Phased-MIMO Radar a Tradeoff Between Phased Array and MIMO Radars

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 6, JUNE 2010 3137

Phased-MIMO Radar: A Tradeoff BetweenPhased-Array and MIMO Radars

Aboulnasr Hassanien, Member, IEEE, and Sergiy A. Vorobyov, Senior Member, IEEE

Abstract—We propose a new technique for multiple-input mul-tiple-output (MIMO) radar with colocated antennas which we callphased-MIMO radar. The new technique enjoys the advantages ofthe MIMO radar without sacrificing the main advantage of thephased-array radar which is the coherent processing gain at thetransmitting side. The essence of the proposed technique is to parti-tion the transmit array into a number of subarrays that are allowedto overlap. Then, each subarray is used to coherently transmit awaveform which is orthogonal to the waveforms transmitted byother subarrays. Coherent processing gain can be achieved by de-signing a weight vector for each subarray to form a beam towards acertain direction in space. Moreover, the subarrays are combinedjointly to form a MIMO radar resulting in higher angular reso-lution capabilities. Substantial improvements offered by the pro-posed phased-MIMO radar technique as compared to the phased-array and MIMO radar techniques are demonstrated analyticallyand by simulations through analyzing the corresponding beampat-terns and the achievable output signal-to-noise-plus-interferenceratios. Both analytical and simulation results validate the effective-ness of the proposed phased-MIMO radar.

Index Terms—Coherent processing gain, MIMO radar, phased-array radar, transmit/receive beamforming.

I. INTRODUCTION

R ADAR technology has been continuously developingover the last 70 years starting from the late 1930s when

radar was first invented for defence applications [1]–[3]. Thedesire for new more advanced radar technologies has beendriven and is dictated by radar’s ubiquitous applicabilityranging from micro-scale radars applied in biomedical engi-neering [4], [5] to macro-scale radars used in radioastronomy[6], [7]. To date, previous developments in radar were basedon the idea that the signal can be processed coherently atthe transmit/receive antenna arrays if the signal coherency ispreserved. The corresponding radar technique is well knownunder the name phased-array radar [1], [2].

In the last decade, the development of a new radar para-digm that is best known under the title multiple-input mul-

Manuscript received July 31, 2009; accepted January 29, 2010. Date of pub-lication February 22, 2010; date of current version May 14, 2010. The associateeditor coordinating the review of this paper and approving it for publication wasDr. Xiang-Gen Xia. This work was supported in part by the Natural Sciencesand Engineering Research Council (NSERC) of Canada and in part by the Al-berta Ingenuity Foundation, Alberta, Canada. This work was presented in part atthe IEEE International Conference on Acoustics, Speech and Signal Processing(ICASSP), Taipei, Taiwan, R.O.C., April 2009.

The authors are with the Department of Electrical and Computer Engi-neering, University of Alberta, Edmonton, AB, T6G 2V4, Canada (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2043976

tiple-output (MIMO) radar has become the focus of intensiveresearch [8]–[16]. The essence of the MIMO radar concept is toemploy multiple antennas for emitting several orthogonal wave-forms and multiple antennas for receiving the echoes reflectedby the target. The enabling concept for MIMO radar, e.g., thetransmission of multiple orthogonal waveforms from differentantennas, is usually referred to as the waveform diversity [8],[9]. Consequently, the waveform design and optimization hasbeen the main focus of the research in MIMO radar [17]–[21].

Many approaches to MIMO radar have been developed thatrevolve around the main idea of exploiting the waveform diver-sity. Based on the array configurations used, MIMO radars canbe classified into two main types. The first type uses widely sep-arated transmit/receive antennas to capture the spatial diversityof the target’s radar cross section (RCS) (see [9], and referencestherein). This type assumes an extended target model and, there-fore, takes advantages of the properties of the associated spa-tially-distributed signal model. In this case, the waveform di-versity is similar to the multi-path diversity concept in wirelesscommunications over fading channels. According to this con-cept, signals transmitted over multiple fading links/channels canbe decoded reliably at the receiver due to the fact that it is un-likely that all links/channels undergo unfavorable fading con-ditions simultaneously [22], [23]. Similarly, the waveform di-versity in radars enables reliable detection due to the fact thatit is unlikely that all waveforms suffer from scintillation effectssimultaneously.

The second MIMO radar type employs arrays of closelyspaced transmit/receive antennas to cohere a beam towards acertain direction in space (see [13], and references therein). Inthis case, the target is usually assumed to be in the far-field.Also, it is often assumed that the transmitted signals are narrow-band and, therefore, the point source signal model is commonlypresumed. The waveform diversity, in this case, boils down toincreasing the virtual aperture of the receive array due to thefact that multiple independent waveforms are received by thesame receive array [13].1

Recently, some attempts to exploit jointly the benefits ofthe phased-array and MIMO radars have been also reported[24]–[28]. In [24], a general antenna configuration is consid-ered where several well-separated subarrays are used to forma MIMO radar with each subarray operating in phased-arraymode. In [25], the idea of dividing the aperture of the transmitarray with colocated antennas into multiple disjoint subaper-tures has been introduced. Utilizing the same partitioningstructure, the authors of [26]–[28] have developed closed-form

1It would be, however, more precise to rename the waveform diversity con-cept as the multiple independent waveforms concept in the case of MIMO radarwith colocated antennas. We use the traditional terminology hereafter though.

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3138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 6, JUNE 2010

expressions for MIMO signal sets to achieve wide transmitbeampattern. In particular, it is shown that the height and widthof a rectangular beampattern can be controlled by adjusting thelevel of correlation between the MIMO signals.

In this paper, we consider the case of a radar system withcolocated antennas and introduce transmit array partitioning tointegrate the phased-array radar into the MIMO radar. Indeed, ascompared to the phased-array radar, the use of MIMO radar withcolocated antennas enables improving angular resolution, in-creasing the upper limit on the number of detectable targets, im-proving parameter identifiability, extending the array apertureby virtual sensors, and enhancing the flexibility for transmit/re-ceive beampattern design [13]–[15]. However, the advantagesoffered by the MIMO radar come at the price of losing thetransmit coherent processing gain offered by the phased-arrayradar [16]. Hence, the MIMO radar with colocated antennasmay suffer from beam-shape loss [15] which leads to perfor-mance degradation in the presence of the target’s RCS fading.Moreover, the MIMO radar exhibits loss in the signal-to-noiseratio (SNR) gain [16].

