# Local Degree of Freedom of Clutter for Reduced-Dimension Space-Time Adaptive Processing with MIMO Radar

Authors: Yiduo Guo; Jian Gong; Yu Xiao
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

Degree of freedom (DOF) of clutter in the reduced-dimension (RD) domain, which is called local DOF (LDOF), is of great importance for RD MIMO-STAP (space-time adaptive processing for multiple-input multiple-output radar) algorithms. In this paper, the LDOF equivalence of different RD MIMO-STAP algorithms are firstly proved, and then a generalized LDOF estimation rule under different conditions is developed to estimate the clutter LDOF for MIMO radar effectively. The accuracy of the proposed rule is verified, and how to design RD MIMO-STAP processors under the guidance of the proposed rule is presented through numerical simulations.


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## Body

## 1. Introduction

Airborne radar system explores space-time adaptive processing (STAP) technique to detect slow-moving targets under strong ground clutter [1, 2]. Originally, the theory of STAP is developed for conventional airborne phased array radar system, which can be considered as a single-input multiple-output (SIMO) radar system. However, STAP technique for SIMO radar has been well studied, but its capability of clutter suppression needs to be improved. Compared with SIMO radar, multiple-input multiple-output (MIMO) radar enjoys better spatial resolution because of the much larger virtual array than the physical array [3]. Therefore, MIMO-STAP technique has drawn considerable attention focusing on the significant potentials for clutter suppression in recent years [4, 5].According to the principle of STAP theory, the degree of freedom (DOF) of STAP processors should be larger than that of clutter, so the estimation of clutter DOF for SIMO radar and MIMO radar has been intensively studied. For SIMO radar, the well-known Brennan’s rule is firstly presented in [6] and then formally proved by Ward in [7]. The extended Brennan’s rule to subarray STAP is provided in [8], and DOF estimation in the case of arbitrary arrays is analyzed in [9]. Nonideal radar configurations, such as bistatic arrays and sparse arrays, have been considered in [10, 11]. For MIMO radar, the extension of Brennan’s rule of MIMO radar has been proposed by Chen and Vaidyanathan in [12]; Xie et al. presented the estimation rule for the clutter DOF of the MIMO-phased array radar in [13]; and Li et al. derived the estimation of clutter DOF of wideband MIMO radar in [14]. These references have made great contributions to the successful design of STAP processors.However, the optimal STAP processor is usually difficult to apply in real-time applications because of the large independent identically distributed (IID) training sample requirement and the expensive computational cost. To solve these problems, many algorithms aiming to reduce the complexity and improve the convergence of STAP in SIMO radar have been proposed. Among them, reduced-dimension (RD) STAP algorithms are of great importance for their simplicity and effectiveness [15, 16]. For MIMO radar, because of the added dimension created by the orthogonal waveforms, the two problems mentioned above become even more challenging. Fortunately, with slight modifications, many RD STAP algorithms proposed for SIMO radar can also be applied to MIMO radar. With the transformation of RD STAP algorithms, the clutter data are projected onto a lower-dimensional subspace; the DOF of clutter in this subspace is called local degree of freedom (LDOF) of clutter. Similar with the full STAP algorithm, the DOF of RD STAP processor would also be larger than the clutter LDOF. In [17, 18], Zhang et al. has given the LDOFs estimation formulas of four classical RD STAP algorithms for airborne SIMO radar with uniform linear array and subarray, which play an important role in the development of RD STAP algorithms. Whereas the research on RD MIMO-STAP algorithms is more urgent than that on SIMO-STAP algorithms, the clutter LDOF for airborne MIMO radar is seldom mentioned.In this paper, we develop the estimation rule of clutter LDOF for airborne MIMO radar with linear array, and the complete proof of the proposed rule is also presented. Although there are 23 classes of typical transformation matrix according to the projected domain of the original signal, we only focus on 4 classes of RD MIMO-STAP algorithms, which are extended directly from RD STAP algorithms for SIMO radar named element-space pre-Doppler algorithm, element-space post-Doppler algorithm, beam-space pre-Doppler algorithm, and beam-space pre-Doppler algorithm [7]. This is because these 4 algorithms are usually utilized in practice and the proofs for the other 4 algorithms are actually similar to the previous ones.The rest of the paper is organized as follows. In Section2, the clutter model and clutter DOF for airborne MIMO radar is briefly reviewed and the definition of clutter LDOF of MIMO radar is also presented. In Section 3, the conclusion, which indicates that clutter LDOFs of different RD MIMO-STAP algorithms with the same dimension are actually equivalent, is proved. In Section 4, we develop the clutter LDOF estimation rule for airborne MIMO radar under different conditions. In Section 5, numerical simulations are conducted to demonstrate the effectiveness and applications of the proposed clutter LDOF estimation rule. Finally, Section 6 concludes the paper.


