IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007 167
MIMO Radar Ambiguity FunctionsGeoffrey San Antonio, Student Member, IEEE, Daniel R. Fuhrmann, Senior Member, IEEE, and
Frank C. Robey, Member, IEEE
Abstract—Multiple-Input Multiple-Output (MIMO) radar hasbeen shown to provide enhanced performance in theory and inpractice. MIMO radars are equipped with the ability to choosefreely their transmitted waveforms at each aperture. In conven-tional radar systems Woodward’s ambiguity function is used tocharacterize waveform resolution performance. In this paper weextend the idea of waveform ambiguity functions to MIMO radars.MIMO ambiguity functions are developed that simultaneouslycharacterize the effects of array geometry and transmitted wave-forms on resolution performance. Overall resolution performanceis shown to be governed by a space-time covariance functionthat can be controlled by the system on transmit using waveformdiversity. Visual examples are provided to illustrate the resolutionenhancement possible using MIMO technology.
Index Terms—Ambiguity function, beamforming, mul-tiple-input multiple-output (MIMO), radar.
I. INTRODUCTION
MULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) sys-tems have gained popularity and attracted attention of
late for their ability to enhance all areas of system performance.MIMO ideas are not new, in fact their origin can be traced to thecontrol systems literature. In [1], the idea of optimally selectingmultiple system inputs to enhance parameter estimation was dis-cussed. The early 1990s saw an emergence of MIMO ideas intothe field of communication systems [2]. More recently, one willfind the ideas of MIMO appearing in sensor and radar systems.
A MIMO radar system [3], [4] consists of transmit andreceive sensors, with the transmit sensors having the abilityto transmit arbitrary and independent waveforms. In manyways a MIMO radar is similar to a MIMO communicationsystem. While this analogy might not be incorrect, it does notfully capture the purpose of a radar system. Among the manypossible uses of a radar system, tracking and detecting targets,estimating target model parameters, and creating images oftargets are some of the most common. Various authors have allshown how these system tasks can be enhanced by using MIMOradar. In [4] it is shown how detection performance might beimproved using MIMO techniques. In [5]–[9] the issue of howto make use of the extra degrees-of-freedom (DOF) offered
Manuscript received September 19, 2006; revised January 4, 2007. This workwas supported part by Air Force Contract FA8721-05-C-0002. Opinions, inter-pretations, recommendations, and conclusions are those of the authors and arenot necessarily endorsed by the United States Government. The associate editorcoordinating the review of this manuscript and approving it for publication wasDr. Muralidhar Rangaswamy.
G. San Antonio and D. Fuhrmann are with the Electronic Systems and Sig-nals Research Laboratory, Department of Electrical and Systems Engineering,Washington University, St. Louis, MO 63130 USA (e-mail: [email protected]).
F. C. Robey is with the MIT/Lincoln Laboratory, Lexington, MA 02420 USA.Digital Object Identifier 10.1109/JSTSP.2007.897058
by MIMO is addressed. Designing transmit beampatterns issignificantly enhanced by using MIMO ideas. Also, adaptivetechniques such as MVDR and the GLRT can be enhanced byMIMO [3], [10]. Early work [11]–[13] emphasized some basicperformance gains that might be achievable.
As mentioned above, two of the primary functions of a radarare to find targets and estimate parameters of a model used todescribe those targets. Early radars could distinguish one unam-biguous parameter, range. Continuous wave (CW) radars couldidentify only range-rate. Pulse-Doppler radar can simultane-ously identify range and range-rate. Radar arrays allow for es-timation of angular parameters. The work in this paper it willbe shown how MIMO ideas can enable the unambiguous obser-vation of additional target parameters.
Modern radar systems are designed to be highly accuratefor their intended purpose. Designers and engineers need toknow the level of resolution to expect from a particular systemconfiguration. Some of the tools used to characterize perfor-mance are statistical parameter estimation bounds and ambi-guity functions. Typically parameter estimation bounds such asthe Cramer-Rao (CR) bound depend on the ambiguity function.The CR bound is a local bound; it depends on the shape of theambiguity function in the local region surrounding the param-eter estimate. The classic ambiguity function was introduced byWoodward and is used to characterize the local and global res-olution properties of time-delay and Doppler for narrowbandwaveforms. Other authors have extended Woodward’s ideas tolarger classes of waveforms and whole radar systems.
