Enhanced knowledge-aided space–time adaptive processing exploiting inaccurate prior knowledge of the array manifold

Digital Signal Processing 60 (2017) 262–276

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Digital Signal Processing www.elsevier.com/locate/dsp

Enhanced knowledge-aided space–time adaptive processing exploiting inaccurate prior knowledge of the array manifold Zhaocheng Yang a,∗ , Rodrigo C. de Lamare b a b

College of Information Engineering, Shenzhen University, Shenzhen, Guangdong, 518060, China Department of Electronics, University of York, YO10 5DD, York, UK

a r t i c l e

i n f o

Article history: Available online 11 October 2016 Keywords: Space–time adaptive processing Knowledge-aided Array manifold Inaccurate prior knowledge Clutter suppression

a b s t r a c t The accuracy of the prior knowledge of the clutter environments is critical to the clutter suppression performance of knowledge-aided space–time adaptive processing (KA-STAP) algorithms in airborne radar applications. In this paper, we propose an enhanced KA-STAP algorithm to estimate the clutter covariance matrix considering inaccurate prior knowledge of the array manifold for airborne radar systems. The core idea of this algorithm is to incorporate prior knowledge about the range of the measured platform velocity and the crab angle, and other radar parameters into the assumed clutter model to obtain increased robustness against inaccuracies of the data. It first over-samples the space–time subspace using prior knowledge about the range values of parameters and the inaccurate array manifold. By selecting the important clutter space–time steering vectors from the over-sampled candidates and computing the corresponding eigenvectors and eigenvalues of the assumed clutter model, we can obtain a more accurate clutter covariance matrix estimate than directly using the prior knowledge of the array manifold. Some extensions of the proposed algorithm with existing techniques are presented and a complexity analysis is conducted. Simulation results illustrate that the proposed algorithms can obtain good clutter suppression performance, even using just one snapshot, and outperform existing KA-STAP algorithms in presence of the errors in the prior knowledge of the array manifold. © 2016 Elsevier Inc. All rights reserved.

1. 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 large space–time degrees of freedom (DoFs), the full-rank STAP has a slow convergence and requires about twice of the DoFs of the independent and identically distributed (IID) training snapshots to yield an average performance loss of roughly 3 dB [1]. In real scenarios, it is hard to obtain so many IID training snapshots, especially in heterogeneous environments. Therefore, STAP techniques providing high performance in small training support situations have always been a hot topic in this area. Reduced-dimension and reduced-rank methods have been considered to counteract the slow convergence of the full-rank STAP, such as the extended factored or the multibin element-space postDoppler STAP method [2,3], the principal-components methods [5], the joint-domain localized (JDL) approach [6], the cross-spectral

*

Corresponding author. E-mail addresses: [email protected] (Z. Yang), [email protected] (R.C. de Lamare). http://dx.doi.org/10.1016/j.dsp.2016.10.005 1051-2004/© 2016 Elsevier Inc. All rights reserved.

metric method [7], the multistage Winer filter method [8], the joint iterative optimization of adaptive filters [9] and the joint interpolation, decimation and filtering algorithms [10], etc. These methods can reduce the number of training snapshots to twice of the reduced-dimension, or twice of the clutter rank. The parametric adaptive matched filter (PAMF) based on a multichannel autoregressive model [11] provides another alternative solution to the slow convergence of the full-rank STAP. By exploiting the fact that space–time beamformers do not need all their DoFs to mitigate interference signals, sparse space–time beamformers are designed to improve the convergence for a generalized sidelobe canceler processor in [12] and a direct filter processor [13]. However, there is still need to improve the convergence or reduce the sample support when employing these approaches because the number of required snapshots is large relative to the non-stationarity assumption. Recently developed knowledge-aided STAP (KA-STAP) incorporates 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 the high-fidelity estimates of the clutter covariance matrix and exhibits good performance in heterogeneous environments [14–22,28,29, 23–26]. Among the developed KA-STAP algorithms, they can be

Z. Yang, R.C. de Lamare / Digital Signal Processing 60 (2017) 262–276

categorized into two cases [14]: intelligent training and filter selection; and Bayesian filtering and data prewhitening. In this paper, we focus on the latter case. Colored loading implements the STAP filter in two steps: a prewhitening step using the prior matrix, followed by adaptive filtering [16,17]. Fully automatic methods for combining the matrix with prior knowledge and secondary data are considered in [18,19]. Furthermore, in [20], an automatic combination of the inverse of the matrix with prior knowledge and the inverse of the estimated covariance matrix from secondary data is developed. The authors in [21] 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 low computation complexity approach that is similar to the KAPE is developed in [22]. A modified sample matrix inversion (SMI) through clutter covariance matrix estimated from least squares (LS) to overcome the range-dependent clutter nonstationarity in conformal array configurations is described in [23]. The convergence and the performance of KA-STAP algorithms are analyzed in [25] and [26]. In recently years, to reduce the requirements of the secondary data or the accurate prior knowledge of the clutter statistics, sparsity-based STAP algorithms are developed to compute the clutter covariance matrix by exploiting the sparsity of the clutter in the whole angle-Doppler plane [27–34]. These algorithms discretize the whole angle-Doppler into a large number of grid points and reconstruct the angle-Doppler profile or image using the sparse recovery methods. The corresponding recovered signal’s dimension is very large (16 to 50 timing of the system DoFs), which leads to a high computational complexity. Regardless of any KA-STAP method, the accuracy of prior knowledge will have a great impact on the STAP performance. In this paper, we focus on the mitigation of the impact of inaccurate prior knowledge and the development of a robust enhanced KA-STAP algorithm to estimate the clutter covariance matrix. The proposed method can be divided into three steps. First, it generates the candidates of the clutter space–time steering vectors using prior knowledge of the range of the measured platform velocity and the crab angle, and other radar parameters, unlike the approach in [21–24] which directly uses the array manifold. Then, the proposed method selects the important clutter space–time steering vectors and computes the corresponding eigenvectors and eigenvalues of the clutter subspace from the formulated candidates. Third, it estimates the clutter covariance matrix and computes the STAP filter. Furthermore, we detail several issues about using inaccurate prior knowledge of array manifold and also discuss some extensions of the proposed algorithm with the existing techniques. Compared with sparsity-based STAP, the proposed algorithm only over-samples the potential clutter Doppler and spatial frequencies using some prior knowledge, which significantly reduces the recovered signal’s dimension. Additionally, the proposed algorithm can avoid the target signal cancelations because no desired target component corrupts the assumed model, which is also pointed out by [21]. Finally, simulation results demonstrate the effectiveness of our proposed algorithm. The remaining paper is organized as follows. In Section 2, we introduce the signal model and the review of existing STAP using prior knowledge of array manifold. Then, in Section 3, we detail the proposed enhanced KA-STAP algorithm, discuss some extensions of the proposed algorithm with the existing techniques and also illustrate the implementations and complexity analysis. Simulated airborne radar data are used to evaluate the performance of the proposed algorithm in Section 4. Section 5 provides the summary and conclusions.

263

Fig. 1. Airborne radar geometry with a ULA antenna.