## 2. Problem Formulation

### 2.1. Clutter Model and Clutter DOF

Considering a monostatic side-looking airborne MIMO array radar at the altitudeH with the constant moving velocity V. As is illustrated in Figure 1, there are N receiving elements uniformly spaced by dR=λ/2 and M transmitting elements uniformly spaced by dT=γdR, where λ is the wavelength and γ is the sparse coefficient. The number of pulses at a coherent processing interval (CPI) is K and the pulse repetition frequency (PRF) is fr. The MIMO radar transmits an orthogonal waveform set s1,s2,…,sM, and these waveforms will be extracted by M matched filters at each receiving element. The clutter is modeled as a Gaussian process.Figure 1
The model of monostatic side-looking airborne MIMO radar.
(a)
The geometry of carrier aircraft
(b)
The model of the array and the signalTheith clutter patch on the nth receiving element for the mth waveform and kth pulse can be written as
(1)yim,n,k=αismej2πγm−1+n−1fsi+j2πk−1fdi,where αi is the reflect coefficient of the ith clutter patch, fsi=dRcosψi/λ is the spatial frequency of the ith clutter patch, and fdi=2Vcosψi/λfr is the Doppler frequency of the ith clutter patch; cosψi=cosθicosφ is the directional cosine.Since the MIMO radar transmits orthogonal waveforms, the output of the matched filters can be expressed as(2)cim,n,k=αiej2πn−1fsi+j2πk−1fdi∑q=1MsmsqHej2πγm−1fsi=αiej2πn−1+βk−1+γm−1fsi,where β=fdi/fsi=2V/dRfr.Therefore, for themth waveform and kth pulse, the total clutter echoes on the nth receiving element can be expressed as
(3)xm,n,k=∑i=1Ncαiej2πn−1+βk−1+γm−1fsi,where Nc≫MNK is the number of clutter patches.Define the temporal steering vectorST,i, transmitting spatial steering vector St,i and receiving spatial steering vector Sr,i as
(4)ST,i=1,ej2πβfsi,…,ej2πβfsiK−1T∈CK×1,St,i=1,ej2πγfsi,…,ej2πγfsiM−1T∈CM×1,Sr,i=1,ej2πfsi,…,ej2πfsiN−1T∈CN×1.TheMNK×1 clutter signal vector, which is obtained by stacking the signals in (3) for n=1,2,…,N, m=1,2,…,M, and k=1,2,…,K, can be expressed as
(5)x=∑i=1NcαiSi=UΛ,where Si=ST,i⊗St,i⊗Sr,i is the spatial-temporal steering vector; U=S1,S2,…,SNc and Λ=α1,α2,…,αNcT.Therefore, the clutter covariance matrix is given as(6)Rx=ExxH=UΣUH,where Σ=diagα12,α22,…,αNc2; E· and ·H denote statistical expectation and the conjugate transpose, respectively.The clutter DOF is defined asr=rankRx. Because Σ is a positive definite matrix and spanRx=spanU, the clutter DOF is equal to r=rankU.Letfs=fs1,fs2,…,fsNcT; U can be rewritten as
(7)U=1,…,ej2πn−1+βk−1+γm−1fs,…,ej2πN−1+βK−1+γM−1fs.Ifβ≤N and γ≤N+βK−1 are both integers, some row vectors in U will be exactly the same and there will be at most N+βK−1+γM−1 distinct row vectors in U. Thus, when N+βK−1+γM−1≤MNK, the clutter DOF, which is equated to the rank of U, is approximately
(8)r=rankU≈N+βK−1+γM−1.Equation (8) is the so-called estimation formula of clutter DOF for airborne MIMO radar provided by Chen et al. in [12], which can be viewed as the extension of Brennan’s rule to the MIMO radar case. According to (8), we can see that the clutter DOF increases when β goes up.


### 2.2. RD MIMO-STAP and Clutter LDOF

Generally, RD STAP algorithms apply data-independent transformations to prefilter the received signal and then solve an adaptive filtering operation with the dimension-reduced data. Given the transformation matrixT, the dimension-reduced clutter signal can be expressed as
(9)xr=THx=∑i=1NcαiTHSi=∑i=1NcαiSi,r=UrΛ,where Ur=S1,r,S2,r,…,SNc,r.For airborne MIMO radar, similar with the airborne SIMO radar, the transformation matrixT is commonly defined as
(10)T=TT⊗Tr⊗Tt,where TT∈CK×P is the time-domain transformation matrix and Tr∈CN×Q and Tt∈CM×O are the receiving and transmitting space-domain transformation matrices, respectively.Therefore, the dimension-reduced spatial-temporal steering vector and the dimension-reduced clutter covariance matrix can be written as(11)Si,r=THSi=TTHST,i⊗TrHSr,i⊗TtHSt,i,Rr=ExrxrH=THUΣUHT=UrΣUrH.Definition 1.
One hasrr=rankRr=rankUr as the clutter LDOF for airborne MIMO radar, which demonstrates the DOF of clutter in the dimension-reduced subspace. This is extended directly from the definition of LDOF of airborne SIMO radar proposed by [17].