The purpose of this paper is to extend the ideas of ambiguityanalysis to MIMO radar systems. An ambiguity function willbe developed that expresses the new degrees of freedom offeredby MIMO radar while reducing in form to Woodward’s ambi-guity function for a simple single sensor narrowband system.The functions presented herein are the necessary tools for effec-tively evaluating whole sets or classes of waveforms to be usedin a MIMO radar system. Section II provides an overview ofprevious work done on ambiguity functions. This will provide abackground for establishing the MIMO ambiguity function. InSection III the signal model used to describe a MIMO radar willbe presented. The MIMO ambiguity function will be developedin Section IV. Section V will show some examples of MIMOambiguity functions and highlight some new design problems.
II. BACKGROUND
MIMO radar is in many ways a generalization of traditionalradar technology. As a preliminary step to defining the MIMOradar ambiguity function, we need to establish what previouswork has been done in the area of ambiguity function analysis.The results given later in this paper will reveal that the MIMO
168 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
ambiguity function is a generalization of most of the previouscontributions.
The earliest radar systems were designed to make simplemeasurements. These measurements included estimating time-delay/range or velocity (using continuous wave (CW) radar). Assystems became more complex and precise, the inevitable issuesof accuracy and resolution arose. Early researchers introduced afunction called the ambiguity function that captures some of theinherent resolution properties of a radar system. The ambiguityfunction was first introduced by Ville [14]; however it is gener-ally identified with Woodward because of his pioneering work[15], [16]. Woodward was interested in characterizing how wellone could identify the target parameters of time-delay (range)and Doppler (range-rate) based on the transmission of a knownwaveform . He established his ambiguity function by firstnoting that a good waveform is one which could be used to dis-tinguish between radar returns with different target parameters.He defined a total mean-squared error metric between a knownwaveform and a frequency-shifted and time-delayed ver-sion
(1)
When the square is expanded, the only remaining term de-pending on the parameters is a term which is the inner productbetween the original waveform and the time-delayed/fre-quency-shifted version. Woodward called this the radarambiguity function. It is
(2)
A magnitude-squared operation is usually introduced to theinner product term. In order to minimize the function (1), theambiguity function should be large for andsmall for . An ideal ambiguity functionis one which resembles a thumbtack in the plane.A good description of the properties of this function can befound in [17]. Among its more well-known properties is thefact that there is an inherent ambiguity or duality betweenresolution in time and resolution in frequency. For a giventime-bandwidth product, targets can not be resolved perfectlyin time and frequency simultaneously. In fact, this is why (2) isreferred to as an ambiguity function. In conjunction with thisconcept is the idea that ambiguity or energy (volume under thesurface) can be moved around in the plane but notremoved. An intuitive comparison to make is to think of theambiguity function as analogous to a probability density func-tion. Just as the trace of a covariance matrix bounds the spreadof the density function, the time-bandwidth product boundsthe spread of the ambiguity function. It should be noted thatalternative derivations of the ambiguity function exist and areequally valid. In particular, in [18] a derivation of the ambiguityfunction is provided that starts from a statistical view of thereceived radar signal. This viewpoint is adopted in this paper.
It was recognized that Woodward’s ambiguity functionneeded modification to handle larger bandwidth signals, longduration signals, and targets with high velocity. Other au-thors and researchers have taken up this task by considering
waveforms with larger bandwidths and targets described byhigher-order motion parameters. The hierarchy of generaliza-tions can be confusing since many authors simply call theirfunctions generalized ambiguity functions. Correct modelingof these types of scenarios was carried out in [19]–[21]. Theymodel more accurately the actual physics involved with movingtargets and reflecting signals. As a traveling wavefield reflectsoff a moving target, the field either expands or compresses intime due to the movement of the target. When a narrowbandwaveform is transmitted, this compressive effect is ignored forthe waveform’s complex envelope and only considered for thecarrier. The condition that must be met for this compressiveeffect to be ignored is based on the time-bandwidth productTB, target velocity , and the propagation speed of travelingwaves . The condition is
(3)
The ambiguity function derived for conditions that violate (3) is
(4)
The term specifically accounts for the stretching/compressing in time of the reflected signal. There is a scalar termin front of the integral to account for a change in amplitude ofthe reflected signal as it is stretched. The amplitude scaling isnecessary for the conservation of energy when the waveform isstretched in time. Several authors have suggested similar wide-band ambiguity functions [22]–[26].
