Multidim Syst Sign ProcessDOI 10.1007/s11045-016-0387-2

Sum and difference coarray based MIMO radar arrayoptimization with its application for DOA estimation

Yan Huang1 · Guisheng Liao1 · Jun Li1 · Jie Li1 ·Hai Wang1

Received: 26 May 2015 / Revised: 4 January 2016 / Accepted: 16 February 2016© Springer Science+Business Media New York 2016

Abstract Based on the sumand difference coarrays,multiple-inputmultiple-output (MIMO)radar with minimum redundancy (MR) concept, referred to as MRMIMO, can considerablyincrease the spatial degrees of freedom (DOFs). However, traditional MR MIMO needscomputational search to determine the position of each element. In this paper, a modifiedMR monostatic MIMO configuration is proposed, referred to as MMRM MIMO. In theproposed system, the MMRM MIMO radar is consisted of several levels of uniform lineararray, which brings the advantage that the position of each element can be determinedwithoutcomputational search. Furthermore, it offers more than N 2 DOFs for an N -elemental array.In order to utilize the extended DOFs ofMMRMMIMO radar for direction-of-arrival (DOA)estimation, an average Toeplitz approximation method (TAM) is employed, which achievesrobust performance even under low signal-to-noise ratio, few snapshots and array error.Numerous simulation results are provided to demonstrate the effectiveness of the proposedmethod for DOA estimation.

Keywords Direction-of-arrival (DOA) estimation · Modified minimum redundancymonostatic multiple-input multiple-output (MMRMMIMO) radar · Toeplitz approximationmethod (TAM) · Difference coarray · Sum coarray

1 Introduction

Multiple-input multiple-output (MIMO) radar has drawn considerable attention in the arraysignal processing area for the last decade. Unlike the standard phased-array radar providinglow degrees of freedom (DOFs), MIMO radar can transmit, via its antennas, multiple prob-

This study has been supported by the National Natural Science Foundation of China under contract No.61271292 and No. 61431016 and the Fundamental Research Funds for the Central Universities.

B Yan Huangyellowstone0636@hotmail.com

1 National Lab of Radar Signal Processing, Xidian University, Xi’an, Shaanxi 710071, China

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ing signals that may be chosen quite freely (Bekkerman and Tabrikian 2006). For colocatedantennas,MIMO radar has been used to provide longer aperture withmore DOFs (Li and Sto-ica 2007), higher resolution (Frankford et al. 2014; Hack et al. 2014), and higher sensitivityto slowly moving targets (Ali et al. 2014). However, assume colocated MIMO radar with Ntransmitter sensors and N receiver sensors, N 2 is the upper bound of DOF. In order to extendmore DOFs, nonuniform linear array (NLA) was explored in many previous works: fourth-order cumulants (Suwandi et al. 2006), Khatri-Rao (KR) subspace approach (Ma et al. 2009),nested array (Pal and Vaidyanathan 2010, 2012b, a), co-prime array (Liu and Vaidyanathan2015; Zhang et al. 2013, 2014b; Tan and Nehorai 2014; Tan et al. 2014), gridless method(Pal and Vaidyanathan 2014a, b), sum and difference pattern design (Shnidman 2004; Haupt2006). Actually, two coarray concepts (Hoctor and Kassam 1990) are applied to the MIMOradar and sparse NLA to extend more DOFs: MIMO radar uses sum coarray while sparseNLA utilizes difference coarray. The further introduction of coarray concept will be shownin Sect. 3.

Most previous studies just utilized one type of coarrays to extend aperture. In Chenand Vaidyanathan (2008), a minimum redundancy colocated MIMO radar (MR MIMO)is proposed for extended aperture with both coarrays. However, sum coarray cannot per-form sufficiently in this case due to different transmitter and receiver array. Here, definethe minimum redundancy monostatic MIMO radar as MRMMIMO, which has an identicaltransmitter and receiver array. The MRMMIMO radar may considerably extend DOFs withsufficient use of both coarrays. Due to the high computational complexity of MR search, theMR MIMO and MRM MIMO may hard to determine the position of each element.

Based on the designing NLA, how to apply the extended DOFs for direction-of-arrival(DOA) estimation is another foremost task to solve. Numerous approaches have been pro-posed to solve this problem. In Abramovich et al. (1999) and Abramovich et al. (1998),an iterative Toeplitz reconstruction method was proposed but with multiple iterations. InSuwandi et al. (2006), an approach based on fourth-order cumulants was proposed to sup-press Gaussian noise, but it cannot be applied to Gaussian signal. In Pal and Vaidyanathan(2010), the covariance matrix is reconstructed by a spatial smoothing method. However, itwill need complex procedures and cannot decoherent completely. Therefore, a fast methodwith robust performance is needed for DOA estimation.

In this paper, a modified MR scheme is proposed for monostatic MIMO radar, whichreferred to as modified minimum redundancy monostatic (MMRM) MIMO radar. A newconstraint is added to the optimization problem of the traditional MRMMIMO, which makesseveral levels of uniform linear array (ULA) to compose the MMRM MIMO. The positionof each element in MMRM MIMO radar do not need computational search. Additionally,the N -elemental MMRMMIMO radar can offer more than N 2 DOFs. Moreover, a Toeplitzapproximation method (TAM) is proposed to use the extended DOFs of the MMRMMIMOradar for direction-of-arrival (DOA) estimation. The TAM reconstructs an extended Toeplitzcovariance matrix of MMRM MIMO radar. Furthermore, multiple signal classification(MUSIC) algorithm and estimation of signal parameters via rotational invariance techniques(ESPRIT) can be applied to the covariance matrix directly. Unlike the previous approaches,the proposed TAM is a fast approach with low computational complexity and it can achieverobust performance under few snapshots, low signal-to-noise ratio (SNR) and array error.