## 2.1. Clutter Model and Clutter DOF

Considering a monostatic side-looking airborne MIMO array radar at the altitudeH with the constant moving velocity V. As is illustrated in Figure 1, there are N receiving elements uniformly spaced by dR=λ/2 and M transmitting elements uniformly spaced by dT=γdR, where λ is the wavelength and γ is the sparse coefficient. The number of pulses at a coherent processing interval (CPI) is K and the pulse repetition frequency (PRF) is fr. The MIMO radar transmits an orthogonal waveform set s1,s2,…,sM, and these waveforms will be extracted by M matched filters at each receiving element. The clutter is modeled as a Gaussian process.Figure 1
The model of monostatic side-looking airborne MIMO radar.
(a)
The geometry of carrier aircraft
(b)
The model of the array and the signalTheith clutter patch on the nth receiving element for the mth waveform and kth pulse can be written as
(1)yim,n,k=αismej2πγm−1+n−1fsi+j2πk−1fdi,where αi is the reflect coefficient of the ith clutter patch, fsi=dRcosψi/λ is the spatial frequency of the ith clutter patch, and fdi=2Vcosψi/λfr is the Doppler frequency of the ith clutter patch; cosψi=cosθicosφ is the directional cosine.Since the MIMO radar transmits orthogonal waveforms, the output of the matched filters can be expressed as(2)cim,n,k=αiej2πn−1fsi+j2πk−1fdi∑q=1MsmsqHej2πγm−1fsi=αiej2πn−1+βk−1+γm−1fsi,where β=fdi/fsi=2V/dRfr.Therefore, for themth waveform and kth pulse, the total clutter echoes on the nth receiving element can be expressed as
(3)xm,n,k=∑i=1Ncαiej2πn−1+βk−1+γm−1fsi,where Nc≫MNK is the number of clutter patches.Define the temporal steering vectorST,i, transmitting spatial steering vector St,i and receiving spatial steering vector Sr,i as
(4)ST,i=1,ej2πβfsi,…,ej2πβfsiK−1T∈CK×1,St,i=1,ej2πγfsi,…,ej2πγfsiM−1T∈CM×1,Sr,i=1,ej2πfsi,…,ej2πfsiN−1T∈CN×1.TheMNK×1 clutter signal vector, which is obtained by stacking the signals in (3) for n=1,2,…,N, m=1,2,…,M, and k=1,2,…,K, can be expressed as
(5)x=∑i=1NcαiSi=UΛ,where Si=ST,i⊗St,i⊗Sr,i is the spatial-temporal steering vector; U=S1,S2,…,SNc and Λ=α1,α2,…,αNcT.Therefore, the clutter covariance matrix is given as(6)Rx=ExxH=UΣUH,where Σ=diagα12,α22,…,αNc2; E· and ·H denote statistical expectation and the conjugate transpose, respectively.The clutter DOF is defined asr=rankRx. Because Σ is a positive definite matrix and spanRx=spanU, the clutter DOF is equal to r=rankU.Letfs=fs1,fs2,…,fsNcT; U can be rewritten as
(7)U=1,…,ej2πn−1+βk−1+γm−1fs,…,ej2πN−1+βK−1+γM−1fs.Ifβ≤N and γ≤N+βK−1 are both integers, some row vectors in U will be exactly the same and there will be at most N+βK−1+γM−1 distinct row vectors in U. Thus, when N+βK−1+γM−1≤MNK, the clutter DOF, which is equated to the rank of U, is approximately
(8)r=rankU≈N+βK−1+γM−1.Equation (8) is the so-called estimation formula of clutter DOF for airborne MIMO radar provided by Chen et al. in [12], which can be viewed as the extension of Brennan’s rule to the MIMO radar case. According to (8), we can see that the clutter DOF increases when β goes up.


## 2.2. RD MIMO-STAP and Clutter LDOF

Generally, RD STAP algorithms apply data-independent transformations to prefilter the received signal and then solve an adaptive filtering operation with the dimension-reduced data. Given the transformation matrixT, the dimension-reduced clutter signal can be expressed as
(9)xr=THx=∑i=1NcαiTHSi=∑i=1NcαiSi,r=UrΛ,where Ur=S1,r,S2,r,…,SNc,r.For airborne MIMO radar, similar with the airborne SIMO radar, the transformation matrixT is commonly defined as
(10)T=TT⊗Tr⊗Tt,where TT∈CK×P is the time-domain transformation matrix and Tr∈CN×Q and Tt∈CM×O are the receiving and transmitting space-domain transformation matrices, respectively.Therefore, the dimension-reduced spatial-temporal steering vector and the dimension-reduced clutter covariance matrix can be written as(11)Si,r=THSi=TTHST,i⊗TrHSr,i⊗TtHSt,i,Rr=ExrxrH=THUΣUHT=UrΣUrH.Definition 1.
One hasrr=rankRr=rankUr as the clutter LDOF for airborne MIMO radar, which demonstrates the DOF of clutter in the dimension-reduced subspace. This is extended directly from the definition of LDOF of airborne SIMO radar proposed by [17].


## 3. LDOF Equivalence of Different RD MIMO-STAP Algorithms

In this section, we will prove that clutter LODFs of different RD MIMO-STAP algorithms withTT, Tr, and Tt of the same size are equivalent. This can significantly reduce the complexity of the formulation and proof of the proposed LDOF estimation rule, which will be presented in Section 4. It should be noted that β and γ are simplified to be integers in our proofs, and when β and γ are not integers, the method using Fourier series and significant components proposed by Zhang et al. is also suitable for our situation. The readers are referred to [17, 18] for more detailed descriptions.


