Joint angles of departure and angles of arrival estimation via a spatial-temporal scheme in...

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Published in IET Radar, Sonar and NavigationReceived on 17th September 2013Revised on 27th November 2013Accepted on 19th January 2014doi: 10.1049/iet-rsn.2013.0307

T Radar Sonar Navig., 2014, Vol. 8, Iss. 7, pp. 837–844oi: 10.1049/iet-rsn.2013.0307

ISSN 1751-8784

Joint angles of departure and angles of arrivalestimation via a spatial-temporal scheme inoverloaded bistatic multiple-input multiple-outputradarsCheng-Han Hsu, Hsin-Chin Liu, Meng-Chang Hua

Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan

E-mail: [email protected]

Abstract: A signal subspace-based technique was used to estimate joint angles of departure (AODs) and angles of arrival (AOAs)in an overloaded bistatic multiple-input multiple-output (MIMO) radar system. First, a spatial-temporal scheme is used to expandthe degrees of freedom in the array. Thus, the number of target AODs and AOAs that can be estimated is higher than the numberof the product of the transmit and receive elements. The steering vectors of AODs and AOAs are then obtained from spatial-temporal eigenvectors, which contain the spatial and temporal information of received signals. This enables highly accurateAOD and AOA estimates even in an overloaded system. The results of several simulations are given to illustrate theeffectiveness of the proposed approach.

1 Introduction

Bistatic multiple-input multiple-output (MIMO) radar withmultiple antennas for simultaneously transmitting severalorthogonal waveforms and receiving the reflected signals isunder intensive study [1, 2]. Since the transmitted signalsare orthogonal, they are separable at the receiver by usingmatched filter-banks. Additionally, since MIMO processingprovides high spatial and signal waveform diversity, it hasbetter system capacity and parameter estimation accuracycompared with conventional radar systems [1]. Moreover,MIMO radar can obtain a virtual aperture larger than theactual aperture. Therefore MIMO radar scan achieve ahigher degree of freedom compared with conventionalradars. The degrees of freedom can be exploited forimproved angle estimation resolution and for cluttermitigation.Recently, the use of CDMA signals to increase frequency

diversity in MIMO systems was discussed in [3, 4]. Thespreading code can increase processing gain for a givendata rate, which provides additional degrees of freedom forsuppressing interference in such a MIMO system. Adaptivetechniques for estimating the angles of departure (AODs)and the angles of arrival (AOAs) have been applied inMIMO radars. The conventional minimum variancedistortionless response [5], two-dimensional multiplesignal classification (MUSIC) [6], ESPRIT [7, 8], andreduced-dimension (RD) MUSIC [9] estimators have beenused to estimate AODs and AOAs. However, none of theseestimators are effective in overloaded MIMO systems,where the number of impinging signals is higher than the

number of the product of transmitter elements and receiverelements.Various schemes have been proposed for estimating AOA

in an overloaded system [10, 11]. For overloaded arrayprocessing, the Genetic Algorithm (GA) with soft-biasedinitialisation [10] has demonstrated performancecomparable with that of the optimal maximum likelihoodjoint detection method [11]. The methods presented in[12, 13] exploit the cyclostationarity property ofcommunication signals to extract desired signals withspecific cycle frequencies. Hence, they null out co-channelinterferences and additive noise with different cyclefrequencies. However, after exploiting the cyclostationarityproperty to null out the interferences with different cyclefrequencies, the number of remaining signals with thesame cycle frequency may still not exceed the number ofantenna elements.This study considered the problem of joint AOD and AOA

estimation in overloaded systems. A pseudo noise (PN) codesequence is used to spread the radar signal, and aspatial-temporal scheme is performed to expand the degreesof freedom. This increases the number of targets in whichAODs and AOAs can be estimated, even if the numberexceeds the product of the transmit and receive elements.The steering vectors of the signals reflected from thedifferent targets, which contain the spatial and temporalinformation, can be estimated by signal subspace technique.The AODs and AOAs can also be obtained by applying theconcept of constructing steering vector. Finally, simulationresults are presented to illustrate the effectiveness of theproposed approach.

837& The Institution of Engineering and Technology 2014

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Rest of this paper is organised as follows: Section 2 defines
the signal model and the system structure. Section 3 presentsthe proposed method of estimating AODs and AOAs of thetargets. Computational complexities are discussed inSection 4. Analysis and comparison with other methods areperformed in Section 5. Finally, Section 6 draws theconclusions.