To overcome the aforementioned weakness of the MIMOradar, we develop a new radar technique which combines theadvantages of the MIMO radar (such as waveform diversity)with the advantages of the phased-array radar (such as coherentprocessing).2 In order to enable such an integration, we partitionthe transmit array into a number of subarrays that are allowedto overlap. Then, each subarray is used to coherently transmita waveform which is orthogonal to the waveforms transmittedby other subarrays. Coherent processing gain can be achievedby designing the weight vector of each subarray to form a beamtowards a certain direction in space (the same direction for allsubarrays). In parallel, the subarraysare combined jointly to forma MIMO radar resulting in higher angular resolution capabilities.

The advantages of the new radar technique, that is naturallycalled phased-MIMO radar, over the phased-array and MIMOradars are analyzed in terms of the corresponding beampatternand signal-to-interference-plus-noise ratios (SINRs) expres-sions. Particularly, the new radar technique:

i) enjoys all the advantages of the MIMO radar, i.e., itenables improving angular resolution, detecting a highernumber of targets, improving parameter identifiability,and extending the array aperture;

ii) enables the use of existing beamforming techniques atboth the transmitting and the receiving ends;

iii) provides the means for designing the overall beampatternof the virtual array;

iv) offers a tradeoff between angular resolution and robust-ness against beam-shape loss;

v) offers improved robustness against strong interference.Our paper is organized as follows. Some background on

MIMO radar is presented in Section II, where several conceptsneeded for developing a new phased-MIMO radar techniqueare revisited. In Section III, we give a formulation of a newphased-MIMO radar and highlight its advantages. Some relateddesign problems such as transmit beamforming and transmitsubarray waveform designs are also discussed. Section IVis devoted to the analysis of the phased-MIMO radar with

2An early exposition of this work has been presented in [29].

transmit/receive beamforming. Particularly, we derive thephased-MIMO radar beampattern for the case when conven-tional transmit/receive beamforming is used, and compare itwith the beampatterns of the phased-array and MIMO radars.We also compare the three aforementioned radar techniquesin terms of their achievable output SINRs. The possibility ofusing robust/adaptive beamforming is also discussed. Section Vpresents our simulation results which show significant perfor-mance gains that can be achieved by the phased-MIMO radartechnique as compared to the phased-array and MIMO radars.Section VI contains our concluding remarks.

II. MIMO RADAR: PRELIMINARIES

Consider a radar system with a transmit array equipped withcolocated antennas and a receive array equipped with

colocated antennas. Both the transmit and receive arrays areassumed to be close to each other in space (possibly the samearray) so that they see targets at same directions. The thtransmit antenna emits the th element of the waveform vector

which satisfies the orthogonalitycondition

(1)

where is the radar pulse width, is the time index within theradar pulse, is the identity matrix, and andstand for the transpose and Hermitian transpose, respectively.The total energy transmitted by a MIMO radar system withinone radar pulse is given by

(2)

The snapshot vector received by the receive array canbe modeled as

(3)

where , , and are the independent componentsof the target/source signal, interference/jamming, and sensornoise, respectively. Under point target assumption, the targetsignal can be written as

(4)

where is the target direction, is the complex-valued re-flection coefficient of the focal point , and and arethe actual transmit and actual receive steering vectors associatedwith the direction . Note that without any loss of generality weassume that the first element of and , respectively, isa reference element.

The returns due to the th transmitted waveform can be re-covered by matched-filtering the received signal to each of thewaveforms , i.e.,

(5)

HASSANIEN AND VOROBYOV: PHASED-MIMO RADAR 3139

where denotes the conjugate operator. Then, thevirtual data vector can be written as

(6)

where stands for the Kronker product and accounts forthe interference-plus-noise components. The target signal com-ponent in (6) can be expressed as

(7)

where is the steering vectorassociated with a virtual array of sensors.3

For the special case of uniform linear array (ULA) at thetransmitter and the receiver, the th entry of the virtualarray steering vector is given by

(8)

where and are the interelement spacings measured inwavelength for the transmit and receive arrays, respectively. Itwas shown in the literature that the interelement spacing of thetransmit array can take values higher than a half wavelengthwithout suffering from ambiguity at the receiving end [14]. Inparticular, if is chosen, then the resulting virtualarray is a ULA of elements spaced wavelength apartfrom each other and its steering vector simplifies to [14]

(9)which means that an array with effective aperture canbe obtained by using antennas. Based on the latterfact, higher angular resolution and better performance can beachieved by using in (6) for detection and estimation purposes.Despite the aforementioned advantages, the MIMO radar hasalso significant disadvantage as compared to the phased-arrayradar. Particularly, if orthogonal waveforms are used, the aboveMIMO radar formulation does not allow for coherent processingat the transmit array and, therefore, it lacks robustness againstsensor noise and RCS fading. Moreover, the MIMO radar has

times smaller “clear region” as compared to the phased-arrayradar [30].

III. PROPOSED PHASED-MIMO RADAR FORMULATIONS

In this section, we propose a new formulation for MIMOradar which allows for beamforming at the transmit and receivearrays. The new formulation enables a compromise between thecoherent processing gain offered by the phased-array radar andthe above-mentioned advantages of the MIMO radar, e.g., thewaveform diversity. Since the proposed formulation combinesthe advantages of both phased-array and MIMO radar, we choseto call it phased-MIMO radar.

3The size extension of the resulting virtual array due to the orthogonality ofthe transmitted waveforms is traditionally referred to as waveform diversity forMIMO radar with colocated antennas.

