Knowledge-Aided STAP Using Low Rank and Geometry Properties

This paper presents knowledge-aided space-time adaptive processing (KA-STAP) algorithms that exploit the low-rank dominant clutter and the array geometry properties (LRGP) for airborne radar applications. The core idea is to exploit the fact that the clutter subspace is only determined by the space-time steering vectors, {red}{where the Gram-Schmidt orthogonalization approach is employed to compute the clutter subspace. Specifically, for a side-looking uniformly spaced linear array, the} algorithm firstly selects a group of linearly independent space-time steering vectors using LRGP that can represent the clutter subspace. By performing the Gram-Schmidt orthogonalization procedure, the orthogonal bases of the clutter subspace are obtained, followed by two approaches to compute the STAP filter weights. To overcome the performance degradation caused by the non-ideal effects, a KA-STAP algorithm that combines the covariance matrix taper (CMT) is proposed. For practical applications, a reduced-dimension version of the proposed KA-STAP algorithm is also developed. The simulation results illustrate the effectiveness of our proposed algorithms, and show that the proposed algorithms converge rapidly and provide a SINR improvement over existing methods when using a very small number of snapshots.


I. INTRODUCTION
Space-time adaptive processing (STAP) is considered to be an efficient tool for detection of slow targets by airborne radar systems in strong clutter environments [1]- [4]. However, due to the very high degrees of freedom (DoFs), the full-rank STAP has a slow convergence and requires about twice the DoFs of the independent and identically distributed (IID) training snapshots to yield an average performance loss of roughly 3dB [1]. In real scenarios, it is hard to obtain so many IID training snapshots, especially in heterogeneous environments. Therefore, it is desirable to develop STAP techniques that can provide high performance in small training support situations.
Reduced-dimension and reduced-rank methods have been considered to counteract the slow convergence of the fullrank STAP [1]- [9], [26]- [28]. These methods can reduce the number of training snapshots to twice the reduced dimension, or twice the clutter rank blueif we assume that the degrees of freedom of the reduced dimension correspond to the rank of the clutter. The parametric adaptive matched filter (PAMF) based on a multichannel autoregressive model [29] provides another alternative solution to the slow convergence of the full-rank STAP. Furthermore, the sparsity of the received data and filter weights has been exploited to improve the convergence of a generalized sidelobe canceler architecture in [30]. However, it still fundamental for radar systems to improve the convergence performance of STAP algorithms or reduce their required sample support in heterogeneous environments because the number of required snapshots is large relative to those needed in IID scenarios.
Recently developed knowledge-aided (KA) STAP algorithms have received a growing interest and become a key concept for the next generation of adaptive radar systems [32]- [45]. The core idea of KA-STAP is to incorporate prior knowledge, provided by digital elevation maps, land cover databases, road maps, Global Positioning System (GPS), previous scanning data and other known features, to compute estimates of the clutter covariance matrix (CCM) with high accuracy [32], [33]. Among the previously developed KA-STAP algorithms, there is a class of approaches that exploit the prior knowledge of the clutter ridge to form the STAP filter weights in [42]- [44] and [45]. The authors in [42] introduced a knowledge-aided parametric covariance estimation (KAPE) scheme by blending both prior knowledge and data observations within a parameterized model to capture instantaneous characteristics of the cell under test (CUT). A modified sample matrix inversion (SMI) procedure to estimate the CCM using a least-squares (LS) approach has been described in [43] to overcome the range-dependent clutter non-stationarity in conformal array configurations. However, both approaches require the pseudo-inverse calculation to estimate the CCM and this often requires a computationally costly singular value decomposition (SVD) [42]. redAlthough two weighting approaches with lower computations are discussed in [42], they are suboptimal approaches to the LSE by the SVD and the performance of these approaches relative to the LSE by the SVD depends on bluethe radar system parameters, especially the array characteristics [42]. Moreover, the latter approach has not considered the situation when the prior knowledge has uncertainties. Under the assumption of the known clutter ridge in the angle-Doppler plane, the authors in [44] imposed the sparse regularization to estimate the clutter covariance excluding the clutter ridge. Although this kind of method can obtain a high-resolution even using only one snapshot, it requires the designer to know the exact positions of the clutter ridge resulting in being sensitive to the prior knowledge. Furthermore, the computational complexity caused by sparse recovery is expensive. A data independent STAP method based on prolate spheroidal wave functions (PSWF) has been considered in MIMO radar by incorporating the clutter ridge [45], where the computational complexity is significantly reduced compared with the approaches in [42] and [43]. However, it is highly dependent on the ideal clutter subspace and is not robust against clutter subspace mismatches.