2 Signal model

2.1 Transmitted signal

Consider a bistatic MIMO radar system with CDMA signalsconsisting of a M-element transmitter array and a N-elementreceiver array, both of which are Uniform Linear Arrays.All elements are omni-directional with half-wavelengthinter-element spacing d for both transmitter and receiver. InFig. 1, the coordinates of the M sensors in the transmitterare denoted by Tm = (xT,m, yT,m), m = 1, 2, …, M, and thecoordinates of N sensors in the receiver are denoted byRn = (xR,n, yR,n), n = 1, 2, …, N. A radar signal is spreadwith a periodic spreading sequence c[l ] which is defined asthe first spreading code with length L (spreading factor ofprocessing gain), and Tc is the chip period. The transmittedsignal can be written as

s(t) =∑1n=−1

∑L−1

i=0

c[i]p t − nLTc − iTc( )

,

p(t) = 1, 0 ≤ t ≤ Tc0, otherwise

{

where p(t) is the square function. Fig. 1 shows that thetransmitted signal, which is spread again and emitted fromthe mth transmitted sensor, can be expressed as

sm(t) =∑1n=−1

∑L−1

l=0

∑J−1

j=0

c[i]hm[j]c t − nLTc − lTc − jTh( )

,

c(t) = 1, 0 ≤ t ≤ Th0, otherwise

{(1)

where hm[ j], j = 1, 2,…, J represents the jth second spreadingcode corresponding to the mth sensor, Th denotes the secondchip duration, and c(†) denotes a square function. This

Fig. 1 Bi-static MIMO radar scenario


discussion assumes that transmitted waveforms areorthogonal and that orthogonality is maintained even fordifferent mutual delays, that is,

�si(t)s

∗j (t − t) dt = 0 for all

i≠ j, and for all time delays of interest [14].

2.2 Channel model

The following discussion assumes that the targets are locatedat some points Xk = (xk, yk), k = 1, 2,…, P in space and arestationary during the observation time. For clarity andmathematical tractability, clutter is ignored [15–20], and thetarget range is assumed to be much larger than the aperturesof the transmitter and the receiver array. The radar crosssection (RCS) fluctuations are also assumed to be fixedduring a sensor scan but vary independently from scan toscan. The target model represents a classical Swerling caseI (a slowly moving target) [21].The time-invariant channelimpulse response is modelled by a taped delay line withdelays

hk(t) = ak(t)d(t − tk ) (2)

where the complex-valued fading coefficient αk(t) of the kthtarget, including the RCS, is assumed to be invariant for[t, t + Tc) and where tk is the delay.

2.3 Received signal

The received signal at the nth receiving sensor is thesuperposition of all signals originating from the mthtransmitting sensor plus the additive noise. Denote byrm,n(t) the received signal is

rm,n(t) =∑Pk=1

aksm t − tTm(Xk )− tRn(Xk)( )+ wn(t),

t = (−1, +1) (3)

where wn(t) is the additive noise at nth receiver, tTm(Xk) =d(Tm, Xk)/c is the propagation time delay between the mthtransmitting sensor and kth target and where d(Tm, Xk ) =��(xT ,m − xk )

2 + (yT ,m − yk)2

√. The propagation time tRn(Xk)

from the target to the nth receiving sensor is definedanalogously to tTm(Xk). Additional notation is needed toexpress the received signal in compact-matrix form. Let wk

m=2pfc tTm(Xk)− tT1(Xk)

( )and ck

n = 2pfc tRn(Xk)− tR1(Xk)( )

,

IET Radar Sonar Navig., 2014, Vol. 8, Iss. 7, pp. 837–844doi: 10.1049/iet-rsn.2013.0307

Fig. 2 Structure of receiver

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where fc denotes the carrier frequency. Let fk be the anglebetween the kth target and the transmitter element, and letθk be the angle between the kth target and the receiverelement. Since the inter-element spacing of receiver andtransmitter are both half-wavelengths, wk
m=p(m−1) sinfk ,and ck

n = p(n− 1) sin uk , where λ denotes the wavelength.Since the signal is narrow-band,

rm t − tTm(Xk)− tRn(Xk)( )