### 3.1. Clutter LDOF for Element-Space Pre-Doppler (ES Pre-Dop) RD MIMO-STAP

For ES pre-Dop RD MIMO-STAP, the dimension-reduced spatial-temporal steering vector has the following form:(12)Si,r=ST,ia:a+P−1⊗Sr,ib:b+Q−1⊗St,ic:c+O−1=ej2πfsiq−1+βp−1+γo−1,q=b:b+Q−1,p=a:a+P−1,o=c:c+O−1,where a, b, and c are constants for the selections of pulses, receiving elements, and transmitting elements.Thus,Ur can be expressed as
(13)Ur=ej2πfsa−1+βb−1+γc−1,…,ej2πfsa+Q−2+βb+P−2+γc+O−2T.Since the common coefficientej2πfsa−1+βb−1+γc−1 does not have any impact on the rank of Ur, we could set a=b=c=1, which means that the first P pulses, Q receiving elements, and O transmitting elements are selected for this class of RD MIMO-STAP algorithms. Therefore, according to the same reason of [7], the rank of Ur, that is, the clutter LDOF, is the number of distinct row vectors in Ur, which is equivalent to the number of distinct elements in the following set:
(14)q−1+βp−1+γo−1q=1:Q,p=1:P,o=1:O.


### 3.2. Clutter LDOF for Element-Space Post-Doppler (ES Post-Dop) RD MIMO-STAP

For ES post-Doppler RD MIMO-STAP, the time-domain transformation matrix can be expressed as(15)TT=TT1,…,TTp,…,TTP∈CK×P,where TTp=1,…,e−j2πK−1dp/KT, p=1,2,…,P, dp∈1,2,…,K, is the pth Doppler bin.Thus, the reduced-dimension temporal steering vector can be written as(16)ST,i,r=TTHST,i=ST,i,r1,…,ST,i,rp,…,ST,i,rPT,where ST,i,rp=∑k=0K−1ej2πkβfsi−dp/K=1−ej2πβfsiK/1−ej2πβfsi−dp/K.With the selection of the firstQ receiving elements and O transmitting elements, the transformed spatial-temporal steering vector can be written as
(17)Si,r=ST,i,r⊗Sr,i1:Q⊗St,i1:O=1−ej2πβfsiKej2πfsiq−1+γo−11−ej2πβfsi−dp/Kq=1:Q,q=1:P,o=1:O.Ignoring the common coefficient1−ej2πβfsiK, Ur can be simplified as
(18)Ur=11−ej2πβfs−d1/K,…,ej2πfsQ−1+γO−11−ej2πβfs−dP/KT.GivenΠ=∏p=1P1−ej2πβfs,1−dp/K,…,∏p=1P1−ej2πβfs,Nc−dp/KT, the rank of Ur is equivalent to that of the following matrix:
(19)Ur∗=Π⊙11−ej2πβfs−d1/K,…,Π⊙ej2πfsQ−1+γO−11−ej2πβfs−dP/KT,where ⊙ denotes Hardmard product.ThepQO−1+qO−1+oth row of Ur∗ has the form of
(20)ej2πfsq−1+γo−1⊙hp,0+hp,1ej2πβfs+⋯+hp,P−1ej2πβfsP−1T.We can see from (20) that every row of Ur∗ is the linear combination of monomials.
(21)expj2πfsq−1+βp−1+γo−1,q=1:Q,p=1:P,o=1:O.The proof in [18] indicates that the rank of Ur∗, that is, the clutter LDOF, is equivalent to the number of different expressions of (21). Therefore, it is actually the same as the clutter LDOF of the ES pre-Dop RD MIMO-STAP.


### 3.3. Clutter LDOF for Beam-Space Pre-Doppler (BS Pre-Dop) RD MIMO-STAP

In this case, the receiving and transmitting space-domain transformation matrices can be expressed as(22)Tr=Tr1,…,Trq,…,TrQ∈CN×Q,Tt=Tt1,…,Tto,…,TtO∈CM×O,where Trq=1,…,e−j2πN−1brq/NT, q=1,2,…,Q, brq∈1,2,…,Q, is the qth selected receiving beam; Tto=1,⋯,e−j2πM−1bt,o/MT, o=1,2,…,O, bto∈1,2,…,O, is the oth selected transmitting beam.So the dimension-reduced receiving and transmitting spatial steering vector can be expressed as(23)Sr,i,r=Sr,i,r1,…,Sr,i,rq,…,Sr,i,rQT,St,i,r=St,i,r1,…,St,i,ro,…,St,i,rOT,where Sr,i,rq=∑n=0N−1ej2πnfsi−brq/N=1−ej2πfsiN/1−ej2πfsi−brq/N and St,i,ro=∑m=0M−1ej2πmfsi−bto/M=1−ej2πfsiM/1−ej2πfsi−bto/M.Selecting the firstP pulses, the transformed spatial-temporal steering vector can be written as
(24)Si,r=ST1:P⊗Sr,i,r⊗St,i,r.With the similar derivation to the previous RD algorithm, the rank ofUr is equal to the number of the distinct expressions of
(25)ej2πβfsp−1⋅fq,0+⋯+fq,Q−1ej2πfsq−1⋅go,0+⋯+go,O−1ej2πγfso−1,where p=1,…,P, q=1,…,Q, and o=1,…,O.Equation (25) is also the linear combination of (21). Thus, the clutter LDOF of this algorithm is also equivalent to the two previous algorithms.