A. Phased-MIMO Radar

The main idea behind our formulations is to partition thetransmit array into subarrays which areallowed to overlap. In general, each transmit subarray can becomposed of any number of antennas ranging from 1 to suchthat no subarray is exactly the same as another subarray. Allantennas of the th subarray are used to coherently emit thesignal so that a beam is formed towards a certain directionin space, e.g., direction of the target. Then, the beamformingweight vector can be properly designed to maximize the co-herent processing gain. At the same time, different waveformsare transmitted by different subarrays. In this paper, we adoptthe set of the so-called frequency spread signals used in [31]which is defined as ,where is a pulse shape of duration . It has been shownin [31] that the aforementioned set of signals maintains orthog-onality for a variety of time delays and Doppler shifts, if thefrequency increment between andsatisfies . Note that if the delays and Doppler shiftsare nonzero, the space–time ambiguity function has to be con-sidered [28].

Let us introduce the vector which contains 0 and1 entries. Here 1 means that the antenna with an index corre-sponding to the location of that 1 in belongs to the th sub-array while 0 means that the corresponding antenna does notbelong to that subarray. Thus, the vector defines the th sub-array. Let the th subarray be composed of transmitantennas. Then, the number of 1’s in equals and thenumber of 0’s equals . The steering vector asso-ciated with the th subarray can be found aswhere stands for the Hadamard (element-wise) product.

The complex envelope of the signals at the output of the thsubarray can be modeled as

(10)

where is the unit-norm complex vector which con-sists of beamforming weights corresponding to the activeantennas of the th subarray, i.e., the antennas for which the as-sociated elements of equal to 1, and zero weightscorresponding to other antennas. It is worth noting that the en-ergy of within one radar pulse is given by

(11)

which means that the total transmitted energy for the phased-MIMO radar within one radar pulse is equal to .

The signal reflected by a hypothetical target located at direc-tion in the far-field can be then modeled as

(12)


where is the reflection coefficient4 of the hypotheticaltarget, and are the beamforming vector andsteering vector, respectively, which contain only the elementscorresponding to the active antennas of the th subarray, and

is the time required for the wave to travel across the spa-tial displacement between the first antenna of the transmit arrayand the first antenna of the th subarray. In (12), we assumethat the first element of is taken as a reference element.

Let us introduce the transmit coherent processingvector

(13)

and the waveform diversity vector

(14)

Then, the reflected signal (12) can be rewritten as

(15)

where is the vector of wave-forms.

Assuming that the target of interest is observed in the back-ground of interfering targets with reflection coefficients

and locations , the received complexvector of array observations can be written as

(16)

where, is defined as in (15). By matched-filteringto each of the waveforms (e.g., as in (5)), we can formthe virtual data vector

(17)

where the vector

(18)

is the virtual steering vector associated with direction andis the noise term whose covariance is given by

where is the noise power.It is worth noting that if is chosen, i.e., if the whole

transmit array is considered as one subarray and only one wave-form is emitted, then the signal model (17) simplifies to thesignal model for the conventional phased-array radar [32]

(19)

while the virtual steering vector given in (18) simplifies to

(20)

4The reflection coefficient for each target is assumed to be constant duringthe whole pulse but varies from pulse to pulse.

where is the uplink coherent processing gain of theconventional phased-array radar towards the direction andis the phased-array transmit beamformer weight vector.In this case, the received data vector is of dimensionwhich explains the low angular resolution performance of thephased-array radar.

On the other hand, if is chosen, then the signal model(17) simplifies to (6) which is the signal model for the MIMOradar without array partitioning. In this case, the datavector enables the highest possible angular resolution at theprice of having no coherent processing gain at the transmittingside.

The proposed MIMO radar formulation (17) combines thebenefits/advantages of the phased-array radar (19), (20) andthe MIMO radar (6). In particular, the new formulation enjoysmany interesting characteristics. First, it enjoys all advantagesof MIMO radar, i.e., it enables improving angular resolution,detecting a higher number of targets, improving parameteridentifiability, extending the array aperture by virtual sensors.Second, it allows for uplink beamforming at the transmittingside and, therefore, it enables optimization/maximization ofthe coherent processing gain and controlling/minimizing thetotal transmit power. Third, it provides the means for designingthe overall beampattern of the virtual array. In other words,it empowers optimization of the overall beampattern of thevirtual array by designing the transmit/receive beamformingweights jointly. Fourth, the proposed formulation offers atradeoff between angular resolution and robustness againstbeam-shape loss by properly selecting the number of subarraysused. Finally, it offers a tradeoff between improvements inperformance and the required computational complexity.

By inspecting (10), it is interesting to observe that differentantennas5 transmit linear combinations of the base orthogonalwaveforms . Let be the signals trans-mitted by the transmit antennas. These signals need not beorthogonal and can be expressed as

(21)

where is the vector of trans-

mitted signals, is the weight matrix of di-mension . It is worth noting that (10) and (21) are equiv-alent. Based on (21), the proposed phased-MIMO radar can alsobe interpreted in terms of the MIMO radar with nonorthogonalantenna-wise waveforms, but orthogonal subarray-wise wave-forms. Note that because of the inherent structure (cf. (10)), theoptimal processing of subarray-wise orthogonal waveforms ismaintained at the receive array, where the waveforms orthogo-nality is required for optimal detection [18].

B. Practical Considerations

In modern radar technology, it is desirable that each transmitantenna operates at full power. It, in turns, requires uniformdistribution of power across the transmit array antennas. It is

5Note that some antennas belong to more than one subarray.


also desirable that the signals transmitted by different antennasbe constant-modulus so that nonlinear amplifiers can be used.However, many of the recently developed MIMO radar tech-niques result is arbitrary signal cross-correlation matrices (see,for example, [19] and [20]). In general, such arbitrary signalcross-correlation matrices are hard to achieve using constant-modulus signals.6 Therefore, signals with variable amplitudesare often used. The latter can lead to the requirement of havinga separate amplifier for each transmit antenna, which can be achallenge, especially in large arrays. In this respect, note that in[26]–[28], the signals are designed to fit the currently availablephased-array radar technology by allowing the signals to be par-tially correlated.

