Multichannel adaptive signal detection in space-time colored compound-gaussian autoregressive processes
https://doi.org/10.1186/1687-6180-2012-69
© Xu et al; licensee Springer. 2012
Received: 3 July 2011
Accepted: 20 March 2012
Published: 20 March 2012
Abstract
In this article, we consider the problem of adaptive detection for a multichannel signal in the presence of spatially and temporally colored compound-Gaussian disturbance. By modeling the disturbance as a multichannel autoregressive (AR) process, we first derive a parametric generalized likelihood ratio test against compound-Gaussian disturbance (CG-PGLRT) assuming that the true multichannel AR parameters are perfectly known. For the two-step GLRT design criterion, we combine the multichannel AR parameter estimation algorithm with three covariance matrix estimation strategies for compound-Gaussian environment, then obtain three adaptive CG-PGLRT detectors by replacing the ideal multichannel AR parameters with their estimates. Owing to treating the random texture components of disturbance as deterministic unknown parameters, all of the proposed detectors require no a priori knowledge about the disturbance statistics. The performance assessments are conducted by means of Monte Carlo trials. We focus on the issues of constant false alarm rate (CFAR) behavior, detection and false alarm probabilities. Numerical results show that the proposed adaptive CG-PGLRT detectors have dramatically ease the training and computational burden compared to the generalized likelihood ratio test-linear quadratic (GLRT-LQ) which is referred to as covariance matrix based detector and relies more heavily on training.
Keywords
1 Introduction
In an airborne radar system, space-time adaptive processing (STAP) has been widely used in radar target detection; see [1–4] and references therein. Various well-known STAP based detectors have been extensively investigated under Gaussian assumption [4–7]. However, with the support of measured data, the Gaussian model is no longer suitable for background disturbance in many situations of practical interest. The conventional STAP detectors may suffer severe performance degradation when the disturbance is non-Gaussian. Instead, a compound-Gaussian model can successfully describe the non-Gaussian disturbance as a product of a spatially and temporally "slowly varying" texture and a locally "rapidly varying" Gaussian speckle component [8, 9]. The texture component accounts for random power variations over range cells. This model includes the so-called spherically invariant random processes (SIRPs). Working with the compound-Gaussian model, a multitude of adaptive detectors have been studied in the past few years, for instance, the generalized likelihood ratio test-linear quadratic (GLRT-LQ) which was independently derived in [10, 11], the detectors with Rao and Wald tests [12], the Bayesian optimum radar detector (BORD) [13], and so forth. Notice that all of the aforementioned STAP detectors proposed in Gaussian and compound-Gaussian environment can be considered as covariance matrix based detectors [8, 14]. Implementing these detectors involves estimating and inverting a space-time covariance matrix of the disturbance signal for each cell under test (CUT) utilizing independent and identically distributed (IID) target-free training data (or secondary data). Obviously, when the joint spatial-temporal dimension is large, the training and computational requirements will be quite onerous. Moreover, some practical situations may exacerbate the training data selection and collection problem and limit the amount of appropriate IID training data. The lack of training may lead to ill-conditioned covariance matrix estimate and significant degradation in the covariance matrix based detection procedure.
To overcome the above difficulty caused by large joint spatial-temporal dimension, the structural information about the disturbance space-time covariance matrix can be exploited. More precisely, a multichannel autoregressive (AR) process has been found to be able to model the spatial-temporal correlation of the disturbance efficiently [15–18]. In [15], based on approximating the disturbance spectrum with a multichannel AR model of low order, a parametric adaptive matched filter (PAMF) for STAP detection was presented for multichannel system in Gaussian environment. The PAMF detector, which has been proved to be equivalent to a parametric Rao detector in [16], has dramatically outperformed the conventional adaptive matched filter (AMF) [6, 7] with small training size. Also in Gaussian background, a parametric GLRT [17] and a simplified parametric GLRT [18] have been successively developed by utilizing a parametric model (multichannel AR model) in the GLR principle. Experimental results on simulated and real data show that two parametric GLRT detectors work well with limited or even no range training data [18–20]. However, under such conditions, the AMF detector and Kelly's GLRT cannot be implemented. Moreover, application of the multichannel AR model in non-stationary Gaussian clutter for STAP are investigated in [21–23].