The rest of the sections of this paper are organized as follows. Section 2 shows the sig-nal model of monostatic MIMO radar. In order to reduce the computational complexity ofminimum redundancy search, Sect. 3 introduces coarray concepts and shows the array opti-mization ofMMRMMIMO radar. Section 4 analyzes the relation between difference coarrayand wavepath difference and proposes the TAM on MMRM MIMO radar, following by the

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Table 1 Symbol abbreviations

Abbreviation Definition

MIMO Multiple-input multiple-output

MR MIMO Minimum redundancy MIMO in Hoctor and Kassam (1990)

MMRM MIMO Modified minimum redundancy monostatic MIMO

MRM MIMO Minimum redundancy monostatic MIMO

analysis of array calibration and algorithm. Next, extensive numerical examples of the pro-posed method are provided in Sect. 5. Finally, this paper is concluded in Sect. 6 with a shortdiscussion of future work. The symbol abbreviations are shown in Table 1.

2 Signal model

Consider a monostatic MIMO system with an identical N -elemental NLA as the identicaltransmitter and receiver array (TR array), which is shown in Fig. 1. The first sensor is thereference sensor, and the set of sensor locations is {xn} (n = 1, . . . , N ). Assume K far-fieldtargets impinging on this array from directions θk(k = 1, . . . , K ), respectively. The carrierwavelength is denoted by λ, hence the N × K transmitter and receiver steering vector are

At = [at (θ1) at (θ2) · · · at (θK )

](1)

Ar = [ar (θ1) ar (θ2) · · · ar (θK )

](2)

where

at (θk) = ar (θk) =[e j

2πλx1 sin(θk ) e j

2πλx2 sin(θk ) · · · e j 2πλ xN sin(θk )

]T(3)

and superscriptT denotes transpose without conjugation. Assume that L denotes the pulselength of transmitted signal. The transmitted signal matrix is

S = [s1 s2 · · · sN

]T(4)

where sn = [sn (1) sn (2) · · · sn (L)

]Tdenotes the transmitted signal of the nth transmitter.

Assume that the scattered coefficient of the kth target is βk . Within pth pulse repetitionperiod, the received signal is

Y (p) =K∑

k=1

ar (θk) βk (p) aTt (θk) S + N (p) , p = 1, . . . , P (5)

Fig. 1 Monostatic MIMO radarsignal model

1x 2x 3x Nx

kθ

Target k

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Multidim Syst Sign Process

where Y (p) = [Y1 (p) Y2 (p) · · · YN (p)

]Tis the received signal matrix; Yn (p) ∈

CL×1(n = 1, . . . , N ) is the signal received by the nth sensor; N (p) =[n1 (p) n2 (p) · · · nN (p)

]Tis an N × L noise matrix, and nn (p) ∼ Nc

(0, σ 2

n IN).

Then the sufficient statistic can be obtained by a matched filter (Li and Stoica 2007):temporal matching the received signal to each of the N orthogonal transmitted signals, whichyields

z (p) = vec

(1√LY (p) SH

)= vec

(1√L

(K∑

k=1

ar (θk) βk (p) aTt (θk)S + N (p)

)

SH)

(6)

where superscriptH denotes transpose conjugate. As the transmitted signals of MIMO Radarare orthogonal, which obtains SSH = LI. Define atr (θk) = at (θk) ⊗ ar (θk), hence (6) canbe rewritten as

z (p) =K∑

k=1

√Lβk (p) atr (θk) + w (p) = Aβ (p) + w (p) (7)

where w (p) = vec(

1√LN (p)SH

), A = [

atr (θ1) atr (θ2) · · · atr (θK )]denotes the man-

ifold of MIMO radar virtual array, β (p) = √L

[β1 (p) β2 (p) · · · βK (p)

]denotes the

scattering coefficient vector of K targets within the pth pulse repetition period. Then sum-marize (7) as

Z = AB + W (8)

where Z = [z(1) z(2) · · · z(P)] ∈ CN2×P , B = [β(1) β(2) · · · β(P)] ∈ C

K×P and W =[w(1)w(2) · · · w(p)] ∈ C

N2×P .Assume that the sources are independent, then the covariancematrix ofZ can be expressed

as

RZZ = E{ZZH} = 1

PARBBAH + σ 2

wI (9)

where σ 2w denotes the power of noise.

Under sufficient snapshots and high signal-to-noise-ratio (SNR), RZZ is approximatelyconsidered as Hermite-Toeplitz matrix for ULA. The subspace-based algorithms likeMUSICalgorithm (Zhang et al. 2014a, c), polynomial-rooting of MUSIC (Root-MUSIC) algorithmCao et al. (2011) and ESPRIT algorithm (Zheng and Chen 2015; Sun et al. 2014; Gu et al.2010) can be applied to RZZ for direction-of-arrival (DOA) estimation. In order to resolvemore sources than sensors, the covariance matrix of NLA may be preprocessed before uti-lizing the traditional algorithms.

3 Array optimization of modified minimum redundancy monostaticmultiple-input multiple-output (MMRM MIMO) radar

In this section, a brief introduction about the sum coarray and difference coarray is presented,followed by the array optimization of modified minimum redundancy monostatic multiple-input multiple-output (MMRM MIMO) radar and analysis of the relationship betweendifference coarray and wavepath difference.

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3.1 Sum coarray

In the case of a MIMO array, sum coarray is defined as all unique sums of sensor locations.Define the coarray of a pair of apertures as a set

S = {xT,m + xR,n, xT,m ∈ AT , xR,n ∈ AR

}(10)

where AT denotes the set of transmitter sensor locations and AR is the set of receiver sensorlocations. The unique element in set S is consisted of a set, which referred to as sum coarraySu . Actually, the sum coarray Su is corresponding to the sensor positions of the virtual array.The sensor location set S can be regarded as a Kronecker product, which can be formulatedas:

S = {xT,m

} ⊗ {xR,n

}, xT,m ∈ AT , xR,n ∈ AR,

where ⊗ denotes the Kronecker product.