In this paper, we propose KA-STAP algorithms using prior knowledge of the clutter ridge that avoid the pseudo-inverse calculation, require a low computational complexity, and mitigate the impact of uncertainties of the prior knowledge. redSpecifically, for a side-looking uniformly spaced linear array (ULA), the proposed method selects a group of linearly independent space-time steering vectors that can represent the ideal clutter subspace using prior knowledge of the dominant low-rank clutter and the array geometry properties (LRGP). The orthogonal bases of the clutter ideal subspace are computed by a Gram-Schmidt orthogonalization procedure on the selected space-time steering vectors. Two robust approaches to compute the STAP filter weights are then presented based on the estimated clutter subspace. To overcome the performance degradation caused by the redinternal clutter motion (ICM), we employ a covariance matrix taper (CMT) to the estimated CCM. The array calibration methods discussed in [42] can be applied to our proposed algorithm to mitigate the blueimpact of non-ideal factors, such as channel mismatching. Moreover, a reduced-dimension version of the proposed KA-STAP algorithm is devised for practical applications. Finally, simulation results demonstrate the effectiveness of our proposed algorithms.
The main contributions of our paper are: (i) A KA-STAP algorithm using blueprior knowledge of the LRGP is proposed for airborne radar applications.
(ii) A KA-STAP combining CMT is introduced to counteract the performance degradation caused by ICM redand prior knowledge uncertainty, and a reduced-dimension version is also presented for practical applications. redFurthermore, the proposed algorithm provides evidence for the KAPE approach to directly use the received data and the calibrated space-time steering vectors (only the spatial taper without the temporal taper) to compute the assumed clutter amplitude.
(iii) A detailed comparison is presented to show the computational complexity of the proposed and existing algorithms.
(iv) A study and comparative analysis of our proposed algorithms including the impact of inaccurate prior knowledge and non-ideal effects on the SINR performance, the convergence speed and the detection performance with other STAP algorithms is carried out.
The work is organized as follows. Section II introduces the signal model in airborne radar applications. Section III details the approach of the proposed KA-STAP algorithms and also discusses the computational complexity. The simulated airborne radar data are used to evaluate the performance of the proposed algorithms in Section IV. Section V provides the summary and conclusions.

II. SIGNAL MODEL
The system under consideration is a side-looking pulsed Doppler radar with a redULA consisting of M elements on the airborne radar platform, as shown in Fig.1. The platform is at altitude h p and moving with constant velocity v p . The radar transmits a coherent burst of pulses at a constant pulse repetition frequency (PRF) f r = 1/T r , where T r is the pulse repetition interval (PRI). The transmitter carrier frequency is f c = c/λ c , where c is the propagation velocity and λ c is the wavelength. The number of pulses in a coherent processing interval (CPI) is N . The received signal from the iso-range of interest is represented by a space-time N M × 1 data vector x.
redThe received space-time clutter plus noise return from a single range bin can be represented by [4] x = Na m=1 Nc n=1 σ m,n redv(f s,m,n , f d,m,n ) ⊙ α(m, n) + n, (1) where n is the Gaussian white thermal noise vector with the noise power σ 2 n on each channel and pulse; N a is the number of range ambiguities; N c is the number of independent clutter patches over the iso-range of interest; f s,m,n and f d,m,n are the redspatial and Doppler frequencies of the mnth clutter patch, respectively; redσ m,n is the complex amplitude for the mnth   The space-time steering vector is given as the Kronecker product of the temporal and spatial steering vectors, v(f s,m,n , f d,m,n ) = v t (f d,m,n ) ⊗ v s (f s,m,n ), which are given by [1] v t (f d,m,n ) = [1, · · · , exp(j2π v s (f s,m,n ) = [1, · · · , exp(j2π where () T denotes the transposition operation, f s,m,n = da λc cos θ m,n sin φ m,n , f d,m,n = 2vpTr λc cos θ m,n sin φ m,n , and d a is the inter-sensor spacing of the ULA. If we stack all clutter patches' amplitudes into a vector σ = [σ 1,1 , · · · , σ 1,Nc , · · · , σ Na,1 , · · · , σ Na,Nc ] T , redand assume there are no non-ideal factors, then the clutter plus noise received data denoted by (1) can be reddescribed as where V denotes the clutter space-time steering matrix, given by Thus, the CCM based on (5) can be expressed as where Σ = E[σσ H ]. Under the condition that the clutter patches are independent from each other, Σ = diag(a), a = [a 1,1 , a 1,2 , · · · , a Na,Nc ] T and a m,n = E[|σ m,n | 2 ] (m = 1, · · · , N a , n = 1, · · · , N c ) for the statistics of the clutter patches. Here, E[·] denotes the expectation operator, diag(a) stands for a diagonal matrix with the main diagonal taken from the elements of the vector a and () H represents the conjugate transpose of a matrix. The optimal filter weight vector on maximizing the output SINR for the Gaussian distribution clutter which is given by the full-rank STAP processor can be written as [4] w opt = µR −1 s, where µ is a constant which does not affect the SINR performance, s is the N M × 1 space-time steering vector in the target direction, and R = E[xx H ] = R c + σ 2 n I is the clutter plus noise covariance matrix (I is the identity matrix).