2. Background 2.1. Signal model and problem formulation The system under consideration is a pulsed Doppler radar with a uniformly spaced linear array antenna (ULA) consisting of M elements on the airborne radar platform, as shown in Fig. 1. The platform is at an altitude h p and moving with constant velocity v p . The angle ψ refers to the crab angle between the platform velocity and the array. For a special scatterer point P (which can be the target or the discretized clutter patch) located at range r, the angle variables φ and θ refer to its azimuth and elevation. 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. The transmitter carrier frequency is f c = c /λc , where c is the propagation velocity and λc is the wavelength. The coherent processing interval (CPI) length is equal to N T r . The received signal from an iso-range gate of interest can be presented by a space–time N M × 1 data vector x (also called a snapshot vector). A general model for the space–time clutter plus noise snapshot at the iso-range gate is given by [1]

xk =

Nc Na  

σi,k v(φi,k , θi,k , f i,k ) + n,

(1)

i =1 k =1

where n is the Gaussian white thermal noise vector, with the noise power σn2 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 gate; φi ,k and θi ,k is the azimuth and elevation to the ikth clutter patch; f i ,k is the corresponding Doppler frequency; σi,k is the complex amplitude for the ikth clutter patch with each element proportional to the square-root of the patch clutter–noise ratio (CNR); and v(φi ,k , θi ,k , f i ,k ) is the N M × 1 space–time steering vector for the clutter patch with azimuth φi ,k , elevation θi ,k and Doppler frequency f i ,k . The space–time steering vector is given as the Kronecker product of the temporal and spatial steering vectors, denoted as v(φi ,k , θi ,k , f i ,k ) = vt ( f i ,k ) ⊗ vs (φi ,k , θi ,k ). For the ikth clutter patch, the corresponding temporal and spatial steering vectors are given by [1]

vt ( f i ,k ) = [1, exp( j2π f i ,k T r ), · · · , exp( j2π ( N − 1) f i ,k T r )] T , (2) vs (φi ,k , θi ,k ) j2π d

cos θi ,k sin φi ,k ), · · · , λc j2π ( M − 1)d exp( cos θi ,k sin φi ,k )] T , λc

= [1, exp(

(3)

where [·] T denotes the transposition operation, d is the intersensor spacing of the ULA, and the Doppler frequency f i ,k is

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f i ,k =

2v p

cos θi ,k sin(φi ,k + ψ).

λc

(4)

By stacking all clutter patches’ amplitudes into a vector σ = [σ1,1 , · · · , σ1, N c , σ2,1 , · · · , σ Na , N c ] T , we can rewrite the clutter plus noise snapshot (1) as

x = Vσ + n,

(5)

where V denotes the clutter space–time steering matrix, given by

of interest [21–24]. Ignoring range ambiguities,1 the array mani¯ for the assumed clutter fold or the space–time steering matrix V patches at a special range gate can be computed as [21–24]

¯ = [v(φ1 , θ1 , f 1 ), · · · , v(φ ¯ , θ ¯ , f ¯ )]. V Nc Nc Nc

(10)

Approaches in [21–23] consider an approximation to (5) as given by

¯ σ¯ , xc ≈ V

(11)

where σ¯ = [σ¯ 1 , · · · , σ¯ N¯ c ] denotes the assumed clutter patches’ amplitude vector, and try to estimate σ¯ by solving the following LS problem T

V =[v(φ1,1 , θ1,1 , f 1,1 ), · · · , v(φ1, N c , θ1, N c , f 1, N c ), · · · ,

(6)

v(φ Na ,1 , θ Na ,1 , f Na ,1 ), · · · ,

Under the assumption that returns from different clutter patches are uncorrelated, the clutter covariance matrix based on (5) can be expressed as H

Rc = E [xx ] = V V ,

(7)

where  = diag(P) and P i ,k = E [|σi ,k | ] representing the clutter power distribution. 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 given by the full-rank STAP, on maximizing the output signal-to-interference-plus-noise ratio (SINR) for the Gaussian distribution clutter can be written as [1] 2

wopt = μR−1 s,

(8)

where μ is a constant which does not affect the SINR performance, s denotes the N M × 1 space–time steering vector in the target direction, and R = E [xx H ] = Rc + σn2 I is the clutter plus noise covariance matrix (I is the identity matrix). However, in practice, R is unknown and has to be estimated from the received snapshots. One of the most common ways to compute the clutter plus noise covariance matrix estimate based on the maximum likelihood (ML) criterion in Gaussian clutter assumption is given by [4]

ˆ ML = R

1 L

L

2

σ¯

v(φ Na , N c , θ Na , N c , f Na , N c )].

H



σ¯ = arg min x − V¯ σ¯ 2 ,

xl xlH ,

(9)

l =1

where xl , l = 1, · · · , L are known as secondary or training snapshots and usually come from the adjacent range bins of the CUT. However, this approach requires a large number of training snapshots that are target-free and satisfy the IID conditions with the snapshot in the CUT. 2.2. Review of STAP using prior knowledge of array manifold In practice, some 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 GPS data [21]. In other words, one can know the values of the platform velocity v p , the crab angle ψ , and the elevation angle θ . One class of recently developed KA-STAP algorithms is based on this type of prior knowledge, e.g., the approaches presented in [21–24]. These approaches usually assume a predefined ¯ c and discretize the value of the number of the clutter patches N ¯ c patches for each range bin whole azimuth angle evenly into N

(12)

where  · 2 denotes the l2 -norm. An LS estimate (LSE) of the solution for the above minimal optimization problem is given by



σ¯ = V¯ H V¯

−1

¯ H x. V

(13)

The clutter covariance matrix can be estimated by

¯ ˆ c = Vdiag R (P¯ )V¯ H ,

(14)

where P¯ = [ P¯ 1 , · · · , P¯ N¯ c ] T = [|σ¯ 1 |2 , · · · , |σ¯ N¯ c |2 ] T represents the instantaneous power estimates for the assumed clutter patches. To overcome the drawbacks of errors in prior knowledge and impacts of intrinsic clutter motion (ICM), the KAPE approach provides a filter bank strategy to select the Doppler shift to determine the center of the mainbeam clutter and a subsequent filter bank approach implementing a sequence of covariance matrix tapers (CMTs) to estimate the clutter spectral width [21]. This procedure is complicated and computational expensive. To avoid inversion in (13), a lower computational complexity algorithm is developed in [22]. To avoid the effect of target signal at CUT, adjacent range bins of the CUT are used to estimate σ¯ [23], which is given by

P¯ k =

1 L

L

|σ¯ l;k |2 ,

(15)

l =1

where σ¯ l;k is the estimated kth clutter patch’s amplitude for the lth snapshot. The approach in [23], i.e., the corrected sample coˆ ML with the above estimated clutter covariance variance matrix R ˆ c (namely CSMIECC), does not consider the errors in prior matrix R knowledge and directly computes the clutter plus noise covariance matrix as

ˆ = Rˆ ML + ρ Rˆ c , R

(16)

where ρ is the correcting factor. Finally, the STAP filter weights can ˆ instead of R. be computed according to (8) using R The approach in [24] approximates the clutter subspace by the ¯ shown as array manifold described by V,

  R {Uc } ≈ R V¯ ,

(17)

where R {A} denotes the range space of the matrix A. The final STAP filter weights are calculated by2



ˆ = Uc⊥ Uc⊥ w

H

s,

(18)

where Uc⊥ denotes the orthogonal subspace of Uc . Note that this approach also does not consider the errors in prior knowledge.

1 It is justified for several reasons in [21] to show why the range ambiguities can be ignored. 2 Here, we mainly focus on the clutter suppression and do not consider the jammers in this paper.

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265

3. Proposed enhanced KA-STAP using inaccurate prior knowledge of array manifold In this section, we detail the design of the proposed enhanced KA-STAP algorithm using inaccurate prior knowledge of the array manifold (shortened as InAME-KA-STAP), discuss several extensions of the proposed InAME-KA-STAP with other existing schemes, and also illustrate the implementations and complexity analysis of the proposed algorithm. 3.1. Prior clutter covariance matrix estimate By inspecting (2), (3) and (10), we observe that the accuracy of prior knowledge of the platform velocity v p , the crab angle ψ , the azimuth angle φ and the elevation angle θ decides the bias between the assumed and the true clutter space–time steering vectors. In the following, we allow some errors in the above prior knowledge and present a new approach to estimate the clutter covariance matrix. In the proposed approach, we only have to roughly know a range of possible values that constitute the prior knowledge, which represents a relaxed condition on the prior knowledge. Then we use this range of values which constitute the prior knowledge to form the space–time steering vectors. By incorporating the received data, we first select the most important space–time steering vectors for the clutter and then compute the eigenvectors and eigenvalues of the clutter subspace using the selected space–time steering vectors. In the following, we will detail the proposed prior clutter covariance matrix estimate approach into three steps. Step 1: use a given range of values as prior knowledge to form the clutter space–time steering vectors. For the elevation angle θ and the number of range ambiguities N a , they are functions of the platform altitude and the range to the clutter patches. Considering the iso-range gate at range r from the radar and assuming a spherical earth model with a 4/3 effective radius, we have the elevation angle θ to this iso-range ring as [1]



θ = − arcsin

r 2 + h p (h p + 2ae ) 2r (ae + h p )



(19)

,

where ae = 4/3re is the effective earth radius and re = 6371 km. The number of range ambiguities is given by



Na =

1 for rh ≤ r u , arg maxi {r + (i − 1)r u < rh } for rh > r u

2ae h p + h2p .

