A common problem in signal processing is detecting a point or distributed target in a background of colored noise with unknown statistical properties^[1,2,3,4]. This has been a topic of ongoing interest more than three decades. According to different design criteria,a large number of approaches are proposed under various scenarios. The most famous detectors for point-like target detection include Kelly’s Generalized Likelihood Ratio Test (KGLRT)^[5],Adaptive Matched Filter (AMF)^[6],De Maio’ Rao (DMRao) test^[7],Adaptive Coherence Estimator (ACE)^[8],Adaptive Beamformer Orthogonal Rejection Test (ABORT)^[9],Whitened-ABORT (W-ABORT)^[10],Double-Normalized AMF (DN-AMF)^[11,12],and so on. For these detectors,it usually assumes that exact or rough knowledge about the signal steering vector is known a priori. However,in some practical applications,the aforementioned knowledge of the signal may not be available. Fewer papers consider the problem of adaptive detection with no information about the signal steering vector. In particular,in Ref. ^[13] under the assumption of completely unknown signature of the signal,the GLRT for a point target is proposed,which,for convenience,is denoted as the Adaptive Energy Detector (AED) in this paper. Moreover,the problem of detection of a distributed target with unknown signal steering vector is exploited in Ref. ^[14],and several GLRT-based detectors are proposed therein. The corresponding Rao and Wald tests are both developed in Ref. ^[15].

Another kind of problem usually encountered in practice is detection of a target in the presence of signal mismatch. That is to say,the actual signal steering vector is not aligned with the presumed one. This phenomenon occurs due to imperfect array calibration,spatial multipath,pointing errors,and so on. In order to design selective (less tolerant to signal mismatch) detectors,the ABORT and W-ABORT are proposed in Ref. ^[9] and Ref. ^[10],respectively. Therein,the hypothesis test is modified by adding a deterministic fictitious signal under the null hypothesis. This fictitious signal is assumed to be orthogonal to the nominal signal in the quasi-whitened^[9] or truly whitened space^[10]. Another method to devise selective detectors is adding a noise-like fictitious interference both under the null and alternative hypotheses^[11,12,16]. The corresponding GLRT is found to have the same form as the ACE^[16],while the Rao test and Wald test coincide with each other,referred to as the DN-AMF^[11]. The ACE,ABORT,W-ABORT,and DN-AMF all have enhanced mismatched signal discrimination capabilities. However,they suffer certain performance loss for perfectly matched signals. In order to overcome this problem,the two-stage detectors^[17,18,19] and tunable detectors^[20,21,22] are proposed in the literature. The two-stage detectors^[17,18,19] are formed by cascading two detectors with different abilities in terms of mismatched signals rejection. One declares the presence of a target only when data survive both detection thresholds.

However,the abilities of robustness to and rejection of a mismatched signal of the two-stage detectors are limited by these capabilities of their individual cascading detectors. In contrast,the tunable detectors^{[20,21,22,23]} are of considerable flexibility in controlling the degree to which the mismatched signal is rejected through tuning one or more tunable parameters. However,the tunable detector in Ref. ^[20] has limited abilities of mismatched signal rejection,while the tunable detectors in Refs. ^[21,22,23] have limited robustness.

In this paper,we move a further step toward the design of adaptive detectors. The main contributions lie in the following two aspects. One is that we reconsider the problem of detecting a target of completely unknown signal steering vector. We derive the Rao and Wald tests,which are found to coincide with the GLRT in Ref. ^[13],i.e.,the AED. Moreover,we give the exact statistical distribution of the AED. As a consequence,the Probabilities of Detection (PD) and False Alarm (PFA) are calculated easily. Another contribution is that by inspection of the similarity of the AED with several existing adaptive detectors,we propose a two-parameter tunable detector in an ad hoc manner. The novel tunable detector possesses three distinct features. First,it is very general and covers the KGLRT,AMF,ABORT,W-ABORT,and AED as its five special cases. Second,the detection performance of the novel detector for matched signals is superior to that of existing detectors in some situations. Third,the novel tunable detector has more flexible ability of controlling the degree to which the mismatched signal is rejected. This function of the novel tunable detector is controlled by two tunable parameters.