More recently, connections have been made between thewideband ambiguity function and the continuous-time wavelettransform [27]–[30]. While wideband waveforms have beenshown to enhance resolution, the correct processing of suchsignals can be computationally intensive. The Fast FourierTransform (FFT) can be used with narrowband signals toefficiently compute Woodward’s ambiguity function. Equation(2) is basically a convolution integral. In contrast, the wavelettransform is the tool associated with processing widebandsignals. The wavelet transform describes signals that have beenshifted in time and time dilated. Techniques for computingwavelet transforms over a continuum of time shifts and timedilations are described in [31].
The above description of ambiguity functions was basedon radar systems operating with a single aperture. Someresearchers have also defined ambiguity functions for radarsystems with multiple apertures. The first to do so was Urkowitz[32]. He formulates an ambiguity function that is a functionof azimuth, elevation, range, and Doppler. In his system, thewaveform transmitted at each aperture is the same except fora time-delay or phase shift. His focus is on waveforms that arenarrowband. As a result he shows that his ambiguity functioncan be factored into separable space and time components.It will be shown later that for certain situations, the MIMOambiguity function will also be space-time separable. For afurther discussion of space-time separability see [33], [34].
Formulations of the ambiguity function have also appearedfor radars configured in a bistatic configuration [35]. Usuallythe term bistatic has been used in the literature to refer to a radar
SAN ANTONIO et al.: MIMO RADAR AMBIGUITY FUNCTIONS 169
system with a large separating baseline between transmit and re-ceive apertures. In this configuration the resolution of target pa-rameters such as range and Doppler becomes highly dependenton the location of the target relative to the transmit and receiveapertures. We will use the term bistatic to refer to any systemconfiguration with separate transmit and receive sensors.
Some authors have proposed ambiguity functions that incor-porate the estimation of nuisance parameters. For example, aparticular problem might include the joint estimation of delayand Doppler along with the background noise power, althoughthis is not consistent with Woodward’s original concept of ambi-guity. The authors of [36] propose an ambiguity function basedon the Kullback-Leibler divergence. Their approach to definingthe ambiguity function is based on the general problem of pa-rameter estimation in curved exponential families. In their paperthey are able to show how their ambiguity function reduces toWoodward’s function under special signal model assumptions.In a related problem, the authors of [37] derive the CR boundsfor estimating range, velocity, and angle with an active array.Their definition of the ambiguity function agrees with [36].
III. MIMO SIGNAL MODEL
In this section models will be introduced for each part of theMIMO radar system. These parts include signal transmission,signal propagation, signal reflection, and signal reception.
A. Transmit Signal Model
Currently, proposed MIMO radars consist of coherent net-works of transmit and receive sensors. These sensors could bedistributed apertures or elements of a single phased array. Inthis paper, it will be assumed that all sensors have an isotropicradiation pattern and that no mutual coupling between sensorsoccurs. A MIMO radar has transmit sensors and receivesensors. The th transmit sensor will have Cartesian coordinatesgiven by the column vector and the th receiver will haveCartesian coordinates . The and matrices
and will denote the collection of all transmit and re-ceive sensor locations
Each transmit sensor has the ability to transmit an independentwaveform. The actual signal transmitted from the th sensors is
The bandwidth and time duration are considered to be con-stant for all waveforms . All sensors operate at the samecenter frequency . Moreover, it will be assumed that the signalbandwidth satisfies the condition . Under this assump-tion, the common complex envelope notation can be applied tothe form of the transmitted signal
(5)
where is the complex envelope of the th waveform.
B. Channel and Target Models
Upon transmission, the radar signals as given by the modelin (5) propagate through free space, reflect off objects, and re-turn to the radar receivers. Proper statistical modeling of the re-ceived signals requires a thorough understanding of the propaga-tion channel characteristics and the target reflection process. Foreach transmit/receive sensor pair there exists a forward transmitchannel to the target and a reverse receive channel from the target.These channels will be modeled as lossless time delay and phasedelay channels. All others losses, such as attenuation, will beassumed to be due to the target reflection process.
Several authors have considered various models for targetreflection in MIMO radar systems, with the two extremes beingnoncoherent scattering and coherent scattering. Proper modelselection is based on a careful comparison of target complexity,signal bandwidth, signal duration, target motion, and sensorarray configuration. As we view the ambiguity function toreflect signal properties and not target parameters, these modelvariations will not play a role. The target model applied inthis paper will be a point target moving with constant velocity,and each sensor will “see” the same complex realization ofthe target scattering function. In reality this may be an overlyoptimistic assumption, but it allows for the definition of anambiguity function that is independent of the target scatteringfunction up to a single scalar value. Other ambiguity functionscould be defined for the more general case where the targetscattering function is only partially correlated from sensorto sensor. In this scenario the target scattering plays a morefundamental role and the ambiguity function is no longer afunction of only the transmitted signals.