Our proposed phased-MIMO radar formulation provides flex-ibility in achieving the aforementioned requirements by opti-mizing the transmit beampattern over the weight matrix .7

To illustrate this, let us express the element-wise cross-correla-tion matrix as

(22)

where denotes the expectation operator. It can be seen from(22) that with desirable properties can be obtained by meansof designing . Moreover, the transmit power distribution at ageneric focal point located at the direction can be expressed as

(23)

Therefore, the beampattern can also be designed over .In addition, while optimizing over , the power constraints

per individual transmit antennas can be enforced so that thepower distribution across antennas would be uniform. Once theweight matrix is found, the problem of designingsuch that satisfies desired properties (for example, con-stant-modulus property) can be also addressed.

In this paper, we are rather concerned about proving thebenefits of the phased-MIMO radar over the phased-array andMIMO radars based on the simplest designs for the weightmatrix . Therefore, we further consider the conventionaltransmit/receive beamforming as a design for .

IV. PHASED-MIMO RADAR TRANSMIT/RECEIVE

BEAMFORMING

In this section, we use the conventional transmit/receivebeamforming techniques and analyze the performance of theproposed phased-MIMO radar (17). We also compare thephased-MIMO radar to the phased-array radar (19) and theMIMO radar (6) in terms of their transmit-receive beampatternsand achievable SINRs. We consider in details the case whennonadaptive transmit/receive beamforming techniques are usedfor the proposed phased-MIMO radar. Moreover, we brieflydiscuss the case of adaptive transmit/receive beamforming.

6Note that binary pulse shift keying (BPSK) constant-modulus signals areused in [19]. However, it significantly restricts the set of achievable cross-cor-relation matrices.

7Note that the optimization over� has been recently considered for a dif-ferent application to direction of arrival estimation in MIMO radar in [33].

At the transmit array, existing uplink beamforming tech-niques lend themselves easily to design the weight vectors

for different subarrays such that certain beampatternand/or transmit power requirements are satisfied. As a canon-ical example, we choose to partition the transmit array so thatthe th subarray is composed of the antennas located at the

th up to the th positions, i.e., each subarrayconsists of antennas. We call such partitioning asfully-overlapped, hereafter. Then the corresponding transmitbeamforming weight matrix is given as

......

......

(24)

where is the th weight of the th subarray beamformingweight vector and is the vector of zeros.

Note that we only impose the constraints on the total trans-mitted power, that is, we always assume that the total transmittedenergy within one radar pulse equals to . No constraints onthe power distribution across transmit antennas is imposed. Al-though, in this case, the power distribution across antennas isnonuniform, it enables us to design based on simple conven-tional beamforming techniques. The latter simplifies the anal-ysis significantly, but keeps the conclusions general.

A. Nonadaptive Transmit/Receive Beamforming

The SNR gain of the phased-array radar is proportional to thequantity . Using the Cauchy-Schwarz inequality, wehave , where equality holds when

which is referred to as the conventional nonadaptivebeamformer. In the case when a single source signal is observedin the background of white Gaussian noise, the conventionalnonadaptive beamformer is known to be optimal in the sensethat it provides the highest possible output SNR gain. For thisreason and due to its simplicity, the conventional nonadaptivebeamformer has been widely used in antenna array systems [34].Therefore, we use the conventional nonadaptive beamformingat both the transmit and receive arrays of the phased-MIMOradar system and derive expressions for transmit/receive beam-pattern and output SINR. We also analyze and compare thephased-MIMO radar beampattern and SINR expressions to theanalogous expressions for the phased-array and MIMO radars.

Since all subarrays have equal aperture, the beamformerweight vectors for conventional uplink beamforming, are givenby

(25)

At the receive array, the conventional beamformer is applied tothe virtual array and, therefore, the receive beamformerweight vector is given by

(26)


Let be the normalized phased-MIMO radar beampattern,that is

(27)

To simplify the derivation and analysis of the beampatternand SINR expressions for the phased-MIMO radar, we assumea ULA, i.e.,

(28)

Hence, the beampattern (27) can be reformulated as follows:

(29)

(30)

where (30) is obtained from (29) using the fact that

(31)Let us consider three different cases for partitioning the transmitarray.Case 1) Each subarray is composed of all transmit an-

tennas, i.e., each subarray is the same as the wholearray. In this case, we have and

, where is the vector of ones. In this case,the beampattern (30) boils down to

(32)

which is the beampattern of a phased-array oftransmit and receive antennas.

Case 2) The subarrays are nonoverlapped and, therefore,each subarray consists of antennas. Thistype of partitioning is adopted in [26]–[28]. In thiscase, the th element of vector correspondsto the th element of andvectors , contain eachnonzero elements. Hence, it is easy to show that

.Therefore, the beampattern (30) boils down again to

(33)

Case 3) In this case, we consider the fully-overlapped methodof partitioning. Recall that each subarray consists of

transmit antennas. Due to full-overlap, itis easy to show that ,

where is the th element of the transmitsteering vector . Therefore, the phased-MIMOradar beampattern (30) can be rewritten as

(34)

where the facts that ,, and are used.

Let ,

, and be thetransmit (uplink) beampattern, the waveform diversity beam-pattern, and the receive (downlink) beampattern, respectively.Then, the phased-MIMO radar beampattern can be seen as theproduct of three individual beampatterns, that is,

(35)

By inspecting (35), we draw the following observations.• The first two terms and of the product (35)

are dependent on the number of subarrays while thethird term is independent on . Hence, the beam-pattern analysis for the phased-MIMO radar will focus onthe first two terms.

• The beampattern expression for the phased-array radar canbe deduced form (35) by substituting . Hence, weobtain

(36)

where and .Note that this case is different from Case 1 for array par-titioning mentioned above because only one waveform istransmitted in this case while a mixture of any orthog-onal waveforms is transmitted in Case 1.

• The MIMO radar beampattern expression can be obtainedform (35) by substituting which results in

(37)

where and .Comparing (35), (36), and (37) to each other, we notice thatthe phased-array and the MIMO radar have the same overallbeampattern, i.e.,

(38)

However, they have different uplink and waveform diversitybeampatterns and, therefore, different gains. The phased-arrayradar has the highest possible transmit coherent processing gainat the price of no diversity gain, while the MIMO radar has thehighest waveform diversity gain at the price of no transmit co-herent processing gain. On the other hand, the beampattern ofthe phased-MIMO radar enjoys two interesting properties whichare given in the following two propositions.