### 3.4. Clutter LDOF for Beam-Space Post-Doppler (BS Post-Dop) RD MIMO-STAP

In this case, the transformed spatial-temporal steering vector is(26)Si,r=ST,i,r⊗Sr,i,r⊗St,i,r.Similarly, every row of simplifiedUr has the form of
(27)hp,0+⋯+hp,P−1ej2πβfsP−1⋅fq,0+⋯+fq,Q−1ej2πfsq−1⋅go,0+⋯+go,O−1ej2πγfso−1,where p=1,…,P, q=1,…,Q, and o=1,…,O.Obviously, the rank ofUr, that is, the clutter LDOF, is the number of distinct expressions of (21).Therefore, the equivalence of LDOFs between different RD MIMO-STAP algorithms is proved.


## 3.1. Clutter LDOF for Element-Space Pre-Doppler (ES Pre-Dop) RD MIMO-STAP

For ES pre-Dop RD MIMO-STAP, the dimension-reduced spatial-temporal steering vector has the following form:(12)Si,r=ST,ia:a+P−1⊗Sr,ib:b+Q−1⊗St,ic:c+O−1=ej2πfsiq−1+βp−1+γo−1,q=b:b+Q−1,p=a:a+P−1,o=c:c+O−1,where a, b, and c are constants for the selections of pulses, receiving elements, and transmitting elements.Thus,Ur can be expressed as
(13)Ur=ej2πfsa−1+βb−1+γc−1,…,ej2πfsa+Q−2+βb+P−2+γc+O−2T.Since the common coefficientej2πfsa−1+βb−1+γc−1 does not have any impact on the rank of Ur, we could set a=b=c=1, which means that the first P pulses, Q receiving elements, and O transmitting elements are selected for this class of RD MIMO-STAP algorithms. Therefore, according to the same reason of [7], the rank of Ur, that is, the clutter LDOF, is the number of distinct row vectors in Ur, which is equivalent to the number of distinct elements in the following set:
(14)q−1+βp−1+γo−1q=1:Q,p=1:P,o=1:O.


## 3.2. Clutter LDOF for Element-Space Post-Doppler (ES Post-Dop) RD MIMO-STAP

For ES post-Doppler RD MIMO-STAP, the time-domain transformation matrix can be expressed as(15)TT=TT1,…,TTp,…,TTP∈CK×P,where TTp=1,…,e−j2πK−1dp/KT, p=1,2,…,P, dp∈1,2,…,K, is the pth Doppler bin.Thus, the reduced-dimension temporal steering vector can be written as(16)ST,i,r=TTHST,i=ST,i,r1,…,ST,i,rp,…,ST,i,rPT,where ST,i,rp=∑k=0K−1ej2πkβfsi−dp/K=1−ej2πβfsiK/1−ej2πβfsi−dp/K.With the selection of the firstQ receiving elements and O transmitting elements, the transformed spatial-temporal steering vector can be written as
(17)Si,r=ST,i,r⊗Sr,i1:Q⊗St,i1:O=1−ej2πβfsiKej2πfsiq−1+γo−11−ej2πβfsi−dp/Kq=1:Q,q=1:P,o=1:O.Ignoring the common coefficient1−ej2πβfsiK, Ur can be simplified as
(18)Ur=11−ej2πβfs−d1/K,…,ej2πfsQ−1+γO−11−ej2πβfs−dP/KT.GivenΠ=∏p=1P1−ej2πβfs,1−dp/K,…,∏p=1P1−ej2πβfs,Nc−dp/KT, the rank of Ur is equivalent to that of the following matrix:
(19)Ur∗=Π⊙11−ej2πβfs−d1/K,…,Π⊙ej2πfsQ−1+γO−11−ej2πβfs−dP/KT,where ⊙ denotes Hardmard product.ThepQO−1+qO−1+oth row of Ur∗ has the form of
(20)ej2πfsq−1+γo−1⊙hp,0+hp,1ej2πβfs+⋯+hp,P−1ej2πβfsP−1T.We can see from (20) that every row of Ur∗ is the linear combination of monomials.
(21)expj2πfsq−1+βp−1+γo−1,q=1:Q,p=1:P,o=1:O.The proof in [18] indicates that the rank of Ur∗, that is, the clutter LDOF, is equivalent to the number of different expressions of (21). Therefore, it is actually the same as the clutter LDOF of the ES pre-Dop RD MIMO-STAP.