III. KA-STAP ALGORITHMS USING LRGP
In this section, we firstly review the method that estimates the CCM using a LS technique in [42], [43] and point out the existing problems of this method. Then, we detail the design and the computational complexity of the proposed KA-STAP algorithms using LRGP.

A. CCM estimated by LS
In practice, prior knowledge of certain characteristics of the radar system and the aerospace platform, such as platform heading, speed and altitude, array normal direction, and antenna phase steering, etc., can be obtained from the Inertial Navigation Unit (INU) and the GPS data [39], [42]. In other words, we can obtain the values of the number of range ambiguities N a , the platform velocity v p , and the elevation angle θ. Thus, we can develop KA-STAP algorithms based on redthese prior knowledge, e.g., the methods described in [42]- [44] and [45]. redIn reality, the clutter consists of returns over a continuous geographical region, which we divide into a discrete set of clutter patches for analytical and computational convenience. The rest of the discussion is on the issues associated with choosing the number of clutter patches N c . A possible approach is to assume a value of N c and discretize the whole azimuth angle evenly into N c patches for each range bin [42], [43]. redIn addition, it usually ignores range ambiguities, i.e., N a = 1, where the justification can be seen in [42]. Then, the parameter σ redin (5) can be estimated using the observation data by solving the LS problem as follows [42], [43],σ where redσ = [σ 1 ,σ 2 , · · · ,σ Nc ] T . Herein, the solution for the above problem based on an LS technique is given bŷ Because σ depends only on the clutter distribution, it does not vary significantly with the range under homogeneous clutter environments. Furthermore, to avoid the effect of the target signal at CUT, the near range bins of the CUT are used to estimate σ [43], which is given bŷ where 2L is the total number of the secondary data. Then, the estimated CCM by the LS method (we call it least-squares estimator (LSE) in the following) iŝ redThen the clutter plus noise covariance matrix is estimated asR whereσ 2 n is the estimated noise power level which can be collected by the radar receiver when the radar transmitter operates in a passive mode [2]. Finally, the STAP filter weights can be computed according to (8) using redR instead of R.
However, there are several aspects that should be noted. First, the above approach requires the designer to redchoose the suitable azimuth angle φ and the suitable number N c of the clutter patches, which are difficult to obtain in practice. The redselection of N c and φ will affect the space-time steering vectors of the clutter patches, which affects the estimation accuracy of the estimated CCM. Specifically, if the assumed number of clutter patches N c > N M , then V H V −1 does not exist. redSecond, the computational complexity of the which should be avoided in practice. redTwo weighting approaches with lower computations are discussed in [42]. blue-However, the solutions obtained by the weighting approaches are suboptimal approximations to the LSE obtained by the SVD. The performance of these approaches relative to the LSE computed by the SVD depends on the radar system parameters, especially the array characteristics [42]. blueIn the presence of non-ideal factors in the clutter component and despite the inclusion of the estimated angle-independent channel mismatch in the space-time steering vectors V and the use of the modified V to solve the problem (9), the techniques do not consider the impact of the temporal random taper α d . Nevertheless, the received data vector x is formed by all non-ideal factors. Thus, whether it is suitable to compute the parameter σ only considering the spatial random taper is worth being investigated, as will be discussed in Section III.C.

B. Proposed KA-STAP Algorithm
To overcome the rank-deficiency and the inverse of the matrix V H V, in the following, we will detail the proposed KA-STAP algorithm to estimate the CCM using prior knowledge of LRGP. redIn this subsection, we only consider the ideal case of the received data, i.e., the signal model in (5).
