MIMO Radar Signal Processing || Adaptive Signal Design For MIMO Radars

5ADAPTIVE SIGNAL DESIGN FORMIMO RADARS

BENJAMIN FRIEDLANDER

Department of Electrical Engineering, University of California, Santa Cruz

We consider the problem of signal design for MIMO radars, where the transmitwaveforms are adjusted based on target and clutter statistics. A model for the radarreturns that explicitly incorporates the transmit waveforms is developed. Bothestimation and detection problems are formulated for that signal model. Optimaland suboptimal algorithms are derived for designing the transmit waveforms. Theperformance of these algorithms is illustrated by computer simulation. Theseresults indicate that adaptive design of the radar signal can in some cases provideimproved estimation and detection performance compared to fixed designs such asthe one employing a set of orthogonal waveforms.

5.1 INTRODUCTION

More recent advances in linear amplifier and waveform generation technology, andthe ever-increasing processing power, have spawned interest in the development ofradar systems that attempt to make full use of the spatiotemporal degrees offreedom available to the radar transmitter. These technological advances make itpossible to consider the design of radar systems that allow the transmitter full flexi-bility in selecting the transmitted waveform (within given bandwidth and power con-straints) on a pulse-by-pulse or antenna-by-antenna basis.

The flexibility of using a multiplicity of transmitted waveforms, and of adap-tively adjusting these waveforms, offers significant performance advantages.

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

193

Fundamentally, the additional degrees of freedom afforded by the ability to vary thetransmit waveform can be used to optimize a desired performance criterion. Forexample, the waveform can be adapted to the target signature to enhance detectability,increase clutter or interference rejection, improve the quality of the estimated radarmap, improve spatial resolution, or reduce search time.

A considerable amount of research has been done on waveform design issues suchas work on optimum transmit–receive design [1–3], which assumes a deterministictarget model with a range spread, using a single transmit antenna, or an antennawith multiple polarization modes [4]. In a more recent paper [5] we studiedoptimal optimal waveform design for a single-antenna radar. We presented a signalsubspace framework that made it possible to derive the optimal radar waveform fora given scenario and evaluate the corresponding radar performance. These worksinvolve temporal processing only.

There has been considerable interest in radar systems employing multiple antennasat both the transmitter and receiver, commonly referred to as MIMO radar [6–21].In this case spatiotemporal processing of the radar signals is required. The waveformstransmitted by the radar may be adaptive or nonadaptive. By “adaptive” we mean thatthe waveforms depend on information about the particular scene being observed,such as the target and clutter statistics, or on the actual target returns. By “nonadap-tive” we mean that the waveforms are designed without making use of such infor-mation. However, the waveforms may be different for different antennas and maychange from pulse to pulse. We note that this terminology is by no means standardand that we use it here only in the context of the transmit waveform design. As anexample, the radar may perform adaptive processing at the receiver (e.g., STAP),while using a nonadaptive method to design the transmit waveform.

Most of the work to date on signal design for MIMO radar has focused on the non-adaptive case. A widely studied approach involves the transmission of orthogonalsignals on the different antennas. This makes it possible to separate the signals arriv-ing from the different transmit antennas at the receiver, and to perform any transmitarray processing functions on the receive side “after the fact.” For example, onecan scan the transmit beam across the illuminated area within a single dwell time,or perform adaptive beamforming to reduce interference and improve resolution[6–13]. By employing adaptive processing, it is possible to improve clutter rejectionin ways that are not possible in conventional radar [14,16]. MIMO radar can alsoprovide angular diversity, which is useful in some scenarios [22–24].

Here we consider the waveform design problem for MIMO radars in a rathergeneral setting that allows for any waveform design procedure, either adaptive ornonadaptive. In particular, we develop a procedure to design the optimal waveformthat maximizes the signal-to-interference-plus-noise ratio (SINR) at the output ofthe detector, as well as some suboptimal variations. The key to our approach is useof a model for the received radar signals that explicitly includes the transmitted wave-forms. Most radar systems perform range processing as a first step, employing amatched filter. As is well known, the response to a single point target measuredat the matched filter output does not depend on the shape of the waveform, onlyon its energy (however, the shape of the waveform affects the range–Doppler

ADAPTIVE SIGNAL DESIGN FOR MIMO RADARS194

ambiguity function, which plays an important role when multiple targets arepresent). Thus, the transmitted signal does not appear explicitly in the processingsubsequent to the range processing step. In Section 5.2 we develop the modelfor the signals received at the antenna outputs prior to any processing. Thismodel exposes the dependence on the transmitted waveform and allows thederivation of various estimation algorithms in Section 5.3 and detection algorithmsin Section 5.4.

5.2 PROBLEM FORMULATION

Consider a MIMO radar employing NT antennas at the transmitter and NR antennasat the receiver. The radar operates at a carrier frequency fc and has a bandwidth B.We assume here that the two arrays are collocated, that is, represent a monostaticradar. The extension to the bistatic case is straightforward but requires a differentparametrization of the scatterer locations, which entails significant changes in theequations presented here. In this section we develop a model for the signal receivedby the MIMO radar. We use the convention in which the time-domain variables aredenoted by an overline (e.g., x), to distinguish them from their frequency-domaincounterparts (e.g., x).

Let h(u, t) denote the distribution of scatterer amplitudes in the area illuminated bythe transmit antenna (the antenna footprint), where t is the round-trip delay from areference antenna to the scatterer location. The round-trip delay t can be translatedinto range r by r ¼ ct/2, where c is the speed of light. Consider the scatteringfrom a patch of size du dt, representing a scatterer with amplitude h(u, t)du dt.Let ti(u) denote the delay difference between the ith transmit antenna and thereference antenna along direction u.

The ith transmitter transmits the signal s(t)e jwct, where s(t) is the baseband wave-form. The signal reflected from the scatterer is

h(u, t)du dt s t � t=2� tti(u)

� �e jwc(t�t=2�tt

i (u)) (5:1)

where tti(u) denotes the delay difference between the ith transmit antenna

and the reference antenna along direction u. This signal arrives at kth receiveantenna as

h(u, t)du dt s(t � t� tti(u)� tt

k(u))e jwc(t�t�tti (u)�tr

k(u)) (5:2)

where trk(u) denotes the delay difference between the kth receive antenna and the

reference antenna along direction u. The delay t is the round-trip delay betweenreference antennas.

5.2 PROBLEM FORMULATION 195

In the case where the narrowband assumption holds, we have s(t � t) � s(t).Therefore, the signal arriving at the kth antenna is

h(u, t)du dt s(t � t)e jwc(t�t�tti (u)�tr

k(u)) (5:3)

After downmodulating to baseband, this signal becomes

h(u, t)du dt s(t � t)e�jwc(tþtti (u)þtr

k(u)) (5:4)

Note that so far we have considered only a single waveform s(t). Allowing for thepossibility that each transmit antenna will have a different waveform, we definethe vector

s(t) ¼ [ s1(t), . . . , sNT (t)]T (5:5)

consisting of the waveforms for the different antennas. Let

aT (u) ¼ [e�jwctt1(u), . . . , e�jwct

tNT

(u)]T (5:6)

denote the array manifold of the transmit array, where tt are the transmitter-relateddelay differences, and similarly

aR(u) ¼ [e�jwctr1(u), . . . , e�jwct

rNR

(u)]T (5:7)

is the array manifold of the receiver array, where tr are the receiver-related delaydifferences. The vector of received signals is then

x(t) ¼ aR(u)h(u, t)du dt e�jwctaT (u)T s(t � t) (5:8)

Without loss of generality, we absorb the phase term e�jwct into h(u, t). The arraymanifolds aR(u) and aT (u) were defined above for the case of identical omnidirec-tional antennas. In practice, these array manifolds will include the gain patterns ofthe antennas, effects of gain/phase mismatch, and mutual coupling, and will havea form different from the one defined above.

It is often desired to filter the received signal. Assume that a filter with impulseresponse q(t) is used to filter each element of x(t). The output of the filter y(t) isgiven by

y(t) ¼ aR(u)h(u, t)du dtaT (u)T sq (t � t) (5:9)

where sq (t) ¼ q(t) � s(t). Note that the filtering of the received signal is equivalent toreplacing the transmitted signal by its filtered version.


When the scene consists of many point scatterers, we need to integrate over all oftheir contributions to get

x(t) ¼ð ð

aR(u)h(u, t)aT (u)T s(t � t) dt du (5:10)

Note that we have here a convolution in the t variable so that

x(t) ¼ð

aR(u)aT (u)T h(u, t) � s(t)du (5:11)

Replacing s(t) by sq (t) will give the filtered version of x(t).These continuous-time angle equations can be replaced by their sampled versions

provided that the proper sampling intervals Dt and Du are used. Sampling in time andangle, we get

x[tn] ¼X

m

aR(um)aT (um)T h[um, tn] � s[tn] (5:12)

where h[um, tn] ¼ h(um, tn)DuDt, and where um and tn ¼ nDt are the sampling points.Replacing s[tn] by sq [tn] will give the filtered version of x[tn].