The rest of the paper is organized in the following manner. Section 2 gives the problem formulation,while Section 3 shows that the Rao and Wald tests are equivalent to the AED,and gives its exact statistical distribution as well as its PD and PFA. Section 4 exploits the novel tunable detector. Numerical examples are given in Section 5. Finally,Section 6 concludes the paper.

For simplicity the radar antenna is assumed to be uniformly-spaced linear array antenna,consisting of N elements. We want to discriminate between hypothesis H₀ that the data in the cell under test (primary data),say x,contains only disturbance and hypothesis H₁ that x contains disturbance n and useful signal s_t=as. a is the unknown signal amplitude and s is the presumed signal steering vector. The disturbance n,including clutter and noise,is Gaussian distributed,with a mean zero and a covariance matrix R. R is positive definitive Hermitian but unknown. To estimate R,it is customary to assume that a set of training data (secondary data),x_l,l=1,2,···,L,is available. The secondary data are Independent and Identically Distributed (IID),contain no useful signal but for the noise n_l. n_l shares the same statistical properties as the noise n in the primary data. To sum up,the detection problem can be formulated as the following binary hypothesis test:

For convenience,let S=XX^H,which is L times the sample covariance matrix (SCM),with X=[x₁,x₂,···,x_L].

In this section,we first derive the Rao and Wald tests when the signal steering vector s is unknown. These two detectors are both found to be identical to the AED,which is proposed according to the GLRT criterion and is given by^[13]

Then we give the exact statistical distribution of the AED,according to which we derive the PD and PFA.

Let ${{θ}}$ is the (N+N²)×1-dimensional parameter vector,given by

where${{θ} _{r}}$=s_t,${{θ} _{s}}$=vec(R),the notations (·)^T and vec(·) stand for the transpose and vectorization operators,respectively.

The Fisher Information Matrix (FIM) for the parameter vector ${{θ}}$ is^[24]

where the superscript (·)^* denotes the conjugate. Eq. (4) can be partitioned as

Then the Rao test is found to be^[24]

where ${\hat {θ} _0}$is the Maximum Likelihood Estimate (MLE) of ${{θ}}$ under hypothesis H₀.

To derive the Rao test we need the joint Probability Density Function (PDF) of x and X under hypothesis H₀,which is given by

where the symbols (·)^H,tr(·),and $| \cdot |$stand for the conjugate transpose,trace,and determinant of a square matrix. Taking the derivative of the logarithm of Eq. (7) with respect to (w.r.t.) s_t,yields

Moreover,taking the derivative of Eq. (8) w.r.t. s^H_t produces

Since ${{I}_{{{θ} _r},{{θ} _r}}} = - {{{\partial ^2}\ln f} \mathord{\left/ {\vphantom {{{\partial ^2}\ln f} {\partial {{s}_t}\partial {s}_t^{\rm{H}}}}} \right. } {\partial {{s}_t}\partial {s}_t^{\rm{H}}}}$^[24],we have the result

Substituting of Eq. (8) and Eq. (10) into Eq. (6),along with the fact that s_t under hypothesis H₀ reduces to the null vector,results in the Rao test for given R

It is easy to show that the MLE of R under hypothesis H₀ is${\hat R_0} = $(S+xx^H)/(L+1). Inserting it into Eq. (11),and dropping the constant scalar,after some algebra,yields the final Rao test

which is statistically equivalent to

which in turn is the exact the AED in Eq. (2).