From (5), we know that the clutter return is a linear combination of returns from all clutter patches. Thus, we have Proof: The first equation can be obtained from (7). With regard to the second equation, let us denote the SVD of the matrix V by redV = UCD H . Then, we have Note that the orthogonal basis of the clutter subspace U can be calculated by V, or VV H , herein we will not need to compute that via the CCM. From (14), it also results that the clutter subspace is independent from the power of the clutter patches and is only determined by the clutter spacetime steering vectors. Moreover, from the above subsection, it is seen that the clutter space-time steering vectors can be obtained using the prior knowledge from the INU and GPS data. Therefore, it is easier to compute the orthogonal bases of the clutter subspace U by V, or VV H than that by the CCM due to the unknown power of the clutter patches. The other problem to calculate the clutter subspace arising is that one should know the clutter rank first. Fortunately, some rules for estimating the clutter rank was discussed in previous literature, such as [1], [2], [46] and [47]. Specially, for a sidelooking ULA, the estimated clutter rank is redapproximated by Brennan's rule as where β = 2v p T r /d a and the brackets ⌈⌉ indicate rounding to the rednearest largest integer. In [47], this rule has been extended to the case with arbitrary arrays. Usually, N r ≪ N M and the STAP algorithms can be performed in a low dimensional space so that the computational complexity and the convergence can be significantly improved [45]. After the clutter rank is determined, there are several approaches to compute the orthogonal bases of the clutter subspace. First, we can use the Lanczos algorithm [48] applied to VV H to compute the clutter subspace eigenvectors. The computational complexity of that using the Lanczos algorithm is on . Moreover, the computational complexity can be significantly reduced for the case of ULA and constant PRF by exploiting the Toepliz-block-Toeplitz structure of VV H [48].
Second, an alternative low-complexity approach is to perform the Gram-Schmidt orthogonalization procedure on the space-time steering vectors V, redwhere the implementation steps of the Gram-Schmidt orthogonalization are listed in Table I and interested readers are referred to [49] for further details. Note that this procedure is at the computational cost . It should be also noted that the approach of the Gram-Schmidt orthogonalization can be applied to arbitrary arrays if we can obtain the prior knowledge of the array geometry, some radar system parameters and some information of the platform.
redIn particular, for the case of side-looking ULA, we can further reduce the computational complexity to compute the clutter subspace eigenvectors. Since the dimension of the columns of V should satisfy N c ≫ N r , if we carry out the Gram-Schmidt orthogonalization procedure on the columns of V one by one, this will result in unnecessary computations due to the linear correlation among the columns. Thus, it is desirable to directly find a group of vectors that are linear independent or nearly linear independent (i.e., most of the vectors are linearly independent and only very few vectors are linearly correlated). redFortunately, for a ULA we have the following proposition.
Proposition 1: For the case of side-looking ULA and constant PRF, the clutter subspace belongs to the subspace computed by a group of space-time steering vectors {v p } Nr p=0 , which are given bȳ v p (n, m) = exp(j2πf s (βn + m)), where Proof: Let us stack the above space-time steering vectors into a N r × N M matrixṼ, which is shown as where Note thatṼ is a Vandermonde matrix of dimension N r ×N M . redFor z n,m , n = 0, · · · , N − 1 and m = 0, · · · , M − 1, the number of linearly independent columns ofṼ is determined by the number of distinct values of βn + m. If β is an integer, If β is a rational value (not an integer), the number of distinct values of βn + m is larger than N r = ⌈M + β(N − 1)⌉. Therefore,Ṽ has full rank, which is equal to [1] rank(Ṽ) = min(N r , N M ) = N r .
The dimension of the clutter subspace is also N r . Herein, the clutter subspace shares the same subspace withṼ. We can then compute the clutter subspace by taking the Gram-Schmidt orthogonalization procedure on the rows ofṼ. Moreover, it should be noted that the computational complexity of the second approach is on the order of redO which exhibits a much lower complexity compared with the LSE resulting in a very useful tool for practical applications. It also avoids the procedure to determine the values of the number of clutter patches N c and the azimuth angle φ.