The sampling intervals are determined as follows. For temporal sampling weassume that s(t) is bandlimited with bandwidth B. The return signal will then be simi-larly bandlimited. Sampling above the Nyquist rate, we get Dt , 1=2B. For spatialsampling, assume that the aperture of the composite array is D. We want to havethe return from an area whose angular extent is Du to be correlated across the arrayaperture. The beamwidth of the radiation from the patch is l=DuR. We want the aper-ture to be within this beamwidth, therefore D , (l=DuR)R or Du , l=D. If thiscondition is satisfied, we can represent the patch by a single amplitude h(u, t)du dtfor all the antennas.

Note the presence of the convolution h[um, tn] � s[tn] in this model. It is convenientto reformulate the model in the frequency domain, replacing the convolution by mul-tiplication. Therefore we replace the sampled time sequences x[tn], h[um, tn], and s[tn]by their discrete Fourier transforms x[ fn], h[um, fn], and s[ fn].

The received signal at frequency fn is an NR � 1 vector given by

x [fn] ¼XNa

m¼1

aR(um)aT (um)T h[um, fn]s[ fn] (5:13)

where Na is the number of azimuth cells in the antenna footprint. This can be writtenin a more compact matrix form as

x [ fn] ¼ AR diag{h[ fn]}ATT|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

As[ fn]

s[ fn] (5:14)


where AR ¼ [aR(u1), . . . , aR(uNa )] is the NR � Na receive array manifold matrix,AT ¼ [aT (u1), . . . , aT (uNa )] is the NT � Na transmit array manifold matrix, andh[fn] ¼ [h[u1, fn], . . . , h[uNa , fn]]T is the Na � 1 vector of scatterer amplitudes atfrequency fn. Assembling these equations for all frequencies into a single matrixequation, we get

X ¼ [x[ f1], . . . , x[ fNf ]] ¼ [As[ f1]s[ f1], . . . , As[ fNf ]s[ fNf ]] (5:15)

where X is the NR � Nf received data matrix.An alternative way of writing the received data vector is given by

x[ fn] ¼ ARdiag{h[ fn]}ATT s[ fn] ¼ AR diag{g[ fn]}|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

Ah[ fn]

h[ fn] (5:16)

where g[ fn] ¼ ATT s[ fn] is the Na � 1 illumination vector at frequency fn. Assembling

these equations for all frequencies into a single matrix equation, we get

X ¼ [x[ f1], . . . , x[ fNf ]] ¼ [Ah[ f1]h[ f1], . . . , Ah[ fNf ]h[ fNf ]] (5:17)

Note that Eq. (5.15) expresses the received data as a linear function of the transmitwaveform s[ fn] while Eq. (5.17) expresses the received data as a linear function ofthe scatterer amplitudes h[ fn].

It is sometimes convenient to write these equations for the Nf NR � 1 data vectorx ¼ vec{X}, instead of for the data matrix X. It is straightforward to show that

x ¼ AR diag{h}ATT|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

As

s (5:18)

where AR is a Nf NR � Nf Na block diagonal matrix with blocks AR. More precisely,AR ¼ INf � AR, where INf is an Nf � Nf identity matrix, and� denotes the Kroneckerproduct. Similarly, AT is a Nf NT � Nf Na block diagonal matrix with blocks AT , orAT ¼ INf � AT . Note that As is block diagonal with Nf blocks of size NR � NT.

The vectors h and s are obtained by stacking the vectors h[ fn] and s[ fn], respecti-vely, for all frequencies. More precisely, let H ¼ [h[ f1], . . . , h[ fNf ]] be the Na � Nf

matrix representing the distribution of scatters in azimuth and frequency. Thenh ¼ vec{H}. Similarly, let S ¼ [s[ f1], . . . , s[ fNf ]] be a NT � Nf matrix representingthe transmitted waveforms. The ith row of this matrix is the DFT of the waveformtransmitted by the ith antenna. Then s ¼ vec{S}.

The received data vector can be written alternatively as

x ¼ AR diag{g}|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}Ah

h (5:19)


where g ¼ ATT s is the stacked NaNf � 1 azimuth–frequency illumination vector.

Equivalently, g ¼ vec{G}, where

G ¼ ATT S (5:20)

is the Na � Nf illumination function in azimuth and frequency. The matrix Ah is ablock diagonal matrix with Nf blocks of size NR � Na.

Using the illumination matrix G, we have

X ¼ AR(G�H) (5:21)

Equation (5.21) is the fundamental model for the received signal in terms of the scat-tering scene represented by H, the radar illumination G, and the receive array manifoldmatrix AR. This equation can be rewritten in terms of the time-domain counterparts ofX, H, G as

X ¼ XF� ¼ AR(G�H)F� ¼ AR((GFT )� (HFT ))F� (5:22)

where F is the Nf � Nf DFT matrix (i.e., Fz is the DFT of z).Note that if we define Gq ¼ AT

T Sq, where Sq is the filtered version of S, then we getthe filtered output Z ¼ AR(Gq �H). The filtered version of S is given bySq ¼ S� (1� q) where q is a 1 � Nf vector of the filter frequency response.

5.2.1 Signal Model with Reduced Number of Range Cells

In the discussion above we assumed that the scattering map H and the received data Xhave the same number Ns of delay/range samples. In practice, the number of resol-vable range cells is often smaller than the number of samples of the received data.In that case it is convenient to represent H as a Na � Nr matrix, where Nr � Ns isthe number of resolvable range cells. This requires some modification of the signalmodel, as is shown next.

As before, we convert H into a frequency-domain Na � Nf matrix H, where

H ¼ H FT, with F equivalent to the Nf � Nr Fourier transform matrix generating Nf

frequencies from Nr time samples. The equations introduced earlier still hold, exceptthat the square Nf � Nf DFT matrix F is replaced by the nonsquare DFT matrix F.The received data matrix X can be written in terms of the reduced dimension H as

X ¼ AR(G� (H FT)) (5:23)

Next we define the Na Nf � Na Nf permutation matrix P, which converts vec{HT } intovec{H}:

vec{H} ¼ P vec{HT } (5:24)


Similarly we define the NaNr � NaNr permutation matrix P

vec{H} ¼ P vec{HT} (5:25)

From H ¼ H FT

it follows that HT ¼ F HT

and

vec{HT } ¼ (INa � F) vec{HT} (5:26)

Multiplying by P, we obtain

vec{H} ¼ P(INa � F) vec{HT} (5:27)

Converting vec{HT} to vec{H} using the permutation matrix P, we get

vec{H} ¼ P(INa � F) PT

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}F

vec{H} (5:28)

and finally

x ¼ Ahh ¼ (AhF)|fflffl{zfflffl}A

h

h (5:29)

This equation relates the received data x to the reduced dimension H. Note that x isa NR Nf � 1 vector, and h is a Na Nr � 1 vector. Thus, the scattering map h can beestimated by least-squares solution of the equation above, provided that NR Nf Na Nr.

Note that in the model introduced earlier the data could be computed frequency byfrequency, that is, that X can be generated column by column. Here x needs to becomputed in its entirety, requiring significantly more computations. This added com-plexity is related to the fact that the reduced-dimension model involves an inter-polation step. To see this more clearly note that

HF� ¼ H FT

F�|fflffl{zfflffl}U

(5:30)

where U is an interpolator operating on the rows of the reduced dimension H, togenerate Ns samples from Nr � Ns samples.

5.2.2 Multipulse and Doppler Effects

So far we considered the radar return for a single pulse. Extending this to the multi-pulse case is straightforward, but requires factoring in Doppler effects.


First, consider the case where the scatters are stationary and Doppler is inducedonly by the motion of the platform carrying the radar system. Assume a platformmoving at velocity v and that the transmit and receive arrays are aligned with the velo-city vector of the platform. This assumption is not essential, and the results can bemodified in an obvious manner to handle arrays with arbitrary angles relative tothe platform velocity vector. A scatterer at azimuth u relative to the line of flightwill have a radial velocity of v cos u and a Doppler shift of fd ¼ v cos ufc=c, wherefc is the center frequency of the radar and c is the speed of light. Alternatively,fd ¼ v cosu/l, where l is the radar wavelength. This Doppler frequency introducesa phase shift of 2pv cos upTp=l at pulse number p, where Tp is the pulse repetitioninterval. Denoting a ¼ 2pvTp=l, this phase shift can be written as ap cos u.

Let AD be a P � Na matrix representing the Doppler effect where the elements ofAD are given by

[AD] p,m ¼ e japcos um (5:31)

Let ADR be the spacetime array manifold

ADR ¼ AD AAR (5:32)

where A denotes the Khatri–Rao product where AAB ¼ [a1 � b1, . . . , aN � bN],where an and bn are the columns of A and B, respectively.

The spacetime manifold consists of P submatrices, each corresponding to oneradar pulse. In other words

ADR ¼

ADR[1]ADR[2]

..

.

ADR[P]

2

6664

3

7775(5:33)

It is straightforward to show that the radar return matrix for pulse p is given by

X[ p] ¼ ADR[ p](G�H) (5:34)

Assembling all the radar returns into a single matrix X

X ¼

X[1]X[2]

..

.

X[P]

2

6664

3

7775(5:35)

we get

X ¼ ADR(G�H) (5:36)


This equation is very similar to Eq. (5.21) for the single-pulse case, where thespatial array manifold AR is replaced by the spacetime manifold ADR. In the STAPliterature X is often called the “radar data cube.”