The Wald test is given by^[24]

where${\hat {θ} _{{r_1}}}$is the MLE of${{θ} _{r}}$under hypothesis H₁,and${{θ} _{{r_0}}}$is the value of${{θ} _{r}}$under hypothesis H₀. Moreover,

One can verify that${{I}_{{{θ} _r},{{θ} _s}}}$is a null matrix. It follows ${\left\{ {{{[{{I}^{ - 1}}({{\hat {θ} }_1})]}_{{{θ} _r},{{θ} _r}}}} \right\}^{ - 1}} = {{I}_{{{θ} _r},{{θ} _r}}}$. Setting Eq. (8) to be zero,we have the MLE of s_t under hypothesis H₁ as${\hat {s}_t} = {x}$. Moreover,${{θ} _{{r_0}}} = \mathbf{0}$. Gathering these results,along with Eq. (10),we have the Wald test,for fixed R

Clearly,the MLE of R under hypothesis H₁ is ${\hat {R}_1} = {S} {\left/ {\vphantom {{S} {(L + 1)}}} \right. {(L + 1)}}$. Plugging ${\hat {R}_1}$into Eq. (16),and dropping the constant scalar,leads to the final Wald test

which is exactly the AED in Ref. ^[13].

Note that the derivations of the Rao test and Wald test above are parallel to those in Ref. ^[15]. Actually,the Rao and Wald tests in this paper are special cases of the corresponding Rao and Wald tests in Ref. ^[15]. Specifically,the Rao test in Eq. (12) is the Rao test in Eq. (11) in Ref. ^[15] when M=1 therein,while the Wald test in Eq. (17) is the Wald test in Eq. (17) in Ref. ^[15] for the case of M=1. However,we would like to present the detailed derivations above,since utilization of the Rao or Wald test to design a detector is not as common as the GLRT criterion. Moreover,it is also noteworthy that the two-step GLRT,two-step Rao test,and two-step Wald test are all identical to Eq. (13). For the sake of brevity,the derivations are not given here. Finally,it is also worthy to point out that the AED is Uniformly Most Powerful Invariant (UMPI)^[25].

Eq. (13) can be recast as

where $\bar {x} = {{R}^{ - 1/2}}{x}$and $\bar {S} = {{R}^{ - 1/2}}{S}{{R}^{ - 1/2}}$. R^1/2 is the square-root matrix of R and R^-1/2=( ${{\partial \ln {f_1}({x},{X})} \mathord{\left/ {\vphantom {{\partial \ln {f_1}(x,X)} {\partial {s_t}}}} \right. } {\partial {{s}_t}}} = {[{({x} - {{s}_t})^{\rm{H}}}{{R}^{ - 1}}]^{\rm{T}}}$ (8)

Moreover,taking the derivative of Eq. (8) w.r.t. s^H_t produces

Since ${{I}_{{{θ} _r},{{θ} _r}}} = - {{{\partial ^2}\ln f} \mathord{\left/ {\vphantom {{{\partial ^2}\ln f} {\partial {{s}_t}\partial {s}_t^{\rm{H}}}}} \right. } {\partial {{s}_t}\partial {s}_t^{\rm{H}}}}$^[24],we have the result

Substituting of Eq. (8) and Eq. (10) into Eq. (6),along with the fact that s_t under hypothesis H₀ reduces to the null vector,results in the Rao test for given R

It is easy to show that the MLE of R under hypothesis H₀ is${\hat R_0} = $(S+xx^H)/(L+1). Inserting it into Eq. (11),and dropping the constant scalar,after some algebra,yields the final Rao test

which is statistically equivalent to

which in turn is the exact the AED in Eq. (2).

The Wald test is given by^[24]

where${\hat {θ} _{{r_1}}}$is the MLE of${{θ} _{r}}$under hypothesis H₁,and${{θ} _{{r_0}}}$is the value of${{θ} _{r}}$under hypothesis H₀. Moreover,

One can verify that${{I}_{{{θ} _r},{{θ} _s}}}$is a null matrix. It follows ${\left\{ {{{[{{I}^{ - 1}}({{\hat {θ} }_1})]}_{{{θ} _r},{{θ} _r}}}} \right\}^{ - 1}} = {{I}_{{{θ} _r},{{θ} _r}}}$. Setting Eq. (8) to be zero,we have the MLE of s_t under hypothesis H₁ as${\hat {s}_t} = {x}$. Moreover,${{θ} _{{r_0}}} = \mathbf{0}$. Gathering these results,along with Eq. (10),we have the Wald test,for fixed R

Clearly,the MLE of R under hypothesis H₁ is ${\hat {R}_1} = {S} {\left/ {\vphantom {{S} {(L + 1)}}} \right. {(L + 1)}}$. Plugging ${\hat {R}_1}$into Eq. (16),and dropping the constant scalar,leads to the final Wald test

which is exactly the AED in Ref. ^[13].