Point targets will be described by a parameter vector con-sisting of a position vector component and velocity vectorcomponent . The position vector will be defined in the samecoordinate system as the transmit and receive sensor arrays.The velocity vector will also be defined in the array coordinatesystem. Depending on the array configuration, each target canbe identified by up to six unambiguous parameters, three forposition and three for velocity. We will now show how theseparameters appear in the model. In general if the signal istransmitted from sensor , reflects off a target, and is receivedby sensor , the response can be described by
(6)
The function is a time-varying function thatdepends on the target parameters and sensor locations. Using aTaylor series expansion we find
(7)
The term is simply the two-way time delay due to a targetlocated at with transmit and receive sensors at and
(8)
denotes the usual Euclidean vector norm.
170 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
The term is the frequency shift caused by a targetmoving with velocity vector and position . The frequencyshift is caused by an instantaneous change in path lengthbetween transmitter, target, and receiver. For simplificationassume the radar is fixed, then is defined as
(9)
where and are the transmit and receive path lengths.Each derivative is the projection of the target velocity vectoronto either the transmitter or receiver line-of-sight (LOS)
Now the frequency shifts can be explicitly written in terms of atransmit component and receive component
(10)
Under certain assumptions concerning array geometry, simpli-fications can be made to both the formulas for time-delay andfrequency shift. In a later section, a hierarchy of assumptionswill be outlined which will have implications for the form ofthe MIMO ambiguity function. It is also convenient to definethe stretch factor
C. Received Signal Model
The received signal at the th receive sensor before demodu-lation to baseband can be written as
Here the term represents background noise received atsensor . The noise will be modeled as white noise with knownpower . The amplitude coefficient is the complex real-ization of the target scattering function for the transmit-re-ceive channel. Under the coherent scattering model, all the scat-tering function realizations are identical, . After com-plex demodulation, the received signal signal is
(11)
IV. MIMO AMBIGUITY FUNCTION
In this section the ambiguity function for the MIMO radarsystem described in the previous section will be developed.
Fig. 1. Visual interpretation of covariance function.
At each of the receive sensors, the optimal detector is a filtermatched to specific set of target parameters.
(12)
Because the received signal consists of a sum of transmittedsignals, a bank of matched filters can be constructed at eachreceiver. The subscript will refer to each of thesefilters. Each of these filters has a kernel function given by oneof the transmitted waveforms. (In a real-world system, it may bedesirable to use a modified version of the transmit waveforms.This is commonly referred to as a mismatched filter.) At the threceiver the output of the filter matched to the th transmittedwaveform is
(13)
In the equation above and in the rest of the paper the notationrefers to a specific waveform that propagates through space.
The notation refers to transmitted waveforms used for matchedfiltering on receive. Fig. 1 illustrates graphically what each ofthe inner product terms represents for specific set of parame-ters . These inner product terms can be thoughtof as space-time covariance functions between the waveformsent at sensor reflecting off a target with parameters andreceived at sensor , and a waveform sent at sensor reflectingof a target with parameters and received at sensor . The am-biguity function is the coherent sum of all matched filter outputs
SAN ANTONIO et al.: MIMO RADAR AMBIGUITY FUNCTIONS 171
( pairs). We have assumed the coherent scattering model andthus the MIMO ambiguity function is
(14)
Following the practice in the literature, the noise term in (13)is dropped. Therefore, the ambiguity function tells us how wellthe set of transmitted signals can distinguish between two setsof target parameters in the absence of noise. Another interpre-tation is that under this signal model, the ambiguity function isthe inner product of received signals under two sets of target pa-rameters. The above definition is not arbitrary: it is exactly theterm that appears in the no-noise data-loglikelihood function.
In the case of a single collocated transmitter and receiver, theexpression in (14) elegantly reduces to Woodward’s ambiguityfunction
(15)
(16)
The approximation in (15) is the standard narrowband assump-tion; the frequency shift due to target motion does not affect thecomplex envelope for small time-bandwidth signals. We see that(14) does indeed reduce to (16) [same as (2)] which is solely afunction of relative time shift and Doppler shift .
In the rest of this section, we will examine (14) further andexplore aspects of signal design in determining radar resolutioncapability. In particular we will see the importance of the MIMOtransmit beampattern. Additionally, the MIMO ambiguity func-tion will be specialized to a few scenarios of interest.
