, the resulting compressive effect onthe complex envelope in time cannot be ignored. If the target velocity is slowenough that the condition in Eq. (3.3) holds, then the signal compression can beignored in the covariance function. In this simplification and all to follow, modifiedversions of various components of the cross-correlation function are denoted by


superscript numbers. The covariance function elements under the abovementionedassumption can be written as follows:

R1i, i

(u1, u0, j) ¼ð

si(t � ti, j(p1))s�i(t � ti, j(p0))e j2p(fi, j(u1)�fi, j(u0))tdt (3:34)

The cross-correlation function then becomes

c1(u1, u0) ¼XNR

j¼1

aHTR(u1, j)R1(u1, u0, j)aTR(u0, j) (3:35)

Thus far no assumptions have been made concerning the location of the target withrespect to the array. The model used is one in which components of the target velocityvector can be identified. This identifiability results from the placement of the sensorsin relation to the target. The locations are such that different sensors might see differ-ent frequency shifts due to a single velocity vector. A common model simplificationthat can be applied is to assume that all the sensors are close enough to each other sothat, for a given target velocity vector, the projection of that vector onto each sensor’sLOS is nearly identical. Essentially, this assumption is valid only if the the bistaticangle b for the transmit–receive sensor pair with the largest separation falls withina predefined threshold. Theoretically, any nonzero bistatic angle would allow forthe resolution of velocity vector components; however, there would appear to be apractical limit. Under this assumption, target velocity vector components cannotbe resolved unambiguously; therefore, the velocity vector parameter should bereduced to a scalar parameter corresponding to the radial velocity along the radar’sLOS. The frequency shift terms in the ambiguity function now become independentof individual sensors, allowing the array steering vectors to be decomposed into atransmit vector and receive vector. We can apply this simplification by first rewritingthe terms in the covariance function:

R2i, i(u1, u0, j) ¼

ðsi t � ti, j(p1)� �

s�i (t � ti, j(p0))e j2p( fv1�fv0 )tdt (3:36)

The transmit array steering vectors are

aT (u) ¼ e j2pt1(p)( fcþfv); e j2pt2(p)( fcþfv); . . . ; e j2ptNT (p)( fcþfv)� T

(3:37)

The receive array steering vectors are

aR(u) ¼ e j2pt1(p)( fcþfv); e j2pt2(p)( fcþfv); . . . ; e j2ptNR (p)( fcþfv)� T

(3:38)


Now we can write the cross-correlation function as

c2(u1, u0) ¼XNR

j¼1

aHT (u1)R2(u1, u0, j)aT (u0)

e�j2ptj( p1)( fcþfv1 )e j2ptj( p0)( fcþfv2 ) (3:39)

A third simplification to the model could be that the target appears in the far field ofthe array. Under this assumption propagating waves appear planar. Furthermore,equations derived for actual time delays due to target position can be approximated.First, the components of the position vector should be changed to (range, azimuth,elevation) from (x, y, z). Now the most sensible coordinate system is an array-centeredcoordinate system with one of the elements chosen as the phase center and origin.Range is the distance to the target as measured from the phase center. Azimuth andelevation are also referenced to the phase center. For a target in the far field, thetime delays can be approximated as

ti, j(p) ¼ ti(p)þ tj(p)

� r

c� uT (uaz, uel)xi,T

cþ r

c� uT (uaz, uel)x j,R

c

¼ 2r

c� xi,T

cþ x j,R

c

h iuT (uaz, uel) (3:40)

The function uT (uaz, uel) is a unit vector in the direction specified by (uaz, uel):

uT (uaz, uel) ¼ cos(uaz) sin(uel); sin(uaz) sin(uel); cos(uel)½ �T (3:41)

At this point the cross-correlation function will not be rewritten since notationally theexpression does not become much simpler. However, the cross-correlation function,denoted as c3(u1, u0), will refer to the case when the target is in the far field andtarget velocity is represented by a scalar. The target parameter vector has four com-ponents, u ¼ (r, uaz, uel, v). Some of the examples will refer to this form of the cross-correlation function. Similarly, R3

i,i(u1, u0, j) will refer to elements of the covariance

function in this case.The last model simplification reduces the bandwidth significantly so that the

narrowband assumption can be applied to waveforms. Under this assumption, thewaveforms are considered sufficiently narrowband that actual intersensor timedelays can be ignored in the delay of the complex envelope. This simplifies thecovariance function. Under the narrowband assumption, we obtain