= e−jp(m−1) sinfk e−jp(n−1) sin uk sm t − tT1(Xk)− tR1(Xk)( )

(4)

and (4) for the received signal model can be further simplifiedas follows

rm,n(t) =∑Pk=1

ake−jp(m−1) sinfk e−jp(n−1) sin uk

× sm t − tT1(Xk )− tR1(Xk )( )+ wn(t) (5)

Let tT1(Xk) + tR1(Xk) = tk; (5) can then be represented as

rm,n(t) =∑Pk=1

akan(uk)bm(fk)sm(t − tk)+ wn(t) (6)

where bm(fk) = e−jp(m−1) sinfk represents the phasedifferences between the signals transmitted from the mthsensor and the first sensor corresponding to the kth target.The phase differences in all transmitting sensors can thenform a transmitting steering vector [b1(fk), b2(fk),…,bM(fk)]

T = b(fk). The receiving steering vector [a1(θk),a2(θk),…, aN(θk)]

T = a(θk) is defined analogously to b(fk).Assume that the travel time of all reflected signalscorresponding to different targets is longer than Tc but notlonger than LTc, that is, Tc≤ tk≤ LTc. After stacking the Lfirst chips and J second chips into a vector of one CoherentProcessing Interval, (6) can be rewritten as

Rm,n =∑Pk=1

akan(uk )bm(fk )ckhTm +W n (7)

where ck = [ck(1), ck(2), …, ck(L)]T with size L × 1 is the

spreading code sequence corresponding to a different traveltime, hm = [hm(1), hm(2), …, hm(J )]

T with size J × 1 is thesecond spreading code sequence, (·)T denotes the transposeoperator, and

W n

=

wn(1) wn(2) · ·· wn(J )

wn(J+1) wn(J+2) · ·· wn(2J )

..

. ... . .

. ...

wn((L−1)J+1) wn((L−1)J+2) · ·· wn(LJ )

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦[CL×J

is the noise matrix. All reflected signals received from nthreceived sensor are then given by

Rn =∑Pk=1

akan(uk )ckbT(fk)H

T +W n (8)


where H = [h1, h2, …, hM] with size J ×M. The receivedsignals are then sent through a matched filter bank with thespread sequence matrix H. Since the transmitted waves areassumed to be orthogonal and since orthogonality isassumed to be maintained even for different mutual delays(⟨hi, hj⟩ = 0 and ||hi||

2 = 1, i≠ j = 1, 2, …, M ), the signalreceived after the matched filter bank shown in Fig. 2 canbe written as

�Rn =∑Pk=1

akan(uk)ckb(fk)HTH +W nH

=∑Pk=1

akan(uk)ckb(fk)+ �W n (9)

where HTH with size M ×M is an identity matrix. The outputof all received signals is Z = �R1, �R2, . . . , �RN

[ ], and (9) can

be written as

Z =∑Pk=1

akck a1(uk), a2(uk), . . . , aN (uk)[ ]

⊗ bT(fk )+W

=∑Pk=1

akckaT(uk)⊗ bT(fk)+W

=∑Pk=1

akckgT(uk , fk)+W

= CLGT(u, f)+W

(10)

where C = [c1, c2,…, cP] with size L × P is the spreading codematrix corresponding to different targets, Λ = diag[α1, α2,…,αP]∈ CP × P is the reflected coefficients matrix,⊗ denotes thekronecker product operator, G(θ, f) = [g(θ1, f1), g(θ2, f2), ·s,

g(θP, fP)]∈CMN × P is the AOD-AOA steering matrix, and

W = �W 1, �W 2, . . . , �WN

[ ].


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3 Proposed approach
3.1 AOD/AOA estimation scheme withspatial-temporal scheme

The degrees of freedom of (10) can be increased by using thefollowing spatial-temporal scheme

z =∑Pk=1

akck ⊗ g(uk , fk)+ w

=∑Pk=1

ak f (uk , fk)+ w

= F(u, f)a+ w

(11)

where α = [α1, α2, …, αP]T∈CP × 1 is the reflected

coefficient vector, F(θ, f) = [f(θ1, f1), f(θ2, f2), ·s, f(θP,fP)]∈CMNL × P is the spatial temporal steering matrix; andf(θk, fk) = ck⊗ g(θk, fk) is the spatial-temporal steeringvector of the kth reflected signal.Additionally,wp = W 1,1, W 1,2, . . . ,

[W 1,MN , W 2,1,

W 2,2, . . . , W 2,MN , . . . , W L,1, . . . , W L,MN ]T, where Wi, j

represents the (i, j)th entry of W.This discussion establishes that, for a bistatic MIMO radar

system with ULA in the transmitter and receiver, allspatial-temporal steering vectors with distinct AODs, AOAsand spreading sequences are linearly independent. Thelinear independence of every two steering vectors withdistinct AODs and AOAs can be demonstrated by intuitivearguments. A mathematical proof is given for a detaileddiscussion.