Proposition 1: If the phased-MIMO radar is formed by par-titioning a ULA into fully-overlapped subarrays, then thetransmit-receive beampattern equals the transmit-receive beam-pattern of the phased-MIMO radar formed by partitioning thesame ULA into subarrays, that is,

(39)

Proof: Noting that is of dimensionand is of dimension , the proof is readily obtainedby substituting in (34) and exchangingfor and for .

Proposition 2: The transmit-receive beampattern of thephased-MIMO radar with subarrays has lower highestsidelobe level than the transmit-receive beampattern of thephased-array radar, that is,

(40)

where is a continuum of all spatial angels within the sidelobearea.

Proof: See Appendix A.Based on Proposition 2, the phased-MIMO radar enjoys

better robustness against interfering targets located in the side-lobe area as compared to the phased-array and MIMO radars.To illustrate this, let us examine the optimal output SINR for allthree radar techniques. The output SINR of the phased-MIMOradar can be defined as

(41)

where is the variance of the target reflectioncoefficient and the interference-plus-noise covariance matrix isgiven by

(42)

where is the variance of the th interferencereflection coefficient.

Substituting in (41), the output SINR for thephased-MIMO radar can be rewritten as

(43)

Then, substituting in (43), we obtain the output SINRfor the phased-array radar as

(44)Similarly, the output SINR for the MIMO radar can be obtainedby substituting in (43) as

(45)For the sake of comparison, we analyze the output SINRs

given in (43)–(45) for the following two cases.1) Dominant Noise Power: If the target is observed in the

background of few weak interferers which are well separatedfrom the target, then the interference-to-noise power can be at-tributed to the noise term only. In such a case, the SINR for thephased-array radar simplifies to

(46)

while the SINR for the MIMO radar simplifies to

(47)

Comparing (46) and (47), we observe that

(48)

which means that the phased-array radar is more robust to back-ground noise as compared to the MIMO radar. On the otherhand, the SINR expression for the phased-MIMO radar boilsdown to

(49)

where is the ratio of the phased-MIMOradar SINR to the phased-array radar SINR. It is worth notingthat . The dependence of on shows thatthe SINR gain of the phased-MIMO radar linearly decreases byincreasing . At the same time, larger provides larger di-mension of the extended virtual array. This depicts the tradeoffbetween the SINR gain and the high angular resolution capabil-ities.

2) Dominant Interference: If the target is observed in thebackground of strong interference, then we can fairly considerthe noise power to be negligible as compared to the interferencepower. By neglecting the noise term in (44) and (45), we obtain

(50)


which means that both the phased-array and MIMO radars havethe same robustness against interference. On the other hand, theSINR of the phased-MIMO radar can be analyzed by reformu-lating (43) as follows:

(51)

(52)

(53)

where (52) is obtained by multiplying the numerator and de-nominator of (51) by .

Using (28) and the equality (31), the SINR expression (53)can be rewritten as given by expression (54), shown at thebottom of the page.

It is worth noting that because. Using this

fact and the fact that the phased-MIMO radar has lower highestsidelobe level than the phased-array radar (see Proposition 2),we conclude that

(55)

where is the worst case SINR achievable by thephased-MIMO radar, i.e., SINR for the case when the interfer-ence leaks to the output of the beamformer through the highestsidelobe, and is the worst case SINR achievable by thephased-array radar. It follows from (55) that the phased-MIMOradar is capable of providing better SINR performance as com-pared to the phased-array and MIMO radars. This observationwill be verified further based on simulation examples.

B. Robust/Adaptive Beamforming

To control/minimize the transmitted power, we can resortto robust uplink beamforming. One meaningful approach isto minimize the norm of the beamformer weight vector whileupper-bounding the sidelobe levels. Mathematically, this robustuplink beamformer can be formulated as follows:

(56)

where is the parameter of user choice used to upper-bound thesidelobe levels.

It is worth noting that transmit coherent processing gainoffered by either (25) or (56) will be smaller than the transmitcoherent processing gain of the phased-array radar. This is dueto the fact that the effective array aperture of each transmitsubarray is smaller than the effective array aperture of thewhole array. This also makes the main beam associated witheach subarray wider than the main beam associated with thephased-array radar. However, this natural drop in performanceof transmit beamforming is the price paid for the many benefitsthat are gained at the receiving end.

It is also possible to use adaptive processing techniques whichaim at maximizing the output SINR at the receive array. Hence,we resort to the famous minimum variance distortionless re-sponse (MVDR) beamformer [34]. The essence of the MVDRbeamformer is to minimize the interference-plus-noise powerwhile maintaining a distortionless response towards the direc-tion of the target of interest. This can be expressed as the fol-lowing optimization problem

(57)

where is the receive beamforming weight vector.The solution to (57) is given by [34]

(58)

In practice, the matrix is unavailable and, therefore, thesample covariance matrix is used, where

are data snapshots which can be collected fromdifferent radar pulses within a coherent processing interval. Itis worth noting that the target signal component is present in

. An alternative way to obtain a target signal-free sample co-variance matrix is to collect the data snapshots fordifferent range bins [14], [35]. The latter is used in our simula-tions. Also note that the avenue for using robust adaptive beam-forming techniques [36]–[38] is also opened.