For the corresponding problem in compound-Gaussian environment, a non-Gaussian parametric adaptive matched filter (NG-PAMF) has been derived in [24]. However, this test involves explicit knowledge of the disturbance statistics, which are not always available. Unlike the NG-PAMF, the normalized parametric adaptive matched filter (NPAMF) reported in [8, 14] requires no a priori knowledge about the disturbance statistics. This feature is rather important in real-time operation. However, Michels et al. [14] still combined the multichannel AR identification algorithm with the sample covariance matrix (SCM) even in compound-Gaussian background. The sample matrix is the maximum likelihood estimate (MLE) of the covariance matrix for Gaussian disturbance, but is no longer the MLE for compound-Gaussian disturbance. The covariance matrix estimation in compound-Gaussian environment is generally intractable. Conte et al. have advocated the use of a normalized sample covariance matrix (NSCM) in [25]. Also, considering the texture component as an unknown deterministic quantity, an approximate ML (AML) estimator has been derived by Gini et al. [26].
Motivated by the previous studies, the main purpose of this article is to derive a parametric GLRT (PGLRT) for detecting a multichannel signal in the presence of compound-Gaussian disturbance modeled as a multichannel AR process. Without any knowledge about the disturbance statistics, we resort to a suboptimal GLRT algorithm considering the texture components as unknown deterministic parameters. In further derivation, to get round the difficulty in performing the joint maximization for all the unknown parameters, a two-step GLRT design criterion is adopted in this article. We first derive the model-based parametric GLRT in compound-Gaussian environment (CG-PGLRT), which possesses the perfect knowledge about the multichannel AR parameters. We apply three covariance matrix estimation strategies, i.e., sample covariance matrix (SCM), normalized sample covariance matrix (NSCM) and approximate ML (AML) estimator, to the multichannel AR parameter estimation procedure for estimating the unknown AR coefficient matrices. Then three adaptive versions of the CG-PGLRT detector: CG-PGLRT-SCM, CG-PGLRT-NSCM, and CG-PGLRT-AML are obtained. Finally, the performance assessments are presented. Numerical results indicate that the CG-PGLRT-SCM detector has no texture CFAR property, while all of the CG-PGLRT, CG-PGLRT-NSCM, and CG-PGLRT-AML detectors ensure CFAR property with respect to the texture probability density function (PDF). Compared to the covariance matrix based detector, all of the proposed adaptive detectors can handle the training-limited case and significantly decreased the computation complexity in compound-Gaussian environment.
This article is organized as follows. The problem statement and the signal and disturbance models are presented in Section 2. The parametric GLRT detector for compound-Gaussian environment (CG-PGLRT) and adaptive CG-PGLRT detectors are derived in Section 3. Then the performance assessment of the proposed detectors is displayed in Section 4. Finally, conclusions are given in Section 5.
Notation: Vectors (matrices) are denoted by boldface lower (upper) case letters, all vectors are column ones, superscripts (•)
T
and (•)
H
denote transpose and complex conjugate transpose, respectively, denotes the multivariate complex Gaussian distribution with mean μ and covariance matrix R.
denotes the complex number field and det{•} takes the determinant of a matrix.
2 Problem statement and signal model
2.1 Problem statement
Consider the scene that the radar transmits a coherent train of N pulses and receives the signal with a uniform linear array with J sensors. The received data collected over K range cells is organized in a J × N × K data cube. For the range cell under test (CUT), a binary hypothesis test is applied to the JN-dimensional complex baseband space-time vector of primary data . Typically, x0 contains an unwanted additive disturbance signal with unknown space-time covariance matrix and may contain a target signal α s with deterministic but unknown complex amplitude, α, and known target space-time steering vector . The space-time steering vector takes the form of the Kronecker product of the normalized spatial and temporal steering vectors. K IID complex baseband space-time vectors of target-free training data exist for assisting the signal detection.
The disturbance signals lump clutter, jamming, and thermal noise, and may be correlated in space and time.
2.2 Signal model
where denotes the unknown J×J coefficient matrices for AR process d k (n) of known order P, and ε k (n) denotes the J × 1 spatial noise driving vectors that are temporally white but spatially colored.