After computing the orthogonal basis of the clutter subspace, we try to design the STAP filter weights by two different kinds of methods. One is to use the minimum norm eigencanceler (MNE) derived in [5] to form the filter weights. Specifically, the MNE method is a linearly constrained beamformer with a minimum norm weight vector appearing orthogonal to the clutter subspace, which is described by [5] min w w H w, subject to U H w = 0 and w H s = 1, The solution to the above optimization problem in (22) is provided by [5] The other method tries to design the filter weights using both the orthogonal bases of the computed clutter subspace and the observation data. Let us first calculate the root-eigenvalues by projecting the data on the clutter subspace U, formulated aŝ Then, the clutter plus noise covariance matrixR can be estimated byR whereΓ = diag(γ ⊙γ * ) and ⊙ denotes the Hadamard product. Finally, the STAP filter weights can be computed bŷ where we use the fact thatR −1 = U Γ −1 + 1 σ 2 n I U H . The whole procedure of the proposed KA-STAP algorithm is summarized in Table I.

C. Proposed KA-STAP Employing CMT
In practice, there are many non-ideal effects, such as the internal clutter motion (ICM) and bluethe channel mismatch [3], which result in mismatch between the actual clutter subspace and that computed by our proposed algorithm. In this case, the performance of our proposed algorithm will
redFor the angle-dependent channel mismatch under normal circumstances, the transmit and receive antenna patterns bluepoint in the same direction and have a significant maximum in the look direction. blueThe energy from the sidelobes is generally several orders of magnitude lower than that from the mainbeam. This will lead to clutter subspace leakage mainly coming from the main beam [3]. Thus, the angle-dependent channel mismatch can be approximated by spatial random tapers only related to the main beam. Since the main beam is usually fixed in a CPI, then this random tapers can be seen as angle-independent. For the angle-independent channel mismatch, we assume the spatial taper α s is a random vector but stable over a CPI due to the narrowband case considered in the paper. Herein, when in presence of channel mismatch, the clutter plus noise bluereceived data vector is given by [3] x = (V ⊙ Ξ s )σ + n, where bluethe columns of Ξ s are all equivalent to 1 N ⊗ α s and 1 N denotes the all 1 vector with dimension N . When considering ICM, the received data can be represented as [3] x where V s = V ⊙ Ξ s and α d is the temporal taper accounting for the ICM. Then, the clutter plus noise covariance matrix is where where T d denotes the space-time CMT accounting for the ICM and 1 M,M is the M × M all 1 matrix. In order to obtain the clutter plus noise covariance matrix, we should estimate R s and T d in (29). redRegarding the estimation of R s , we can firstly use the array calibration methods discussed in [42] to redestimate the spatial taper (denoted asα s ), which will not be discussed here due to space limitations. The reader is bluereferred to the literature [42] for further details. redThen, substitutingα s into V s , we obtain the blueestimateV s . On the other hand, since the elements of α d do not equate blueto zeros, we assumē If we multiply both side of (28) byᾱ d ⊗ 1 M and use the estimateV s instead of V s , then it becomes where n s = n ⊙ (α d ⊗ 1 M ). In this situation, similarly as the analysis in Section III.B, we can blueemploy the Gram-Schmidt orthogonalization procedure to compute bluea matrix with eigenvectors ofV s , which is denoted asÛ s . Then the root-eigenvalues γ s can be calculated bŷ blueWe can then estimate R s aŝ Here, it uses the fact that the amplitude of the temporal taper caused by the ICM is one. This fact can be seen the ICM models bluereported in [1], [3], which will be also detailed afterwards. From (35), we observe thatR s can be estimated using the received data x directly without α d . It also provides evidence for the KAPE approach to directly use the received data and the calibrated space-time steering vectors (only the spatial taper without the temporal taper) to compute the parameter σ.