A. MIMO Ambiguity Function Composition
When the ambiguity function (14) is expanded by substitutingfor the definition of , the result is a triple sum over
(17)
One observation that can be made is that the exponential termsare components in transmit and receive array steering vectors. Inits present form with no simplifications based on array geometryand target location, there is very little if anything to that can bedone to simplify this expression. One notational simplificationthat can be made is to collect all the inner product terms into asingle matrix. Define the matrix , wherethe element is equal to
(18)
Every element of the matrix has a functional dependence onand . Using this notation we can rewrite the ambiguity
function as
(19)
One further simplification would be to write out a combinedtransmit and receive steering vector. Define the following
vector
(20)
Using the notation in (20) the ambiguity function can be writtenas a sum over a series of quadratic forms
(21)
Intuitively some simple observations can be made about the am-biguity function when written in this form. The transmit signalsonly play a role in the covariance function. By changing the sig-nals that are sent, one can control the covariance function overthe parameter space . In the case of transmit-ting arbitrary wideband signals while observing high-velocitytargets using a nonsimple array geometry, all target parametersbecome coupled in the covariance function. As will be shownlater, under the assumption of narrowband signals there is aspace-time separability that falls out of the ambiguity function.
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In the next sections a few specific scenarios will be examined.These scenarios were chosen to highlight certain key factors atplay in the ambiguity function. There will be a hierarchy to thesimplifications in that the progression will be from more to lesscomplexity.
B. MIMO Ambiguity Function Simplifications
The first simplification that will be made concerns the im-pact of the target velocity. As was shown in a previous section,when high-velocity targets are observed with high time-band-width product transmit signals, the resulting compressive effecton the complex envelope in time cannot be ignored. If the targetvelocity is slow enough that the condition in (3) holds, then thesignal compression can be ignored in the covariance function.In this simplification and all to follow, modified versions of var-ious components of the ambiguity function will be denoted bysuperscript numbers. The covariance function elements underthe above mentioned assumption can be written as
(22)
The ambiguity function then becomes
(23)
Thus far, no assumptions have been made concerning the loca-tion of the target with respect to the array. The model used sofar is one in which components of the target velocity vector canbe identified. This identifiability results from the placement ofthe sensors in relation to the target. The locations are such thatdifferent sensors might see different frequency shifts due to asingle velocity vector. A common model simplification that canbe applied is to assume that all the sensors are close enough toeach other so that for a given target velocity vector, the projec-tion of that vector onto each sensors LOS is nearly identical.Willis discusses this simplification in great detail [38]. Essen-tially this assumption is valid only if the the bistatic angle forthe transmit-receive sensor pair with the largest separation fallswithin a certain threshold. Normally the threshold is taken to be
. Under this assumption, target velocity vector compo-nents can not be resolved unambiguously, therefore one shouldreduce the velocity vector parameter to a scalar parameter cor-responding to the radial velocity along the radar’s LOS. Thefrequency shift terms in the ambiguity function now become in-dependent of individual sensors allowing the array steering vec-tors to be decomposed into a transmit vector and receive vector.Apply this simplification by first rewriting the terms in the co-variance function
(24)
The transmit array steering vectors are
(25)
The receive array steering vectors are
(26)
Now we can write the ambiguity function as
(27)
A third simplification to the model could be that the target ap-pears in the far-field of the array. Under this assumption propa-gating waves appear planar. Furthermore, equations derived foractual time delays due to target position can be approximated.First, the components of the position vector should be changedto (range, azimuth, elevation) from . Now the most sen-sible coordinate system is an array centered coordinate systemwith one of the elements chosen as the phase center and origin.Range is the distance to the target as measured from the phasecenter. Azimuth and elevation are also referenced to the phasecenter. For a target in the far field, the time delays can be ap-proximated as
(28)
The function is a unit vector in the direction spec-ified by
(29)
At this point the ambiguity function will not be rewritten sincenotationally things do not become much simpler. However theambiguity function denotated as will refer to thecase when the target is in the far-field and target velocity is rep-resented by a scalar. The target parameter vector has four com-ponents, . Some of the examples will refer tothis form of the ambiguity function. Similarly,will refer to elements of the covariance function in this case.