Next we consider the case where the scatterers are moving (an airborne target,moving vehicles, etc.). In the most general case the radar return matrix is given by

X[ p] ¼ AR(G�H[ p]) (5:37)

where H[ p] is the scattering map and the phase of each scatterer varies from pulse topulse in accordance with the radial velocity of the scatterer relative to the radarplatform.

The case where only a single scatterer (or a rigid collection of scatterers) is inmotion is of special interest because we are often interested in detecting a movingtarget in the presence of stationary clutter clutter. In this case the data model canbe written as follows. Let Ht denote the zero-Doppler scattering map of a target atazimuth ut with velocity component vt in the radar direction. Then the target radarreturn is given by

Xt[p] ¼ e jbtp AR(G�Ht) (5:38)

where bt ¼ 2pvtTp cos ut=l: Let

dt ¼ [e jbi1, . . . , ejbtP]T (5:39)

be the target Doppler vector representing the target related phase shift for all pulses.Then, stacking the matrices X t[ p] into X t, we get

Xt ¼ dt � (AR(G�Ht)) (5:40)

Multiple targets can be handled similarly so that the total return from all targets willbe a sum of terms having the form of Eq. (5.40):

X ¼X

i

dt[i]� (AR(G�Ht[i])) (5:41)

In the multipulse model discussed so far, it was assumed that the same waveformswas used on all pulses. We note that a more general formulation would allowtransmission of different waveforms on different pulses, in which case the modelwould be

X[ p] ¼ ADR[ p](G[ p]�H[ p]) (5:42)

where G[ p] ¼ ATTs[ p], where s[ p] is the waveform transmitted during pulse p.

However, we do not explore the case of different waveforms for different pulses here.


5.2.3 The Complete Model

So far we considered the case where the scattering scene H is noise-free. In practice,these data will be contaminated by noise assumed to be zero-mean complex Gaussianand independent across measurements. Additionally, it is sometimes useful to decom-pose the scattering scene H into a component due to targets of interest H t and acomponent due to clutter H c. We denote the corresponding components of thereceived data by X t and Xc respectively. The complete signal model for the receivedradar signal Y is then given by

Y ¼ Xt þ Xc þ Xn (5:43)

where Xn is the noise matrix, X t ¼ AR(G�H t) and X c ¼ AR(G�H c). This can alsobe written in vectorized form as y ¼ x t þ x c þ xn, where xn CN (0, Rxn ). Thenoise term may include receiver noise and any external interference. In the casewhere only receiver noise is present, the noise covariance matrix has the simplerform Rxn ¼ sn

2I, where sn2 is the noise variance.

5.2.4 The Statistical Model

The clutter return x c changes randomly over time and can be represented as arandom process whose statistics are determined by the characteristics of the scatterersHc. In the following we assume that h c ¼ vec{Hc} is a multivariate complex Gaussianvector with zero mean and covariance Rhc , that is, h c CN(0, Rhc ). Recalling that theclutter return is related to the scatters by xc ¼ Ahhc, it follows that Rxc ¼ AhRhcA

Hh .

The clutter covariance matrix can always be decomposed as Rxc ¼ Vxc VHxc

whereVxc is an NRNf � rc matrix, where rc is the rank of Rxc . If all the clutter scatterers h c

are assumed to be uncorrelated, R xcwill be of full rank: rc ¼ NRNf. If the clutter is

correlated, then rc , NRNf.The target return x t is also assumed to have a similar statistical model where h t ¼

vec{H t} is a multivariate complex Gaussian vector with zero mean and covarianceRhi specifically, CN (0, Rht ), and Rxt ¼ AhRhtA

Hh . The rank of Rxt will be denoted rt.

The covariance matrix of the target is usually of low rank, in which caseRxt ¼ Vxt V

Hxt

, where Vxt is an NRNf � rt matrix, where rt is the rank of Rxt . Forpoint targets we have rt ¼ 1, while for extended targets we may have rt . 1. Asseen in the following sections, this formulation allows the treatment of extendedtargets in a systematic way.

5.3 ESTIMATION

One of the objectives of a radar system is to estimate the scattering scene H in order todetect, localize, or classify targets of interest. In this section we describe two esti-mation techniques; the first is for the case where the transmitter uses a set of ortho-gonal waveforms for the different antennas, while the second can be used for arbitrarytransmit waveforms.

5.3 ESTIMATION 203

5.3.1 Beamforming Solution

Consider a receiver where the output of each antenna is passed through a bankof filters with frequency responses q i, i ¼ 1, . . . , NT, matched to the transmitwaveforms. These matched filters q i are the rows of S�. The match-filtered outputscan be written as

Z ¼

Z1

Z2

..

.

ZNT

2

6664

3

7775¼

AR((ATT Sq1 )�H)

AR((ATT Sq2 )�H)

..

.

AR((ATT SqNT

)�H)

2

66664

3

77775(5:44)

where Z i is an NR � Nf matrix containing the outputs of all receive antennas filteredby q j, and Sqi ¼ S � (1NT � qi), where 1NT is an NT � 1 vectors of 1 2 s.

Next we pass the NTNR outputs through a beamformer that uses a weight vectorV(u) equal to the steering vector of the composite transmit and receive arrays. Inother words, V(u) ¼ aT (u)� aR(u). The beamformer output b(um), a 1 � Nf

vector, can be written as

b(um) ¼ V(um)HZ ¼ (aR(um)H AR)XNT

n¼1

aT [n]�((ATT Sqn )�H) (5:45)

where aT[n] is the nth element of aT (u) and um is the direction at which the steeringvector is pointed. Next we rewrite the illumination function as

ATT Sqn ¼ AT [n]T Sqn [n] ¼

XNr

k¼1,k=n

AT [k]T Sqn [k] (5:46)

where AT [k] is the kth row of AT. Inserting this into Eq. (5.45), we get

b(um) ¼ (aR(um)HAR)XNT

n¼1

aT [n]�(AT [n]T Sqn [n])�Hþ ~b(um) (5:47)

where

~b(um) ¼ (aR(um)HAR)XNT

n¼1

aT [n]�XNT

k¼1,k=n

(AT [k]T Sqn [k])�H (5:48)

Note that Sqn [n] are the transmit signals passed through their own matched filterswhile Sqn [k] for k = n are the signals passed through filters matched to the other

signals. In other words, ~b(um) contains all the cross-terms. Assuming that the transmitwaveforms are uncorrelated (or have sufficiently low correlation), the elements of~b(um) will be small relative to the elements of the first term in Eq. (5.47), and canbe neglected. Note also that Sqn [n] is the range response of the nth waveform.


Assume that the waveforms are designed so they all have the same range response sq,namely, Sqn [n] ¼ sq. Then Eq. (5.47) can be written as

b(um) � (aR(um)HAR)(((aT (um)HAT )T sq)�H) (5:49)

We see here that the rows of H are weighted by the product of the transmit and receivebeampatterns aT (u)HAT , aR(u)HAR and the columns by the range response sq, as isexpected. To see this more clearly, let

p(um) ¼ (aR(um)HAR)� (aT (um)HAT ) (5:50)

be the row vector denoting the beamformer response vector when it is pointed indirection um. Equation (5.49) can be rewritten as

b(um) ¼ (p(um)H)� sq (5:51)

An alternative derivation of these results for the scenario with a single scatterer of unitstrength at azimuth um, proceeds as follows. The receive array response for trans-mission from antenna n at the output of the corresponding matched filter is givenby aR(um)aT (um)[n]sq. If we assume that the signals from different transmitters arecompletely decoupled at the matched filter outputs, the response to this signal atthe outputs of the other matched filters will be zero. Therefore the stacked responsevector for all the matched filters is given by (aT (um) � aR (um))sq. The correspondingbeamformer weight vector is V(um), which was defined above.

Next we consider briefly the case where the transmitted signals are not orthogonal.Then the transmitted waveform matrix can be written as

S ¼ CT ~S (5:52)

where ~S is an orthogonal set of waveforms and C is an NT � NT matrix containing thecoefficients of the representation of the columns of S in the orthogonal basis ~S. Thereceived data matrix for a scene with a single scatterer of unit strength at azimuth um isgiven by

X ¼ aR(um)aT (um)T CT ~S ¼ aR(um)(CaT (um))T ~S (5:53)

Passing this through a bank of filters matched to the rows of ~S, we get

Z ¼ ((CaT (um))� aR(um))~sq (5:54)

provided that all of these orthogonal signals have the same range response ~sq.In principle, the effect of the correlation can now be removed by filtering Z with

C21 � INR

(C�1 � INR )((CaT (u))� aR(u))~sq ¼ (C�1CaT (u))� (INR aR(u))~sq

¼ aT (u)� aR(u)~sq (5:55)

provided that C is a well-conditioned nonsingular matrix.

5.3 ESTIMATION 205

In the more general case where the range responses corresponding to the rows of ~Sare different, it may still be possible to decorrelate the outputs by “equalizing” theresponses. This can be accomplished by dividing the rows of Z i by ~sq/~sq[i], where~sq[i] is the range response of the ith row of ~S and ~sq is a nominal range response(e.g., the average of the range responses). This will work only if the responses~sq[i] are not too different from each other.