Note that the derivations of the Rao test and Wald test above are parallel to those in Ref. ^[15]. Actually,the Rao and Wald tests in this paper are special cases of the corresponding Rao and Wald tests in Ref. ^[15]. Specifically,the Rao test in Eq. (12) is the Rao test in Eq. (11) in Ref. ^[15] when M=1 therein,while the Wald test in Eq. (17) is the Wald test in Eq. (17) in Ref. ^[15] for the case of M=1. However,we would like to present the detailed derivations above,since utilization of the Rao or Wald test to design a detector is not as common as the GLRT criterion. Moreover,it is also noteworthy that the two-step GLRT,two-step Rao test,and two-step Wald test are all identical to Eq. (13). For the sake of brevity,the derivations are not given here. Finally,it is also worthy to point out that the AED is Uniformly Most Powerful Invariant (UMPI)^[25].

Eq. (13) can be recast as

where $\bar {x} = {{R}^{ - 1/2}}{x}$and $\bar {S} = {{R}^{ - 1/2}}{S}{{R}^{ - 1/2}}$. R^1/2 is the square-root matrix of R and R^-1/2=(R^1/2)^-1.

One can easily verify that $\bar {x}$,under hypo-thesis H₁,is ruled by a complex circular Gaussian distribution,with a mean aR^-1/2s and a covariance matrix I_N,written symbolically as $\bar {x} \sim {\cal C}{{\cal N}_N}(a{{R}^{ - 1/2}}{s},{{I}_N})$. Moreover,$\bar {S}$is distributed as a central complex Wishart distribution,with N Degrees Of Freedom (DOFs) and a scale matrix I_N,both under hypotheses H₁ and H₀,denoted as$\bar {S} \sim {\cal C}{{\cal W}_N}(L,{{I}_N})$. Moreover,$\bar {x}$and$\bar {S}$are statistically independent on each other. Using the theorem 3.2.13 of Ref. ^[26],we can obtain the distribution of t_AED,which is a complex noncentral F-distribution,with a noncentrality parameter

and N and L-N+1 DOFs,denoted as

where r is defined as the Signal-to-Clutter-plus-Noise Ratio (SCNR).

Hence,according to the Cumulative Distribution Function (CDF) of the noncentral complex F-distribution^[27],we can obtain the PD of the AED,described as

where IG_k+1(a) is the incomplete Gamma function,with the expression

IG_k+1(a)=e^-a $ \cdot \sum\limits_{m = 0}^k $a^m/m!

Note that Eq. (21) is equal to the PD of the AED derived in Ref. ^[13]. Precisely,replacing k and$\eta$by i-1 and y₀-1,respectively,in Eq. (21) does not change the result but leads to a form identical to the PD of the AED in Eq. (10) in Ref. ^[13].

Setting r=0 in Eq. (21) yields the PFA

Remarkably,the PFA in Eq. (22) is more convenient than the PFA given in Eq. (3) in Ref. ^[13] to numerically calculate the detection threshold,since Eq. (22) only contains finite-sum term,whereas Eq. (3) in Ref. ^[13] is involved in a hypergeometric function.