Lemma: For a bistatic MIMO radar with ULA in thetransmitter and receiver, all spatial-temporal steering vectorswith distinct AODs or AOAs are linearly independent.Here, the lemma given in [22] is applicable: rank(A⊗B) =rank(A)·rank(B).Let ai and aj denote the ith and jth columns of A, respectively,which are linearly independent for i≠ j, and let bl and bkdenote the lth and kth column of B, respectively, which arealso linearly independent for l≠ k. According to lemma 1,all columns in A⊗B are linearly independent.Therefore if the numbers of transmitter elements and the

number of receiver elements are lower than the number oftargets, that is, P >M, and P > N, there exists a set of Pspatial-temporal steering vectors with distinct AODs orAOAs. The rank of the spatial-temporal steering matrix isthen rank(Γ) = P

G = a(u1)⊗ b(f1)⊗ c1, a(u2)⊗ b(f2)⊗ c2,[. . . , a(uP)⊗ b(fP)⊗ cP

](12)

The linear independence of all steering vectors in the AODsteering matrix B = [b(f1), b(f2), …, b(fP)] and in theAOA steering matrix A = [a(θ1), a(θ2), …, a(θP)] is wellestablished. Next, consider the case where the number oftransmitter elements and receiver elements exceeds thenumber of targets, P >M and P >N, rank(B) =M, rank(A) =N. According to Lemma, all AOD-AOA steering vectorsa(θi)⊗ b(fi), i = 1, 2, …, P, for P distinct AODs or AOAs[(θ1, f1), (θ2, f2), …, (θP, fP)] are linearly independent ifMN > P. The rank of AOD-AOA steering matrix is rank(G)


= P, where G = [a(θ1)⊗ b(f1), a(θ2)⊗ b(f2), …, a(θP)⊗ b(fP)]∈CMN × P.

Next consider another situation in which the number oftargets exceeds the product of the transmitter and thereceiver elements, that is, P >MN, The rank of theAOD-AOA steering matrix is rank(G) =MN. Consider a setof P linear dependent spatial-temporal steering vectorsf (θi, fi) = a(θi)⊗ b(fi)⊗ ci, i = 1, 2,…, P. Since ci, i = 1, 2,…, P is a shift version of the PN sequence, and since allshift versions are linearly independent [23], a(θi)⊗ b(fi)⊗ci, i = 1, 2,…, P are also linearly independent. For thespatial-temporal steering matrix F = [f(θ1, f1), f(θ2, f2),…,f(θP, fP)], rank(F) = P. Next, the autocorrelation matrix Kz

of z is given by

Kz = E zzH{ } = F(u, f)KaF

T(u, f)+ s2nI (13)

where E{·} denotes the expectation and Kα = E{ααT}.Assume that the number of targets P is known and that P≤ Land L≥MN. Equation (13) can then be eigen decomposed as

Kz =∑MNL

i=1

lieiei = EsEn

[ ]L

EHs

EHn

[ ](14)

where l1 ≥ l2 ≥ · · · ≥ lP ≥ lP+1≃ · · ·≃ lMNL = s2n are the

eigenvalues of Kz, ei is the eigenvector associated with λi fori = 1, 2,…, MNL and Λ = diag{λ1, λ2,…, λMNL}. Moreover,Es = [e1, e2,…, eP] and En = [eP + 1, eP + 2…, eMNL] areorthogonal and span the signal subspace and noise subspace,respectively, corresponding to Kz. Since the channel isassumed to be ergodic, the ensemble autocorrelation matrix Kz

can be replaced by Kz if the pulse size is sufficient.Since the signal subspace spans the same range space as the

vectors in spatial-temporal steering vector, that is, R{Es} = R{F(θ, f)}, the signal subspace contains the AOD and AOAinformation and the different delay versions of c (ck for k =1, 2, …, P). The mathematical calculation is simplified byassuming that the spreading code corresponding to thetarget is known in this subsection; therefore, F(θ, f) ispartially known. Thus, the calculation of g(θk, fk) for thekth target associated with the estimated AOD uk and AOAfk is now reduced to a constrained estimation problem