Note that for the purpose of the analysis above the matched filters were shown as ifapplied directly to the transmitted waveforms. In reality, the matched filtering isapplied to the received signals, not the transmitted signals. The bank of matchedfilters is implemented at the receiver by

~Z ¼ (1NR � S�)� (X� 1NT ) (5:56)

where 1NR , 1NT are NR � 1 and NT � 1 column vectors of ones, respectively, and

~Z ¼

~Z1~Z2

..

.

~ZNR

2

6664

3

7775(5:57)

where ~Zi is the output of a matched filterbank processing the signal from the ithreceive antenna (whereas Zi was the ith matched filter applied to all antennas). ~Z is a(NTNR) � Nf matrix. Note that ~Z and Z are equivalent up to permutation of rows.The beamformer weight vector applied to ~Z is aR(u) � aT(u) instead of aT (u) � aR(u).

Computing the beam former outputs b(um) for all angles um of interest, andarranging them in an Na � Nf matrix B, we get an estimate of the scattering distributionH in the azimuth–frequency domain. The beamformer output can be written as

B ¼ VH(1NR � S�)� (X� 1NT ) ¼ VH ~Z (5:58)

where V ¼ [V(u1), . . . , V(uNa )]:Applying the beamformer to the time-domain received data instead of the

frequency-domain data, we get an Na � Ns matrix B that provides an estimate ofthe scattering distribution H in the azimuth–range domain.

In the multipulse case the beamformer will be applied to the receiver outputsgenerated for multiple pulses: ~Z[1], . . . , ~Z[P]. These outputs will be stacked into asingle data matrix ~Z. In the case of a stationary scattering scene the signal modelsfor the single-pulse and multipulse cases have exactly the same form, except thatthe spatial manifold AR is replaced by the spacetime manifold ADR. It follows thatthe weight vectors for the multipulse case are the columns of

V ¼ AD AAR AAT (5:59)

while in the single-pulse case they are the columns of AR A AT. As before, B ¼ VH~Z.


In the case where there are moving targets, we need to calculate the beamformeroutput for all possible Doppler frequencies, because the target Dopplers areunknown. Let

ad(b) ¼

e j2pb0

e j2pb1

..

.

e j2pb(P�1)

2

6664

3

7775(5:60)

be the Doppler manifold, where 20.5 � b� 0.5 is the normalized Dopplerfrequency. The spacetime steering vector now has the form

V(u, b) ¼ ad(b)� aR(u)� aT (u) (5:61)

The beam former output will be calculated for a finite set of angles um and Dopplerfrequencies bn. Let

Ad ¼ [ad(b1), . . . , ad(bNd)] (5:62)

be the Doppler manifold matrix. Then the beam former weight vectors are thecolumns of

V ¼ Ad � (AR AAT ) (5:63)

and B ¼ VH~Z. Comparing Eq. (5.63) to Eq. (5.59), we note that here we have allcombinations of Doppler frequencies and angles, whereas before each angle had asingle specific Doppler frequency associated with it. This was because the scenewas stationary and motion was induced only by the motion of the radar platform.

When the scene consists of both stationary (e.g., clutter) and moving (e.g., target)components, one can choose to use the steering vector defined in either (5.59) or(5.63), depending on what is being estimated. Usually it is the moving target thatis of interest. However, it is sometimes necessary to estimate the stationary back-ground, which can be accomplished at the smaller computational cost associatedwith (5.59). Note that in the multipulse case the beamformer outputs can be arrangedin a NdNa � Nf matrix B, or equivalently in a Nd � Na � Nf “cube” representing anestimate of the scattering distribution in the Doppler–azimuth–frequency domain.Applying the beamform er to the time-domain received data instead of thefrequency-domain data, we get a Nd � Na � Ns cube that provides an estimate ofthe scattering distribution in the Doppler–azimuth–range domain.

To illustrate the beamforming estimation, we present an example for a MIMOradar with NT ¼ NR ¼ 10, where the element spacing of the transmit and receivearrays was dT ¼ 5l and dR ¼ 0.5l, respectively. Other system parameters wereNs ¼ 1024, Nf ¼ 1024, and Na ¼ 256.

Figure 5.1 depicts the beampatterns of the transmit, receive, and compositetransmit-receive arrays, for rectangular windowing. Note the high angular resolutionachieved here — a 3 dB beamwidth of 18 — compared to a SIMO system employing

5.3 ESTIMATION 207

a 10-element array that has a beamwidth of approximately 108. This high resolution isa consequence of the large aperture of the transmit array where the antenna spacingwas chosen to be much larger than a half-wavelength. More specifically, we let dT ¼

NRdR, which results in the composite transmit–receive array behaving like aNRNT -element array with l/2 spacing. Note that the transmit beampattern has mul-tiple beams (grating lobes). However, the receive beampattern eliminates all exceptone of these beams, thus resolving the possible ambiguity. The composite trans-mit–receive beampattern does not change if we reverse the spacing of the transmitand receive arrays, specifically, letting dT ¼ 0.5l and dR ¼ 5l in this example.

Figure 5.2 depicts the range response of the system. The transmit waveforms usedhere were random sequences of +1, independent from antenna to antenna. These arenot orthogonal but have low cross-correlation. It is possible, of course, to designbetter sequences that are orthogonal and have good autocorrelation properties (see,e.g., Ref. 26). Figure 5.3 depicts the beamformer output matrix B in azimuth–range for a simple test case with 10 point targets and SNR of 0 dB. Note the highspatial resolution achieved by the combination of the transmit and receive arraybeampatterns.

Repeating the experiment related in Fig. 5.3 for the multipulse case using the steer-ing vector (5.59) yields a very similar figure except that the signal-to-noise ratio isimproved by a factor of P, due to the coherent integration across pulses. In this experi-ment we assumed both the clutter and the targets to be stationary.

Figure 5.1 Beampatterns for NT ¼ NR ¼10 with dT ¼ 5l and dR ¼ 0.5l.


Next we consider the case where the targets are moving. We used the steeringvector (5.63) to estimate the Doppler–azimuth–range distribution of the scatterers.Figure 5.4 depicts an azimuth–Doppler “slice” through the range cell of a targetlocated at azimuth of 308 and normalized Doppler frequency of 0.25. The targetcan be clearly seen as well as the clutter ridge.

Figure 5.2 Range response using a random 1024-point transmit waveform.

Figure 5.3 Estimated scattering distribution H using the beamformer output B (intensityin dB).

5.3 ESTIMATION 209

5.3.2 Least-Squares Solutions

A more direct estimation of the scattering distribution follows from the fact that thereceived data x is a linear function of h, suggesting the possibility of estimating husing the least-squares solution of x ¼ Ahh. Note that x is a NRNf � 1 vector andh is an NaNf � 1 vector. For the equation to be solvable, we must have Na � NR,which implies poor angular resolution. When this condition is not satisfied, we canuse the reduced dimension version where x ¼ Ahh, where h is a NaNr � 1 vector.If Na � NRNf/Nr, we have a unique solution for h. Note that by decreasing Nr, wecan increase the maximum value of Na, which satisfies this inequality. In otherwords, we increase the number of azimuth cells at the cost of reducing the numberof range cells. The least-squares solution can be used for arbitrary transmit waveformsand does not require orthogonality of the transmit waveforms.

To illustrate the least-squares solution, we present an example for a MIMO radarwith NT ¼ NR ¼ 10, where the element spacing of the transmit and receive arrays was5l and 0.5l, respectively. Other system parameters were Ns ¼ 128, Nf ¼ 128, Na ¼

32, and Nr ¼ 32. Note that Na ¼ 32 , NRNf/Nr ¼ 40. Figure 5.5 depicts the esti-mated scatterer distribution H in azimuth–range for a simple test case with 10point targets and SNR of 0 dB. All 10 targets are visible in their correct locations.

5.3.3 Waveform Design for Estimation

In the absence of noise and interference, the estimation of the scatterer distribution His affected relatively little by the choice of the transmit waveform, as long as the radar

Figure 5.4 Azimuth–Doppler distribution for a range bin containing a single target andclutter, using orthogonal waveforms (intensity in dB).


illuminates the entire area of interest, and as Ah is of sufficient rank. In the presenceof noise and interference the accuracy of the estimates will be affected by the radarillumination G, which is controlled by the transmitted waveforms S.

Different performance criteria can be used to design the transmitted waveform,depending on the application at hand. In this section we consider two cases: (1) max-imizing the total radar return and (2) matching the illumination to the scene. In thefirst case we design S so as to maximize the noise-free received signal x ¼ Ass. Inother words, we maximize kxk2 ¼ sHA

Hs A

Hs Ass. The solution is given by making

s equal to the eigenvector of AsH Ass corresponding to its largest eigenvalue. This

approach tends to focus the transmit energy on the dominant scatterer or group ofscatterers, which may not be desirable, because the rest of the scene may contain fea-tures of interest that are not be adequately illuminated.

An alternative way of controlling the effective illumination is to match the illumi-nation to the scene. In other words, we want to make the projection of g on h as largeas possible. This means maximizing gHPhg ¼ sHA

�T PhA

TT s. Since Ph is of unit rank,

the solution is to have s ¼ (ATATT)21ATh up to normalization of the transmit power.

The total received energy is smaller than in the case described above, but it is morelikely that all features of interest will be illuminated.