## 3.3. Clutter LDOF for Beam-Space Pre-Doppler (BS Pre-Dop) RD MIMO-STAP

In this case, the receiving and transmitting space-domain transformation matrices can be expressed as(22)Tr=Tr1,…,Trq,…,TrQ∈CN×Q,Tt=Tt1,…,Tto,…,TtO∈CM×O,where Trq=1,…,e−j2πN−1brq/NT, q=1,2,…,Q, brq∈1,2,…,Q, is the qth selected receiving beam; Tto=1,⋯,e−j2πM−1bt,o/MT, o=1,2,…,O, bto∈1,2,…,O, is the oth selected transmitting beam.So the dimension-reduced receiving and transmitting spatial steering vector can be expressed as(23)Sr,i,r=Sr,i,r1,…,Sr,i,rq,…,Sr,i,rQT,St,i,r=St,i,r1,…,St,i,ro,…,St,i,rOT,where Sr,i,rq=∑n=0N−1ej2πnfsi−brq/N=1−ej2πfsiN/1−ej2πfsi−brq/N and St,i,ro=∑m=0M−1ej2πmfsi−bto/M=1−ej2πfsiM/1−ej2πfsi−bto/M.Selecting the firstP pulses, the transformed spatial-temporal steering vector can be written as
(24)Si,r=ST1:P⊗Sr,i,r⊗St,i,r.With the similar derivation to the previous RD algorithm, the rank ofUr is equal to the number of the distinct expressions of
(25)ej2πβfsp−1⋅fq,0+⋯+fq,Q−1ej2πfsq−1⋅go,0+⋯+go,O−1ej2πγfso−1,where p=1,…,P, q=1,…,Q, and o=1,…,O.Equation (25) is also the linear combination of (21). Thus, the clutter LDOF of this algorithm is also equivalent to the two previous algorithms.


## 3.4. Clutter LDOF for Beam-Space Post-Doppler (BS Post-Dop) RD MIMO-STAP

In this case, the transformed spatial-temporal steering vector is(26)Si,r=ST,i,r⊗Sr,i,r⊗St,i,r.Similarly, every row of simplifiedUr has the form of
(27)hp,0+⋯+hp,P−1ej2πβfsP−1⋅fq,0+⋯+fq,Q−1ej2πfsq−1⋅go,0+⋯+go,O−1ej2πγfso−1,where p=1,…,P, q=1,…,Q, and o=1,…,O.Obviously, the rank ofUr, that is, the clutter LDOF, is the number of distinct expressions of (21).Therefore, the equivalence of LDOFs between different RD MIMO-STAP algorithms is proved.


## 4. Estimation of Clutter LDOF under Different Conditions

In this section, we develop the clutter LDOF estimation rule for airborne MIMO radar with differentβ and γ. Due to the conclusion in Section 3, the generalized formula of clutter LDOF for one of the four reduced-dimension algorithms is only required to be given. In this section, we choose ES pre-Dop RD algorithms to show how to get the clutter LDOF. According to the analysis of Section 3.1, we notice that the clutter LDOF is equivalent to the number of distinct elements in the set Θ=q−1+βp−1+γo−1, where p=1,…,P, q=1,…,Q, and o=1,…,O. Therefore, what we only need to do is find out the number of distinct elements in Θ.


### 4.1. Clutter LDOF withβ≤Q

Whenβ≤Q, there are only Q+βP−1 different elements of set q−1+βp−1∣q=1:Q,p=1:P. Thus, Θ can be rewritten as
(28)Θ∗=l−1+γo−1l=1:Q+βP−1,o=1:O.Now, the influence ofγ should be taken into account. If γ≤Q+βP−1, with the same analysis, the number of different elements in Θ∗, that is, the clutter LDOF, is
(29)kr=Q+βP−1+γO−1.Ifγ>Q+βP−1, Θ∗ has no repeat element, and the clutter LDOF is thus given by
(30)kr=Q+βP−1O.


### 4.2. Clutter LDOF withβ>Q

In this subsection, four different situations for LDOF estimation are analyzed.(1)
Q<β≤Q+γO−1 and γ≤Q.Similar with the caseβ≤Q and γ≤Q+βP−1, the clutter LDOF is given as
(31)kr=Q+βP−1+γO−1.
(2)
Q<β≤Q+γO−1 and γ>Q+βP−1.In this case, the elements of setΘ are all different from each other, and the clutter LDOF is thus
(32)kr=OPQ.
(3)
β>Q+γO−1 and γ≤Q.Corresponding toβ≤Q and γ>Q+βP−1, the clutter LDOF is
(33)kr=Q+γO−1P.
(4)
β>Q+γO−1 and Q<γ≤Q+βP−1.Parallel with situation 2, the clutter LDOF is equal to(34)kr=OPQ.


### 4.3. LDOF Estimation Rule Discussions

Remark 1.
The generalized estimation rule of clutter LDOF withQ<β≤Q+γO−1 and Q<γ≤Q+βP−1 is not analyzed in this paper, which needs further study.Remark 2.
IfQ=N,P=K,andO=M, which means that the dimension of clutter has not been reduced, the proposed estimation rule can be viewed as the generalized form of the extended Brennan’s rule to the MIMO radar case [12], in which β≤N and γ≤N+βK−1 are considered to be sufficient.Remark 3.
In the case ofM=1, the MIMO radar will be degenerated to the SIMO radar case, and thus, our estimation rule is automatically reduced to the generalized form of clutter LDOF formulas proposed in [17], where the influence of β is not necessary to be analyzed.


## 4.1. Clutter LDOF withβ≤Q

Whenβ≤Q, there are only Q+βP−1 different elements of set q−1+βp−1∣q=1:Q,p=1:P. Thus, Θ can be rewritten as
(28)Θ∗=l−1+γo−1l=1:Q+βP−1,o=1:O.Now, the influence ofγ should be taken into account. If γ≤Q+βP−1, with the same analysis, the number of different elements in Θ∗, that is, the clutter LDOF, is
(29)kr=Q+βP−1+γO−1.Ifγ>Q+βP−1, Θ∗ has no repeat element, and the clutter LDOF is thus given by
(30)kr=Q+βP−1O.