redRegarding the estimation of T d , it can be obtained via a rough knowledge of the interference environment (e.g., forest versus desert, bandwidth, etc.) [6]. One common model, referred to as the Billingsley model, is suitable for a land scenario. The only parameters required to specify the clutter Doppler power spectrum are essentially the operating wavelength and wind speed. The operating wavelength is usually known, while the wind speed should be estimated. Another common model, presented by J. Ward in [1], is suitable for a water scenario. The temporal autocorrelation of the fluctuations is Gaussian in shape with the form: where σ v is the variance of the clutter spectral spread in m 2 /s 2 . In the following simulations, we consider the CMT model of the latter one. redAfter computing the estimatesR s andT d , we can compute the CCM asR c =R s ⊙T d . SinceR c is still of low rank, we adopt the Lanczos algorithm to compute the clutter subspace U, where the computational complexity is on the order of O((N M ) 2 N ′ r ) (N ′ r is the clutter rank ofR c ). Finally, the STAP filter weights are computed according to (23) or (26). The whole procedure can be seen in Table I. redPrior knowledge uncertainty impact. In the proposed algorithms, the prior knowledge uncertainty, such as velocity misalignment and yaw angle misalignment, will have a great impact on the performance. However, the scheme that employs the CMT will mitigate this impact. To illustrate this, we take a typical airborne radar system for example. The parameters of the radar system are listed at the beginning of Section IV. Consider a far field scenario, the elevation angle will be close to zero resulting in cos θ ≈ 1. Let v pu and φ u denote the velocity deviation and the yaw angle deviation, respectively. Then, for a discretized azimuth angle φ, the spatial frequency f s and Doppler frequency f d can be represented as From (37) and (38), we see that the prior knowledge uncertainty will affect the position and shape of the clutter ridge, which leads to the mismatch between the exact and the assumed space-time steering vectors. Fig.2 provides a more direct way to illustrate the impact of prior knowledge uncertainty to the clutter ridge in the spatio-temporal plane. By employing a CMT, the clutter spectra will become wider along the clutter ridge in the figure including the exact clutter ridge. From this point of view, the impact of prior knowledge uncertainty is mitigated. Because the methods in [43], [44] and [45] do not consider any strategies to mitigate the impact of prior knowledge uncertainty, the performance will depend highly on the accuracy of the prior knowledge. The KAPE approach in [42] also adopts the CMT and can mitigate the impact of prior knowledge uncertainty in a sense. But the differences between the proposed algorithm and the KAPE approach lie in three aspects. First, the KAPE approach estimates the CCM using the LS or some approximate approaches. While the proposed algorithm estimates the CCM using the Gram-Schmidt orthogonalization procedure (that is not an approximate approach) by exploiting that the clutter subspace is only determined by the space-time steering vectors. Furthermore, for a side-looking ULA radar, the proposed algorithm directly selects a group of linearly independent space-time steering vectors using the LRGP and then takes the Gram-Schmidt orthogonalization procedure to compute the clutter subspace.
Second, the proposed algorithm shows evidence bluethat is feasible to directly use the received data vector and the calibrated space-time steering vectors (only the spatial taper without the temporal taper) to compute the parameter σ. Third, the proposed algorithm with an RD version in the following section is presented to further reduce the complexity.

D. Proposed Reduced-Dimension (RD) KA-STAP Algorithms
From the above discussions, one aspect to be noted is that it is impractical to use all the DoFs available at the ULA for reasons of computational complexity when N M is too large. In such situations, a common approach is to break the full DoFs problem into a number of smaller problems via the application of an M N × D (with D ≪ M N ) transformation matrix S D to the data [1]. Our proposed KA-STAP algorithms can be easily extended to this kind of approach. By applying the reduced-dimension transformation matrix S D to the data and the space-time steering vectors, we obtain where¯denotes the results after the transformation. Then, the reduced-dimension CCMR c becomes In a manner similar to that of the proposed full-DoF KA-STAP algorithm described in Section III.B, we compute the orthogonal bases of the clutter subspaceŪ, estimate the CCM R c =ŪΓŪ H , and then calculate the STAP filter weights according to (23) or (26). When employing a CMT to the ideal clutter covariance matrix, the final RD clutter covariance matrix can be estimated aŝ redwhereÛ s is computed by taking the Gram-Schmidt orthogonalization procedure toV s = S H DV s ,Γ s is calculated via (35) usingx andÛ s instead of x andÛ s , andT d denotes the estimated RD CMT. Again, the STAP filter weights can be computed according to (23) or (26). By inspecting (40) and (41), we find that the computational complexity of our proposed RD-KA-STAP algorithm is related to D instead of N M (D ≪ N M ), which leads to great computation savings.
In this paper, we focus on the reduced-dimension technique known as extended factored (EFA) algorithm or multibin element-space post-Doppler STAP algorithm [1]. The simulations with this technique will show the performance of our proposed RD-KA-STAP algorithm.

E. Complexity Analysis
Here we illustrate the computational complexity of the proposed algorithms (shortened as LRGP KA-STAP and LRGP RD-KA-STAP) and other existing algorithms, namely, the sample matrix inversion algorithm (SMI), the EFA algorithm in [1], the joint-domain-localized (JDL) algorithm in [7], the CSMIECC algorithm in [43], and the KAPE algorithm in [42]. In Table II, D denotes the size of the reduced dimension. We can see that the computational complexity of our proposed algorithms is significantly lower than the CSMIECC and the KAPE algorithms red (N r ≪ N c , N M ), which require the pseudo-inverse of the matrix V H V. With regard to the SMI algorithm, our proposed algorithms also show a lower computational complexity because the number of snapshots used for training the filter weights of the SMI is in the order of 2N M .