3.2 Difference coarray

In the case of aMIMOarraywith discrete elements, difference coarray is all unique differencebetween sensor locations. We define a set as

D = {su − s′

u

}, su, s

′u ∈ Su (11)

where Su denotes the sum coarray, i.e., the set of sensor locations in the virtual array ofMIMO radar. The unique element in set D is defined as difference coarray Du . Then thenumber that element du (du ∈ Du) occurs in set D is defined as weight function wd (du).Actually, the elements in difference coarray denote the position intervals between any twosensors in the virtual array. Additionally, the weight function of the difference coarray canbe recognized as a specific convolution operation as following:

wd (du) = Su (du) ∗Su (−du) ,

where ∗ denotes the convolution operation.

3.2.1 Comparison of two types of colocated MIMO radar

There are two types of colocated MIMO radar: one system has the transmitter and receiverarray separated in parallel and the other system uses the same sensors as TR array.The minimum redundancy multiple-input multiple-output (MR MIMO) radar in Chen andVaidyanathan (2008) is the former colocated MIMO radar, the minimum redundancy mono-static multiple-input multiple-output (MRM MIMO) radar and proposed MMRM MIMOradar belong to the latter colocated MIMO radar.

3.3 Array optimization of modified minimum redundancy monostatic MIMOradar

The aim of minimum redundancy is to gain the most degrees of freedom (DOFs) with limitedsensors, i.e., longest difference coarray. The MRM MIMO radar may achieve the optimalperformance, which is better than the MR MIMO radar in Chen and Vaidyanathan (2008).The MRM MIMO can be found by the following optimization problem:

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Multidim Syst Sign Process

maxN

DOF (N )

s.t. {−Lu, . . . , Lu} ⊂ {Du} , Lu = max {Du} (12)

where N is the number of sensors, Du is the difference coarray determined by locationsof N sensors and the constraint condition {−Lu, . . . , Lu} ⊂ {Du} denotes that differencecoarray has all the integer locations from −max{Du} to max{Du}, i.e., difference coarrayis consecutive. For solving (12), the MRM MIMO needs extensive computational search(Ruf 1993) with at most O

((2Lu + 1)2N−1) computational complexity in each search. The

computational complexity is determined by the randomsample,which is extremely huge if thelocations of sensors are varying for specific application. In order to decrease computationalcomplexity, fixed position of each element in the TR array is necessary. Therefore, we wouldlike to get most DOFs by several levels of ULA with N sensors. This can be cast as anoptimization problem below:

maxN

DOF (N )

s.t. {−Lu, . . . , Lu} ⊂ {Du} , Lu = max {Du} (13a)

x j,N j − x j,N j−1 = · · · = x j,i − x j,i−1 = · · · = x j,N2 − x j,N1 , (13b)

J∑

j=1

N j = N , N j ∈ N+, (13c)

where N j denotes number of sensors in the jth level and x j,i is the position of the ith sensor inthe jth level. By solving (13), the modified minimum redundancy monostatic MIMO radar isput forward, which is calledMMRMMIMO radar. TheMMRMMIMO is a heuristic solutionof (13) and can be recognized as the tradeoff of the low complexity and maximum of DOF.The advantage of this array is its certain construction without computational complexity oncegiven the number of sensors, however, it will lose some DOFs.

Assume that the least spacing between two elements is d = λ/2. The TR array ofMMRMMIMO is consisted of two parts: Part I is a two-level nested array and Part II is a ULA (e.g.,in Fig. 2). Part I is easily constructed with two levels, and the locations of sensors in Part Iis shown in Pal and Vaidyanathan (2010), that is

SI = {S1, S2} = {{0d, . . . , (N1 − 1) d} , {N1d, . . . , [N2 (N1 + 1) − 1] d}} ,

where N1 and N2 is the number of sensors in the first and second level of nested array,respectively. Two-level nested array is a regular array whose sum coarray provides a longconsecutive location set starting from the reference location 0d (e.g., in Fig. 2b the set ofconsecutive locations is Scon = {0d, 1d, 2d, 3d, 4d, 5d, 6d, 7d}). The set of consecutivelocations can be presented as

Scon = {0d, 1d, . . . , [N2 (N1 + 1) − 1] d, . . . , [N2 (N1 + 1) + N1 − 1] d} (14)

Part II is established as aULAwith spacing being [N2 (N1 + 1) + N1] d which determinedby the length of Scon . The separation of elements in Part II is 8d shown in Fig. 2a. Assumethat the number of sensors in Part II is N3, which infers that the set of sensor locations isS3 = {[2N2 (N1 + 1) + N1 − 1] d, . . . , [N2 (N3 + 1) (N1 + 1) + N1N3 − 1] d}.

Above all, the set of sensor locations in TR array is {S1, S2, S3} and the consecutivedifference coarray of the MMRM MIMO radar can be expressed as

Du = {−Md, . . . , Md} , M = 2 [N2 (N3 + 1) (N1 + 1) + N1N3 − 1] (15)

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Multidim Syst Sign Process

0d 1d 2d 5d 13d 21d

Part IIPart I

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

s(element position in Sum coarray)

conS

-40 -30 -20 -10 0 10 20 30 4002468

101214161820

Wd(

d)

d(element position in Difference coarray)

(a)

(b) (c)

Fig. 2 aMMRAwith 6 sensors; b sum coarray (i.e., virtual array); c the weight function of difference coarray

The difference coarray in Fig. 2c is {−42d,−41d, . . . , 0d, . . . , 41d, 42d}. Given N sen-sors, we would like to know the number of sensors per Part, which will maximize the DOFs.This can be simplified from (13) as the following optimization problem:

maxN1,N2,N3

DOF (N1, N2, N3)

s.t.3∑

j=1

N j = N , N j ∈ N+ (16)

To solve this problem, the relationship between N1, N2 and N3 should be confirmed. Thesolution to this problem is given by the following two corollaries.

Corollary 1 If NA = N1 + N2 is even, the MMRMMIMO radar will provide the maximumdegree of freedom when N1 = N2; if NA = N1 + N2 is odd, the MMRM MIMO radar willoffer the maximum degree of freedom when N2 = N1 + 1.

Proof See “Appendix 1”. Corollary 2 Assume that the whole number of sensors in MMRA is N. Then the optimal NA

is given by

(i) if N = 3N1 − 1 or 3N1, then NA = 2N1(i.e., N1 = N2) and N3 = N1 − 1 or N1;(ii) if N = 3N1 + 1, then NA = 2N1 + 1(i.e., N2 = N1 + 1) and N3 = N1.