## 4.2. Clutter LDOF withβ>Q

In this subsection, four different situations for LDOF estimation are analyzed.(1)
Q<β≤Q+γO−1 and γ≤Q.Similar with the caseβ≤Q and γ≤Q+βP−1, the clutter LDOF is given as
(31)kr=Q+βP−1+γO−1.
(2)
Q<β≤Q+γO−1 and γ>Q+βP−1.In this case, the elements of setΘ are all different from each other, and the clutter LDOF is thus
(32)kr=OPQ.
(3)
β>Q+γO−1 and γ≤Q.Corresponding toβ≤Q and γ>Q+βP−1, the clutter LDOF is
(33)kr=Q+γO−1P.
(4)
β>Q+γO−1 and Q<γ≤Q+βP−1.Parallel with situation 2, the clutter LDOF is equal to(34)kr=OPQ.


## 4.3. LDOF Estimation Rule Discussions

Remark 1.
The generalized estimation rule of clutter LDOF withQ<β≤Q+γO−1 and Q<γ≤Q+βP−1 is not analyzed in this paper, which needs further study.Remark 2.
IfQ=N,P=K,andO=M, which means that the dimension of clutter has not been reduced, the proposed estimation rule can be viewed as the generalized form of the extended Brennan’s rule to the MIMO radar case [12], in which β≤N and γ≤N+βK−1 are considered to be sufficient.Remark 3.
In the case ofM=1, the MIMO radar will be degenerated to the SIMO radar case, and thus, our estimation rule is automatically reduced to the generalized form of clutter LDOF formulas proposed in [17], where the influence of β is not necessary to be analyzed.


## 5. Simulation Results

In this section, simulations with different conditions are conducted to verify the effectiveness of the proposed clutter LDOF estimation rule and demonstrate its guidance to the design of RD STAP processor. First of all, we assume the following simulated parameters: side-looking airborne 8-element monostatic MIMO array radar, whose wavelength is 0.23 m, is located at the altitude 6000 m with constant velocity 140 m/s; the receiving elements are uniformly spaced by 0.115 m and the transmitting elements are uniformly spaced bydT=0.115⋅γ; CNR = 60 dB; and the number of pulses is 8.For element-space RD algorithms, the first 3 receiving elements and 3 transmitting elements are selected; for beam-space RD algorithms, received beams with indices 1, 3, and 5 and transmitted beams with indices 3, 5, and 7 are selected, which means thatQ=O=3. P=3 denotes that the first 3 pulses are selected for pre-Doppler algorithms, and Doppler bins with indices 1 to 3 are selected for post-Doppler algorithms. P=4 denotes that the first 4 pulses are selected for pre-Doppler algorithms, and Doppler bins with indices 1 to 4 are selected for post-Doppler algorithms. In all figures, the LDOFs estimated using the proposed rule is plotted by vertical solid black lines.


### 5.1. Effectiveness of the Proposed Estimation Rule

Firstly,P=3 is considered. The clutter eigenspectrums with β=1,γ=2 and β=1,γ=4 are shown in Figure 2(a), which is used to verify the estimated results derived from (29). As is illustrated in Figure 2(a), the eigenspectrums exhibit sharp cutoffs along the calculated ranks, which indicates that (27) can give an accurate estimation of clutter LDOF. Figure 2(b) shows clutter eigenspectrums with β=1,γ=6 and β=1,γ=9 to verify the accuracy of (30). One can see from Figure 2(b) that the clutter LDOF does not change with the increasing γ, and the estimated LDOF is exactly the same as the eigenvalue decomposition method.Figure 2
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (29) and (30).
(a)
Estimated results derived from (29)
(b)
Estimated results derived from (30)Secondly,P=4 is considered. The eigenspectrums with β=4,γ=1 and β=5,γ=1 are plotted in Figure 3(a), which is utilized to verify the effectiveness of (31). The eigenspectrums of clutter in Figure 3(b) are exploited to validate the accuracy of (33), where we set β=7,γ=1 and β=8,γ=1. From Figure 3(a), we can see that the numbers of eigenvalues are 17 and 20, which are exactly the same as the numbers calculated by (31). Then, we can observe from Figure 3(b) that the eigenspectrums display sharp cutoffs at the same position of estimated LDOF for different β, which indicates that the LDOF does not increase as β goes up, so the LDOF could be accurately estimated by (33).Figure 3
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (31) and (33).
(a)
Estimated results derived from (31)
(b)
Estimated results derived from (33)Finally, (32) with P=3 and (34) with P=4 are verified by Figures 4(a) and 4(b), respectively. In Figure 4(a), β=4,γ=12 and β=4,γ=13. In Figure 4(b), β=12,γ=4 and β=13,γ=4. We can observe that all eigenspectrums decrease gradually, which means that the clutter covariance matrix is nonsingular.Figure 4
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (32) and (34).
(a)
Estimated results derived from (32)
(b)
Estimated results derived from (34)