R4i, i(u1, u0, j) ¼

ðsi tð Þs�i t � 2

c(r1 � r0)

� �e j2p(fv1�fv0 )tdt

¼ð

si tð Þs�i t � Dtð Þe j2pDftdt (3:42)

; R(Dt, Df ) (3:43)


As expected, each element of the covariance function is the simple Woodward ambi-guity function depending on Dt and Df only. We call R(Dt, Df ) the narrowbandcovariance function. When this function is substituted into the cross-correlationfunction, one can see that there is a separability now between space (angle) andtime (range):

c4(u1, u0) ¼XNR

j¼1

e�j2ptj(p1)( fcþfv1 )e j2ptj(p0)( fcþfv0 )aHT (u1)R4(u1, u0, j)aT (u0)

¼XNR

j¼1

e�j2ptj(p1)( fcþfv1 )e j2ptj(p2)( fcþfv0 )aHT (u1)R(Dt, Df )aT (u0) (3:44)

Careful inspection of Eq. (3.44) shows that the cross-correlation function is now inde-pendent of the absolute range terms that appear in the complex exponentials. Thesecomplex multipliers disappear when the magnitude squared is applied to the sum.Also, the steering vectors are now the same as the standard narrowband steeringvectors with a frequency shift applied to the carrier frequency. Equation (3.44) canbe written compactly as

c4(u1, u0) ¼ aHT (u1)R(Dt, Df )aT (u0)aH

R (u1)aR(u0) (3:45)

where

aT (u) ¼ exp j2px1,TuT (uaz,uel)

c( fc þ fv)

� �; exp j2px2,T

uT (uaz,uel)c

( fc þ fv)

� �;

. . . ; exp j2pxNT ,TuT (uaz,uel)

c(fc þ fv)

� ��T

(3:46)

aR(u) ¼ exp j2px1,RuT (uaz,uel)

c( fc þ fv)

� �; exp j2px2,R

uT (uaz,uel)c

( fc þ fv)

� �;

. . . ; exp j2pxNR,RuT (uaz,uel)

c( fc þ fv)

� ��T

(3:47)

3.5.3 Autocorrelation Function and Transmit Beampatterns

In previous papers [5,7,8] it was shown how the choice of transmit waveform signalcorrelation could affect the transmit beampattern of a MIMO radar. The major resultsare as follows. For an array independently transmitting narrowband wide-sense-stationary waveforms, the transmit beampattern function is

PN(u) ¼ aH(u)Ra(u) (3:48)


Here aH(u) is the narrowband transmit array steering vector parameterized by theangle u and R is the zero-lag signal correlation matrix. Similarly, for an array trans-mitting wideband signals, the transmit beampattern is

PW (u) ¼ð

BaH(u, fc þ f )S( f )a(u, fc þ f )df (3:49)

Again, aH(u, fc þ f ) is the narrowband transmit steering vector parameterized by theangle u, but now specifically calculated at the frequency fc þ f. The matrix S( f ) is thecross-spectral density matrix (CSDM) of the transmitted waveforms. It is defined asthe elementwise Fourier transform of the signal correlation matrix:

S( f ) ¼ðT

0R(t)e�j2pftdt (3:50)

Returning to the definition of the MIMO ambiguity function, we can identify the roleof the transmit beampattern. To begin, consider the second model simplification. InEq. (3.39) the term aH

T (u1)R2(u1, u0, j)aT (u0) appears. If the two target parametersare equal, u1 ¼ u0, this term becomes the transmit beampattern. The dependence ofthe covariance function on j drops out:

R2i,i(u1, u1, j) ¼

ðsi(t � ti, j(p1))s�i (t � ti, j(p1))dt

¼ð

si(t)s�i(t � [ti, j(p1)� ti, j(p1)])dt

¼ð

si(t)s�i(t � [ti(p1)� ti(p1)])dt

; R2i,i(ti(p1)� ti(p1)) (3:51)