Proof See “Appendix 2”. After the values of N1, N2 and N3 are confirmed, the full establishment of MMRM

MIMO radar is determined. To be noted here, the DOF of the non-uniform MIMO radar

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Multidim Syst Sign Process

Table 2 The DOFs of the MR MIMO radar, MMRM MIMO radar and MRM MIMO radar

Element number MR MIMO MMRM MIMO MRMMIMO

4 21 33 49

8 127 165 357

9 181 225 461

10 221 289 589

-70

-60

-50

-40

-30

-20

-10

0-80

-60

-40

-20

0

20

40

60

80

-80 -60 -40 -20 0 20 40 60 80

Received Angle(degree)

Tran

smitt

ed A

ngle

(deg

ree)

-45

-40

-35

-30

-25

-20

-15

-10

-5

0-80

-60

-40

-20

0

20

40

60

80

-80 -60 -40 -20 0 20 40 60 80

Received Angle(degree)

Tran

smitt

ed A

ngle

(deg

ree)

-55

-50-45

-40

-35

-30

-25-20

-15

-10

-50

-80

-60

-40

-20

0

20

40

60

80

-80 -60 -40 -20 0 20 40 60 80

Received Angle(degree)

Tran

smitt

ed A

ngle

(deg

ree)

(a) (b) (c)

Fig. 3 The transmit-receive beampattern. a Of 5-elemental MMRM MIMO radar. b Of 27-elemental ULA.c Of 5-elemental MR MIMO radar

is calculated by the difference coarray of the virtual array, i.e., the DOF of MIMO radar iscorresponding to the number of sensors in the difference coarray. The DOF of 6-elementalMMRM MIMO radar is 85 since its difference coarray has 85 sensors, which is shownin Fig. 2c. For comparison, the DOFs of the MR MIMO radar in Chen and Vaidyanathan(2008), theMMRMMIMO radar and theMRMMIMO radar are listed in Table 2 for differentnumbers of element.

In Table 2, the MMRMMIMO radar can get more than N2 DOFs when given N sensors.It is important that the DOFs in Table 2 be the upper bound provided by the 3 arrays. Dueto the positive and negative symmetric elements in difference coarray, half of the DOFs maybe utilized finally.

3.4 Transmit-receive beampattern of MMRM MIMO radar

After giving the DOFs of the proposed MMRM MIMO radar, the transmit-receive beam-pattern is shown to further demonstrate its excellent performance. Assume that the expecteddirection is 10◦. Since the 5-elemntal MMRM MIMO radar has 53 DOFs, here the 27-elemental ULA and 5-elemental MR MIMO are chosen for comparison. As can be seen inFig. 3, all the beampatterns have focused on the expected direction. Furthermore, the main-lobe of the proposedMMRMMIMO radar is as wide as that of ULA due to the same aperture,while the mainlobe of MR MIMO is wider as its available DOFs are 41.

4 Toeplitz approximation method (TAM) for DOA estimation

As discussed in the Sect. 1, different methods were put forward to utilize the extended DOFsfor DOA estimation. In this section we put forward a Toeplitz approximation method (TAM)

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Multidim Syst Sign Process

working on the MMRM MIMO radar. This method is a fast algorithm for DOA estimationwith no need for complicated iteration and cumulants.

4.1 Relations between difference coarray and wavepath difference

For monostatic MIMO radar, the elements of sum coarray correspond to the sensor locationsin virtual array. The manifold of virtual array is atr (θ) = [1, . . . , e− j2π sin θ(xm+xn)/λ, . . . ,

e− j2π sin θ(xN+xN )/λ]. The elements of difference coarray correspond to the uniquely differ-ent locations between the sensors in virtual array, which correspond to the unique wavepathdifference in the covariance matrix. Hence, the uniquely different locations in the differ-ence coarray is corresponding to the wavepath difference in the covariance matrix. Actually,the unique elements in the Toeplitz covariance matrix represent the consecutive wavepathdifference. Consider that the covariance matrix of (M+1)-elemental ULA has 2M + 1unique wavepath difference, including positive and negative difference [see (19)]. If the2M+1unique wavepath difference are gotten in Du of N -elemental MMRMMIMO, we canresolve the original N×N covariancematrix of virtual array to an extended (M+1)×(M+1)covariance matrix [see (22)–(23)]. In order to get the whole (2M + 1) unique wavepath dif-ference, the locations in difference coarray should be consecutive from −M to M . MUSICalgorithm can be applied to the extended covariance matrix to resolve more sources thansensors. According to the analysis above, the specific TAM is shown as follows.

4.2 Toeplitz approximation method (TAM) on MMRM MIMO radar

The covariance matrix of MMRM MIMO radar can be expressed as

RZZ (i, j) = zizHj (17)

where zi denotes the i th row of the received signal matrix. Assume that signal and noise areuncorrelated and each row vector in noise matrix is independent, hence (17) can be rewrittenas

RZZ (i, j) = zizHj ={AiRSSAH

i + wiwHi , i = j

AiRSSAHj , i �= j

(18)

where Ai denotes the ith row of the manifold matrix and wi denotes the ith row of thenoise matrix. Under enough snapshots and high SNR, the covariance matrix of receivedsignal on ULA is approximately considered as a Toeplitz matrix, where the element in eachdiagonal is corresponding to the consecutive wavepath difference. Define the covariancematrix of received signal on (M+1)-elemental ULA as RULA, where M = 2xN . Then notethat RULA(i, j) = R(i − j), the RULA is expressed as

RULA =

⎡

⎢⎢⎢⎣

R (0) R (1) · · · R (M)

R (−1) R (0) · · · R (M − 1)...

.... . .

...