(21)

From (19), we note that small errors in the measured platform height will lead to negligible changes in the elevation angle θ . To have an intuitive sense to understand this point, we take two typical cases as examples: typical radar I, a UHF-band radar with f r = 300 Hz and h p = 9 km; and typical radar II, an L-band radar with f r = 1984 Hz and h p = 3.488 km. Because the space–time steering vectors are related to the cosine values of the elevation angles, we plot the mean square error (MSE) of cos θ versus the iso-range ring’s range r in Fig. 2, where we consider the measured platform height error follows uniform distribution within [−50, 50] m corresponding to a large error in the measured platform height. The MSE estimate is computed by using the Monte Carlo simulations, given as



MSE = 10 lg

 |cos θ − cos θtrue |2 N mc



,

Table 1 Parameters for typical radar systems. Parameter

Value

Typical radar I: UHF-band Antenna array spacing Carrier frequency Transmit pattern PRF Platform velocity Platform height Range resolution Antenna elements number Pulse number in one CPI

Half wavelength 450 MHz Uniform 300 Hz 50 m/s 9000 m 75 m 10 10

Typical radar II: L-band Antenna array spacing Carrier frequency Transmit pattern PRF Platform velocity Platform height Range resolution Antenna elements number Pulse number in one CPI

Half wavelength 1240 MHz Uniform 1984 Hz 100 m/s 3488 m 120 m 10 10

(20)

where r u = c /(2 f r ) denotes the radar’s unambiguous range, and rh denotes the radar horizon range, given by

rh =

Fig. 2. The MSE of cos θ versus the iso-range ring’s range r.

(22)

where N mc is the number of Monte Carlo simulations. From the figure, we observe that if the iso-range ring’s range is larger than 9.5 km (for typical radar I, h p = 9 km) and 4.2 km (for typical radar II, h p = 3.488 km), the MSE is lower than −40 dB. That is to say, we can ignore the errors in the platform height to the elevation angle in nearly the whole observation range. For the azimuth angle φ and the number of discretized clutter patches N c , we discuss the settings of them with a large number of simulations in the above two typical radar systems. The radar parameters for these two typical radar systems are shown in Table 1. We assume the unitary thermal noise power σn2 = 1 for a single channel and single pulse, and single channel, single pulse clutter-to-noise (CNR) ratio is 40 dB, defined as

CNR =

tr (Rc )

σn2 N M

,

(23)

where tr(·) denotes the trace operation of a matrix. For the actual clutter model (1), we assume the number of clutter patches in an iso-range ring is N c = 361, the azimuth angle for the ikth clutter −1− N c patch is φi ,k = π 2k and the crab angle is ψ = 0◦ . It is known 2( N c −1) that, for a ULA with M antenna elements, the spatial frequency

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Z. Yang, R.C. de Lamare / Digital Signal Processing 60 (2017) 262–276

Fig. 3. The covariance error against the range r with different settings of N c . (a) Typical Radar system I and (b) Typical Radar system II.

resolution is 1/ M. We consider five different cases of N c in the assumed clutter model: case 1, N c = M; case 2, N c = 2M; case 3, N c = 3M; case 4, N c = 4M; and case 5, N c = 5M. The clutter covariance matrix using the assumed model is computed according to (14) with the actual power of the clutter patches. To characterize the impact of the performance when using the assumed clutter model, we assess the covariance error of the inversion between the covariance matrix of the actual clutter model and the one of the assumed clutter model, namely3

 − 1  − 1    2  , ˆ c + σ 2I = R − R + σ I c  

(24)

F

where  ·  F denotes the l F -norm of a matrix. Figs. 3(a) and (b) plot the covariance error against the range r with different settings of N c for the above described typical radar systems. From the figures, we observe that when N c ≥ 4M, the covariance error becomes very small and further enlarging N c leads to a small improvement of the assumed clutter model compared with the actual clutter model. This conclusion coincides with that in the [2]: it is a good rule to set the angular spacing to about twenty to thirty percent of the diffraction-limited resolution. For the platform velocity v p and the crab angle ψ , on one hand, the typical measured accuracies using the INU and GPS are about 0.1–0.2 m/s for v p and around 1–1.5◦ /s for ψ . On the other hand, it is hard to keep the platform velocity and crab angle constants during a period because of the aircraft controls. If we form the assumed space–time steering vectors for every platform velocity and crab angle for each CPI,4 it will lead to a high computational complexity and impractical of the proposed algorithm. However, if we can allow the formed space–time steering vectors suitable for a set of platform velocities and crab angles (even different CPIs), it will lead to a significant reduction of the computational complexity. This can be realized by knowing beforehand an appropriate range of values of those parameters. Specifically, we assume the measured velocity of the radar platform v p is among the range [ v p − v pm , v p + v pm ], and the measured crab angle ψ  belongs

to the range [ψ − ψm , ψ + ψm ], where v p and ψ are the true platform velocity and the true crab angle, v pm and ψm denote the maximum measured errors of the radar platform velocity and the crab angle, respectively. For a radar system equipped with a ULA, we note that the spatial frequency ϑi ,k with the azimuth angle φi ,k , the elevation angle θi ,k and at the ith range ambiguity ring is given by

ϑi ,k =

d

λc

Here, we use the error of the inversion of the covariance matrix instead of itself. This is because the STAP filter weight vector is also a function of the inversion of the covariance matrix, e.g., (8). 4 In the STAP using prior knowledge of array manifold, we compute the assumed clutter model under the assumptions that the platform velocity and crab angle are constant during the CPI.

(25)

From the above discussions, we know that antenna inner-spacing d, the operating wavelength λc and the azimuth angle are deterministic, and the measured error of the elevation angle can be ignored. Therefore, we can decide the spatial frequency with a high accuracy. Additionally, from (4), we note the Doppler frequency is correlated with the velocity of the radar platform v p and the crab angle ψ . For prior knowledge of the measured velocity of the radar platform v p and the measured crab angle ψ  , the Doppler frequency is calculated as

f i,k =

2v p

λc

cos θi ,k sin(φi ,k + ψ  ),

(26)

where f i,k denotes the Doppler frequency using the inaccurate prior knowledge. Thus, the difference between f i,k and the actual Doppler frequency f i ,k is

f i ,k = f i,k − f i ,k = −

2v p

λc

cos θi ,k sin(φi ,k + ψ  )

2v p

cos θi ,k sin(φi ,k + ψ) λc 2( v p + v ) = cos θi ,k sin(φi ,k + ψ + ψ ) λc

− =

3

cos θi ,k sin φi ,k .

2v p

λc

2v p

cos θi ,k sin(φi ,k + ψ)



cos θi ,k sin(φi ,k + ψ) cos ψ

λc  + cos(φi ,k + ψ) sin ψ − sin(φi ,k + ψ)  2 v cos θi ,k sin(φi ,k + ψ) cos ψ + λc  + cos(φi ,k + ψ) sin ψ

Z. Yang, R.C. de Lamare / Digital Signal Processing 60 (2017) 262–276

≈

2v p

λc





cos θi ,k sin(φi ,k + ψ) 1 −



2ψ 2



2 v

+



cos θi ,k sin(φi ,k + ψ) 1 −

λc



v(φi ,k , θi ,k , f i ,k, g ) = vt ( f i ,k, g ) ⊗ vs (φi ,k , θi ,k ).

2ψ





2( v p + v ) ψ 2

+ =

v −

λc

2v p ψ

λc 2

+ =

λc 

2

2

 ( v p + v ) cos θi ,k sin(φi ,k + ψ)

2

cos θi ,k cos(φi ,k + ψ)

 v −

λc

cos θi ,k cos(φi ,k + ψ)

2ψ



2ψ



v p cos θi ,k sin(φi ,k + ψ)

2

    (i 1 , k1 , g 1 ) ∈ arg max v H (φi ,k , θi ,k , f i ,k, g )b0  ,

cos θi ,k sin(φi ,k + ψ + ϕ )

λc

   

×

v p ψ

2

+ v −

2ψ 2

i ,k, g ,h

v p

(27)

,

v(φi 1 ,k1 , θi 1 ,k1 , f i 1 ,k1 , g1 )  , v(φi ,k , θi ,k , f i ,k , g )

uc ;1 = 

v / v p −

2ψ

(28)

.