Although the computational complexity of the EFA and JDL algorithms is lower than our proposed LRGP KA-STAP algorithm, two aspects should be noted. One is that the number of snapshots used for training filter weights is much larger than our proposed algorithms. The other is that the computational complexity of EFA and JDL is proportional to the number of Doppler frequencies of interest (we only list the computation complexity for one Doppler frequency). While our proposed algorithms only have to compute the CCM once for different Doppler frequencies of interest. Besides, the computational complexity of our proposed LRGP RD-KA-STAP is lower than the EFA since L in EFA is in the order of 2D, where D is usually larger than N r .

IV. PERFORMANCE ASSESSMENT
In this section, we assess the proposed KA-STAP algorithms by computing the output SINR performance and probability of detection performance using simulated radar data. The output SINR is defined by redThroughout the simulations, unless otherwise stated, the simulated scenarios use the following parameters: side-looking ULA, uniform transmit pattern, M = 8, N = 8, f c = 450MHz, f r = 300Hz, v p = 50m/s, redd a = λ c /2, β = 1, N r = ⌈M + β(N − 1)⌉ = 15, h p = 9000m, signal-tonoise ratio (SNR) of 0dB, the target located at 0 • azimuth with Doppler frequency 100Hz, clutter-to-noise ratio (CNR) of 50dB, and unitary thermal noise power. All presented results are averaged over 100 independent Monte Carlo runs.

A. Impact of ICM on the SINR Performance
In this subsection, we evaluate the impact on the SINR performance with different ICM for our proposed algorithms.
In the examples, we consider four different ICM cases with σ v = 0, σ v = 0.05, σ v = 0.1 and σ v = 0.5. The number of snapshots for training is 4. In Fig. 3(a), (b), (c) and (d), we show the SINR performance against the target Doppler frequency of our proposed LRGP KA-STAP algorithm both with and without a CMT. From the figures, we observe the following conclusions. (i) When there is non-ICM, the proposed LRGP KA-STAP algorithm without a CMT can obtain the optimum performance since the computed clutter subspace is exact. However, it degrades the SINR performance with the increase of σ v resulting in extra sensitivity to the ICM. That is because the computed clutter subspace can not represent the true clutter subspace. (ii) Our proposed LRGP KA-STAP algorithm with a CMT illustrates a robust characteristic to the ICM. When the estimated parameter σ v of CMT is correct, we can achieve the optimum SINR performance. Furthermore, it is demonstrated the range of values of CMT mismatch in which the estimated spreading exhibit acceptable SINR performance, which can be useful in applications. This can be interpreted as that the computed clutter subspace via the application of the CMT to the ideal clutter subspace, spans a similar space to the true clutter subspace.

B. Impact of Inaccurate Prior Knowledge on the SINR Performance
In this subsection, we focus on the impact of inaccurate prior knowledge on the SINR performance of our proposed algorithms. In the first example, we consider the impact of the velocity misalignment by showing the SINR performance against the target Doppler frequency, as shown in Fig.4. Consider three different cases: the velocity misalignments of prior knowledge are (a) 0.5m/s; (b) 1m/s; (c) 2m/s, compared with true platform velocity. The potential Doppler frequency space from −150 to 150Hz is examined and 4 snapshots are used to train the filter weights. The plots show that the proposed LRGP KA-STAP algorithm without a CMT is sensitive to the velocity misalignment, while the LRGP KA-STAP algorithm with a CMT is robust to that. The reason for this is that the velocity misalignment of prior knowledge will lead to the mismatch between the computed clutter subspace and the true clutter subspace. Although the computed clutter subspace via the CMT can not avoid this situation, it can mitigate this impact. Because the velocity misalignment between the clutter patches and the platform can be seen as the Doppler spreading of the clutter patches. Moreover, the results also show that a slightly larger value of the estimated parameter σ v will result in an improved SINR performance for the velocity misalignment case.
The evaluation of the impact caused by the yaw angle misalignment is shown in Fig.5, where we also consider three different cases: the yaw angle misalignments of prior knowledge are (a) 0.2 • ; (b) 0.5 • ; (c) 1 • . The curves also indicate that: (i) the proposed LRGP KA-STAP algorithm without a CMT is sensitive to the yaw angle misalignment, while the LRGP KA-STAP algorithm with a CMT is robust to that; (ii) a slightly larger value of the estimated parameter σ v will result in an improved SINR performance. The misalignment of the yaw angle will lead to a Doppler frequency mismatch between the radar platform and the clutter patches. While the CMT mainly aims at mitigating the performance degradation caused by the clutter Doppler spreading, the CMT will lead to an improved estimated clutter subspace and will exhibit robustness against the yaw angle misalignment.