R (−M) R (−M + 1) · · · R (0)

⎤

⎥⎥⎥⎦

(M+1)×(M+1)

(19)

where {R(du)|du = −M, . . . , M} represents 2M + 1 unique consecutive wavepath dif-ference. Under enough snapshots and high SNR, the elements in RZZ are approximatelyconsidered to be equal to the elements in RULA at the relevant location, which can be for-mulated as

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Multidim Syst Sign Process

RZZ (i, j) ≈ RULA(xi , x j

) = R(xi − x j

)(20)

where xi corresponds to the sensor position (e.g., xi = 1 corresponds to the sensor position0d). As analyzed in Sect. 5.1, RZZ can be extended to a (M + 1) × (M + 1) covariancematrix like RULA. Denote R̂ZZ ∈ C

(M+1)×(M+1) as the extended covariance matrix of theMMRM MIMO radar, which is defined as

R̂ZZ(xi , x j

) = RZZ (i, j) ≈ RULA(xi , x j

)(21)

Hence, R̂ZZ is partly filled with (N × N ) nonzero elements at R̂ZZ(xi , x j )(i, j =1, . . . , N ) and the other locations in R̂ZZ are zeros. Additionally, R̂ZZ has 2M + 1 uniquenonzero elements which represent all consecutive wavepath difference in RULA due to theconsecutive difference coarray of MMRM MIMO radar. Hence, R̃ZZ can be fully filled bythe average Toeplitz approximation below

R̃ZZ =

⎡

⎢⎢⎢⎣

R̃Z Z (0) R̃Z Z (1) · · · R̃Z Z (M)

R̃Z Z (−1) R̃Z Z (0) · · · R̃Z Z (M − 1)...

.... . .

...

R̃Z Z (−M) R̃Z Z (−M + 1) · · · R̃Z Z (0)

⎤

⎥⎥⎥⎦

(22)

R̃Z Z (du) = 1

wd (du)

wd (du)∑

i=1

Ri (du) (23)

where Ri (du) denotes the ith nonzero elements in the diagonal corresponding to R(du) andwd (du) is the total number of nonzero elements in the diagonal, which is the weight functiondefined in Sect. 3.2.

Therefore, the steps of the TAM can be summarized as below:

Step 1: Calculate RZZ by (17);Step 2: Partly fill R̂ZZ by (21);Step 3: Fully fill R̃ZZ by (22) and (23).

After the TAM, the MUSIC algorithm can be applied to the extended covariance matrixfor DOA estimation with less sensors than sources.

4.3 Analysis of the TAM

Actually, the SSM chooses only one element in each diagonal of covariance matrix with theloss of some samples. However, with the TAM, all the nonzero elements in the covariancematrix are utilized. Both the TAM and the SSM loses half of the DOFs when using thepositive and negative symmetrical elements in difference coarray. Forward/Backward spatialsmoothing (Piliai and Kwon 1989) cannot extend more DOFs because of the conjugatesymmetric property of difference coarray. The elements in R̃ZZ is the mean value of thenonzero elements in the diagonal of R̂ZZ, hence the covariance matrix is a positive-definiteToeplitz matrix under low SNR and few snapshots, which will bring better performance.

4.4 Application for DOA estimation

The reconstructed covariance matrix can be directly applied for DOA estimation by MUSICalgorithm with less sensors than sources, which referred to as T-MUSIC. The computationalcomplexity of the proposed T-MUSIC method is calculated below:

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Multidim Syst Sign Process

Table 3 Computationalcomplexity of the proposedT-MUSIC, SS-MUSIC andWS-MUSIC

T-MUSIC SS-MUSIC WS-MUSIC

Computationalcomplexity

O(PN +(M + 1)3)

O(P(M+1)+2(M + 1)3)

O(P(M+1)+(M + 1)3)

Table 4 Computationalcomplexity of T-ESPRIT,B-ESPRIT and A-ESPRIT

T-ESPRIT B-ESPRIT A-ESPRIT

Computationalcomplexity

O(PN+(M+1)3 + M3)

O(P(M + 1) +(2M)3) + M3

O(P(M + 1) +(M+1)3+M3)

(1) Construct the original covariance matrix, then the complexity of this step is O(PN ).(2) Reconstruct the partly filled and fully filled covariance matrix, no multiplication is

needed.(3) Eigen-decomposition of the MUSIC algorithm, the complexity is O

((M + 1)3

).

Sum the 3 steps above, the computational complexity of the proposed T-MUSIC methodis O

(PN + (M + 1)3

). For comparison, the computational complexity of the MUSIC algo-

rithm with spatial smoothing method (SS-MUSIC) and without spatial smoothing method(WS-MUSIC) are listed with the proposed T-MUSIC in Table 3.

Obviously M > N , hence the computational complexity of the T-MUSIC method is theleast among the three methods. Similarly, the ESPRIT algorithm can also be applied on thereconstructed covariance matrix, which is referred to as T-ESPRIT. Based on the SSM, twokinds of ESPRIT algorithm were employed to solve the coherent single-snapshot signals(Thakrc et al. 2009). One ESPRIT method divides two subarrays before spatial smoothingand the other ESPRIT method after spatial smoothing. The former one is referred to as B-ESPRIT and the latter one isA-ESPRIT. The computational complexities of the three ESPRITmethods are compared as Table 4 shows. Similar to the Table 3, the T-ESPRIT method hasleast computational complexity compared with the other two ESPRIT methods.

4.5 The calibration of array location error

Assume the array location errors are [�x1,�x2, . . . ,�xN ], then the real steering vector in(3) can be revised as

at (θk) = ar (θk) =[e j

2πλ (x1+�x1) sin(θk ) e j

2πλ (x2+�x2) sin(θk ) · · · e j 2πλ (xN+�xN ) sin(θk )

]T.

The occurrence of array location error will influence the correct steering vector, which mayresult in the false DOA estimation. As analysis in the former section V-D, the proposedTAM averages the nonzero elements in the covariance matrix, which may not only get thefull-rank covariance matrix but also smooth the array location error in the steering vector.Hence the proposed TAMmay have more robust performance under array location error. Theperformance of TAM for DOA estimation will be shown in Sect. 5.