The non-Gaussian driving process ε k (n) is modeled as a compound-Gaussian process. It follows that ε k (n) can be thought of as zero-mean spherically invariant random vectors (SIRVs), i.e., they can be written in the form . Here the speckle z k (n) are J × 1 complex, zero-mean, Gaussian vectors with unknown covariance matrix Q. The texture component τ k is a positive random variable over range, but constant over time when it has long temporal coherent. The texture PDF f τ (τ) is defined to be the characteristic PDF of the complex SIRV. Given a specific value of τ k , we have . The covariance matrix of the driving process is E{ε k (n)ε k (n) H } = E(τ k )Q where the mean value E(τ k ) is also the average disturbance power.
It is worth noting that the P th-order linear prediction coefficients are identically equal to the AR(P) process coefficients.
respectively, where α = 0 under H0 and α ≠ 0 under H1. Equations (7) and (8) imply that and , where SIRV[μ,Q,f τ (τ)] denotes a complex SIRV obtained by sampling a SIRP with mean μ, covariance matrix Q, and a characteristic PDF f τ (τ). As the distribution f τ (τ) is unknown, the temporally whitened vectors can be modeled as conditionally Gaussian with the unknown variance τ k , i.e., and .
where τ = [τ0, τ1, . . . , τ K ] T and . Note that A H (p) is expressed in terms of the Hermitian operation for notational convenience, but is not necessarily a Hermitian matrix.
3 Parametric GLRT derivation in compound-gaussian environment
In the Neyman-Pearson sense, the optimum solution for the composite hypothesis testing problem (1) is the likelihood ratio test (LRT). But for the case at hand, it cannot be implemented due to total ignorance of the signal parameter α, the multichannel AR parameters Q and A H and the texture PDF f τ (τ). Hence, we resort to a suboptimal GLRT algorithm where the τ k s are modeled as unknown deterministic parameters, and perform the maximum likelihood estimation (MLE) for all the unknown parameters under each hypothesis. Unfortunately, the exact maximization with respect to the unknown parameters is rather difficult and does not exist a close-form expression. Therefore, to get round the above difficulty, the two-step GLRT design criterion is adopted. We first assume that the AR parameters Q and A H are perfectly known, and derive the CG-PGLRT detector based on the primary data. The adaptive versions of CG-PGLRT detector are then obtained by substituting the unknown AR parameters with their estimates based on the training signals only.
3.1 CG-PGLRT detector for known AR parameters
respectively.
and tr{·} denotes the trace operator.
where ηCG-PGLRT is the appropriate modification of the original threshold in (11).
Note that the normalized parametric adaptive matched filter (NPAMF), originally developed in [8, 14] for compound-Gaussian environment, is closely related to the CG-PGLRT detector but replaces true Q and A H with their estimates obtained from the training signals. In [8, 14, 29], several multichannel parameter estimation algorithms are considered in the NPAMF detector, including the Nuttall-Strand and the multichannel least squares methods [30]. However, we obtain herein adaptive version of the CG-PGLRT detector relying on maximum likelihood (ML) parameter estimation criterion. Specifically, our multichannel parameter estimation approach is similar to that proposed in [16, 17] for Gaussian parametric model-based STAP detectors, whereas we utilize only training signals for parameter estimation.
3.2 Adaptive CG-PGLRT detector for unknown AR parameters
To make the derived detector (22) fully adaptive, suitable estimates of the multichannel AR parameters Q and A H must be acquired. Employing the target-free training signals to estimate the multichannel AR parameters allows to decouple from estimation of target parameters, such as complex amplitude α and the texture component τ0. However, derivation of the MLEs of Q and A H with K unknown textures is still a more challenging task. The similar covariance estimation problem in compound-Gaussian environment has been solved in [25, 26, 31], where three different estimation strategies are introduced, including SCM, NSCM, and AML estimators.
In our opinion, the derivation of these three covariance estimators depends upon two different clutter models, that is, the dependent interference model[31] and the independent interference model[26], respectively. The former assumes that the textures of training signals are completely correlated, i.e., τ1 = τ2 = ··· = τ K , and adopts the SCM estimator. The latter, where the NSCM and AML estimators are usually applied, deals with the IID instead.
where and is a diagonal matrix whose diagonal entries are τ1, . . . , T K . We, next, use (23) to derive the MLEs of Q and A H .
where and L = (N - P) × K. Substituting the above back in (23), we find that maximizing (23) with respect to A H reduces to minimizing . Therefore, the MLE of the AR coefficients matrix A H can be obtained by minimizing with respect to A H .