Note that both of these methods require knowledge of the scene h that they aretrying to estimate and that the scene is not known a priori. This suggests an iterativeprocedure where we start with some initial waveform s0 (e.g., a set of orthogonalwaveforms) and obtain an initial estimate of the scene h0. A new waveform s1 isdesigned on the basis of the current estimate h0. This waveform is used to obtain anew estimate h1. This process may be repeated to further refine the estimate of h.However, some care needs to be taken to make this process converge.

Figure 5.5 Estimated image H using the least-squares solution; intensity is in decibels.

5.3 ESTIMATION 211

To illustrate the matched illumination method, we present an example of a MIMOradar with NT ¼ NR ¼ 10, where the element spacing of the transmit and receivearrays was 5l and 0.5l, respectively. Other system parameters were Ns ¼ 128,Nf ¼ 128, Na ¼ 32, and Nr ¼ 32. Figures 5.6 and 5.7 depict the estimated scattererdistribution H in azimuth–range for the random waveform used in the previousimages, and the matched illumination waveform discussed above. The scene consists

Figure 5.6 Estimated image for orthogonal waveforms.

Figure 5.7 Estimated image for matched illumination waveforms.


of a closely spaced cluster of four point targets. The matched illumination waveformyields a less noisy image; the SNR observed in Figs. 5.6 and 5.7 is 16 and 25 dB,respectively. The effect of using different waveforms is quantified more preciselyin the next section.

Figure 5.8 Azimuth–Doppler distribution for a range bin containing a single target andclutter, using a target-matched waveform, dT ¼ 5l, dR ¼ 0.5l.

Figure 5.9 Azimuth–Doppler distribution for a range bin containing a single target andclutter, using a target-matched waveform, dT ¼ 0.5l, dR ¼ 5l.

5.3 ESTIMATION 213

Figure 5.8 depicts an example where the multiple pulses are transmitted. This isthe same case depicted in Fig. 5.4, except that the target-matched waveform isused instead of the orthogonal waveform. In this case the illumination function hasthe shape of the transmit beampattern and the clutter is only partially illuminated,as can be clearly see from the appearance of the clutter ridge. The horizontal stripsindicate the directions where the transmit beam is pointing. Figure 5.9 depicts thesame example, except that the transmit and receive airays were swapped.

5.4 DETECTION

In this section we consider target detection using a MIMO radar. The radar may employa fixed set of transmit waveforms, or it may adapt the waveforms on the basis ofpreviously measured data. We start by developing the structure of the optimal detectorunder the assumption that both target and clutter statistics are known.

5.4.1 The Optimal Detector

Given the signal model presented in Section 5.2, we formulate the target detectionproblem as the following Gauss–Gauss binary hypothesis testing problem. Weassume that the received data contain either clutter and noise, hypothesis H0, ortarget clutter and noise, hypothesis H1

H0: x CN (0, Rxcn ) (5:64)

H1: x CN (0, Rxt þ Rxcn (5:65)

where CN denotes the multivariate complex Gaussian distribution andRxcn ¼ Rxc þ Rxn . The optimal detector for this problem is known to have the quad-ratic form [28]

d ¼ xHQx (5:66)

where d is the detection statistic, and the kernel Q is given by

Q ¼ R�1xcn� [Rxt þ Rxcn ]�1 (5:67)

This can also be written as

Q ¼ R�1xcn

Vxt [VHxt

R�1xcn

Vxt þ I]�1VHxt

R�1xcn¼WWH (5:68)

where Rxt ¼ Vxt VHxt

and

W ¼ R�1xcn

Vxt [VHxt

R�1xcn

Vxt þ I]�(1=2)

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}�Vxt

(5:69)


Then the detection statistic is given by

d ¼ xHWWHx ¼ kWHxk2 (5:70)

In the case of a unit rank target case (rt ¼ 1), Vxt is a vector. To emphasize this fact,we denote it by the lowercase symbol vxt . In this case

W ¼ R�1xcn

vxt (5:71)

where we discarded a scalar multiplier term that can be absorbed into the detectionthreshold. This is the well-known minimum variance distortionless response(MVDR) detector [30]. The more general case of equations (5.69)–(5.70) wherethe rank of the target covariance is greater than unity, is the generalized MVDR(GMVDR) presented in Ref. 29.

5.4.2 The SINR

Because both the target and clutter statistics Rxt , Rxc depend on the transmit signal s,the matrix W defining the optimal detector is a function of the transmit waveform s.We want to select this waveform so as to maximize the performance of the detector.This can be achieved by maxim izing the signal-to-interference-plus-noise ratio(SINR), at the output of the detector.

It follows from Eq. (5.70) that the SINR is given by

SINR ¼ tr{WHRxt W}

tr{WHRxcn W}(5:72)

or equivalently

SINR ¼tr{~V

Hxt

R�1xcn

Rxt R�1xcn

~Vxt }

tr{~VHxt

R�1xcn

~Vxt }(5:73)

It is sometimes convenient to rewrite this expression as

SINR ¼tr{VH

xtR�1

xcnVxt (V

Hxt

R�1xcn

Vxt þ I)�1VHxt

R�1xcn

Vxt }

tr{VHxt

R�1xcn

Vxt (VHxt

R�1xcn

Vxt þ I)�1}(5:74)

which makes it clear the SINR is a function of a single matrix C ¼ VHxt

R�1xcn

Vxt

SINR ¼ tr{C(Cþ I)�1C}

tr{C(Cþ I)�1}(5:75)

This can be rewritten in terms of the eigenvalues li of C as

SINR ¼Prt

m¼1 l2i =(li þ 1)

Prtm¼1 li=(li þ 1)

(5:76)

5.4 DETECTION 215

When Rxt is of unit rank, C is a scalar and all of these equations reduce to

SINR ¼ C ¼ vHxtR�1xcn

vxt (5:77)

The equations above were derived under the assumption that the detector weightvector W was designed using the true target vector vxt . It is of interest to also considerthe case where the detector is designed using the target vector at an assumed targetlocation that may be different from the actual one. Therefore we consider two versionsof the target signature: vxt — the signature corresponding to the actual target locationand vd

xt— the signature corresponding to the target location assumed by the detector.

In the case where rt . 1, we have Vxt and Vdxt

.Recall that vxt ¼ Aht

ht. In the case of a point target, ht is a vector of zeros with asingle nonzero element corresponding to the target location, so that the assumedtarget signature vd

xtmay be any one of the columns of Aht

, while the true target sig-nature vxt is one particular column.

The SINR equations need to be modified as follows. For rt . 1, replace ~Vxt inequation (5.73) by

~Vdxt¼ Vd

xt[(Vd

xt)HR�1

xcnVd

xtþ I]�(1=2) (5:78)

leaving Rxt unchanged. In the unit rank rt ¼ 1 case

SINR ¼(vd

xt)HR�1

xcn(vxt v

Hxt

)R�1xcn

vdxt

(vdxt

)HR�1xcn

vdxt

(5:79)

Both the target and clutter covariance matrices are functions of the transmitted wave-form s. It is straightforward to show that

Rxt ¼ AR diag{ATT s}

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}Aht

Rht diag{ATT s}HA

HR|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

AHht

(5:80)

and therefore

Vxt ¼ AR diag{ATT s}Vht (5:81)

Similarly, we have

Rxc ¼ AR diag{ATT s}

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}Ahc

Rhc diag{ATT s}HA

HR|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

AHhc

(5:82)

Inserting these into Eq. (5.73) gives an explicit expression for the SINE in terms of s.In the reduced-dimension model these equations are modified as follows

Vxt ¼ AR diag{ATT s}~FV�ht

(5:83)


and in equations (5.80), (5.82) we replace R�htby ~FR�ht

, ~FH

, and R�hcby ~FR�hc

~FH

. Inthe multipulse case the SINR is defined as before

SINR ¼ vHxt

R�1xcn

vxt (5:84)

where

Rxcn ¼ AhRhcAHh þ s2I (5:85)

However, Ah is now defined as

Ah ¼ ADR diag{g} (5:86)

where ADR ¼ INf � ADR, and vxt ¼ vec{Xt}, where

Xt ¼ dt � (AR(G�Ht)) (5:87)

Note that in the case of a point target H t has a single nonzero row corresponding tothe target direction.

5.4.3 Optimal Waveform Design

An optimal transmit waveform s can be designed by maximizing the SINR over allpossible choices of s. Because of the nonlinear dependence of SINR on s, aclosed-form solution does not seem to be available and we must resort to numericaloptimization. Various optimization techniques can be used to solve for the transmitwaveform s, which maximizes the SINR. Here we consider a gradient descent optim-ization method [31] that requires knowledge of the derivatives of SINR with respectto the elements of s. The update equation of the gradient descent method is given by

s � sþ m@SINR@s

� �H (5:88)

where m is a constant controlling the convergence rate of the algorithm. At each stepof the algorithm we also rescale s to have a fixed norm, that is, to obey the averagepower constraint.

Taking the derivative of the SINR with respect to s i, we get

@ SINR@si

¼ 2sHAHst R�1xcnAstei � 2sHA

Hst R�1xcn

@Rxcn

@siR�1

xcnAsts (5:89)

where Ast ¼ AR diag{ht}ATT , si is the ith entry of s, and e i is vector of zeros with a 1

at the ith position. (See Refs. 32 and 33 for examples of calculating this type ofderivative.)