C. Comparison With Conventional STAP Algorithms
To provide further investigation about the performance of our proposed algorithms, we compare the SINR performance versus the snapshots of our proposed LRGP KA-STAP and LRGP RD-KA-STAP algorithms with the Loaded SMI (LSMI), the EFA algorithm (3 Doppler bins), the 3 × 3 JDL algorithm, redStoica's scheme in [36] (the prior knowledge covariance matrix is computed in the same way as the CSMIECC bluealgorithm), and the CSMIECC algorithm (the combination parameter is set to 0.6) in [43], where the simulation results are shown in Fig. 6. Here, we consider a scenario of ICM with σ v = 0.5, and assume the diagonal loading factors for all algorithms are set to the level of the thermal noise power. The parameter σ v for our proposed algorithms is redassumed to 1. The curves in the figure illustrate that our proposed algorithms have a very fast SINR convergence speed which only needs three snapshots for training, and offer significant better SINR steady-state performance compared with the LSMI, EFA, JDL, redStoica's scheme and CSMIECC algorithms. This is because the proposed algorithms provide a much better estimation of the CCM by using prior knowledge of the data, the low clutter rank property, the geometry of the array and the interference environment. It should be noted that the SINR performance of the LRGP RD-KA-STAP algorithm is worse than that of LRGP KA-STAP with full-DOFs. This is due to the fact that the reduced DOFs will lead to lower computational complexity at the cost of performance degradation.
The results in Fig.7 illustrate the SINR performance versus the target Doppler frequency. The number of snapshots used for training in the LSMI, EFA, JDL, redStoica's scheme and CSMIECC algorithms is set to 48, while 4 in our proposed algorithms. It is found that our proposed LRGP KA-STAP algorithm provides the best SINR performance among all algorithms, and forms the narrowest clutter null resulting in improved performance for the detection of slow targets. It is also shown that the performance of the proposed LRGP RD-KA-STAP algorithm is worse than that of LRGP KA-STAP with full-DOFs, but better than other algorithms in most Doppler bins. Note that although the LRGP RD-KA-STAP algorithm performs slightly worse than other algorithms in Doppler range of −60 to 60Hz, it requires much smaller snapshots for training filter weights.
In the next example, as shown in Fig.8, we present the probability of detection performance versus the target SNR for all algorithms. The false alarm rate is set to 10 −3 and for simulation purposes the threshold and probability of detection estimates are based on 10, 000 samples. We suppose the target is injected in the the boresight with Doppler frequency 100Hz. We note that the proposed algorithms provide suboptimal  SINR performance versus the target Doppler frequency. The number of snapshots used for training in the LSMI, EFA, JDL and CSMIECC algorithms is set to 48, while we only use 4 snapshots for our proposed algorithms. detection performance using very short snapshots, but remarkably, obtain much higher detection rate than other algorithms at an SNR level from −8dB to 0dB.

V. CONCLUSIONS
In this paper, novel KA-STAP algorithms have been proposed by using prior knowledge of LRGP to obtain an accurate estimation of the CCM with a very small number of snapshots. By exploiting the fact that the clutter subspace is only determined by the space-time steering vectors, we redhave developed a Gram-Schmidt orthogonalization approach to compute the clutter subspace. In particular, for a sidelooking ULA, we have proposed a scheme to directly select a group of linearly independent space-time steering vectors to compute the orthogonal bases of the clutter subspace. Compared with the LSE algorithm, it has not only exhibited a low complexity, but also shown a simple way to compute the CCM. To overcome the performance degradation caused by the non-ideal effects redand the prior knowledge uncertainty, the proposed KA-STAP algorithm that combines the CMT has been presented and a reduced-dimension version has been devised for practical applications. blueThis has also provided evidence that is feasible to directly use the received data vector and the calibrated space-time steering vectors (only the spatial taper without the temporal taper) to compute the assumed clutter amplitude. The simulation results have shown that our proposed algorithms outperform other existing algorithms in terms of SINR steady-state performance, SINR convergence speed and detection performance for a very small number of snapshots, and also exhibit robustness against errors in prior knowledge.