Obviously, once the MLE of τ t = [τ1, . . . , τ K ] T , namely , is available, the exact MLEs of A H and Q can be readily obtained according to (30) and (31). However, so far the explicit solutions for are still inaccessible. Therefore, in the second step of the person-by-person maximization, we introduce the aforementioned two clutter models to motivate the use of the three different covariance estimation strategies.
Notice that an explicit solution for λ is not necessary since the AR coefficients estimate is λ-independent and the adaptive version of detector (22) only requires knowledge of (λ) within a scale multiplicative constant.
Thus, we have and .
Actually, the dependent interference model can be considered as a simple extension of the partially-homogeneous environment under the assumption of the deterministic texture component. Recently, two different parametric Rao tests, referred to as the normalized parametric Rao (NPRao) test and the scale-invariant parametric Rao (SI-PRao) test, respectively, have been developed in [32] for the partially-homogeneous environment. The above parametric Rao tests can be candidates for multichannel parametric detection in the dependent interference model. Different to the CG-PGLRT-SCM detector, the NPRao and SI-PRao tests use both training and test signals for parameter estimation. However, the above techniques are no longer applicable to the independent interference model because of the model mismatch.
where is the sample estimate of the local disturbance power in the reference range cells.
where the matrices and consist of the temporally whitened versions of and , respectively, using the coefficient estimate .
for i = 0, 1, 2, . . . , Nit, where Nit is the number of iterations. Equation (52) guarantees the constraint to be satisfied at each iteration. Note that the covariance matrices , and have the same structure as those in (44)-(46), the only difference being that the normalization factor is replaced by which is refreshed at each iteration. Here, and are used as the initialization matrix for this recursive estimator, i.e., and .
where the original GLRT-LQ detector given in [10, 11] has been reintroduced in (54). The space-time covariance matrix estimate can be also obtained by exploiting the SCM, NSCM, and AML estimators, respectively. However, taking the SCM estimator as an example, we at least need K ≥ JN training signals to ensure a full-rank estimate of the JN × JN matrix R. Obviously, the large JN spatio-temporal product will impose excessive training and computational burdens to the detector.
3.3 Complexity issues
Complexity of the CG-PGLRT-SCM detector, CG-PGLRT-NSCM detector, and CG-PGLRT-AML detector for model order P (suppose KN >JP)
Step | Task | Flops | Task | Flops | Task | Flops |
---|---|---|---|---|---|---|
S 1 |
| O(KJ2(N-P)) |
| O(KJ2P2(N-P)) |
| O(2J2 P 2K(N-P)) +O(N it KJ(N-P)2) |
S 2 |
| O(KJ 2P2(N-P)) |
| O(2KJ2(N-P)) |
| O(2N it KJ 2(N-P)) |
S 3 |
| O(KJ 2P(N-P)) |
| O(2KJ 2P2(N-P)) |
| O(2N it KJ 2P2(N-P)) |
S 4 |
| O(J3(P3+P2)) |
| O(2KJ 2P(N-P)) |
| O(2N it KJ 2P(N-P)) |
S 5 |
| O(J3P) |
| O(J3(P3+P2) - |
| O(N it J 3(P3 + P 2)) |
S 6 |
| O(J3) |
| O(J3P) |
| O(N it J 3P) |
S 7 |
| O(J 2P(N-P)) |
| O(J3) |
| O(N it J 3) |
S 8 |
| O(J 2P(N-P)) |
| O(J 2P(N-P)) |
| O(J 2P(N-P)) |
S 9 |
| O(J 2(N-P)) |
| O(J 2P(N-P)) |
| O(J 2P(N-P)) |
S 10 |
| O(J(N-P)2) |
| o(j2(n-p)) |
| O(J 2(N-P)) |
S 11 |
| O(J(N-P)2) |
| O(J(N-P)2) |
| O(J(N-P)2) |
S 12 |
| O(J(N - P)(J+N-P)) |
| O(J(N-P)2) |
| O(J(N-P)2) |
S 13 | - | - |
| O(J(N-P)(J+N - P)) |
| O(J(N - P)(J+N-P)) |
Total | O(KJ 2 P 2N) + O(JN2) | O(2KJ 2 P 2N) + O(JN2) | O(2N it KJ 2P2N) + O(N it KJN2) |
Complexity of the GLRT-LQ detector with SCM, NSCM, and AML estimators (at least K ≥ JN)
Step | Task | Flops | Task | Flops | Task | Flops |
---|---|---|---|---|---|---|
S 1 |
| O(KJ2 N 2) |