V. SIMULATION RESULTS

In our simulations, we assume a ULA of omnidirec-tional antennas used for transmitting the baseband waveforms

where . We alsoassume a ULA of omnidirectional antennas spacedhalf a wave length apart from each other at the receiving end.The additive noise is modeled as a complex Gaussian zero-meanspatially and temporally white random sequence that has iden-tical variances in each array sensor. We assume two interfering

(54)


Fig. 1. First example: transmit beampatterns using conventional beamformer(� � �� wavelength).

targets located at directions and except in Example4 where one spatially distributed interference is assumed. Thetarget of interest is assumed to reflect a plane-wave that im-pinges on the array from direction . In all our simu-lation examples we compare the proposed phased-MIMO radar(17) with the phased-array radar (19) and the MIMO radar (6).For the phased-MIMO radar, we always used subarrayswhich are assumed to be fully overlapped. In some examples, wecompare different radar techniques to each other in terms of theirtransmit/receive beampatterns, while in other examples, the per-formance of the aforementioned radar techniques is comparedin terms of the output SINRs. The sample covariance matrix iscomputed based on data snapshots (i.e., 100 rangebins) for all methods tested. Note that for the MIMO radar tech-nique, the sample covariance matrix is of size 100 100. Toavoid the effect of low sample size, the diagonal loading (DL)of is used when solving (58). Note that this DL is used notonly for the MIMO radar but also for the other two radar tech-niques tested for the reason of fair comparison. In all examples,output SINRs are computed based on 100 independent simula-tion runs for all methods tested.

A. Nonadaptive Transmit/Receive Beamforming

Example 1: Nonadaptive Transmit/Receive BeampatternWithout Spatial Transmit Aliasing: In the first example, weexamine the transmit/receive beampattern of the transmit/re-ceive beamformer (25)–(26), for the case when the transmitantennas are located half a wavelength apart, i.e.,wavelength. Figs. 1 and 2 show the transmit beampatternsand the waveform diversity beampatterns, respectively, for allthree radar techniques tested, while Fig. 3 shows the overalltransmit/receive beampatterns for the same techniques.

From Fig. 1, we can see that the phased-array radar hasthe typical conventional beampattern with mainlobe (of width

) centered at while the MIMO radar has flat (0 dB) trans-mitting gain. On the other hand, the phased-MIMO transmitbeampattern is characterized by the aperture (actual size) ofthe individual subarrays. Since the aperture of the subarrays is

Fig. 2. First example: waveform diversity beampatterns using conventionalbeamformer (� � �� wavelength).

Fig. 3. First example: overall beampatterns using conventional transmit/receivebeamformer (� � �� wavelength).

always smaller than the aperture of the whole array, the transmitbeampattern of the phased-MIMO radar represents a tradeoffbetween the beampatterns of the MIMO and phased-arrayradars. As we can see in Fig. 1, the reduction in the subarrayaperture results in the beampattern of the phased-MIMO radarwith a wider main beam and a little higher sidelobe levels ascompared to the beampattern of the phased-array radar. Thissmall loss in beampattern shape is repaid at a greater gain inthe waveform diversity beampattern as shown in Fig. 2. It isnoted from Fig. 2 that the phased-array radar has no waveformdiversity gain (0 dB flat pattern), while the waveform diversitybeampatterns of the MIMO and phased-MIMO radars areequivalent to conventional beampatterns offered by an and

elements virtual arrays, respectively. Because , thewaveform diversity beampattern of the phased-MIMO radarhas a wider mainlobe and higher sidelobe levels as comparedto the waveform diversity beampattern of the MIMO radar.However, one can see in Fig. 3 that the overall transmit/receive


Fig. 4. Second example: overall beampatterns using conventional transmit/re-ceive beamformer (� � �� wavelength).

beampattern shape for the proposed phased-MIMO radar issignificantly improved as compared to the beampatterns of thephased-array and MIMO radars. Particularly, it is worth notingthat the overall beampattern of the proposed phased-MIMOradar is proportional to the multiplication of the transmit andthe waveform diversity beampatterns (i.e., proportional to thesummation of the curves in Figs. 1 and 2 in decibels). We alsocan observe from Fig. 3 that the phased-array and MIMO radarshave exactly the same overall transmit/receive beampatterns.At the same time, the phased-MIMO radar has lower sidelobelevels as compared to both the phased-array and MIMO radars.

Example 2: Nonadaptive Transmit/Receive BeampatternWith Spatial Transmit Aliasing: In this example, we investigatethe nonadaptive transmit/receive beampattern for the case whenthe transmit antennas are located more than a half wavelengthapart from each other. In particular, the case (0.5 wavelength) is chosen and the corresponding nonadaptivebeamforming based beampatterns for the phased-array, MIMO,and phased-MIMO radars are plotted.

In this case, the transmit beampatterns for all aforementionedtechniques have similar trends to their counterparts in Fig. 1except that, due to spatial aliasing in the transmit mode, eachbeampattern is repeated five times within the spatial domain

. The reason is that the interelement spacing isfive times half a wavelength, i.e., five times the critical spatialsampling spacing. Similarly, the diversity beampatterns alsoexhibit spatial aliasing for all the techniques tested. The overalltransmit/receive beampattern is shown in Fig. 4. It can beseen from this figure that the phased-array and MIMO radarshave exactly the same transmit/receive beampatterns, while theproposed phased-MIMO radar beampattern enjoys much lowersidelobe levels as compared to the beampatterns of the otherradar techniques. Hence, the proposed phased-MIMO radar isshown to offer a much better overall performance. The spatialaliasing in the transmit and the waveform diversity beampat-terns has an effect on the overall transmit/receive beampattern.As a result, the sidelobe levels for all radar techniques exhibit

Fig. 5. Third example: nonadaptive transmit/receive output SINRs versus SNRat fixed �� 30 dB.

large variations over the spatial range. This phenomenon war-rants the necessity for imposing some sort of sidelobe controlon the design of transmit/receive beamforming for MIMO radarsystems. However, due to space limitations, this opportunity isnot considered here.

Example 3: Nonadaptive Output SINR: In this example, thenonadaptive beamformer output SINR is tested versus SNR fordifferent INR values. Fig. 5 shows the output SINR versus SNR(INR is fixed to 30 dB) for the phased-array, MIMO, and phased-MIMO radars. It can be seen from the figure that the outputSINR for the phased-array and MIMO radars are almost thesame. This observation agrees with the fact that both techniqueshave the same sidelobe attenuation level and, therefore, have thesame interference rejection capabilities. On the other hand, thephased-MIMO radar has a much higher output SINR as com-pared to both the phased-array and MIMO radars.