¼ð

BSi,i(f )e j2p(ti(p1)�ti(p1))f df (3:52)

Taking the result of Eq. (3.52) and substituting back into the quadratic form, one findsthe expression for the wideband transmit beampattern:

aHT (u1)R2(u1, u1, j)aT (u1) ¼

XNT

i¼1

XNT

i¼1

ðB

Si,i( f )e j2p(ti(p1)�ti(p1))f

e j2pfc(ti(p1)�ti(p1))df

¼ð

BaH

T (p1, fc þ f )S( f )aT (p1, fc þ f )df : (3:53)

Equation (3.53) is the same as Eq. (3.49) except for a slight notation difference. Wefind the presence of the parameter p1 instead of u. The difference is that p1 is a true


spatial parameter, whereas u is only an angle parameter. If the far-field simplificationswere applied, then the angle parameter u would appear. Regardless, this equation istelling us how much gain one can expect for targets located at different pointsspatially. It is much easier to identify the transmit beampattern in the narrowbandsimplification of the cross-correlation function as given by Eq. (3.44). The quadraticform aH

T (u1)R(Dt, Df )aT (u0) becomes the narrowband transmit beampattern whenu1 ¼ u0(Dt ¼ 0, Df ¼ 0).

Now, if we return to Eq. (3.39), we see that letting u1 ¼ u0 results in the cross-correlation function becoming a scaled version of the transmit beampattern. This isactually just the autocorrelation function defined in Eq. (3.25):

c2(u1, u1) ¼XNR

j¼1

aHT (u1)R2(u1, u1, j)aT (u1)e�j2ptj(p1)(fcþfv1 )e j2pitj(p1)( fcþfv1 )

¼XNR

j¼1

aHT (u1)R2(u1, u1, j)aT (u1)

¼ NR

ðB

aHT (p1, fc þ f )S( f )aT (p1, fc þ f )df

¼ f(u1) (3:54)

3.6 RESULTS AND EXAMPLES

In this section some visual examples of MIMO ambiguity functions are presented toillustrate new signal design possibilities that arise by using MIMO radar. It is beyondthe scope of this chapter to actually undertake the design of MIMO radar waveforms;the results in this section simply provide motivation for the usefulness and necessityof the MIMO ambiguity function. They also show how the MIMO ambiguity func-tions can be used to evaluate sets of waveforms for use in a particular radar systemconfiguration.

The previous sections showed how the ambiguity function is composed of twoconstituent functions: an autocorrelation function (equivalent to the transmit beam-pattern) and a cross-correlation function. In each of the following examples webegin by showing both of these functions separately. They are then be combinedto show the MIMO radar ambiguity function. An interesting result that occurs isthat two different choices of transmit signals may result in similar cross-correlationfunctions but different autocorrelation functions and thus will have greatly differentambiguity functions.

3.6.1 Orthogonal Signals

In this first example we examine the MIMO ambiguity function for a system thattransmits nearly orthogonal signals from each transmit aperture. The system consists

3.6 RESULTS AND EXAMPLES 143

of three transmit/receive sensors located on the x axis at [233l 0 20l]. Theoperating frequency is set to fc ¼ 2 GHz, so the spacing works out to be[25 m 0 m 3 m]. Each sensor transmits and receives with an omnidirectionalpattern. A single point target is placed in the far field. The target is moving withconstant velocity. Each waveform has a bandwidth B ¼ 500 MHz and time dur-ation T ¼ 20 ms; the time-bandwidth product is therefore TB ¼ 10,000. Thetarget is in the far field and assumed to be moving slow enough that waveformtime-compression can be ignored; hence form 3 of the cross-correlation functionwill be used. Under all these assumptions, the ambiguity function will be a func-tion of 3 unambiguous arguments, range, azimuth, and Doppler frequency,u0 ¼ {r0, uaz,0, fv0 }, for a fixed set of target parameters u1 ¼ {r1, uaz,1, fv1 }.