| O(2KJ2 N 2) |
| O(N it KJ 3 N 3) |
S 2 |
| O(J 3 N 3) |
| O(J3 N 3) |
| O(J 3N3) |
S 3 |
| O(J2N2) |
| O(J2 N 2) |
| O(J2 N 2) |
S 4 |
| O(JN) |
| O(JN) |
| O(JN) |
S 5 |
| O(JN) |
| O(JN) |
| O(JN) |
S 6 |
| O(JN(JN+1)) |
| O(JN(JN+1)) |
| O(JN(JN+1)) |
Total | O(KJ2N2)≥O(J3N3) | O(2KJ 2 N 2)≥O(2J 3 N 3) | O(N it KJ 3 N 3)≥O(N it J 4 N 4) |
Note that the adaptive parametric implementations differ mainly in parameter estimation, they share identical steps in signal whitening and calculating the test statistic. The CG-PGLRT-NSCM detector is slightly more complex than the CG-PGLRT-SCM detector since it requires evaluating by utilizing for normalization. However, the CG-PGLRT-AML detector is the most complex detector among the three. Likewise, similar conclusion can be made for the adaptive versions of the GLRT-LQ detector. For a quick comparison, suppose KN > JP for the parametric detectors in Table 1. Clearly, it can be seen from Tables 1 and 2 that the parametric detectors can reduce the computational complexity of their non-parametric counterparts, especially when the spatial-temporal dimension JN is large.
4 Performance assessment
This section is devoted to the performance analysis of the proposed detectors: CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM, and CG-PGLRT-AML in terms of the probability of false alarm (Pfa) and the probability of detection (P d ). For compound-Gaussian environment, the closed-form expressions for both Pfa as well as P d are not available. Hence, we carry out the analysis via Monte-Carlo techniques based on 100/Pfa and 100/P d independent trials, respectively. In order to limit the computational burden, we set the probability of false alarm Pfa = 10-3 and the number of sensors J = 4 throughout the section.
Moreover, the following statements hold for this section:
where Г(·) is the gamma function, μ = E{τ} denotes the mean of the distribution, and ν is the shape parameter which provides a measure of noise spikiness. Lower ν means more spikes will appear in disturbance. Without loss of generality, we set μ = 1.
where ρ relates to the disturbance power, ρ s decides the spatial correlation, ρ t controls the temporal correlation, and {(m - n + l)ω} defines the phase of the correlation function (56). Here, we choose the value of ρ to satisfy tr{Q} = J.
where f ts and f td denote the target normalized spatial and Doppler frequencies, respectively, and the target spatial steering vector s(f ts ) is defined as . In simulation, we set f ts = 0 and f td = 0.25.
Relevant test parameters for the simulations
Parameters | Value |
---|---|
PFA | 10-3 |
P | 2 |
J | 4 |
N | 16, 32 |
v | 0.1, 0.5, 1, 4.5, 10 |
μ | 1 |
ρ i | 0.3 |
ρ s | 0.99 |
f ts | 0 |
f td | 0.25 |
N it | 3 |
Threshold versus shape parameter ( ν ) for fixed P fa .
P fa versus threshold for the CG-PGLRT with J = 4, N = 16, ρ s = 0.99, ρ t = 0.3, and ν = 0.1, 0.5, 1, 4.5.
P fa versus threshold for the CG-PGLRT-SCM with J = 4, N = 16, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 0.1, 0.5, 1, 4.5.
P fa versus threshold for the CG-PGLRT-NSCM with J = 4, N = 16, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 0.1, 0.5, 1, 4.5.
P fa versus threshold for the CG-PGLRT-AML with J = 4, N = 16, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 0.1, 0.5, 1, 4.5.
Figures 2, 3, 4, and 5 depict plots of Pfa versus threshold corresponding to several shape parameter values (ν = 0.1, 0.5, 1, 4.5), for the proposed CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM, and CG-PGLRT-AML, respectively. The curves for the CG-PGLRT-SCM in Figure 3 show much higher variability compared to the CG-PGLRT, CG-PGLRT-NSCM, and CG-PGLRT-AML, which confirms its lack of CFAR with respect to the texture variations. The Pfa plots for the CG-PGLRT, CG-PGLRT-NSCM, and CG-PGLRT-AML also validate their robust texture CFAR performance.