2

and use the fact that ψ is a small angle leading to sin ψ ≈ ψ and cos ψ ≈ 1 − 2ψ /2. From (27), since | v | ≤ v pm and ψ ≤ ψm , we have

     f i ,k  =  2 cos θi ,k sin(φi ,k + ψ + ϕ ) λ c  

2   2   2 ψ   ×  v p ψ + v − vp   2  =





×

v p 2

2ψ + v p − 1

 2  cos θi ,k  ≤ λc

 

v p 2

2

×





v p 2

= f max,i ,k .

1

1

1

1

(32)

2

and

 2   λ1 = | z1 |2 = ucH;1 b0  .

(33)

The residual vector in the first iteration can be updated as

b1 = b0 − z1 uc ;1 .

(34)

i ,k, g ,h

(35)

ϒ p = ϒ p −1 ∪ {(i p , k p , g p )}. The eigenvector uc ; p is given by

2  − v p − 1 + 2v

2ψ + v p − 1

( ψm )2 + v p − 1

1

    (i p , k p , g p ) ∈ arg max v H (φi ,k , θi ,k , f i ,k, g )b p −1  ,

2



− v p − 1

uc ; p =

2

+ 2v

v(φi p ,k p , θi p ,k p , f i p ,k p , g p ,h p ) −

 p −1  q =1



ucH;q v(φi p ,k p , θi p ,k p , f i p ,k p , g p ,h p ) uc ;q

  ,  p −1    v(φi p ,k p , θi p ,k p , f i p ,k p , g p ,h p ) − q=1 ucH;q v(φi p ,k p , θi p ,k p , f i p ,k p , g p ,h p ) uc ;q  2

 2  cos θi ,k  ≤ λc

1

At the pth (p ≥ 2) iteration, it is supposed that we have obtained the index set ϒ p −1 , and the residual vector b p −1 . The most important space–time steering vector’s index in the dictionary  at the pth iteration is

 2  cos θi ,k sin(φi ,k + ψ + ϕ )

λc

Then the corresponding eigenvector uc ;1 and eigenvalue λ1 are computed by

1

ψ

(31)

ϒ1 = {(i 1 , k1 , g 1 )}.

2

where we define that v p = v p + v , ψ  = ψ + ψ and

ϕ = arctan

(30)

Step 2: select the most important space–time steering vectors and compute the corresponding eigenvectors and eigenvalues. We stack all possible space–time steering vectors v(φi ,k , θi ,k , f i ,k, g ), i = 1, · · · , N a , k = 1, · · · , N c , g = 1, · · · , N f into a dictionary . Because we consider all possible space–time steering vectors with Doppler frequencies f i ,k ∈ [ f i,k − f max,i ,k , f i,k + f max,i ,k ], there will be only some of them corresponding to the accurate or approximated accurate space–time steering vectors.5 Thus, the basic idea of the proposed algorithm is to incorporate the received snapshot x in the CUT and develop an approach that is similar to the orthogonal matching pursuit (OMP) method to select the most important space–time steering vectors from the dictionary  and compute the corresponding eigenvectors Uc = [uc ;1 , uc ;2 , · · · ] and eigenvalues λ = [λ1 , λ2 , · · · ] T . Let the initial index set of the selected space–time steering vectors as ϒ0 = ∅ and the residual b0 = x. We find the first most important space–time steering vector in the dictionary  as

+ ψ cos(φi ,k + ψ) =

N f

f i ,k, g g =1 . Herein, we can compute the clutter space–time steering vector with the Doppler frequency of f i ,k, g , the azimuth angle of φi ,k and the elevation angle of θi ,k as

+ ψ cos(φi ,k + ψ) − sin(φi ,k + ψ) 

267

(36)

2

and the eigenvalue λ p is calculated as



− v p − 1

2

+ ( v pm )2 (29)

From (29), we can compute the range of Doppler frequencies, i.e., f i ,k ∈ [ f i,k − f max,i ,k , f i,k + f max,i ,k ], due to the inaccurate prior knowledge. Then, we divide the Doppler frequency span [ f i,k − f max,i ,k , f i,k + f max,i ,k ] evenly into N f samples

 2   λ p = | z p |2 = ucH; p b p −1  .

(37)

It is well known that the effective rank of the clutter covariance matrix is much lower than the number of DoFs [1,3,4]. That

5 The approximations are because of the errors when sampling the Doppler frequencies.

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Table 2 Compute ϒ , Uc and λ¯ using multiple snapshots. Initialization: B0 = [x1 , · · · , x L ], ϒ0 = ∅, Stopping parameter: p max , , Find the first index:    (i 1 , k1 , g1 ) ∈ arg maxi ,k, g 1L lL=1 v H (φi ,k , θi ,k , f i ,k, g )bl;0 , ϒ1 = {(i 1 , k1 , g1 )}, v(φi 1 ,k1 ,θi 1 ,k1 , f i 1 ,k1 , g1 )

uc ;1 =  v(φ 

λl;1 =

 , 

i 1 ,k1 ,θi 1 ,k1 , f i 1 ,k1 , g 1 )

ucH;1 bl;0 ,

L

2

bl;1 = bl;0 − λl;1 uc ;1 , l = 1, · · · , L, p = 2.

H While 1L and p − 1 ≤ p max l=1  bl; p −1 ∞ >  L  H   1 (i p , k p , g p ) ∈ arg maxi ,k, g 1L l=1 v (φi ,k , θi ,k , f i ,k, g )bl; p −1 , ϒ p = ϒ p −1 ∪ {(i p , k p , g p )}

2

 p −1 H q=1 uc ;q v(φi p ,k p ,θi p ,k p , f i p ,k p , g p ) uc ;q  ,  p −1  H  ,θ , f )− i p ,k p i p ,k p i p ,k p , g p q=1 uc ;q v(φi p ,k p ,θi p ,k p , f i p ,k p , g p ) uc ;q 

v(φi p ,k p ,θi p ,k p , f i p ,k p , g p )−

uc ; p =  v(φ 

2

3 λl; p = ucH; p bl; p −1 , l = 1, · · · , L, 4 bl; p = bl; p −1 − λl; p uc ; p , l = 1, · · · , L, 5 p = p + 1, End

The final solution: ϒ = ϒ p , Uc = [uc;1 , · · · , uc; p ], and λ¯ = [λ¯ 1 , · · · , λ¯ p ]T , where λ¯ q =

is to say the number of columns of the clutter subspace Uc is much smaller than N M. The key question that arises is how to determine the size of the clutter rank or to terminate the above iterative process. This can be achieved when certain criteria are satisfied, e.g., when the iteration number achieves a preset limit p max , or when no column explains a significant amount of energy in the residual:  H b p ∞ ≤ (where  · ∞ denotes the l∞ -norm and is a positive number). The final selected index set, the clutter eigenvectors and eigenvalues are given by ϒ = ϒ p , Uc = [uc ;1 , · · · , uc ; p ] and λ = [λ1 , · · · , λ p ] T . To improve the accuracy of the selected index set, we also can employ multiple snapshots from the adjacent range bins to compute the index set ϒ , the clutter eigenvectors Uc and the averaged eigenvalues λ¯ . Let X = [x1 , · · · , x L ], B = [b1 , · · · , b L ] and ¯ = [λ¯ 1 , · · · , λ¯ L ]) denote the multiple snapshots  = [λ1 , · · · , λ L ] ( matrix, the corresponding residual matrix and the eigenvalue matrix. Then, the whole procedure of the proposed approach using multiple snapshots is detailed in Table 2. Step 3: estimate the clutter covariance matrix and design the filters. After we obtain the clutter eigenvectors Uc and eigenvalues λ ˆ c can be estimated by (or λ¯ ), the clutter covariance matrix R

ˆc = R

p  q =1

λq uc;q ucH;q , or =

p  q =1

λ¯ q uc;q ucH;q

(38)

Then, the STAP filter weights can be computed by



−1

ˆ = μ Rˆ c + σˆ n2 I w

s=μ

1 

σˆ n2



I − Uc  UcH s,

(39)

where



λp λ2 λ1  = diag , ,··· , λ1 + σˆ n2 λ2 + σˆ n2 λ p + σˆ n2

 ,

(40)

,

(41)

or



λ¯ p λ¯ 1 λ¯ 2  = diag , ,··· , ¯λ1 + σˆ n2 λ¯ 2 + σˆ n2 ¯λ p + σˆ n2 ˆ n2



and σ is the estimated receiver thermal noise power which can be collected by the receiver when the radar transmitter operates in passive mode [2]. It should be noted that there is an avoidance of the matrix inversion in the developed filter weight vector (39), compared with the conventional filter weight vector (8).