The last model simplification that will be made will be toreduce the bandwidth significantly so that the narrowband as-sumption can be applied to waveforms. Under this assumption,the waveforms are considered narrowband enough so that actual
SAN ANTONIO et al.: MIMO RADAR AMBIGUITY FUNCTIONS 173
inter-sensor time delays can be ignored in the delay of the com-plex envelope. This simplifies the covariance function. Underthe narrowband assumption
(30)
(31)
As expected, each element of the covariance function is thesimple Woodward ambiguity function depending on andonly. We will call the narrowband covariance func-tion. When this function is substituted into the MIMO ambiguityfunction one can see that there is a separability now betweenspace (angle) and time (range)
(32)
Careful inspection of (32) will show that the ambiguity functionis now independent of the absolute range terms that appear inthe complex exponentials. These complex multipliers disappearwhen the magnitude squared is applied to the sum. Also, thesteering vectors are now the same as the standard narrowbandsteering vectors with a frequency shift applied to the carrier.Equation (32) can be written compactly as
(33)
where
(34)
(35)
C. Connections to Previous Work
In previous papers [5], [7], [8], it was shown how the choiceof transmit waveform signal correlation could effect the transmitbeampattern of a MIMO radar. The major results are as fol-lows. For an array independently transmitting narrowband wide-sense-stationary waveforms, the transmit beampattern functionis
(36)
Here is the narrowband transmit array steering vector pa-rameterized by the angle and is the zero-lag signal corre-lation matrix. Similarly, for an array transmitting wideband sig-nals, the transmit beampattern is
(37)
Again, is the narrowband transmit steering vectorparameterized by the angle , but now specifically calculated atthe frequency . The matrix is the cross-spectral den-sity matrix (CSDM) of the transmitted waveforms. It is definedas the element-wise Fourier Transform of the signal correlationmatrix
(38)
Returning to the definition of the MIMO ambiguity function wecan identify the role of the transmit beampattern. In some ofthe simplifications outlined it is easier to identify than others.To begin, consider the second simplification. In (27) the term
appears. If the two target pa-rameters are equal, , this term becomes the transmitbeampattern. The dependence of the covariance function ondrops out:
(39)
(40)
Taking the result of (40) and substituting back into the quadraticform one finds the expression for the wideband transmit beam-pattern
(41)
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Equation (41) is the same as (37) except for a slight notation dif-ference. We find the presence of the parameter instead of .The difference is that is a true spatial parameter, where as isonly an angle parameter. If the far-field simplifications were ap-plied, then the angle parameter would appear. Regardless, thisequation is telling us how much gain one can expect for targetslocated at different points spatially. It is much easier to identifythe transmit beampattern in the narrowband simplification of theMIMO ambiguity function as given by (32). The quadratic form
becomes the narrowband transmitbeampattern when .
Now if we return to (27), we see that letting resultsin the ambiguity being a scaled version of the transmit beampat-tern
(42)
For the situation when the quadratic form in ques-tion could be called the cross-transmit beampattern. However,since the receiver sensor dependence does not drop out of thecovariance function in this scenario, they are more correctlycross-transmit/receive beampatterns. Regardless of their correctnomenclature, these terms indicate how large a response one ex-pects to see when there is a mismatch in target parameters.
V. RESULTS AND EXAMPLES
In this section, some visual examples of MIMO ambiguityfunctions will be shown illustrating new signal design possibil-ities that arise by using MIMO radar. As it is beyond the scopeof this paper to actually undertake the design of MIMO radarwaveforms, the results in this section will simply provide moti-vation for the usefulness and necessity of the MIMO ambiguityfunction.
A. Orthogonal Signals
In this first example, we will examine the MIMO ambiguityfunction for a system that transmits nearly orthogonal signalsfrom each transmit aperture. The system consists of 3 transmit/receive sensors located on the axis at . Theoperating frequency is set to GHz, so the spacing worksout to be . Each sensor transmits and receiveswith an omnidirectional pattern. A single point target is placedin the far field. The target is moving with constant velocity. Eachof the waveforms has a bandwidth MHz and timeduration us; the time-bandwidth product is therefore
. Since the target is in the far field and assumed to
be moving slow enough so that waveform time-compression canbe ignored, form three of the MIMO ambiguity function will beused. Under all these assumptions, the ambiguity function willbe a function of three unambiguous arguments, range, azimuth,and Doppler frequency, , for a fixed set oftarget parameters .
Fig. 2 shows a single slice of the MIMO ambiguity func-tion. It is a range versus angle slice taken at zero Doppler. Theunits used in this figure and others are dBis. This means thatthe peak of the ambiguity function is referenced to the levelexpected by a sensor isotropically transmitting a unit energywaveform. There are many ways to compute and visualize thesefunctions. In this example the target has been placed in spacewith parameters , and the param-eter has been varied over the parameterspace.