P d versus SINR of CG-PGLRT for P fa = 0.001, J = 4, N = 16, ρ s = 0.99, ρ t = 0.3, and ν = 0.1, 0.5, 1, 4.5, 10.
In Section 3.2, we combined the multichannel AR parameter estimation algorithm with three covariance matrix estimation strategies: SCM, NSCM, and AML estimators, and then gave three adaptive CG-PGLRT detectors, where the true multichannel AR parameters Q and A H are substituted with the estimated ones. Now various simulations are performed to evaluate the detection performance loss of the adaptive CG-PGLRT detectors with respect to the CG-PGLRT detector which possesses the perfect knowledge about the multichannel AR parameters Q and A H .
P d versus SINR of CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM, and CG-PGLRT-AML for P fa = 0.001, J = 4, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 0.5.
P d versus SINR of CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM and CG-PGLRT-AML for P fa = 0.001, J = 4, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 0.1.
P d versus SINR of CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM and CG-PGLRT-AML for P fa = 0.001, J = 4, K = 2, ρ s = 0.99, ρ t = 0.3, and ν = 4.5.
P d versus SINR of CG-PGLRT-SCM, CG-PGLRT-NSCM, CG-PGLRT-AML and GLRT-LQ with R_scm for P fa = 0.001, J = 4, N = 16, ρ s = 0.99, ρ t = 0.3, and ν = 0.5.
P d versus SINR of CG-PGLRT-SCM, CG-PGLRT-NSCM, CG-PGLRT-AML and GLRT-LQ with R_nscm for P fa = 0.001, J = 4, N = 16, ρ s = 0.99, ρ t = 0.3, and ν = 0.5.
P d versus SINR of CG-PGLRT-SCM, CG-PGLRT-NSCM, CG-PGLRT-AML when model order P of multichannel AR process used for computing test statistic is true ( P = 2) and overestimated (assuming P = 3) for P fa = 0.001, J = 4, N = 16, K = 3, ρ s = 0.99, ρ t = 0.3, and ν = 0.5.
5 Conclusions
In this article, multichannel signal detection problem in space-time colored compound-Gaussian environment is discussed. By exploiting the structural information about the disturbance space-time covariance matrix, we model the disturbance signal as a multichannel AR process to ease the training and computational burdens. Modeling the texture as an unknown deterministic parameter, we first derive the CG-PGLRT detector under the assumption that the multichannel AR parameters Q and A H are perfectly known. For the two-step GLRT design criterion, we combine the multichannel AR parameter estimation algorithm with three covariance matrix estimation strategies, i.e., sample covariance matrix (SCM), normalized sample covariance matrix (NSCM) and approximate ML (AML) estimator, and then obtain the adaptive versions of the CG-PGLRT detector by substituting the true multichannel AR parameters with their estimates. Finally, we show the CFAR behavior and detection performances of the proposed detectors: CG-PGLRT, CG-PGLRT-SCM, CG-PGLRT-NSCM, and CG-PGLRT-AML by Monte Carlo trials.
-
The CG-PGLRT-SCM detector has no texture CFAR property, while the CG-PGLRT, CG-PGLRT-NSCM, and CG-PGLRT-AML detectors ensure texture CFAR property.
-
The detection probability of the CG-PGLRT detector increases with increasing disturbance spikes (decreasing ν) at low SINR. However, at large SINR, we observe a degradation in performance as ν decreases.
-
For the limited-training case, the detection performance loss of the adaptive CG-PGLRT detectors with respect to the CG-PGLRT detector can be remedied by increasing temporal dimension N.
-
The CG-PGLRT-AML detector has the best detection performance in the heavy tailed disturbance (ν → 0). Thus, the CG-PGLRT-AML detector is the most suitable detector to implement the adaptive detection in the realistic spiky disturbance.
-
Compared to the covariance matrix based detector, the proposed model-based adaptive detectors have significantly decreased the training requirements and the computation complexity.
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their careful reading, helpful comments and constructive suggestions on improving the exposition of this article. This study is supported by the National Natural Science Foundation of China under Grant No. 11074270 and No. 61172166.
Authors’ Affiliations
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