Fig. 6 shows the output SINR versus SNR while the INR isfixed to 30 dB, i.e., the interference power can be neglected ascompared to the noise power. It can be seen from the figure thatthe phased-array radar output SINR is ten times higher than theMIMO radar output SINR. This gain is attributed to the transmitcoherent processing gain that the phased-array radar enjoys. Itis also observed from this figure that the phased-MIMO radarhas an output SINR that is very close to the output SINR of thephased-array radar yet it enjoys the waveform diversity benefitsoffered by the MIMO radar. This depicts the tradeoff offered bythe phased-MIMO radar.

Example 4: Nonadaptive Output SINR in the Presence ofSpatially Distributed Interference: In this example, we assumeone spatially distributed interference source which is uniformlyspread over the spatial sector . The spatial powerdensity of the distributed source is normalized such that the totalpower of the interference is equivalent to the required INR. Weassume that the target has the same power as the interference,i.e., . Fig. 7 shows the output SINR versus SNRwhere the INR is varied. It can be seen from the figure that atlow SNR the phased-array radar output SINR is higher than


Fig. 6. Third example: nonadaptive transmit/receive output SINRs versus SNRat fixed �� 30 dB.

Fig. 7. Fourth example: nonadaptive transmit/receive output SINRs versus�� ; spatially distributed interference.

the MIMO radar output SINR. However as the SNR increases,the difference between the SINRs of both techniques tend todecrease and the two curves eventually coincide. On the otherhand, the proposed phased-MIMO radar is shown to have anoutput SINR that outperforms both techniques atlarger than 10 dB. At low SNR values, the output SINR ofthe phased-MIMO radar is comparable to the output SINR ofthe phased-array radar which coincides with our theoreticalfounding.

B. Adaptive Transmit/Receive Beamforming

Example 5: MVDR Beamforming Employing MultipleTransmit Multiple Receive Antennas: In this example, theMVDR receive beamforming (58) is used for the phased-array,MIMO, and phased-MIMO radars. All simulation parametersare the same as in Example 1 except that the target power isfixed to 0 dB while the interference power is fixed to 50 dB. Atthe transmit array, the conventional beamformer is used and,

Fig. 8. Fifth example: overall beampatterns using MVDR beamformer (� �

�� wavelength).

Fig. 9. Fifth example: output SINRs versus SNR, � � �� receive antennasspaced half wavelength apart.

therefore, the transmit beampatterns for all radar techniquestested are the same as those given in Fig. 1. The receive MVDRbeampattern is shown in Fig. 8 for all radar techniques tested. Itcan be observed from this figure that all radar techniques exhibitnulls at the locations of the powerful interference. Moreover,the proposed phased-MIMO radar as well as the phased-arrayradar have lower sidelobe levels as compared to the MIMOradar. This means that the phased-MIMO radar has almost thesame robustness against sensor noise as the phased-array radar.At the same time, it enjoys the advantages of the MIMO radar,e.g., waveform diversity.

Fig. 9 shows the optimal SINR as well as the MVDR outputSINR versus SNR (INR is fixed to 30 dB) for all radar tech-niques tested. From this figure, we can see that the phased-arrayradar outperforms the MIMO radar that can be attributed to ro-bustness of the phased-array radar against sensor noise due tothe use of transmit coherent processing. On the other hand, thephased-MIMO radar exhibits SINR performance that is very


Fig. 10. Sixth example: overall beampatterns using MVDR beamformer,� �

� receive antenna.

close to the phased-array SINR performance. It is interestingto note that the proposed phased-MIMO radar offers a substan-tially better MVDR output SINR as compared to the MIMOradar. This gain is expected since the proposed MIMO radarcombines the advantages of both the phased-array and MIMOradars.

Example 6: MVDR Beamforming Employing MultipleTransmit Single Receive Antennas: In the last example, weinvestigate an interesting case of a bistatic radar system whichemploys multiple transmit antennas and a single receive an-tenna. The same scenario as in Example 5 is considered hereexcept that the number of receive antennas is . In thiscase, the dimensions of the received data by the phased-array,MIMO, and phased-MIMO radar systems will be 1 1, ,and , respectively. The MVDR beamformer (58) is usedfor the MIMO and phased-MIMO radar techniques and theMVDR beampatterns are plotted in Fig. 10 for dB.We observe from this figure that the phased-array radar hasfailed to reject the powerful interference. In fact, the adaptivebeamforming is not helpful with the phased-array radar becausethe dimension of the received data is just 1 1. The main beamexhibited by the phased-array radar is attributed to the transmitcoherent processing only. Contrary to the phased-array radar,the MIMO and phased-MIMO radars lend themselves easily toadaptive techniques at the receiving end. The virtual extendedarray of larger dimensionality at the received end is the result ofusing waveform diversity at the transmit array. As we can seefrom Fig. 10, both the MIMO and phased-MIMO radars exhibitexcellent adaptive interference rejection capabilities. However,the MIMO radar suffers from poor performance because ofits lack of robustness against sensor noise. On the other hand,the phased-MIMO radar enjoys much lower sidelobe levelsand, therefore, higher robustness against sensor noise as com-pared to the MIMO radar. Indeed, Fig. 10 demonstrates thatthe proposed phased-MIMO radar enjoys the advantages ofphased-array and MIMO radars and, therefore, is superior toboth.

Fig. 11. Sixth example: output SINRs versus SNR, � � � receive antenna.

Fig. 11 shows the optimal SINRs as well as the MVDR outputSINRs versus SNR (INR is fixed to 30 dB) for all radar tech-niques tested. From this figure, we can see that the phased-arrayradar performs very poorly due to its inability to reject stronginterference. On the other hand, the MIMO radar is able to adap-tively reject interference and, therefore, it has better SINR per-formance as compared to the phased-array radar. Moreover, thephased-MIMO radar exhibits the SINR performance that is su-perior to the performance of both the phased-array and MIMOradars. This gain is attributed to the ability of the phased-MIMOradar to reject interference combined with its robustness againstsensor noise.