Figure 3.1 shows a single slice of the cross-correlation function. It is a range–angle slice taken at zero Doppler. The units used in this figure and others aredBi’s. This means that the peak of the cross-correlation function is referenced tothe level expected by a sensor isotropically transmitting a unit energy waveform.There are many ways to compute and visualize these functions. In this example thetarget has been placed in space with parameters u0 ¼ (1000 km, 08, 0 m/s), andthe parameter u1 ¼ (r1, uaz,1, fv1 ) has been varied over the parameter space. Beforeproceeding further, some important comments must be made concerning the cross-correlation ambiguity function shown in Fig. 3.1. A visual inspection of this figureshows the presence of six distinct lines. Actually, there are nine lines; in three ofthe lines there are two overlapping lines. The reason for seeing nine lines is asfollows. Three orthogonal waveforms reflect off the target and are received by thethree receivers, at each receiver a filterbank is implemented in which each filterresponds to only one transmitted waveform due to orthogonality, and at each receiverwe see three responses: three receivers with three responses each, resulting in ninelines. These lines are ridges of ambiguity. In this figure they are range–angle

Figure 3.1 Range–angle cross-correlation — orthogonal signals — u0 ¼ (0 m, 08, 0 m/s).


ambiguity ridges for each transmit waveform/receive sensor filter pair. The numberof ridges varies depending on the type of waveform transmitted, the specific arraygeometry, and the target location.

Another interesting characteristic of Fig. 3.1 is the interference patterns producedwhen the ambiguity ridges overlap. The cause of these patterns is the unique phaseramp possessed by each ambiguity ridge. As the ridges overlap, large constructiveand destructive nulls are produced. The width of each ridge is inversely proportionalto the waveform bandwidth B. In a radar with closely spaced sensors the bandwidth isnot large enough for the ridges to bend away from one another. Instead, the ridgeswould lie on top of each other, resulting in a single ridge.

In addition to the range–angle slice shown in Fig. 3.1, we can make other 2Dslices or show the cross-correlation function in 3D. Figure 3.2 is a visualization ofthe cross-correlation function in 3D, where different 2D slices have been placed intheir 3D location. One of the important properties of transmitting orthogonalsignals from a MIMO radar is the resolution performance enhancement. Thisperformance enhancement is a consequence of the ability to obtain uniform targetparameter resolution simultaneously at multiple points in space and time. To illustratethis concept, the position of the point target in Fig. 3.1 is changed while the trans-mitted waveforms and array geometry are held fixed. Figure 3.3 shows a range–angle slice of the cross-correlation function for this scenario. As is evident fromthe figure, the cross-correlation function retains its previous shape, but is nowshifted to the true target position. This figure illustrates the fact that with careful selec-tion of the transmitted waveforms, good resolution can be obtained over large regionsof space and time. Figure 3.4 shows the autocorrelation function for the arraydescribed above transmitting orthogonal signals and demonstrates that equal poweris applied across all azimuth angles and ranges on transmit. The actual ambiguity

Figure 3.2 Cross-correlation function — orthogonal signals — u0 ¼ (0 m, 08, 0 m/s).


function is found by computing the function given by the expression (3.24). Becausethe autocorrelation function is relatively invariant across the entire parameter space,the ambiguity function is simply a scaled version of the cross-correlation function.There is a slight dependence on the chosen values of s2

s and s2n, which were selected

as (s2s , s2

n) ¼ (10, 1). Figure 3.5 shows the ambiguity function using orthogonalsignals. The 23 dB contour levels have been circled. This is done for comparisonwith a similar figure using coherent signals.

Figure 3.4 Autocorrelation — orthogonal signals.

Figure 3.3 Range–angle cross-correlation — orthogonal signals — u0 ¼ (0 m, 11.58, 0 m/s).


3.6.2 Coherent Signals

In the results shown above, orthogonal waveforms were transmitted from each sensor.Now the other waveform extreme is examined, transmitting coherent waveforms.These waveforms are coherent in the sense that each sensor transmits the same wave-form up to a time delay or phase shift. The waveforms in this scenario focus at 08azimuth on transmit, so no time delays or phase shifts are applied.