1 L

L

l=1

λl;q , q = 1, · · · , p

3.2. Discussions It is worth noting that our proposed algorithm is also easy to extend to nonlinear arrays if we could compute the space–time steering vectors using prior knowledge. For the spatial frequency ϑi ,k in (25), it can be computed by the following formulation if the array is non-uniform:

ϑi ,k =

dm

λc

cos θi ,k sin φi ,k ,

(42)

where dm denotes the distance between the mth array element and the first array element. The Doppler frequency f i,k in nonlinear arrays with measured platform velocity v p and the measured crab angle ψ  can also be given by (26). Observing (27) and (29), the derivations are independent to the assumption of the ULA. In other words, we can obtain the assumed Doppler frequencies in the same way as that described in step 1. Thus, the assumed space–time steering vectors can be computed by a different spatial steering vector with ϑi ,k in (42). Furthermore, it also can be used to estimate the clutter covariance matrix by combining the ML estimated clutter covariance matrix, in a similar way to that introduced in [18,19,23], i.e.,

ˆ = α Rˆ ML + β Rˆ c , R

(43)

where α > 0 and β > 0 are constrained to be positive numbers, ˆ c is the clutter covariance matrix estimate computed by the and R proposed algorithm as (38). How to adaptively set the parameters α and β is discussed in [18,19] and will not be shown for space limitation. When in presence of the channel mismatching, we can firstly apply the array calibration methods discussed in [21] to estimate the angle-independent channel mismatching factors. Using these estimated factors, we can employ the proposed algorithm to form the space–time steering vectors and to compute the clutter covariance matrix estimate (38) just as the way of the KAPE approach using the LS technique. Let tˆs denote the estimated angle-independent channel mismatching vector, then the space– time steering vectors in (30) can be written as





v(φi ,k , θi ,k , f i ,k, g ) = vt ( f i ,k, g ) ⊗ vs (φi ,k , θi ,k )  tˆs ,

(44)

where  denotes the Hadamard product. When in presence of the ICM, we can also apply the developed scheme of the KAPE approach to add CMTs to overcome the performance degradation caused by the ICM, i.e.,

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269

Fig. 4. Values of ξpower and ξcosine with the variations in range and PRF when h p = 3 km, (a) ξpower , (b) ξcosine .

⎞ ⎛ p   ˆc = ⎝ R λq uc;q ucH;q ⎠  Tˆ ,

(45)

q =1

or

⎞ ⎛ p   ˆc = ⎝ R λ¯ q uc;q ucH;q ⎠  Tˆ ,

Fig. 5. The received data in different CPIs.

(46)

q =1

where Tˆ represents the estimated CMTs. Since this paper mainly focus on how to use inaccurate prior knowledge of the obtained array manifold to compute the clutter covariance matrix, the above mentioned factors will not be considered in the paper and the authors are referred to [21] for details. In the following simulation part, we will show the performance with inaccurate prior knowledge of the proposed algorithm extended to the CSMIECC, the Stoica’s scheme and the KAPE. In addition, in many cases, the impact of the range ambiguities is not significant to the performance of the proposed algorithm. Except for the causes discussed in [21], we further show the reasons for the above point. From (20) and (21), the lower altitude of the radar platform and the higher PRF will lead to more range ambiguities. We also know that the impact of range ambiguities comes from the cosine of elevation angle and the received returns’ power. To have an intuitive sense to understand this impact, we assume that a typical low radar platform altitude is 3 km, and the received returns’ powers in the iso-range ring of interest and the first range ambiguity iso-range ring are both units without considering the attenuation due to the range. Since the radar received returns’ power is proportional to 1/r 4 , the relative received returns’ power and cosine of elevation angle between the iso-range ring of interest and the first range ambiguity iso-range ring are respectively given by

ξpower =

r4

(r + c /(2 f r ))4

.

(47)

and

ξcosine =

cos θr +c /(2 f r ) cos θr

,

(48)

where θr and θr +c /(2 f r ) are elevation angles with the form of (19) at the range r and r + c /(2 f r ) respectively. Fig. 4 plots the values of ξpower and ξcosine as the variations in range and PRF. From

the figures, we see that: when r ≥ 9 km, ξcosine ≥ 0.95 for all the four cases, i.e., PRF = 2000, 4000, 6000 and 10000, which means similar angle-Doppler responses for range ambiguities; when r < 9 km, ξpower < −38 dB for PRF = 2000, ξpower < −28 dB for PRF = 4000 and ξpower < −22 dB for PRF = 6000. This leads to the range ambiguities returns not significant impact to the total clutter returns. Although the impact of range ambiguities for PRF = 10000 is greater than other three cases, but there is still a large degradation of clutter power from range ambiguities, e.g., when 4.6 km ≤ r ≤ 9 km, ξpower < −17 dB. Above all, we can ignore the range ambiguities in the assumed clutter model in many cases when forming the space–time steering vectors. Specifically, we ignore the range ambiguities when forming the assumed space–time steering vectors in the following simulations, where it also will show that the performance of the proposed algorithm suffers a negligible degradation because of range ambiguities. 3.3. Implementations and complexity analysis As shown in Fig. 5, it is general that the platform velocity and the crab angle for different CPIs are different because of the fluctuations caused by the aircraft controls. The existing KA-STAP algorithms using array manifold compute the clutter steering vectors exploiting the needed knowledge for every single CPI. On the other hand, the proposed algorithm requires the knowledge of the range values of the platform velocity and the crab angle, and allows measured errors for them. The following simulations show that the proposed algorithm outperforms the existing KA-STAP algorithms not only for typical measured errors but also for larger errors. Herein, the proposed algorithm can compute the assumed clutter steering vectors once for different CPIs, where the measured platform velocity, v  ( p , k), and the measured crab angle ψk in the k-th CPI are within some range, i.e., v  ( p , k) ∈ [ v ( p , 1) − v pm , v ( p , 1) + v pm ], ψk ∈ [ψ1 − ψm , ψ1 + ψm ]. Here, v ( p , 1) and ψ1 denote the true platform velocity and the true crab angle in the first CPI, and v pm and ψm are the so called prior

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knowledge of range values of platform velocity and the crab angle. After computing the assumed steering vectors, the left task of the proposed algorithm is to select the most important steering vectors and compute the filter weight according to the received data. Observing the Table 2, we note that the complexity of the proposed approach in Table 2 is similar to the OMP algorithm and on the order of O pN M N c N a N f [35], where p is the iteration number. If we ignore the range ambiguities in the proposed approach, the complexity becomes O pN M N c N f . In addition, the LSE can also use the proposed approach. If we assume there are K CPIs,  then the complexity of LSE is O pN M N c K . In the following simulations, we will show that the value of N f can be 5–12, which is generally much smaller than the value of K . In this case, the proposed InAME-KA-STAP algorithm has lower computational complexity than the LSE. Furthermore, the complexity of the proposed algorithm is much lower than that of the method with pseudoinverse since p , N f  N M. 4. Numerical examples In this section, we assess the performance of the proposed InAME-KA-STAP algorithms using simulated radar data, and compare them with the JDL, the PAMF and the existing KA-STAP algorithms, namely, the CSMIECC, the Stoica’s scheme and the KAPE, by showing the probability detection (PD) performance and the output SINR loss performance, which is defined by

 H 2 w ˆ s  , SINRloss = = NM ˆ H Rw ˆ N M w SINR

(49)