5 Numerical examples

In this section, numerical examples are provided to illustrate that the MMRM MIMO radarcan resolvemore sources than sensors by using the proposed TAM. TheMMRMMIMO radarwith 5 physical elements (N = 5) is applied with sensor positions at {0d, 1d, 2d, 5d, 13d}

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Multidim Syst Sign Process

and the sensors of 5-elemental MRM MIMO radar are located at {0d, 3d, 8d, 21d, 22d}.Consider 21 far-field targets (K = 21) impinging on the MMRMMIMO from directions ofarrival (DOAs) {−60◦,−54◦, . . . ,−6◦, 0◦, 6◦, . . . , 54◦, 60◦}, respectively. The SNR usedbelow is defined as the received SNR on the array elements.

5.1 MUSIC spectrum

Figure 4a shows the representative MUSIC spectrum of the 5-elemental MMRM MIMOradar by using the proposed T-MUSIC method and the SS-MUSIC method. Both methodsuse a total of P = 500 snapshots and SNR = 0 dB. As can be observed, both approachesresolve 21 sources sufficiently well. Figure 4b shows the representative MUSIC spectra bythe twomethods under few snapshots. The SNR is fixed to 0 dB and the snapshots are reducedto P = 30. It is obvious that the T-MUSIC can still resolve 21 sources correctly, while theSS-MUSIC suffers due to lack of snapshots. When considering the array location errors, theMUSIC spectrum of the proposed T-MUSIC and the SS-MUSIC are shown in Fig. 5. As can

-80 -60 -40 -20 0 20 40 60 800

51015

202530

35404550

T-MUSICSS-MUSIC

theta/degree

P/dB

-80 -60 -40 -20 0 20 40 60 800

51015

202530

35404550 T-MUSIC

SS-MUSIC

theta/degree

P/dB

(a) (b)

Fig. 4 MUSIC spectrum of T-MUSIC and SS-MUSIC. a SNR = 0 dB, P = 500; b SNR = 0 dB, P = 30

T-MUSICSS-MUSIC

0

51015

202530

35404550

-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 800

51015

202530

35404550 T-MUSIC

SS-MUSIC

theta/degree theta/degree

(a) (b)

P/dB

P/dB

Fig. 5 MUSIC spectrum of T-MUSIC and SS-MUSIC when considering array location error. a SNR =0 dB, P = 500; b SNR = 0 dB, P = 60

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Multidim Syst Sign Process

-80 -60 -40 -20 0 20 40 60 800

510

152025

3035

404550

MMRM MIMOMRM MIMO

MMRM MIMOMRM MIMO

0

5101520

25

3035404550

-80 -60 -40 -20 0 20 40 60 80

theta/degree theta/degree

(a) (b)

P/dB

P/dB

Fig. 6 MUSIC spectrum of MMRM MIMO radar and MRM MIMO radar with 5 sensors by the proposedT-MUSIC method. a SNR = 0 dB, P = 500; b SNR = 0 dB, P = 60

be seen in Fig. 5a, the T-MUSIC can resolve all the DOA correctly while the SS-MUSICmissed oneDOA since it is influenced by the array location error. And in Fig. 5b the T-MUSICstill estimate 21 sources while the SS-MUSIC missed more sources with few snapshots.

Figure 6 shows theMUSIC spectrumof 5-elementalMMRMMIMOradar and5-elementalMRMMIMOradar by using the proposedT-MUSICmethod.As can be seen in both Fig. 6a, b,all theDOAs are accurately estimated based onMMRMMIMOradar andMRMMIMOradar.However, the estimated peaks ofMRMMIMO radar are sharper than those ofMMRMMIMOradar, since the DOF of MRM MIMO radar are 89 while the DOF of MMRM MIMO radarare 53. It can be concluded that the increased DOF determines more accurate performanceof detection.

5.2 Root-mean-square error (RMSE)

The definition of RMSE is

RMSE = 1

K

K∑

k=1

√√√√ 1

J

J∑

j=1

(θ̂k j − θk

)2,

where K denotes the number of targets, J denotes the number of Monte Carlo simulations,θ̂k j denotes the DOA estimate of the kth target and the jthMonte Carlo simulation, θk denotesthe real DOA of the kth target.

In this section, the RMSE of 5-elemental MMRM MIMO radar and the 27-elementalULA are compared with various methods. The 27-elemental ULA with uniform spacingd has been chosen for comparison because of 27 available DOFs provided by 5-elementalMMRM MIMO radar. In Fig. 7, the T-MUSIC, SS-MUSIC, WS-MUSIC, T-ESPRIT, B-ESPRIT and A-ESPRIT are based on the 5-elemental MMRM MIMO radar and the ULAMUSIC and ULATAM are performed on the 27-elemental ULA. Figure 7a shows the RMSEas a function of snapshots for SNR = 0 dB after 1000 Monte Carlo simulations. In Fig. 7b,the RMSE of the four situations for P = 500 snapshots by varying the SNR is illustrated. Itcan be seen that the T-MUSIC and T-ESPRIT have more precise estimations than the othermethods onMMRMMIMOradar.Additionally, the performance of bothmethods approachesthat of the ULA with Toeplitz approximation. Furthermore, the RMSE of T-MUSIC and T-

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Multidim Syst Sign Process

50 100 150 200 250 300 350 400 450 500Snapshots

RM

SE(d

egre

es)

RM

SE(d

egre

es)

SS-MUSICT-MUSICWS-MUSICULA MUSICULA TAMT-ESPRITB-ESPRITA-ESPRIT

-8 -6 -4 -2 0 2 4 6 8 10 1210-3

10-2

10-1

100

101

10-3

10-2

10-1

100

101

SNR(dB)

SS-MUSICT-MUSICWS-MUSICULA MUSICULA TAMT-ESPRITB-ESPRITA-ESPRIT

(a) (b)

Fig. 7 RMSE of the 5-elemental MMRM MIMO radar and the 27-elemental ULA when comparing theproposed methods with the previous methods. a SNR = 0 dB; b P = 500

ESPRIT are still robust under few snapshots and low SNR.However, since the noise subspacecannot be seen as Gaussian distribution after Toeplitz approximation or spatial smoothing,the small eigenvalues are not low enough compared to the large eigenvalues, i.e., the realsignal subspace and noise subspace are not orthogonal well. Therefore, the RMSEs of themethods with Toeplitz approximation or spatial smoothing decrease to an error floor whenthe SNR increases. To be noted here, the Cramer-Rao bound (CRB) is not given since thenumber of estimated sources are more than the number of sensors, i.e., the CRB is based onan underdetermined function. Therefore, the performance of ULA is chosen for comparisoninstead of CRB.