V (uk , fk)= gH(uk , fk)⊗ cHk[ ]

EsEHs g(uk , fk)⊗ ck[ ]

= gH(uk , fk) IMN ⊗ cHk[ ]

EsEHs IMN ⊗ ck[ ]

g(uk , fk)

= gH(uk , fk)Qg(uk , fk)

(15)

where Q = IMN ⊗ cHk[ ]

EsEHs IMN ⊗ ck[ ]

. Equation (15),which is subject to some linear constraints, is the problemof optimising a quadratic function. The most commonmethod of solving this problem is by using the Lagrangemultiplier. The simple method used here is to replacethe Lagrange multiplier with the uni-norm eigenvectorv1(θk,fk) of matrix Q associated with the largest eigenvalue.This vector spans the same range space as the vector g(θk,fk), that is, R{v1(θk, fk)} = R{g(θk, fk)}. To normalise theestimated AOD-AOA steering vector such that its first

element equals, define vector �v1 uk , fk

( ) = v1,1 uk , fk

( )( )−1

v1 uk , fk

( ), where v1,1 uk , fk

( )is the first element of

v1 uk , fk

( ). Let �v1,n uk , fk

( )be the nth element of


Fig. 3 Structure of the efficient code detector

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�v1 uk , fk

( ). The resulting steering vector may not have unity

norm for each element as required of a true steering vector.Therefore the optimisation procedure is used to obtain thefinal estimated steering vector

�v1 uk , fk

( ) = 1,�v1,2 uk , fk

( )�v1,2 uk , fk

( )∣∣ ∣∣ , . . . , �v1,MN uk , fk

( )�v1,MN uk , fk

( )∣∣ ∣∣[ ]

(16)

from which the incident AOD fk and AOA angles uk can becalculated from each element. However, to reduce theestimation error caused by noise, the following averagingprocedure is performed

fk

= 1

N (M − 1)

∑Nn=1

∑M−1

m=1

sin−1 ln�v1,(n−1)M+m+1 fk

( )�v1,(n−1)M+m fk

( )( )

/jp

[ ]

(17)

and

uk

= 1

M (N − 1)

∑Mm=1

∑N−1

n=1

sin−1 ln�v1,(n)M+m uk

( )�v1,(n−1)M+m uk

( )( )

/jp

[ ]

(18)

An interesting issue in MIMO systems is about themaximising number of detectable sources Pmax. Generally,Pmax must be smaller than or equal to MN−1 when theautocorrelation matrix Rs is full rank. Since all delay timesfor received signals are assumed to differ, Pmax≤ L is easilyobtained even when L≥MN (overloaded MIMO system).

Table 1 Short descriptions of the proposed method

Proposed method Complex multiplication

autocorrelation matrix QM2N2L2

eigen-decomposition 12M3N3L3

AOD and AOA estimation byunit-norm eigenvector

PM2N2L2 +M3N3L3 +M3N3L + 12M3N3

efficient code detection MNL2

Table 2 The computational complexities

Estimators Total complex multiplication

MUSIC 13M3N3 + (Q + FtFr−p)M2N2 + FtFrMNESPIRIT QM2N2 + 12M3N3 + 3(M− 1)NP2 + 3(N− 1)MP2

+ 28P3

RD-MUSIC 13M3N3 + (Fr + P)(M3N2 +M3N + 2M3) +QM2N2

proposedmethod

(12L3 + L2 + L + 12)M3N3 + (QL2 + PL2)M2N2 +MNL2

3.2 Efficient code detection

The above subsection shows how the proposed estimationmethod obtains the AODs and AOAs of the signalsreflected from different targets by using the known PN codesequence. In practice, PN code information is unavailablesince the delay time of each reflected signal is unknown. Todetect the P spreading codes related to the reflected signals,an efficient code detection method is presented. Based onthe cyclic property, Fig. 3 shows how the code �ci generatedfor i = 1, 2, …, L can be extended to a code matrix �C. Theinput signals are then passed through the code matchedfilter bank �C. Since the received signals correlate with thecode matrix, the efficient codes of targets are given by

argmax�ci

M (�ci) = ZH�ci∥∥ ∥∥2

2 (19)

where ||·|| denotes the vector 2-norm. This code-matched filterdetects code sequences from the Pmaxima of this function bysearching over �ci for i = 1, 2, …, L. Assume that theprocessing gain is large enough to ignore the near-fareffect. Applying the estimated code sequencesc1, c2, . . . , cP in (15) obtains the AODs and AOAs.