Recalling Rxc from Eq. (5.82) and noting that (@Rxcn=@si) ¼ (@Rxc=@si), we get

@Rxcn

@si¼ AR diag{AT

T ei}Rhc diag{ATT s}HA

HR (5:90)

5.4 DETECTION 217

Note that ATT ei is the ith column of AT

T or the ith row of AT .Then

@ SINR@si

¼ 2pi � 2qi (5:91)

where pi is the ith element of the row vector

p ¼ sHAHst R�1xcnAst (5:92)

and

qi ¼ sHAHst R�1xcnAR

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}qa

diag{ATT ei} Rhc diag{AT

T s}HAHR R�1

xcnAsts

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}qT

b

(5:93)

where qa and qb are row vectors defined in the equation above. DenotingQc ¼ AH

R R�1xcnAst, we have qa ¼ sHQH

c , qTb ¼ Rhc diag{AT

T s}HQcs. Next, let

ATT [i] ¼ AT

T ei be the ithe column of ATT . Then

qi ¼ qadiag{ATT [i]}qT

b ¼ (qa � qb)ATT [i] (5:94)

and assembling all qi values into a row vector q, we get

q ¼ (qa � qb)ATT (5:95)

Finally

@ SINR@s

¼ 2(p� q) (5:96)

Equations (5.88), (5.92), (5.95), and (5.96) define the gradient descent algorithm forcomputing the SINR-maximizing waveform s. To initialize the algorithm, we let s beone of the suboptimal waveforms described in Section 5.4.4.

Note that in the case of the reduced-dimension model Rhc in Eq. (5.93) needs to be

replaced by ~FR~hc~F

H.

5.4.4 Suboptimal Waveform Design

In this section we present two suboptimal waveform design algorithms that are com-putationally less complex than the algorithms for designing the optimal waveform.

The first involves matching the radar illumination to the target in a manner similarto that considered in Section 5.3.3. Let vht denote the unit rank target signature vectorand design the illumination vector g so that it matches the target as closely as poss-ible. In other words, let g ¼ AT

T s � vht . The solution is given by s ¼ (ATATT )�1AT vht

followed by normalization of s to meet the transmit power constraint. We will refer tothe resulting waveform as the “target-matched waveform.”


The second method designs the waveform using both the target signature and theclutter covariance matrix. Let Pc be the Na � Nf matrix representing the power distri-bution of clutter in range and frequency. In vectorized form we have pc ¼ vec{Pc},which equals the diagonal of the clutter covariance matrix Rhc . The illumination willbe designed so that it matches as closely as possible the target : clutter ratio. In otherwords, let g ¼ AT

T s � vht=ffiffiffiffiffipcp

, where both the division and the square-root oper-ations are element-by-element (elementwise) operations. The solution is given bys ¼ (AT AT

T )�1AT (vht=ffiffiffiffiffipcp

), followed by normalization of s to meet the transmitpower constraint. We will refer to the resulting waveform as the “target-to-clutter-matched waveform.”

5.4.5 Constrained Design

In the discussion above we allowed the waveform s to assume any shape, subject onlyto a power constraint. It is of interest to also consider the case where s is a linearcombination of a set of L predetermined waveforms ci. In other words, assumethat s ¼ Cf, where C ¼ [c1, . . . , cL] is an NT Nf � L matrix and f is the L � 1vector of coefficients of the representation of s in this basis. It is straightforward toextend the waveform design algorithms presented here to this case, by replacing swith cf in all the equations.

Consider, for example, the target matched waveform. Replacing s with cf, we get

g ¼ ATTCf � ht (5:97)

and the least-squares solution is

f ¼ (CHA�TA

TTC)�1CH

A�T ht (5:98)

Different choices can be made for the spacetime basis C, including

† C ¼ INT Nf — this is the unconstrained case, where the spacetime basis coversthe entire available space.

† C ¼ INf � aT (u) — here we force the transmitter to point a beam in direction u.The design of the waveform will determine the temporal signal properties, whilethe spatial properties are fixed a priori.

† C ¼ [INf � aT (u1), . . . , INf � aT (uM)] — this is similar to the previous case,except that we predefine M different beam directions and let the designprocess determine the spatial distribution of the transmit energy, as well asthe temporal distribution. In other words, we perform unconstrained temporalwaveform design and constrained spatial design.

† C ¼ q� INT — here we force the transmitter to use a single waveform q on allantennas, but allow the design procedure to determine the spatial response. Inother words, we have a fixed temporal design, while the spatial design isdone adaptively.

5.4 DETECTION 219

† C ¼ [q1 � INT , . . . , qN � INT ] — this is similar to the previous case, except thatwe predefine N different beam temporal waveforms and let the design processdetermine which combination of waveforms to use. In other words, weperform unconstrained spatial design and constrained temporal design.

† C ¼ Ct �Cs — here we select any desired temporal basis function Ct and adesired spatial basis function Cs. In other words, we constrain both the temporaland the spatial designs.

In a separate paper [36] we studied the impact of these choices on the various wave-form design procedures discussed here and the resulting performance of the estimatorand detector. Our results show that proper selection of the basis is useful for obtainingstable and “well-behaved” spatial and temporal response, that is, range response andbeampattern. Imposing the basis constraints does, of course, decrease performancebecause of the reduction in the number of degrees of freedom available in thedesign. However, this performance loss can be made arbitrarily small by increasingthe dimension of the subspace spanned by these basis functions.

5.4.6 The Target and Clutter Models

As was shown above, the optimal detector is based on the second-order statistics ofthe target, clutter, and noise. Different target and clutter models arise in differentapplications. In this section we describe the models used in this work.

We consider both point targets — targets smaller than a single range–azimuthcell, and extended targets covering multiple range–azimuth cells. A pointtarget has a unit rank (rank 1) covariance matrix Rht and Rxt . An extended targetcovering nc cells has a covariance matrix with rank 1 � rt � nc, depending on thedegree of correlation between the radar returns from the different cells. Here wefocus on the case of highly correlated returns that yield a unit rank covariance.This corresponds, for example, to a rigid target with fixed orientation relative tothe radar. Motion relative to the radar will introduce the same random phase fluctu-ations to the returns from all of the scatterers constituting the target so that the targetscatters can be represented by vht e

jf where vht is a deterministic vector and f israndom phase. In this case Rht ¼ vht v

Hht

where vht equals h t up to an arbitraryunit magnitude complex scalar. A target that is not a rigid collection of pointscatterers, or is one that is rotating relative to the radar, will induce differentrandom variations along the vector ht, and the corresponding covariance matrixRht will have an effective rank rt . 1.

We assume that the clutter is represented by a random scattering map H c whereclutter returns from different range–azimuth cells are uncorrelated. The cluttercovariance matrix is therefore diagonal and the elements of the diagonal representthe clutter power distribution pc. The rank of the clutter covariance matrix Rhc is rela-tively large, and the matrix may in general be of full rank.

In order to implement the detector, knowledge of the target and clutter covariancematrices Rht and Rhc is required. The clutter covariance can be estimated by


collecting multiple radar returns, estimating Hc using the methods described inSection 5.3, and forming the sample covariance matrix.

The target statistics are computed for an assumed target position and spectralcharacteristics. In the case of a point target the signature vht is a function of thetarget azimuth. In this case we design the transmit waveform using the constrainedapproach discussed earlier with C ¼ INf � aT (u). In other words, we force the trans-mitter to point a beam in an assumed direction u. The assumed target direction isscanned across the area of interest over subsequent transmission. In the case ofan extended target, the waveform can be designed to match its spectral signature,provided that this signature is known a priori.

5.4.7 Numerical Examples

To illustrate the performance of the various detectors, we calculated the SINR fora MIMO radar with NT ¼ NR ¼ 10, where the element spacing of the transmitand receive arrays was dT ¼ 5l and dR ¼ 0:5l, respectively. Other system para-meters were Ns ¼ Nf ¼ Na ¼ 128. The clutter power pc is assumed to be uniformover all range/azimuth cells in the scene. The SINR values are summarized inTables 5.1 and 5.2 for an orthogonal set of waveforms and a target-to-clutter-matched waveform. The theoretical SINR results, also shown in these tables, arederived in Appendix 5A, where it is shown that for uniform illumination (single

TABLE 5.1 Experimental and Theoreticala SINR (in dB) for a MIMO Radarwith Orthogonal Waveforms

SNR ¼ 0 dB SNR ¼ 20 dB

CNR ¼ 220 dB 10.8 [10.0] 30.8 [30.0]CNR ¼ 0 dB 10.7 [10.0] 30.7 [30.0]CNR ¼ 20 dB 7.0 [7.5] 27.0 [27.5]CNR ¼ 40 dB 210.1 [29.0] 9.8 [11.0]

aTheoretical SINR values are given in square brackets.

TABLE 5.2 Experimental and Theoreticala SINR (in dB) for a MEMO Radarwith Target-to-Clutter-Matched Waveforms

SNR ¼ 0 dB SNR ¼ 20 dB

CNR ¼ 20 dB 20.0 [20.0] 40.0 [40.0]CNR ¼ 0 dB 20.0 [20.0] 40.0 [40.0]CNR ¼ 20 dB 16.8 [17.5] 36.8 [37.5]CNR ¼ 40 dB 0.3 [1.0] 20.3 [21.0]

aTheoretical SINR values are given in square brackets.