ˆ where R is the true clutter plus noise covariance matrix and w is the estimated STAP filter weight vector. The parameters of the typical radar systems are shown in Table 1. In all examples, we assume the thermal noise power is unitary and N c = 361 clutter patches are evenly distributed into the whole azimuth with the CNR of 40 dB. Furthermore, we suppose the CNR in different ranges are the same. For the proposed InAME-KA-STAP algorithm, we set N c = 4M and N f = 11, and ignore the range ambiguities when forming the assumed clutter space–time steering vectors. All presented results are averaged over 500 independent Monte Carlo runs. 4.1. Performance of the proposed InAME-KA-STAP algorithm Because the proposed algorithm requires to know the roughly range values of the measured errors of platform velocity v p and the crab angle ψ , we study the SINR performance against target Doppler frequency using different estimated range values of v p and ψ under several scenarios. In the simulation, the target with signal-to-noise (SNR) of 0 dB and the azimuth of 0◦ is located at the range of 120 km for radar I and 29 km for radar II. Then, since R h < R u for radar I, there is no range ambiguity. While for radar II, at the range of 29 km, the number of range ambiguities is N a = 3. Besides, we assume that side-looking configurations (the true crab angle ψ = 0◦ ) are for radar I and radar II, L = 10 adjacent snapshots are used to estimate the clutter covariance matrix for both algorithms and the measured values of v p and ψ  follow uniformly distribution within v p ∈ [ v p − v pm , v p + v pm ] and ψ  ∈ [ψ − ψm , ψ + ψm ], respectively. Specifically, in the simulated scenarios, we consider six different cases of the inaccurate prior knowledge of v p and ψ , i.e., (a) ψm = 0.5◦ and v pm = 1 m/s; (b) ψm = 1◦ and v pm = 1 m/s; (c) ψm = 2.5◦ and v pm = 1 m/s; (d) ψm = 0.5◦ and v pm = 2 m/s; (e) ψm = 0.5◦ and v pm = 3 m/s; and (f) ψm = 0.5◦ and v pm = 4 m/s. We begin with a small measured error of v p and focus on the

impact of the estimated crab angle errors’ range to the SINR performance with an increasedly measured errors in the crab angle (for cases of (a), (b) and (c) in both radar I and radar II), shown in Figs. 6(a–c) and Figs. 7(a–c). Similarly, we begin with a small measured error of ψ and focus on the impact of the estimated platform velocity errors’ range to the SINR performance with an increasedly measured errors in the platform velocity (for cases of (e), (d) and (f) in both radar I and radar II), shown in Figs. 6(d–f) and Figs. 7(d–f). In the plots, 0.5 ψm (0.5 v pm ), ψm ( v pm ) and 1.5 ψm (1.5 v pm ) denote the estimated crab angle (platform velocity) errors’ range value 0.5, 1 and 1.5 timing of the true crab angle (platform velocity) errors’ range value ψm ( v pm ). Additionally, the parameters used in the proposed algorithm are set to N f = 15, N c = 4M, p max = 50 and = 0.01. From the figures, we note that the proposed InAME-KA-STAP algorithm provides significantly better performance than the LSE algorithm (corresponding to the estimated crab angle (platform velocity) error’s range setting to the value of 0) in presence of inaccurate prior knowledge. This is because the space–time steering matrix used in LSE can not represent the exact space–time steering vector due to the measured errors in v p and ψ . In addition, we observe that 0.5–1.5 timing of the true range value ψm ( v pm ) provides good performance, which indicates that the proposed algorithm is robust to the estimated crab angle and platform velocity errors’ range values. Compared with the true range values, slightly smaller estimated range values provide even better performance. This can be understood that in the proposed algorithm, we have enlarged the errors by (29). Although the range ambiguities present in the scenarios of radar II, the SINR performance of both the LSE algorithm and the proposed InAME-KA-STAP algorithm do not affect much. The reasons for this have been detailed in Section 3.2. Regarding to the parameter settings, e.g., N f and , of the proposed algorithm, we conduct the corresponding simulations in terms of SINR performance for the typical radar I for simplicity, as shown in Fig. 8(a) and Fig. 8(b). The ranges of platform velocity v p and crab angle ψ are set to ψm = 1◦ and v pm = 2 m/s. From the curves, we see that the proposed algorithm is robust to the parameters N f and . Regarding to the effects of the estimation accuracies of the range values of the platform velocity and the crab angle, we depict the SINR against the corresponding estimation accuracies for the typical radar I, as shown in Fig. 9(a) and Fig. 9(b). The target normalized Doppler frequency is fixed to 0.2. To model the estimation accuracies, we assume the range estimates of v p (ψ ) follows Gaussian distribution with the mean v pm ( ψm ), and the 2 covariance σ vp (σψ2 ). Additionally, the horizontal axis of Fig. 9(a) (Fig. 9(b)) is defined by the ratio of the standard deviation and the mean, i.e., σ vp / v pm (σψ / ψm ). From the curves, we observe that the SINR degrades as the increases of the ratio σ vp / v pm or σψ / ψm , which accords with the general observations of KA algorithms. However, the performance of the proposed algorithm is not sensitive to the estimation accuracies, e.g., the SINR degrades little when the ratio σ vp / v pm (σψ / ψm ) is less than 0.4. These estimation accuracies can be easily achieved in practical. Therefore, compared with existing KA-STAP using array manifold, the proposed algorithm is more robust in practical. To study the effect of various parameters on the SINR, we use the average value of the SINR over all Doppler frequencies

0.5 SINR =

SINR( f )df .

(50)

− 0 .5

In the examples, we consider both side-looking and forwardlooking configurations for the two typical radar systems. The target

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271

Fig. 6. SINR performance against the target Doppler frequency for the typical radar I with different measured errors of prior knowledge of ψ and v p . (a) ψm = 0.5◦ and v pm = 1 m/s; (b) ψm = 1◦ and v pm = 1 m/s; (c) ψm = 2.5◦ and v pm = 1 m/s; (d) ψm = 0.5◦ and v pm = 2 m/s; (e) ψm = 0.5◦ and v pm = 3 m/s; (f) ψm = 0.5◦ and v pm = 4 m/s.

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Fig. 7. SINR performance against the target Doppler frequency for the typical radar II with different measured errors of prior knowledge of ψ and v p . (a) ψm = 0.5◦ and v pm = 1 m/s; (b) ψm = 1◦ and v pm = 1 m/s; (c) ψm = 2.5◦ and v pm = 1 m/s; (d) ψm = 0.5◦ and v pm = 2 m/s; (e) ψm = 0.5◦ and v pm = 3 m/s; (f) ψm = 0.5◦ and v pm = 4 m/s.

Z. Yang, R.C. de Lamare / Digital Signal Processing 60 (2017) 262–276

Fig. 8. SINR performance against different values of parameter N f and

for the typical radar I. (a) setting of N f ; (b) setting of .

Fig. 9. SINR against the estimation accuracies of the range values of the platform velocity and the crab angle. (a) SINR versus

SNR is 0 dB and the azimuth is 0◦ for side-looking configuration and 90◦ for forward-looking configuration. In addition, for both radar systems and both configurations, we consider three different cases of the inaccurate prior knowledge of v p and ψ , i.e., (a) ψm = 0.5◦ and v pm = 1 m/s; (b) ψm = 1◦ and v pm = 2 m/s; and (c) ψm = 2.5◦ and v pm = 4 m/s. The estimated platform velocity and crab angle errors’ range values are set to 0.5 timing of the true platform velocity and crab angle range values, respectively. For both side-looking and forward-looking configurations, we use L = 4 snapshots to estimate the clutter covariance matrix. Figs. 10(a–b) and Figs. 11(a–b) depict SINR versus the detection range. Again, from the figures we find similar conclusions: (i) the proposed algorithm shows a robustness to the inaccurate prior knowledge and significantly outperforms the LSE algorithm; (ii) the SINR performance of both the LSE algorithm and the proposed InAME-KA-STAP algorithm do not affect much in presence of range ambiguities. We note that for short ranges of the radar II, the performance of the LSE algorithm is better than that for long ranges. This can be explained by the fact that the cosine of elevation angle in short ranges is smaller than that in long ranges

273

σ vp / v pm ; (b) SINR versus σψ / ψm .

resulting in a lower clutter Doppler frequency bandwidth and then the same level inaccurate prior knowledge will lead to smaller impact in short ranges than the long ranges. We also studied the performance of the proposed algorithm (ignoring range ambiguities in the assumed clutter model) in a much higher PRF radar system, (e.g., f r = 10 kHz), the range ambiguities leaded to about 1–2 dB SINR loss in short ranges. This SINR loss is acceptable especially for high requirements of computational complexity. Furthermore, we also can apply the proposed algorithm considering range ambiguities in the assumed clutter model to the situation of range ambiguities as described in Section 3.1. But this will lead to additional computational complexity. 4.2. Comparison with existing algorithms In this subsection, we compare both the SINR performance and the PD performance of the proposed algorithms, the 4 × 3 (4-Doppler bins and 3-spatial bins) JDL[6], the PAMF, the CSMIECC, the Stoica’s scheme and the KAPE. Specifically, the prior clutter covariance matrix used in the CSMIECC, the Stoica’s scheme and the KAPE is computed in the same way as the LSE algorithm.