Before proceeding further, some important comments mustbe made concerning the MIMO ambiguity function shown inFig. 2. A visual inspection of Fig. 2 will show that six distinctlines are present. In actuality there are nine lines; in three of thelines there are two overlapping lines. The reason for seeing ninelines is as follows: three orthogonal waveforms reflect off thetarget and are received by the three receivers, at each receiver afilter bank is implemented in which each filter responds to onlyone transmitted waveform due to orthogonality, at each receiverwe see three responses, three receivers with three responses eachresults in nine lines. These lines are ridges of ambiguity. In thisfigure they are range-angle ambiguity ridges for each transmitwaveform/receive sensor filter pair. The number of ridges variesdepending on the types of waveforms transmitted, the specificarray geometry, and the target location.
Another interesting characteristic of Fig. 2 is the interferencepatterns produced when the ambiguity ridges overlap. The causeof these patterns is the unique phase ramp possessed by eachambiguity ridge. As the ridges overlap large constructive anddestructive nulls are produced. The width of each ridge is in-versely proportional to the waveform bandwidth . In a radarwith closely spaced sensors the bandwidth is not large enoughfor the ridges to bend away from one another. Instead, the ridgeswould lie on top of each other resulting in a single ridge.
SAN ANTONIO et al.: MIMO RADAR AMBIGUITY FUNCTIONS 175
Fig. 3. MIMO ambiguity function-orthogonal signals-� = (0 m; 0 ; 0 m=s).
In addition to the range-angle slice shown in Fig. 2, we canmake other 2-D slices or show the MIMO ambiguity function in3-D. Fig. 3 is a visualization of the MIMO ambiguity functionin 3-D, where different 2-D slices have been placed in their 3-Dlocation.
One of the important properties of transmitting orthogonalsignals from a MIMO radar is the resolution performance en-hancement. This performance enhancement is the property ofbeing able to obtain uniform target parameter resolution simul-taneously at multiple points in space and time. To illustrate thisconcept the position of the point target in Fig. 2 is changedwhile not changing the transmitted waveforms or array geom-etry. Fig. 4 shows a range-angle slice of the MIMO ambiguityfunction for this scenario. As is evident from the figure, theambiguity function retains its past shape, but is now shifted towhere the true target position is. This figure is illustrating thefact that with careful selection of the transmitted waveforms,good resolution can be obtained over large regions of space andtime.
Lastly, we would like to provide an example of how the theambiguity function is related to the transmit beampattern. Fig. 5shows the MIMO ambiguity function for the array describedabove transmitting orthogonal signals, but now the target param-eters and have been set equal to one another. In this sit-uation, the ambiguity function reduces to a function of only az-imuth angle. Fig. 5 illustrates that equal power is applied acrossall azimuth angles on transmit.
B. Coherent Signals
In the results shown above, orthogonal waveforms were trans-mitted from each sensor. Now the other waveform extreme willbe examined, transmitting coherent waveforms. These wave-forms are coherent in the sense that each sensor transmits thesame waveform up to a time delay or phase shift. The wave-forms in this scenario will focus at azimuth on transmit, sono time delays or phase shifts will be applied.
Fig. 6 shows a range-angle slice of the MIMO ambi-guity function for the case were a target is located at
varied to produce the ambiguity function. This figure showsthat when the target is in the same location as the focus pointof the coherent waveforms, the same resolution is achieved asin the orthogonal waveforms case above. Careful inspection ofthe figure shows that there is a 9 dB increase of the peak of theambiguity function. This is the result of the coherent combina-tion of the transmitted waveforms at the target. Another wayto describe this result is that now there are really 27 ambiguityridges in the MIMO ambiguity function. For every one ridge inthe previous Fig. 2, there are three in Fig. 6.
As before the target location can be varied and a new am-biguity function can be produced. The target is moved to thesecond position as used in the orthogonal signal case. Again,the array geometry and waveforms are left unchanged. Fig. 7shows a range-angle slice of the MIMO ambiguity function.Now there are many ambiguity ridges everywhere with con-siderable constructive and destructive interference. This servesas an example of what coherent waveforms cannot accomplish.That is, when perfectly coherent waveforms are transmitted, theability to focus uniformly on receive for different values
176 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
is lost. This is in contrast to the orthogonal waveform scenariowhere the ambiguity function appears to simply be shifted.