5.3 Angle resolution performance

Assume two signals impinging on the array from θ1 and θ2. During each Monte Carlo sim-ulation, the DOA estimations are θ̂1 and θ̂2. The two directions can be distinguished as long

as∣∣∣θ̂1 − θ1

∣∣∣ < |θ1 − θ2|/2 and∣∣∣θ̂2 − θ2

∣∣∣ < |θ1 − θ2|/2. After over 1000 times of Monte

Carlo simulations, the probability curves of resolution as a function of SNR and snapshotsare plotted.

Consider the MMRMMIMO radar with 5 sensors and ULA with 27 sensors. Assume thetwo sources placed at 10◦ and 11◦. Compare the resolution performances of the proposedmethods with the previous methods. The estimated probability of resolution as a functionof snapshots is shown in Fig. 8a. Figure 8b shows the corresponding performance as afunction of SNR. In conclusion, the T-MUSIC and T-ESPRIT have better performance onthe probability of resolution under few snapshots and lowSNR.Additionally, the performanceof T-MUSIC and T-ESPRIT on the MMRM MIMO radar is close to the ULA with Toeplitzapproximation. Figure 9 shows the representative MUSIC spectrum of the 4 situations onthe performance of resolution under low SNR and few snapshots. It is obvious that the T-MUSIC on MMRM MIMO radar and ULA with Toeplitz approximation can resolve thetwo closely spaced sources while the SS-MUSIC and ULA with no Toeplitz approximationcannot.

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Multidim Syst Sign Process

-8 -6 -4 -2 0 2 4 6 8 10 120

0.10.20.30.40.50.60.70.80.9

1

SNR(dB)

Prob

abili

ty o

f Res

olut

ion

T-MUSICSS-MUSICULA MUSICULA TAMWS-MUSICT-ESPRITB-ESPRITA-ESPRIT

50 100 150 200 250 300 350 400 450 5000

0.10.20.30.40.50.60.70.80.9

1

Snapshots

Prob

abili

ty o

f Res

olut

ion

T-MUSICSS-MUSICULA MUSICULA TAMWS-MUSICT-ESPRITB-ESPRITA-ESPRIT

(a) (b)

Fig. 8 The probability curve of resolution when comparing the proposed methods with the previous methods.a SNR = 0 dB; b P = 300

Fig. 9 MUSIC spectrum of four situations when two sources are placed closely at 10◦ and 11◦. SNR =0 dB, P = 100

6 Conclusion

In this paper, we have used the properties of sum coarray and difference coarray sufficientlyand proposed modified minimum redundancy monostatic (MMRM) MIMO radar with cer-tain expression. We finally obtain much more than N 2 DOFs with N physical elements. Theelements in difference coarray are corresponding to the elements in each diagonal of thecovariance matrix, which represent the consecutive wavepath difference. Therefore, in orderto utilize the extended DOFs, we employed a Toeplitz approximation method (TAM) work-ing on MMRM MIMO radar. This method is a fast algorithm for DOA estimation withoutcomplicated iteration and computation of cumulants. Moreover, this method is more robustunder low SNR, few snapshots and array error against the spatial smoothing method (SSM).However, it can only utilize half of the extended DOFs provided by theMMRMMIMO radar,and the performance of DOA estimation drops down when the number of sources approaches

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Multidim Syst Sign Process

the upper bound. The performance is verified through extensive simulations. Future researchin this area will be toward finding an optimum array to resolve more DOFs. A robust methodto achieve the maximum DOFs is still needed.

Appendix 1

Proof of Corollary 1 From (15), the last sensor location in MMRA is

x = [N2 (N3 + 1) (N1 + 1) + N1N3 − 1] d

= [(N3 + 1) N1N2 + N2 + NAN3 − 1] d (24)

where NA = N1 + N2 is fixed.(i) NA = N1 + N2 is even:If N1 = N2, the last sensor location in MMRA with NA being even is

xE = [(N3 + 1) N 2

1 + N1 + NAN3 − 1]d (25)

Hold NA invariant, then change the number of sensors in first level and second level of PartI as N ′

E,1 = N1 − n and N ′E,2 = N1 + n respectively, where 0 < n < N1. Then (26) can be

rewritten as

x ′E = [

N ′E,2 (N3 + 1)

(N ′E,1 + 1

) + N ′E,1N3 − 1

]d

= [(N3 + 1)

(N 21 − n2

) + (N1 + n) + NAN3 − 1]d

= [(N3 + 1) N 2

1 + (n − n2

) + (N1 − n2N3

) + NAN3 − 1]d < xE (26)

Similarly hold NA invariant, then change the number of sensors in first level and secondlevel of Part I as N ′′

E,1 = N1 + n and N ′′E,2 = N1 − n respectively, where 0 < n < N1, then

it can be obtained that

x ′′E < x ′

E < xE (27)

In conclusion, if NA = N1 + N2 is even, the MMRA will extend the maximum degree offreedom when N1 = N2.