Figure 3.6 shows a range–angle slice of the cross-correlation function for the casewere a target is located at u0 ¼ (1000 km, 08, 0 m/s). Again, the parameter u1 is

Figure 3.5 Range–angle ambiguity function — orthogonal signals — u0 ¼ (0 m, 08, 0 m/s).

Figure 3.6 Range–angle cross-correlation function — coherent signals — u0 ¼ (0 m, 08, 0 m/s).


varied to produce the ambiguity function. This figure shows that when the target is inthe same location as the focus point of the coherent waveforms, the same cross-correlation function is achieved as in the orthogonal waveforms case above.

Careful inspection of the figure shows that there is a 9 dB increase in the peak ofthe ambiguity function. This is the result of the coherent combination of the trans-mitted waveforms at the target. Another way to describe this result is that nowthere are really 27 ambiguity ridges in the MIMO ambiguity function. For everyone ridge in Fig. 3.1, there are three in Fig. 3.6. As before, the target location canbe varied and a new cross-correlation function can be produced. The target is

Figure 3.7 Range–angle cross-correlation — coherent signals — u0 ¼ (0 m, 11.58, 0 m/s).

Figure 3.8 Autocorrelation function — coherent signals.


moved to the second position as used in the orthogonal signal case. Again, the arraygeometry and waveforms are left unchanged. Figure 3.7 shows a range–angle slice ofthe cross-correlation function. Now there are many ambiguity ridges with consider-able constructive and destructive interference. This serves as an example of whatcoherent waveforms cannot accomplish. Specifically, when perfectly coherent wave-forms are transmitted, the ability to focus uniformly on receive for different u1 valuesis lost. This is in contrast to the orthogonal waveform scenario where the cross-correlation function appears to simply be shifted. Similar to the orthogonal signalscenario, the autocorrelation function can be calculated. The result is the plot inFig. 3.8. Clearly this illustrates that with the use of coherent waveforms, there is anonuniform illumination of space with respect to angle. When the cross-correlationand autocorrelation functions are combined to form the MIMO ambiguity function,the result is as shown in Fig. 3.9. As in the case with orthogonal signals, the23 dB contour levels have been circled. In this ambiguity function the 23 dB con-tours are wider and more numerous as compared to the case using orthogonal signals.

3.7 CONCLUSION

This chapter has presented a comprehensive study of ambiguity and resolution inmodern MIMO radar systems and has shown how the ideas of radar ambiguity func-tions developed since the mid-1950s can be extended to the newly proposed class ofMIMO radar systems. These systems are characterized by independent but coherentsensors possibly distributed over large baselines, transmitting waveforms withlarge fractional bandwidths. As a result of additional degrees of freedom, an ambigu-ity function is defined, which, with the proposed parametric model, is a function of12 parameters. We have shown how this function could be simplified under various

Figure 3.9 Range–angle ambiguity function — coherent signals — u0 ¼ (0 m, 08, 0 m/s).

3.7 CONCLUSION 149

scenarios, and even reduced to Woodward’s ambiguity function for simplesingle-sensor systems. The key result presented here is that there is a spacetimecovariance function produced by the transmitted waveforms that governs the resol-ution performance over the parameter space. Connections were made between theMIMO ambiguity function and past work done in the area of transmit beampatternsynthesis.

The results presented in this chapter show how the ideas of ambiguity functionscan be applied to the new class of radars that use MIMO technology. Just asWoodward’s ambiguity function is used as a tool for individual waveform design,the MIMO ambiguity function should be used for designing good MIMO waveforms.There are several areas of MIMO radar design that need further research. To date, theauthors know of little, if any, work that has been done on actually producing radarwaveforms capable of achieving the middle ground between orthogonality andperfect coherence. Clearly, for the results presented in this chapter to be of any prac-tical use, this issue must be resolved. Another line of research that would complementthis study would be to examine a more global parameter estimation bound such as theBarakin bound or Weiss–Weinstein bound for these MIMO systems. In many of thevisual plots shown, large ambiguities were present that might cause errors whendetecting targets at low SNR. These global bounds should capture the SNR thresholdeffects of parameter estimation problems.

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MIMO Radar Signal Processing || Generalized MIMO Radar Ambiguity Functions

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