5.4 DETECTION 221

transmit antenna, or MIMO with orthogonal waveforms)

SINR ¼ NRSNRCNR=Nf þ 1

(5:99)

and for nonuniform illumination produced by the adaptive waveforms discussedearlier (target-matched, target-to-clutter matched, optimal waveforms),

SINR ¼ NRNT SNRCNR=Nf þ 1

(5:100)

The CNR (SNR) are defined as the total clutter (target) power received during thecollection time, divided by the noise power.

A number of observations can be made from these tables. There is a good matchbetween the theoretical and experimental results. The factor of NR (NRNT ) is due tothe receive (transmit and receive) array gain. As expected, the transmitter array gain islost when using orthogonal waveforms that provide uniform illumination of thescene. The adaptively designed waveforms provide a nonuniform illumination andrecapture the lost gain. Note, however, that the orthogonal waveforms make itpossible to do the transmit beam forming at the receiver, after the fact. This makesit possible to scan the entire area in the antenna footprint in a single pulse period.The adaptive waveforms illuminate only a portion of the radar footprint and willgenerally require NR pulses to scan the entire area.

The SINR discussed above was the SINR when the detector uses the targetlocation, or vd

xt¼ vxt . It is of interest to plot the SINR at the detector output as

the assumed target location is scanned over the entire range/azimuth scene, fora given target. Figures 5.10 and 5.11 depict this SINR for a MIMO radar with

Figure 5.10 SINR for orthogonal waveforms.


NT ¼ NR ¼ 10, where the element spacing of the transmit and receive arrays wasdT ¼ 5l and dR ¼ 0.5l, respectively. Other system parameters were Ns ¼ Nf ¼

256, Na ¼ 128. In Fig. 5.10 the transmitter employed a set of orthogonal waveforms,while in Fig. 5.11 a target-to-clutter-matched waveform was used. The highly con-centrated shape of the SINR surface makes it possible to localize the target withhigh resolution. The improved SINR when using an adaptively designed waveformcan be observed by comparing the two figures.

To gain some insight into the shape of the waveform designed by theadaptive design procedure, we present two examples for the same MIMO radar con-sidered above. Figure 5.12 depicts the illumination function G for four waveforms:orthogonal, target-matched, target-to-clutter-matched, and optimal, for the case ofa point target.

Note that the adaptive waveforms and the corresponding illumination function arespectrally flat, corresponding to a narrow pulse in the time domain. This results in anarrow range response. In the spatial domain the waveforms form a narrow transmitbeam focused on the target. The processing gain results from the high spatial resol-ution and the small range cell that contain the target, but contains only a fraction ofthe clutter energy. In the case of uniform clutter the SINR at the detector output isgiven by Eq. (5.100) for the adaptive waveforms, and by Eq. (5.99) for the orthogonalwaveform.

Figure 5.13 depicts the illumination function for the case of an extended target,where the target occupies a single azimuth cell, but is spread along all range cells.In this case the adaptive waveforms are spectrally narrow, resulting with a widerange response so as to contain the extended target. In the spatial domain the wave-forms form a narrow transmit beam focused on the target. In the case of uniform

Figure 5.11 SINR for target-to-clutter-matched waveforms.

5.4 DETECTION 223

clutter the SINR is approximately the same as in the point target case describedabove. The processing gain results from the high spatial resolution and the rangeresponse, which is narrow in the frequency domain and thus picks up only a smallfraction of the clutter energy.

So far we have considered examples where the clutter was uniform. When theclutter is nonuniform, there are opportunities for additional temporal processinggain beyond the gains captures by Eqs. (5.99) and (5.100). As an example, considerthe case of an extended target with nonuniform clutter. By designing the shape of thewaveform in frequency (or time) to illuminate parts of the target where the clutter isrelatively small, we get potentially significant improvement relative to uniformillumination (see e.g., Refs. 34 and 35).

Next we present an example of the multipulse case. Figure 5.14 depicts theSINR for different Doppler frequencies in a single range cell containing a target.Results are shown for the orthogonal waveform and the target-matched waveform.As expected, the SINR for the target-matched waveform is larger by (approximately)a factor of NT than the SINR for the orthogonal waveform because of the transmitarray gain.

Figure 5.12 Illumination function G for four waveforms: orthogonal, target-matched, target-to-clutter-matched, and optimal, for the case of a point target.


Figure 5.13 Illumination function G for four waveforms: orthogonal, target-matched, target-to-clutter-matched, and optimal, for the case of an extended target.

Figure 5.14 SINR versus Doppler for the orthogonal waveform and the target-matchedwaveform.

5.4 DETECTION 225

5.5 MIMO RADAR AND PHASED ARRAYS

The use of multiple antennas on both transmit and receive is, of course, not new.Phased arrays that form beams on both transmit and receive have been around fora long time. (See Refs. 37 and 38 and references cited therein for a historical overviewof phased-array radar technology.) Phased arrays transmit a single waveform that isfed to the different antennas with different phases (delays). In other words, the wave-forms at the different antennas are perfectly correlated.

The more recent use of the term “MIMO radar” refers to the case where the trans-mit array uses waveforms that may differ from antenna to antenna. These waveformsare usually assumed to be orthogonal or uncorrelated [39]. In fact, some referencesdefine MIMO radar as a radar employing orthogonal waveforms at the transmitter.In this chapter we considered a more general class of transmit waveforms.However, for the moment we focus on the case of orthogonal waveforms.

The current literature on MIMO radar discusses its various performance character-istics such as having a high angular resolution and the ability to search an extendedarea with that high resolution in a single dwell [7]. It is useful to note that many of thecapabilities of MIMO radar can be replicated by a phased-array radar, which uses thesame physical antenna array and the same transmit power. The spatial resolution ofthe radar system relates primarily to the array geometry and the directivity of theantenna elements that it employs. The combined transmit–receive array operates asa virtual array with NRNT elements, as discussed in Section 5.3.1, resulting inhigher resolution than a radar with a single transmit antenna. In other words, boththe MIMO radar and the phasedarray have the same combined transmit–receivearray manifold aT (u) � aR(u).

To see this more clearly, we consider the signal received from a point target withunit radar cross section located at azimuth ut and delay (range) tt. In the case of aphased-array radar, the received signal matrix X is given by

X ¼ aR(ut)aTT S (5:101)

where the transmitted signal consists of a single waveform s fed through a transmitbeamformer with weight vector WT:

S ¼W�T s (5:102)

The transmit power is normalized to unity so we assume that ksk2 ¼ 1 andkWTk2 ¼ 1. The rows of X are passed through a matched filter to produce thematch-filtered signal matrix Z

Z ¼ aR(ut)aT (ut)T W�

T sq (5:103)

where sq is the match-filtered waveform s. Assuming that the matched filter isnormalized to unit power gain, we have ksqk2 ¼ 1.


The filtered signal is then beamformed using a beamformer whose weightvector is WR:

y ¼WHR Z ¼ (WH

R aR(ut))(WHT aT (ut))sq (5:104)

We assume that the beamformer is normalized to have unit noise power gain, specifi-cally, kWRk2 ¼ 1.

We note that the beamformer weight vectors WR,WT can be designed in differentways. These could be fixed weights designed to achieve a desired beampattern, orthey may be dependent on the data, leading to an adaptive beamformer on receive,transmit, or both.

Next consider a MIMO radar that makes use of the same transmit and receivearrays and therefore has the same array manifolds aR (u),aT (u). The receive signalmatrix X is given by

X ¼ aR(ut)aT (ut)T S (5:105)

where the rows of S are orthogonal waveforms. The total transmit power is unity asbefore so that kSk2 ¼ 1. This means that each row of S has norm 1/NT. Passing thisthrough a filter matched to the ith transmit waveform, we get

Zi ¼ aR(ut)aT (ut)[i]sq (5:106)

where aT(ut)[i] is the ith element of the vector aT (ut). Assuming that the matchedfilter is normalized to unit power gain, we have ksqk2 ¼ 1/NT. We assume thatthe contributions of the other waveforms can be neglected because oforthogonality. Assembling the outputs of the different matched filters into anoutput matrix Z, we get

Z ¼ (aT (ut)� aR(ut))sq (5:107)

The filtered signal is then beamformed using a beamformer whose weight vector isWT �WR:

y ¼ (WHT �WH

R )Z ¼ (WHR aR(ut))(WH

T aT (ut))sq (5:108)

Comparing equations (5.104) and (5.108), we conclude that both systems produceexactly the same outputs if the same beamformer weights WT,WR are used, exceptthat the signal power in the phasedarray is NT times larger than in the MIMOradar. As we have noted before, the MIMO radar has an SNR that is NT timessmaller than that of the phased-array system, all other things being equal.

Next we consider the multipulse case, which leads to spacetime adaptive proces-sing (STAP). STAP radar has been extensively studied for airborne applications [27].