3.3 Computational complexity analysis

Table 1 shows the computational multiplication (CM)procedure used in the proposed method. Assuming Ptargets, M transmitting elements, N receiving elements, andQ transmitted pulses for each test, the computationalcomplexity of the eigen-decomposition and matrix inversionsteps for an MN ×MN correlation matrix in Table 1 requireapproximately 12M3N3 and 2M3N3CMs, respectively.Table 1 shows that, depending on the scenario and arrayparameters, the eigen-decomposition term 12M3N3L3 in theproposed method dominates all steps.Table 2 compares computational complexity in the

proposed method, the spectral search-based method(MUSIC) [6], ESPRIT [7], and reduced dimension (RD)MUSIC [9]. In the spectral search-based method, Ft and Fr

are the searching times, which depend on the scanning gridsize within search regions f and θ, respectively. Table 2also shows that the eigen-decomposition term dominates inESPRIT and RD-MUSIC and that the spectral searchingterm dominates in MUSIC. Depending on the scenario and


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array parameters, the computational complexity of theproposed techniques is almost L3 times higher than those ofconventional methods. That is, the proposed methodsacrifices computational simplicity to increase the degreesof freedom in the MIMO radar system. Therefore theproposed method is applicable to systems with a smallnumber of transmitter and receiver elements.
Fig. 4 RMSE of AODs and AOAs against number of pulses inwidely spaced targets case

4 Simulation results

Several simulation results are presented to demonstrate theperformance of the proposed technique. The evaluation isbased on two criteria: first, the accuracy of the proposedmethod is compared with those of MUSIC [6], ESPRIT [7],RD-MUSIC [9] algorithms, and Cramer Rao Bound (CRB)[24], either closely or widely spaced, in the multi-targetscase. For widely spaced targets, MUSIC spectra exhibit apeak at each target AOD-AOA angle, the magnitude ofwhich depends on the power of the signal returned. In closecases, MUSIC is not compared since the peaks are difficultto distinguish in the spectra. Second, a demonstratedadvantage of this method is its estimation of signal number.As an index of evaluation, the normalised root mean squareerror (RMSE) was defined as

RMSE(u)

= 1

P

∑Pk=1

��1

P

∑Pg=1

fk,g − fk,g

( )2+ uk,g − uk,g

( )2[ ]√√√√

where ∏ = 1000 is the number of Monte Carlo runs.The models of the bi-static MIMO radar system and thechannels are simulated in MATLAB version 2010a. Theadditive background noise is assumed to be spatially andtemporally zero mean white Gaussian processes withvariance s2

n = 1.

Fig. 5 RMSE of AODs and AOAs estimation against SNR in widelyspaced targets case

4.1 Angle accuracy simulation- widely spacetargets

In the case of widely spaced targets, the two targets withSNR = 10 dB are located at AODs and AOAs that areuniformly distributed in a small range (θt,1, θr,1) = ([−50°,−52°], [−35°, −37°]), and (θt,2, θr,2) = ([15°, 17°], [3°, 5°]).Consider a bistatic MIMO radar consisting of M = 4 elementsin a transmitter array and N = 3 elements in a receiverarray. All elements were assumed to be identical andomnidirectional with a unit gain. The transmitter emits radarsignals, which are spread by a PN code sequence of lengthL = 31 (L≥MN).To demonstrate the effect of finite sampleproperties on the AOD-AOA estimated error, Fig. 4 plotsRMSE against the number of pulses. In the simulations, thenumber of pulses in each signal ranged from 0 to 2000.Since the spatial-temporal technique expands the degrees offreedom, the proposed method is more accurate in estimatingAODs and AOAs. The proposed method obtains significantlylower RMSE compared with MUSIC, RD-MUSIC, andESPRIT at different pulse numbers, and it is lower than 0.1°for Q≥ 400. The ESPRIT method has consistently provensuperior to the MUSIC method, which has large scanningsize. The RMSE error in MUSIC does not decrease when thenumber of pulses increases because of the large scanningsize selected in this simulation. Because the dimension of