### 5.2. Guidance to the Dimension-Reduced STAP Processor Designing

To suppress clutter effectively, the DOF of RD STAP processors should be larger than that of the clutter. Thus, in this subsection, we demonstrate how to utilize the proposed LDOF estimation rule to guide the design of STAP processors. Without loss of generality, BS post-Dop RD algorithm is only considered. Assumingβ=4, γ=12, O=3, Q=3, and P=3, the clutter LDOF obtained from (32) is 27, which is equal to the DOF of system. In this case, the clutter will not be well suppressed. Therefore, with the instruction of (29), Q is changed to 5, then the clutter LDOF is 37, which is smaller than the DOF of the system. Furthermore, with the instruction of (30), Q and P are changed to 5 and 2, respectively, and the clutter LDOF is 27; it is also smaller than the DOF of the system. Thus, the clutter could be suppressed effectively. The above analysis can be confirmed by improvement factor (IF) against the target-normalized Doppler frequency presented in Figure 5, where IF that illustrates the clutter suppression performance is defined as the ratio of output signal-clutter-noise ratio (SCNR) to the input SCNR.Figure 5
Improvement factor against target Doppler frequency for different RD MIMO-STAP processors.


## 5.1. Effectiveness of the Proposed Estimation Rule

Firstly,P=3 is considered. The clutter eigenspectrums with β=1,γ=2 and β=1,γ=4 are shown in Figure 2(a), which is used to verify the estimated results derived from (29). As is illustrated in Figure 2(a), the eigenspectrums exhibit sharp cutoffs along the calculated ranks, which indicates that (27) can give an accurate estimation of clutter LDOF. Figure 2(b) shows clutter eigenspectrums with β=1,γ=6 and β=1,γ=9 to verify the accuracy of (30). One can see from Figure 2(b) that the clutter LDOF does not change with the increasing γ, and the estimated LDOF is exactly the same as the eigenvalue decomposition method.Figure 2
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (29) and (30).
(a)
Estimated results derived from (29)
(b)
Estimated results derived from (30)Secondly,P=4 is considered. The eigenspectrums with β=4,γ=1 and β=5,γ=1 are plotted in Figure 3(a), which is utilized to verify the effectiveness of (31). The eigenspectrums of clutter in Figure 3(b) are exploited to validate the accuracy of (33), where we set β=7,γ=1 and β=8,γ=1. From Figure 3(a), we can see that the numbers of eigenvalues are 17 and 20, which are exactly the same as the numbers calculated by (31). Then, we can observe from Figure 3(b) that the eigenspectrums display sharp cutoffs at the same position of estimated LDOF for different β, which indicates that the LDOF does not increase as β goes up, so the LDOF could be accurately estimated by (33).Figure 3
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (31) and (33).
(a)
Estimated results derived from (31)
(b)
Estimated results derived from (33)Finally, (32) with P=3 and (34) with P=4 are verified by Figures 4(a) and 4(b), respectively. In Figure 4(a), β=4,γ=12 and β=4,γ=13. In Figure 4(b), β=12,γ=4 and β=13,γ=4. We can observe that all eigenspectrums decrease gradually, which means that the clutter covariance matrix is nonsingular.Figure 4
Clutter eigenspectrums of RD MIMO-STAP algorithms to verify (32) and (34).
(a)
Estimated results derived from (32)
(b)
Estimated results derived from (34)


## 5.2. Guidance to the Dimension-Reduced STAP Processor Designing

To suppress clutter effectively, the DOF of RD STAP processors should be larger than that of the clutter. Thus, in this subsection, we demonstrate how to utilize the proposed LDOF estimation rule to guide the design of STAP processors. Without loss of generality, BS post-Dop RD algorithm is only considered. Assumingβ=4, γ=12, O=3, Q=3, and P=3, the clutter LDOF obtained from (32) is 27, which is equal to the DOF of system. In this case, the clutter will not be well suppressed. Therefore, with the instruction of (29), Q is changed to 5, then the clutter LDOF is 37, which is smaller than the DOF of the system. Furthermore, with the instruction of (30), Q and P are changed to 5 and 2, respectively, and the clutter LDOF is 27; it is also smaller than the DOF of the system. Thus, the clutter could be suppressed effectively. The above analysis can be confirmed by improvement factor (IF) against the target-normalized Doppler frequency presented in Figure 5, where IF that illustrates the clutter suppression performance is defined as the ratio of output signal-clutter-noise ratio (SCNR) to the input SCNR.Figure 5
Improvement factor against target Doppler frequency for different RD MIMO-STAP processors.


## 6. Conclusions

In this paper, a generalized rule to estimate the clutter LDOF is proposed for MIMO radar. Four reduced-dimension MIMO-STAP algorithms are introduced, and then, the LDOF equivalence of the four algorithms is proved. Finally, the proposed rule is considered with differentγ and β. The simulated results indicate the accuracy of the proposed rule. With the instruction of the proposed rule, the minimal DOF requirement could be determined and more appropriate system parameters can be chosen. Therefore, better clutter suppression performance of RD MIMO-STAP algorithms could be achieved. The estimation rule presented in this paper enjoys a good potential to guide the practical design of MIMO-STAP processors.



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*Source: 6026251-2018-03-18.xml*