Similar to the orthogonal signal scenario, the MIMO ambi-guity function can be calculated under the assumption that thetarget parameters and are equal. The result is the plot inFig. 8. Clearly this illustrates that by using coherent waveformsthere is a nonuniform illumination of space.
VI. CONCLUSION
In this paper, a comprehensive study of ambiguity and res-olution in modern radar systems was presented. It was shownhow the ideas of radar ambiguity functions developed over thepast fifty years can be extended to the newly proposed class ofMIMO radar systems. These systems are characterized by in-dependent but coherent sensors possibly distributed over largebaselines, transmitting waveforms with large fractional band-widths. As a result of additional degrees-of-freedom, an ambi-guity function is defined that is a function of 12 parameters. Itwas shown how this function could be simplified under variousscenarios, and even reduced to Woodward’s ambiguity func-tion for simple single sensor systems. The key result presented
Fig. 8. Transmit pattern-coherent signals.
herein is that there is a space-time covariance function producedby the transmitted waveforms that governs the resolution perfor-mance over the parameter space. Connections were made be-tween the MIMO ambiguity function and past work done in thearea of transmit beampattern synthesis. When the arguments inthe ambiguity function are set equal to each other, the resultingfunction is equivalent to a transmit beampattern.
The results presented in this paper show how the ideas ofambiguity functions can be applied to the new class of radarsthat use MIMO technology. Just as Woodward’s ambiguityfunction is used as a tool for individual waveform design, theMIMO ambiguity function should be used for designing goodMIMO waveforms. There are several areas of MIMO radardesign that need further research. To date, the authors know oflittle if any work that has been done on actually producing radarwaveforms capable of achieving the middle ground betweenorthogonality and perfect coherence. Clearly for the resultspresented in this paper to be of an practical use, this issue mustbe solved. Another line of research that would complement thisstudy would be to examine a more global parameter estimationbound such as the Barakin bound or Weiss-Weinstein boundfor these MIMO systems. In many of the visual plots shown,large ambiguities were present that might cause errors whendetecting targets at low SNR. These global bounds shouldcapture the SNR threshold effects of parameter estimation.
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Geoffrey San Antonio received the B.S. degree inelectrical engineering from Worcester PolytechnicInstitute, Worcester, MA, in 2003 and the M.S.degree in electrical engineering from WashingtonUniversity, St. Louis, MO, in 2005. He is currentlypursuing the Ph.D. degree in electrical engineeringfrom Washington University.
His research interests lie in the area of statisticalsignal processing, including sensor array signal pro-cessing, radar systems, adaptive algorithms, and in-formation geometry.
Daniel R. Fuhrmann (SM’95) received the B.S.E.E.degree (cum laude) in 1979 from Washington Uni-versity, St. Louis, MO, the M.A. and M.S.E. degreesin 1982, and the Ph.D. degree in 1984, all fromPrinceton University, Princeton, NJ.
From 1979 to 1980, he was employed by TelexComputer Products, Tulsa, OK. Since 1984, he hasbeen with the Department of Electrical Engineering,now the Department of Electrical and SystemsEngineering, Washington University, where he iscurrently Professor. His research interests lie in
various areas of statistical signal and image processing, including sensor arraysignal processing, radar systems, space-time adaptive processing, and imageprocessing for genomics applications.
Dr. Fuhrmann is an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL
PROCESSING and was the Technical Program Chairman for the 1998 IEEE SPWorkshop on Statistical Signal and Array Processing. In the 2000–2001 aca-demic year, he was a Fulbright Scholar visiting at the Universidad Nacional deLa Plata near Buenos Aires, Argentina. He was the General Chairman for the2003 IEEE Workshop on Statistical Signal Processing.
Frank C. Robey (M’90) received the B.S.E.E.degree (summa cum laude) in 1979 and the M.S.E.E.degree in 1980 from University of Missouri, Co-lumbia. He received the D.Sc. degree in 1990 fromWashington University, St. Louis, MO.
From 1980 to 1984, he was a Member of TechnicalStaff at Hewlett-Packard, Loveland, CO and LakeStevens, WA, and from 1984 to 1988 he was withEmerson Electric, St. Louis. He joined MIT LincolnLaboratory in 1991 where he has held assignmentsin Lexington, MA, and Kwajalein, RMI, and is
currently Associate Group Leader. His field of interest is in instrumentationsystems for surveillance and control with a focus on the development anddemonstration of enabling technologies including adaptive and sensor-arraysignal processing, open signal and data processing, and collaborative coherentsensor networking.