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Fig. 10. Average SINR performance versus range for the typical radar I under different measured errors of prior knowledge of v p and ψ with both side-looking and forwardlooking configuration. (a) Side-looking configuration; (b) forward-looking configuration.

Fig. 11. Average SINR performance versus range for the typical radar II under different measured errors of prior knowledge of v p and ψ with both side-looking and forward-looking configuration. (a) Side-looking configuration; (b) forward-looking configuration.

We shortened the CSMIECC scheme, the Stoica’s scheme and the KAPE using the prior clutter covariance matrix formed by our proposed approach as InAM-CSMIECC, InAM-Stoica’s scheme and InAM-KAPE. For simplicity, we consider side-looking configuration of the Typical radar I, as shown in Table 1. Furthermore, we consider the scenario with the ICM, which is formulated as a general model that proposed by J. Ward in [1]. The corresponding autocorrelation of the fluctuations is Gaussian in shape with the form:

 8π 2 σ v2 T r2 2  ζ (m) = exp − m , (51) λc2 where σ v is the velocity standard deviation (in the example, we set σ v = 0.5 corresponding to a moderate clutter Doppler spectral spread situation). For the PAMF, the autoregressive model order is 2 and the autoregressive model coefficients are estimated using

the LS technique. For the CSMIECC and the InAM-CSMIECC, the combination parameter is set to 0.6, recommended in [23]. For the KAPE and the InAM-KAPE, N 1 = 11 filters (corresponding to the difference in Doppler frequency setting to −20 : 4 : 20 Hz) for selecting the Doppler shift and N 2 = 11 filters (corresponding to σ v = 0 : 0.1 : 1) for selecting the CMT. For all KA-STAP algorithms, we consider the case of the inaccurate prior knowledge of v p and ψ as ψm = 0.5◦ and v pm = 1 m/s. For the proposed prior clutter covariance matrix estimate, the estimated platform velocity and crab angle errors’ range values are set to 0.5 timing of the true platform velocity and crab angle range values, respectively, and the number of snapshots used is set to L = 4. In all algorithms, the diagonal loading factor is set to noise power level. In the first example, we evaluate the SINR performance against the number of snapshots of the proposed algorithms and compare

Z. Yang, R.C. de Lamare / Digital Signal Processing 60 (2017) 262–276

Fig. 12. SINR versus the number of snapshots of all algorithms.

275

Fig. 13. SINR versus the target Doppler frequency of all algorithms.

it with other algorithms. It is assumed that the target is injected in the boresight with the normalized Doppler frequency of 0.25. As plotted in Fig. 12, the curves show that

• The InAME-KA-STAP provides the highest SINR performance and can even obtain −2 dB SINR compared with optimum performance using just one snapshot.

• The performance of the InAM-CSMIECC, the InAM-Stoica’s scheme and the InAM-KAPE are significant better than those of the CSMIECC, the Stoica’s scheme and the KAPE. This is because the prior clutter covariance matrix used in the InAMCSMIECC, the InAM-Stoica’s scheme and the InAM-KAPE has higher accuracy than that in the CSMIECC, the Stoica’s scheme and the KAPE. • The convergence of the InAME-KA-STAP, the InAM-CSMIECC, the InAM-Stoica’s scheme and the InAM-KAPE are much faster than other algorithms and 1–2 snapshots can obtain steadystate performance. It should be noted that the convergence of the CSMIECC and the Stoica’s scheme are slower than the JDL and the PAMF. This is because, the accuracy of the prior clutter covariance matrix used in the CSMIECC and the Stoica’s scheme is low and the convergence of them mainly rely on ˆ ML . that of the ML clutter covariance matrix estimate R In the second example, we depict the SINR performance against the target Doppler frequency, as shown in Fig. 13. The numbers of snapshots for training used in the JDL, the PAMF and the ML clutter covariance matrix estimate in KA-STAP algorithms are 24, 20 and 48, respectively. The other parameters are the same as the first example. From the Fig. 11, we find that the performance of the LSE algorithm is the worst because of the errors in prior knowledge of the array manifold. It is also found that the SINR performance of the InAME-KA-STAP, the InAM-Stoica’s scheme and the InAM-KAPE are much better than those of the Stoica’s scheme, the KAPE, the JDL and the PAMF. One should note that the performance of the CSMIECC is only slightly worse than that of the InAM-CSMIECC, but much better than that of the Stoica’s scheme. That is because the way to estimate the final clutter covariance matrix is different. In the third example, we plot the PD performance against the SINR (defined by |α |2 s H Rs, where α is the target complex amplitude constant) for all algorithms, as shown in Fig. 14. The false alarm rate is set to 10−3 and for simulation purposes the threshold and PD estimates are based on 10000 samples, respectively. The other parameters are the same as those in the second ex-

Fig. 14. PD performance versus the SINR of all algorithms.

ample. We note that the InAME-KA-STAP, the InAM-CSMIECC, the InAM-Stoica’s scheme and the InAM-KAPE provide higher detection rate than the CSMIECC, the Stoica’s scheme and the KAPE, and also higher than the conventional STAP algorithms, i.e, the JDL and the PAMF. 5. Conclusions In this paper, a novel InAME-KA-STAP algorithm has been proposed considering inaccurate prior knowledge of array manifold for airborne radar systems. It first incorporates the knowledge about the range values of the measured parameters into the assumed clutter model and enlarges the clutter Doppler frequency bandwidth. By over-sampling the corresponding space–time subspace and selecting the important clutter space–time steering vectors from the over-sampled candidates, we can obtain more accurate clutter subspace than that of the LSE algorithm in presence of measured errors in prior knowledge. Moreover, some extensions of the proposed algorithm with the existing techniques are discussed in detail. The simulation results show that the proposed algorithms require much less snapshots for training than the conventional STAP algorithms, and can even obtain good clutter suppression per-