5.5 MIMO RADAR AND PHASED ARRAYS 227

MIMO radar can be used in STAP as was discussed by various authors (see,e.g., Refs. 14 and 15). A comparison of STAP using transmit beamforming toSTAP that uses orthogonal waveforms follows closely the earlier discussion. In thecase of a phasedarray the received data matrix X, obtained by stacking the datamatrices for a sequence of pulses, is given by

X ¼ (ad(bt)� aR(ut))aTT W�

T s (5:109)

where ad (bt) is the target Doppler manifold [see Eq. (5.60)]. After matched filtering,we have

Z ¼ (ad(bt)� aR(ut))aT (ut)T W�

T sq (5:110)

Applying a Doppler–azimuth weight vector Wd �WR, we get

y ¼ (WHd �WH

R )Z ¼ (WHd ad(bt))(W

HR aR(ut))(WH

T aT (ut))sq (5:111)

In the case of a MIMO radar, the received data matrix X is given by

X ¼ (ad(bt)� aR(ut))aT (ut)T S (5:112)

and after passing it through a bank of matched filters we get

Z ¼ (ad(bt)� aR(ut)� aR(ut))sq (5:113)

Applying a weight vector Wd � WT �WR, we get

y ¼ (WHd �WH

T �WHR )Z ¼ (WH

d ad(bt))(WHR aR(ut))(WH

T aT (ut))sq (5:114)

Comparing Eqs. (5.111) and (5.114), we again conclude that the two systems gen-erate the same outputs except for an SNR difference.

Even though the two systems produce the same outputs for the same set of beam-former weight vectors (with the exception of the SNR issue), there is an importantdifference. In the phased-array system the weight vector WT is applied on the transmitside, whereas in the MIMO radar system it is applied on the receive side. Thisseemingly small difference has some important implications, as discussed next.

5.5.1 Scan Transmit Beam after Receive

A typical choice for the transmit beamformer weight vector is WT ¼ aT(ut)/kaT (ut)k(possible tapering may be included). This makes the transmit array point a beam inthe assumed target direction. In the phased-array implementation the transmitterrequires multiple dwells to cover the area of interest because in each dwell it illumi-nates only a segment of the area that falls within the transmit beam. The transmit arraycan form NT independent beams and will require (approximately) NT dwell times toscan the entire area.


The situation is different when using a MIMO radar with orthogonal waveform.During the transmission no beam is formed and the entire area is illuminatedduring each dwell. The transmit beam will be formed during the processing of thereceived signal. This beam can be scanned over the entire area of interest. Thus,unlike the phased-array radar, the MIMO radar is able to cover the entire areaduring a single dwell. This comes at the cost of an NT -fold reduction of the receivedSNR. Note that this SNR reduction may not have a significant effect on system per-formance if, for example, the scenario is clutter-dominated (after clutter reduction).However, if it is desired to compare the two systems under the same SNR conditions,the SNR reduction must be considered. In order to have the same SNR as the phased-array radar, the MIMO radar will have to integrate the radar returns from NT dwells.Thus, in an equal SNR comparison, both systems will require a comparable numberof dwells to cover a given area.

5.5.2 Adaptation of Transmit Beampattern

In some applications it is desirable to adjust the transmit beam on the basis ofobserved data. Consider, for example, a situation where it is desired to steer thenulls of the transmit beampattern so as to avoid illuminating directions that mightgive rise to multipath clutter. Information about the desired null steering directionsis not available at the transmitter, and is obtained only by analyzing the receiveddata. This is made possible in the MIMO radar where the transmit beampattern isdetermined “after the fact.” Accomplishing this objective using a phased-arrayradar would be more difficult. It would require a two-step approach: a probingphase where the transmitter illuminates the area of interest to determine the directionsin which the nulls should be steered, and a second phase where the transmit beam-pattern is configured with the appropriate nulls.

In other words, MIMO radar accomodates processing schemes that require data-dependent adaptation of the transmit beampattern. Such schemes are more difficultto implement using a phased-array radar.

5.5.3 Combined Transmit–Receive Beamforming

In a phased-array radar, beamforming is applied separately for the transmit array andthe receive array, using the weight vectors WT and WR, respectively. In the case ofMIMO radar we applied the weight vector WT �WR to the virtual transmit/receive array, which again corresponds to beamforming separately the transmit andreceive array. Note, however, that in this case it is possible to apply an arbitrary beam-forming weight vector W to the entire virtual NTNR � 1 array where W = WT �WR.In other words, in the phased-array radar we must design separately the transmit andreceive beamformers using the NT þ NR degrees of freedom afforded by the numberof coefficients in WT and WR. However, in MIMO radar we can design the beam-former W using the corresponding NTNR degrees of freedom. In a phased-arrayradar the composite transmit–receive beampattern is constrained to be the productof the transmit and receive patterns. In a MIMO radar the composite pattern can be

5.5 MIMO RADAR AND PHASED ARRAYS 229

an arbitrary pattern of an NTNR-element array. Note that to ensure the full dimensionof this virtual array, the transmit and receive arrays must be properly designed, aswas discussed earlier (e.g., one is a critically sampled array and the other is athinned array).

The discussion above focused on the case of MIMO radar with orthogonal wave-forms. Earlier in this chapter we considered the use of a very general class of wave-forms designed to achieve a desired illumination of the radar scene. Assume that wespecify the desired illumination gn at N different angles

aTT (un)S ¼ gn (5:115)

where gn is a 1 � Nf row vector. Assembling the N equations into a single matrixequation, we get

aTT (u1)

aTT (u2)

..

.

aTT (uN)

2

66664

3

77775S ¼

g1g2

..

.

gN

2

6664

3

7775(5:116)

which can be solved for the transmitted signal matrix S. Assuming that the anglesun are approximately a beamwidth apart, the vectors aT (un) are approximately ortho-gonal so that

S �XN

n¼1

a�T (un)gn=ka(un)k2 (5:117)

Consider next a phasedarray that scans the area of interest during N dwells, pointingthe transmit/receive beams at directions un. The desired illumination function at thenth dwell is

aTT (u)WT s ¼ gn (5:118)

Letting WT ¼ a�T (un), we get

s ¼ gn

ka(un)k2 (5:119)

or

S ¼WT s ¼ a�T (un)gn

ka(un)k2 (5:120)

Comparing the illumination produced by the two systems, we note that they have thesame spatiotemporal characteristics, except that the MIMO radar illuminatesthe entire area at each dwell, while the phased-array scans the illumination across


the area during multiple dwells. For this reason, the intensity of the illumination of thephased-array radar exceeds that of the MIMO radar. Note that here the phasedarrayuses a different waveform at each dwell time. We conclude that both the phased-array radar and the MIMO radar can create a general spacetime illumination of thearea of interest.

Another type of MIMO radar that has been proposed is the “statisticalMIMO radar” described in Refs. 22–25. In this system the transmit antennasare assumed to be sufficiently far apart that transmit beamforming of the typeused in a phasedarray is impossible because of the lack of phase coherencebetween antennas; this type of radar system is quite different from the one consideredin this chapter.

APPENDIX 5A THEORETICAL SINR CALCULATION

Evaluating the SINR generally requires a numerical computation. However, for thecase of uniform clutter and uniform illumination, the evaluation is straightforward.Recall that SINR ¼ vH

xtR�1

xcnvxt , where Rxc ¼ AhRhAH

h . Assuming uniform clutterwe have Rh ¼ (CNR/NaNf) INaNf so that trace fRhg ¼ CNR, where CNR is theclutter : noise ratio. In this case

Rxc ¼CNRNaNf

AR diag{g� g�}AHR (5A:1)

Assuming uniform illumination diagfg � g�g ¼ cgINaNf, where cg is a constant to bedetermined. Note that ARAH

R � NaINR , which leads to

Rxc ¼CNRcg

NfINRNf (5A:2)

Also note that

kgk2 ¼ sHA�T ATT s ¼ Naksk2 ¼ Na (5A:3)

Therefore we have

tr{diag{g� g�}} ¼ kgk2 ¼ Na ¼ cgNaNf (5A:4)

and we conclude that cg ¼ 1/Nf. It follows that

Rxcn ¼CNR

N2f

INRNf þ1

NfINRNf (5A:5)

APPENDIX 5A THEORETICAL SINR CALCULATION 231

We can now evaluate the SINR as

SINR ¼ vHxt

R�1xcn

vxt ¼kvxtk

2

CNR=N2f þ 1=Nf

(5A:6)

The norm of the target signature is given by

kvxtk2 ¼ hH

t AHh Ahht ¼ hH

t diag{gH}AHRAR diag{gH}ht

¼ NRhHt diag{g� g�}ht (5A:7)

Note that AHh Ah � NRI and therefore

kvxtk2 ¼ NR

Nfkhtk2 ¼ NR

NfSNR (5A:8)

where SNR is the signal-to-noise ratio. Finally we conclude that

SINR ¼ NRSNRCNR=Nf þ 1

(5A:9)

This result has the following natural interpretation. If Nf is interpreted as the numberof range cells, then CNR/Nf is the amount of clutter in the given range cell. Thereceive array gain is NR. Because we have uniform illumination, the total receivedclutter power in the range cell is independent of the shape of the receive beampattern.In other words, the CNR is unaffected by spatial processing at the receiver. However,the target signal is increased by the array gain, providing an increase of the SNR bythe factor NR.

The uniform illumination corresponds to the case where orthogonal signals areused in the different antennas, and also to the case where only a single antenna isbeing used. If any of the adaptive waveforms described earlier is used, the resultingwaveforms create a narrow beam in the direction of the target providing an additionalarray gain of NT. Therefore

SINR ¼ NRNT SNRCNR=Nf þ 1

(5A:10)

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