RD-MUSIC is lower than that of ESPRIT, the ESPRIT has aslightly lower RMSE compared with RD-MUSIC.To evaluate the effect of noise on the estimated error,

RMSE is calculated at different SNR. Fig. 5 plots theRMSE against SNR from 0 dB to 30 dB. Here, the numberof pulses was set to Q = 500. Again, compared with othermethods, the proposed method decreases RMSE by a factorof at least 5 throughout the full range of SNRs. For allsimulated methods except MUSIC, the RMSE drops by onedecade for every 20-dB increase of SNR. The explanationis as follows. The RMSE is a measure of the estimatedangle error, which is proportional to the noise amplitude.Since SNR is inversely proportional to the square of the

noise, the expectation is RMSEP ∝ SNR−1/2, which isconsistent with the observed behaviour.

4.2 Angle accuracy simulation – closely spacetargets

The parameters are set as described in the subsection aboveexcept that the two targets are located at the AODs andAOAs, which are uniformly distributed in a small range(θt,1, θr,1) = ([20°, 22°], [−35°, −37°]), and (θt,2, θr,2) =


Fig. 6 RMSE of AODs and AOAs against number of pulses inclosed spaced targets case

Fig. 7 RMSE of AODs and AOAs estimation against SNR in closedspaced targets case

Fig. 8 RMSE of AODs and AOAs estimation against SNR indifferent number of targets cases

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([15°, 17°], [−30°, −32°]). Fig. 6 compares the performanceof the above methods. The experiment confirmed that theproposed method is unaffected by the closely spaced targetsand is still more accurate than the other algorithm, and itclearly achieves a better angular resolution. Fig. 6 showsthat global performance severely degrades in the ESPRITand especially in the RD-MUSIC when the targets areclosely spaced. The algebraic algorithms translate closelyspaced targets into an ill-conditioned autocorrelation matrix,which complicates the separation between the signalsubspace and the noise subspace. Fig. 7 shows the RMSEagainst SNR from 0 to 30 dB in closely space targets case.The proposed method gives higher resolution in AOD-AODestimation than the existing methods because of the extradegrees of freedom. Again, we can see that the proposedmethod gives lower RMSE than other methods for all SNRcases as expected.

4.3 Change point in the number of targets

In this subsection, Fig. 8 compares RMSE performanceagainst SNR for different numbers of targets. This


parameter is used to verify the advantage of the method interms of the number of targets that can be estimated. Theparameters are set as in the above subsection except that thenumber of targets is now set at P = 2, P = 5, P = 10, P = 15,and P = 30. The AODs and AOAs are also uniformlydistributed [−60°, 60°]. Unsurprisingly, the AODs andAOAs estimation results improve as the SNR increases.Notably, the results in Fig. 8 show that this method givesthe same RMSE for all case with different number oftargets when SNR is >10 dB. That is, performance of theproposed method is unaffected by the number of targetseven though the number of targets is much larger thannumber of elements P≥MN simply because the steeringvectors of the AOD and AOA are estimated one at a time.These experimental results indicate that the proposed

method obtains more accurate AODs-AOAs estimates sincethe scenarios are very difficult (closely spaced targets, lowSNR, and overloaded system). As noted above, the keyfeature of the proposed method is its expansion of thedegrees of freedom, which enables accurate estimation ofAODs and AOAs, even in an overloaded system.

5 Conclusion

An efficient AOD and AOA estimation approach based onsignal subspace technique was proposed for overloadedbistatic MIMO radars. The proposed approach, which canbe used for AOD and AOA estimation, utilises thespatial-temporal scheme to expand the degrees of freedominstead of using the conventional scheme for bistatic MIMOradar. Therefore it can estimate more target anglescompared with existing algorithms and provides goodestimation performance. Computer simulations confirm theeffectiveness of the proposed approach.

6 Acknowledgment

Ted Knoy is appreciated for his editorial assistance and theanonymous reviewers are appreciated for their valuable andconstructive comments, which helped to improve thepresentation of the paper.


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