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formance using just one snapshot for training. Additionally, compared with existing KA-STAP using array manifold, the advantages of the proposed algorithm lie that (1) it is robust to the inaccurate array manifold because of using the prior knowledge of the range values of the measured platform velocity and the crab angle; (2) it does not require to compute the array manifold according to the knowledge for every single CPI, which results in computational complexity reduction; (3) the range values of the measured platform velocity and the crab angle is easily obtained; (4) it is not sensitive to the prior knowledge of the range values of the measured platform velocity and the crab angle. Acknowledgments This work was supported in part by National Natural Science Foundation of China under Grant 61401478, the Science & Technology Innovation Project of Shenzhen under Grant JCYJ20160307112710376 and the Natural Science Foundation of SZU under Grant 2016056. References [1] J. Ward, Space-time adaptive processing for airborne radar, Technical Report 1015, MIT Lincoln laboratory, Lexington, MA, Dec. 1994. [2] R. Klemm, Principles of Space–Time Adaptive Processing, Institute of Electical Engineering, London, UK, 2006. [3] J.R. Guerci, Space–Time Adaptive Processing for Radar, Artech House, 2003. [4] W.L. Melvin, A STAP overview, IEEE Aerosp. Electron. Syst. Mag. 19 (1) (2004) 19–35. [5] A. Haimovich, M. Berin, Eigenanalysis-based space–time adaptive radar: performance analysis, IEEE Trans. Aerosp. Electron. Syst. 33 (4) (1997) 1170–1179. [6] H. Wang, L. Cai, On adaptive spatial-temporal processing for airborne surveillance radar systems, IEEE Trans. Aerosp. Electron. Syst. 30 (3) (1994) 660–670. [7] J.S. Goldstein, I.S. Reed, Theory of partially adaptive radar, IEEE Trans. Aerosp. Electron. Syst. 33 (4) (1997) 1309–1325. [8] J.S. Goldstein, I.S. Reed, P.A. Zulch, Multistage partially adaptive STAP CFAR detection algorithm, IEEE Trans. Aerosp. Electron. Syst. 35 (2) (1999) 645–661. [9] R. Fa, R.C. de Lamare, Reduced-rank STAP algorithms using joint iterative optimization of filters, IEEE Trans. Aerosp. Electron. Syst. 47 (3) (2011) 1668–1684. [10] R. Fa, R.C. de Lamare, L. Wang, Reduced-rank STAP schemes for airborne radar based on switched joint interpolation, decimation and filtering algorithm, IEEE Trans. Signal Process. 58 (8) (2010) 4182–4194. [11] J.R. Roman, M. Rangaswamy, D.W. Davis, Q. Zhang, B. Himed, J.H. Michels, Parametric adaptive matched filter for airborne radar applications, IEEE Trans. Aerosp. Electron. Syst. 36 (2) (2000) 677–692. [12] Z. Yang, R.C. de Lamare, X. Li, L 1 -regularized STAP algorithms with a generalized sidelobe canceler architecture for airborne radar, IEEE Trans. Signal Process. 60 (2) (2012) 674–686. [13] Z. Yang, R.C. de Lamare, X. Li, Sparsity-aware space–time adaptive processing algorithms with L 1 -norm regularization for airborne radar, IET Signal Process. 6 (5) (2012) 413–423. [14] J.R. Guerci, E.J. Baranoski, Knowledge-aided adaptive radar at DARPA: an overview, IEEE Signal Process. Mag. 23 (1) (2006) 41–50. [15] M.C. Wicks, M. Rangaswamy, R. Adve, T.B. Hale, Space–time adaptive processing: a knowledge-based perspective for airborne radar, IEEE Signal Process. Mag. 23 (1) (2006) 51–65. [16] K. Gerlach, M.L. Picciolo, Airborne/spacebased radar STAP using a structured covariance matrix, IEEE Trans. Aerosp. Electron. Syst. 39 (1) (2003) 269–281. [17] J.S. Bergin, C.M. Teixeira, P.M. Techau, J.R. Guerci, Improved clutter mitigation performance using knowledge-aided space–time adaptive processing, IEEE Trans. Aerosp. Electron. Syst. 42 (3) (2006) 997–1009. [18] P. Stoica, J. Li, X. Zhu, J.R. Guerci, On using a priori knowledge in space–time adaptive processing, IEEE Trans. Signal Process. 56 (6) (2008) 2598–2602. [19] X. Zhu, J. Li, P. Stoica, Knowledge-aided space–time adaptive processing, IEEE Trans. Aerosp. Electron. Syst. 47 (2) (2011) 1325–1336. [20] R. Fa, R.C. de Lamare, V.H. Nascimento, Knowledge-aided STAP algorithm using convex combination of inverse covariance matrices for heterogeneous clutter, in: Proc. IEEE Int. Conf. Acoust. Speech and Signal Process, 2010, pp. 2742–2745.

[21] W.L. Mevin, G.A. Showman, An approach to knowledge-aided covariance estimation, IEEE Trans. Aerosp. Electron. Syst. 42 (3) (2006) 1021–1042. [22] Z. Yang, R.C. de Lamare, X. Li, H. Wang, Knowledge-aided STAP using low rank and geometry properties, Int. J. Antennas Propag. 2014 (2014) 1–14. [23] W. Xie, K. Duan, F. Gao, Y. Wang, Z. Zhang, Clutter suppression for airborne phased radar with conformal arrays by least squares estimation, Signal Process. 91 (7) (2011) 1665–1669. [24] B. Friedlander, A subspace method for space time adaptive processing, IEEE Trans. Signal Process. 53 (1) (2005) 74–82. [25] B. Tang, J. Tang, Y. Peng, Performance of knowledge aided space time adaptive processing, IET Radar Sonar Navig. 5 (3) (2011) 331–340. [26] Y. Wu, J. Tang, Y. Peng, On the essence of knowledge-aided clutter covariance estimate and its convergence, IEEE Trans. Aerosp. Electron. Syst. 47 (1) (2011) 569–585. [27] K. Sun, H. Zhang, G. Li, H. Meng, X. Wang, A novel STAP algorithm using sparse recovery technique, in: Proc. IGARSS, 2009, pp. 336–339. [28] J. Li, X. Zhu, P. Stoica, M. Rangaswamy, High resolution angle-Doppler imaging for MTI radar, IEEE Trans. Aerosp. Electron. Syst. 46 (3) (2010) 1544–1556. [29] Z. Yang, X. Li, H. Wang, W. Jiang, Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Process. 93 (12) (2013) 3567–3577. [30] S. Sen, OFDM radar space–time adaptive processing by exploiting spatiotemporal sparsity, IEEE Trans. Signal Process. 61 (1) (2013) 118–130. [31] Z. Yang, X. Li, H. Wang, L. Nie, Sparsity-based space–time adaptive processing using complex-valued homotopy technique for airborne radar, IET Signal Process. 8 (5) (2014) 552–564. [32] S. Sen, Low-rank matrix decomposition and spatio-temporal sparse recovery for STAP radar, IEEE J. Sel. Top. Signal Process. 9 (8) (2015) 1510–1523. [33] Z. Wang, Y. Wang, K. Duan, W. Xie, Subspace-augmented clutter suppression technique for STAP radar, IEEE Geosci. Remote Sens. Lett. 13 (3) (2016) 462–466. [34] Z. Yang, X. Li, H. Wang, R. Fa, Knowledge-aided STAP with sparse-recovery by exploiting spatio-temporal sparsity, IET Signal Process. 10 (2) (2016) 150–161. [35] J.A. Tropp, J. Wright, Computational methods for sparse solution of linear inverse problems, Proc. IEEE 98 (6) (2010) 948–958.

Zhaocheng Yang received the B.E. degree from Beijing Institute of Technology, Beijing, P.R. China, in 2007. He received the Ph.D. degree from the National University of Defense Technology, Changsha, China, in June 2013. From November 2010 to November 2011, he has been a visiting scholar with the University of York, York, U.K. From June, 2013 to August 2015, he has been a lecturer at the National University of Defense Technology. Since September 2015, he is a lecturer with the College of Information Engineering, Shenzhen University. His research interests lie in the area of statistical signal processing, including array signal processing, adaptive signal processing, and its applications to radar systems. Rodrigo C. de Lamare received the electronic engineering degree from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1998, and the M.Sc. and Ph.D. degrees in electrical engineering from the Pontifical Catholic University of Rio de Janeiro (PUCRIO), Brazil, in 2001 and 2004, respectively. Since January 2006, he has been with the Communications Group, Department of Electronics, University of York, United Kingdom, where he is a Professor. Since April 2013, he has also been a Professor at PUC-RIO. Dr de Lamare has participated in numerous projects funded by government agencies and industrial companies. He received a number of awards for his research work and has served as the general chair of the IEEE 7th International Symposium on Wireless Communication Systems (ISWCS) 2010, held in York, UK in September 2010, as the technical programme chair of ISWCS 2013 and WSA 2015 in Ilmenau, Germany, and as the general chair of the 9th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM) to be held in Rio de Janeiro, Brazil, in July 2016. Dr de Lamare is a senior member of the IEEE and an elected member of the IEEE Signal Processing Theory and Method technical committee. He currently serves as an associate editor for the EURASIP Journal on Wireless Communications and Networking and as a senior area editor for the IEEE Signal Processing Letters. His research interests lie in communications and signal processing, areas in which he has published about 350 papers in international journals and conferences.