(ii) NA = N1 + N2 is odd:If N2 = N1 + 1, the last sensor location in MMRA with NA being odd is

xO = [N1 (N3 + 1) (N1 + 1) + (N1 + 1) + NAN3 − 1] d

= [(N3 + 1) N 2

1 + N1N3 + NAN3 + 2N1]d (28)

Hold NA invariant, then change the number of sensors in first level and second level ofPart I as N ′

O,1 = N1 − n and N ′O,2 = N1 + 1+ n respectively, where 0 < n < N1, then (24)

can be rewritten as

x ′O = [

N ′O,2 (N3 + 1)

(N ′O,1 + 1

) + N ′O,1N3 − 1

]d

= [(N3 + 1) (N1 − n) (N1 + 1 + n) + (N1 + n + 1) + NAN3 − 1] d

= [(N3 + 1) N 2

1 + N1N3 + NAN3 + 2N1 − nN3 − n2 (N3 + 1)]d < xO (29)

Similarly hold NA invariant, then change the number of sensors in first level and secondlevel of Part I as N ′′

O,1 = N1+n and N ′′O,2 = N1+1−n respectively, where 0 < n < N1+1,

then it is obvious that

x ′′O < x ′

O < xO (30)

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Multidim Syst Sign Process

Therefore if NA = N1+N2 is odd, theMMRAwill enhance themaximum degree of freedomwhen N2 = N1 + 1.

Appendix 2

Proof of Corollary 2 According to Corollary 1, if NA = N1 + N2 is even, the MMRA canextend the most DOFs when N1 = N2. Assume thatNA is even and the number of sensorsin each level of Part I is N1, then the number of sensors in Part II is

N3 = N − NA = N − 2N1 (31)

Insert (31) into (25), then the last sensor location in MMRA is

x = [(N − 2N1 + 1) N 2

1 + N1 + NA (N − 2N1) − 1]d

= [(N 21 N − 2N 3

1

) + (2N1N − 3N 2

1

) + N1 − 1]d (32)

Based on Corollary 1, if the number of sensors in Part I is NA + 1 and N1 is fixed, thenN ′2 = N1 + 1. So keep N fixed and the number of sensors in Part II is

N ′3 = N − N ′

A = N − 2N1 − 1 (33)

Insert (33) into (28), then the last sensor location in MMRA is

x ′ = [N ′2

(N ′3 + 1

)(N1 + 1) + N1N

′3 − 1

]d

= [(N 21 N − 2N 3

1

) + (3N1N − 6N 2

1

) + (N − 3N1) − 1]d (34)

Then the difference between (36) and (34) is

�xO−E = x ′ − x

= [(N1N − 3N 2

1

) + (N − 4N1)]d (35)

Insert the known condition into (35), which leads to

�xO−E =⎧⎨

⎩

(−2N1) d < 0, N = 3N1 − 1;(−N1) d < 0, N = 3N1;1, N = 3N1 + 1;

(36)

Next, if the number of sensors in Part I is NA + 2 and N ′2 is fixed, then N ′′

1 = N1 + 1. HenceN is fixed and the number of sensors in Part II is

N ′′3 = N − N ′′

A = N − 2N1 − 2 (37)

Insert (37) into (25), the last sensor location in MMRA is

x ′′ =[(N ′′3 + 1

)N

′′21 + N ′′

1 + N ′′AN

′′3 − 1

]d

= [(N 21 N − 2N 3

1

) + (4N1N − 9N 2

1

) + (3N − 11N1) − 5]d (38)

Then the difference between (38) and (34) is

�xE−O = x ′′ − x ′

= [(N1N − 3N 2

1

) + (2N − 8N1) − 4]d (39)

123

Multidim Syst Sign Process

Insert the known condition into (39), which can be rewritten as

�xE−O =⎧⎨

⎩

0 N = 3 (N1 + 1) − 1;N1 + 2 > 0, N = 3 (N1 + 1) ;2N1 + 4 > 0, N = 3 (N1 + 1) + 1;

(40)

As observed in Sect. 3, the more N1 is, the longer spacing between sensors in Part II is.Compared (36) to (40), we can obtain the conclusion: if N = 3N1−1 or 3N1, then NA = 2N1

and N3 = N1 − 1 or N1; if N = 3N1 + 1, then NA = 2N1 + 1 and N3 = N1.

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Yan Huang received the B.S. degree in Electrical Engineering fromXidian University, Xi’an, China, in 2013. He is currently workingtoward the Ph.D. degree in the National Laboratory of Radar SignalProcessing, Xidian University. His research interests include MIMOradar signal processing, synthetic aperture radar, ground moving targetindication, high-resolution and wide swath imaging.

Guisheng Liao received the B.S. degree from Guangxi University,Guangxi, China, and the M.S. and Ph.D. degrees from Xidian Uni-versity, Xi’an, China, in 1985, 1990, and 1992, respectively. He iscurrently a Professor at the National Key Laboratory of Radar SignalProcessing, Xidian University. He has been a Senior Visiting Scholar inthe Chinese University of Hongkong, Hong Kong. His research inter-ests include space-time adaptive processing, array signal processing,SAR ground moving target indication, and distributed small satelliteSAR system design. Dr. Liao is a member of the National OutstandingPerson and the Cheung Kong Scholars in China.

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Multidim Syst Sign Process

Jun Li received the B.S. degree from University of Electronic Scienceand Technology, Chengdu, China, in 1994 and the M.S. degree fromthe Guilin University of Electronic Technology, Guilin, China, in 2002.He received the Ph.D. degree in Information and Communication Engi-neering from Xidian University, Xi’an, China, in 2005. From 1994 to1999, he was with Research Institute of Navigation Technology, Xi’an.He worked as a Visiting Scholar at the department of Electronic andElectrical Engineering of University College London for one year from2009. Dr. Li is currently an associate professor in the National Labora-tory of Radar Signal Processing. His current research interests includearray signal processing, MIMO radar and quantum information.

Jie Li received the B.S. degree in Electrical Engineering from XidianUniversity, Xi’an, China, in 2013. She is currently working toward thePh.D. degree in the National Laboratory of Radar Signal Processing,Xidian University. Her research interests include space-time adaptiveprocessing, OFDM signal processing, MIMO signal processing.

Hai Wang received the B.S. degree from Anqing Normal University,Anhui, China, and the M.S. degree from Wuhan Institute of Technol-ogy, Wuhan, China, in 2009 and 2012, respectively. He is currentlyworking toward the Ph.D. degree in the National Laboratory of RadarSignal Processing, Xidian University. His research interests includeMIMO radar signal processing, Synthetic aperture radar, Ground mov-ing target indication.

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