VI. CONCLUSION

A new technique for MIMO radar with colocated antennashas been proposed. This technique is based on partitioning thetransmit array to a number of subarrays which are allowed tooverlap. Each subarray is used to coherently transmit a wave-form which is orthogonal to the waveforms transmitted by othersubarrays. Coherent processing gain is achieved by designingthe weight vector for each transmit subarray to form a beamtowards a certain direction in space. The subarrays are com-bined jointly to form a MIMO radar resulting in higher angularresolution capabilities. It is shown that the proposed techniquecombines the advantages of the phased-array and MIMO radarsand, therefore, it has a superior performance. Simulation resultsconfirm our theoretical observations and demonstrate the effec-tiveness of the proposed phased-MIMO radar technique. Theformulation of the new phased-MIMO radar technique opensa new avenue in MIMO radar developments. Some new prob-lems highlighted in the paper are the transmit beamforming andtransmit subarray waveform designs, which satisfy certain de-sired properties.

APPENDIX APROOF OF PROPOSITION 2

In this appendix, we prove that the highest sidelobe of thephased-MIMO radar beampattern is lower than the highest


Fig. 12. Comparison between sidelobe levels of the phased-MIMO and phased-array radars (number of transmit antennas is fixed to � � ��); � �� .

sidelobe of the phased-array radar beampattern, i.e., we provethat

(59)Using Fourier transform analysis, the square-root of the right-

hand side of (59) can be expressed as

(60)

where is the Dirac-delta function, is the interelementspacing, is the wavelength of the propagating wave, is theconvolution operator, and is given by

(61)

for some positive integer . Similarly, the square-root of theleft-hand side of (59) can be expressed as

(62)

where

(63)

The function is plotted in Fig. 12 for different values of. It can be observed from the figure that the case which

corresponds to the phased-array radar has higher sidelobes ascompared to all other cases when .

Let us now state some properties related to the function.

P1) The main lobe occupies the region .P2) For , the main lobe occupies the whole region

while side lobes do not exist.P3) For , there is only one sidelobe whose peak islocated at ; at this point .P4) For , there are multiple sidelobes. The sidelobeclosest to the main lobe occupies the region

.P5) Noting that is monotonically increasingwithin , the highest peak sidelobe is the one closestto the main lobe.P6) The peak of the highest sidelobe occurs whenis approximately at its maximum, i.e., whichoccurs at [34, Ch. 2].

Let , , and be the peaks of the highest sidelobes of, , and

, respectively. Let also , , and be thelocations of , , and , respectively. Then, the maximumsidelobe of the left-hand side of (59) is upper-bounded asfollows

(64)

Therefore, the inequality (59) can be proved by proving that

(65)

Dividing by , we obtain the ratio

(66)

Using Property P6), can be approximated as

(67)

Moreover, noting that, we conclude that . Using Properties P4)–P6),

we find that . Therefore, can be upper-bounded as follows:

(68)

On the other hand, we have . More-over, using Property P4) we conclude that


. Therefore, can be lower-bounded as follows

(69)Similarly, using the same chain of arguments as above,

can be lower-bounded as follows:

(70)

Therefore, using (68), (69), and (70), we obtain

(71)

Substituting (67) and (71) in (66), we prove that. This means that which proves (59) for the case

when and . Using Property P3), forthe case when , we have

. Hence, in this case, we haveyielding . Therefore, (59) is fully proved.

ACKNOWLEDGMENT

The authors would like to thank Dr. D. Fuhrmann of MichiganTechnological University, Dr. Y. Abramovich of Australian De-fence Science and Technology Organisation (DSTO), Adelaide,and Dr. A. Gershman of Darmstadt University of Technologyfor useful discussions, providing some references and clarifica-tions on the existing radar technology.

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Aboulnasr Hassanien (M’08) received the B.Sc. de-gree in electronics and communications engineeringand the M.Sc. degree in communications engineeringfrom Assiut University, Assiut, Egypt, in 1996 and2001, respectively, and the Ph.D. degree in electricalengineering at McMaster University, Hamilton, ON,Canada, in 2006.

From 1997 to 2001, he was with the Departmentof Electrical Engineering, South Valley University,Aswan, Egypt. From May to August 2003, he was avisiting Researcher at the Department of Communi-

cation Systems, University of Duisburg-Essen, Duisburg, Germany. From Aprilto August 2006, he was a Research Associate with the Institute of Telecom-munications, Darmstadt University of Technology, Germany. From September2006 to October 2007, he was an Assistant Professor at the Department of Elec-trical Engineering, South Valley University, Aswan, Egypt. He has been withthe Department of Electrical and Computer Engineering, University of Alberta,Edmonton, AB, Canada, as a Research Associate since November 2007. Hisresearch interests are in MIMO radar, statistical and array signal processing, ro-bust adaptive beamforming, and parameter estimation.

Sergiy A. Vorobyov (M’02–SM’05) received theM.S. and Ph.D. degrees in systems and control fromKharkiv National University of Radioelectronics,Ukraine, in 1994 and 1997, respectively.

Since 2006, he has been with the Department ofElectrical and Computer Engineering, University ofAlberta, Edmonton, AB, Canada, where he has beenrecently promoted to an Associate Professor. Sincehis graduation, he also held various research and fac-ulty positions in Kharkiv National University of Ra-dioelectronics, Ukraine; the Institute of Physical and

Chemical Research (RIKEN), Japan; McMaster University, Canada; Duisburg-Essen and Darmstadt Universities, Germany; and the Joint Research Institute,Heriot-Watt, and Edinburgh Universities, U.K. His research interests includestatistical and array signal processing, applications of linear algebra and op-timization methods in signal processing and communications, estimation anddetection theory, sampling theory, and cooperative and cognitive systems.

Dr. Vorobyov is a recipient of the 2004 IEEE Signal Processing Society BestPaper Award, the 2007 Alberta Ingenuity New Faculty Award, and other re-search awards. He was an Associate Editor for the IEEE SIGNAL PROCESSING

LETTERS from 2007 to 2009. Since 2006, he has served as an Associate Ed-itor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He is a member ofSensor Array and Multi-Channel Signal Processing Technical Committee ofIEEE Signal Processing Society.

Phased-MIMO Radar a Tradeoff Between Phased Array and MIMO Radars

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