Eq. (21) and Eq. (22) are for the case of the deterministic target. For completeness,we also consider the case of the fluctuation target. Precisely,we consider the Swerling I target model. Since the PFA,given in Eq. (22),does not contain the target property. It is the same for the random and deterministic target. Hence,we only need to calculate the PD for the fluctuation target. In this case,the amplitude of the signal is modeled as a complex Gaussian random variable^[13]. Accordingly,the SCNR r has an exponential density function^[13] given by

In light of the identity $\int_0^\infty {{\rho ^n}{e^{ - \rho c}} {d} \rho } ={n!} / { {{{c^{(n + 1)}}}}} $. ,we have

where IG_k+1(a) is the incomplete Gamma function,as mentioned above.

Averaging Eq. (21) over Eq. (23) and using Eq. (24),we have the PD of the AED for the Swerling I target

For comparison purpose,we briefly derive the PDs of the KGLRT,AMF,DMRao,and ACE for the Swerling I target in the following. The analytical expressions for the PDs of the KGLRT,AMF,ACE,and DMRao are well-known (see,e.g.,Refs. ^[7,17]). For simplicity,they are not repeated here. Averaging these expressions over Eq. (23),along with Eq. (24),we have the corresponding conditional PDs

(28)

where

Note that Eq. (25)-Eq. (29) hold both for matched and mismatched signals. The signal mismatch occurs when the nominal steering vector,such as the array steering,s is not aligned with the actual signal steering vector s₀. To quantify the amount of the signal mismatch,we resort to the cosine squared of the angle f between s and s₀ in the whitened space. That is to say,we have

When the signal is deterministic,the PDF of b under hypothesis H₁ is given by (A2-23) in
Ref. ^[27],i.e.,

where$\delta _{{\beta _\phi }}^2 = \rho {\sin ^2}\phi $,sin²f=1-cos²f,and f_x,y(b) is the PDF of b under hypothesis H₀^[27].

When the phenomenon of signal mismatch exists,averaging Eq. (32) over Eq. (23) yields

which is the PDF of the quantity b in Eq. (30) under hypothesis H₁ for the Swerling I target. Finally,averaging Eq. (25)-Eq. (29) over Eq. (33) results in the final PDs for the Swerling I target.

4    Novel Tunable Detector for the Case of Signal Mismatch

The AED does not assume any information about the signal steering vector. Furthermore,it does not contain the actual signal steering. Hence,the AED is not affected by the signal mismatch. According to this characteristic of the AED,as well as the relationship between the AED and other detectors,we introduce a two-parameter tunable detector when signal mismatch exists in this section.

The AMF for the problem of Eq. (1) is found to be^[6]

while the KGLRT is given by^[5]

which has the equivalent form of

(36)

Moreover,the ABORT^[9] and W-ABORT^[10] are found to be

and

respectively.

By comparing the aforementioned detectors with the AED in Eq. (13),we introduce the following novel tunable detector

which is referred to as the MEAWK. The letters in the acronym MEAWK stand for the AMF,AED,ABORT,W-ABORT,and KGLRT,respectively.

Notice that when a=g=0,Eq. (39) degenerates into the AMF; when a=0 and g=1,it reduces to the ABORT; when a=1 and g=0,it is equal to the AED; when a=g=1,it is statistically equivalent to KGLRT; when a=1 and g=2,it becomes the W-ABORT.

A quick comment on the MEAWK is that it possesses the Constant False Alarm Rate (CFAR) property,because the AMF is CFAR,and the quantities b in Eq. (30) and x^HS^-1x are both essentially not related to the covariance matrix R^[6,17]. Moreover,the MEAWK nearly has the same computational complexity with the AMF,AED,ABORT,W-ABORT,and KGLRT,since the main computational burden for them is the same,and it is the matrix inversion of S.

In light of Eq. (35) and Eq. (30),we have the important stochastic representation of the MEAWK

Hence,the PD and PFA of the MEAWK are

and

respectively,where$\eta$ is the detection threshold of the MEAWK to be assigned to ensure a required PFA.

The exact PD and PFA can be calculated ac-cording to the solution of the equation$ \eta $b^1-g =
b+a(1-b) (with b being the sole unknown) and the four cases: 1) 0≤g<1 and 0≤[b+a(1-b)]/$ \eta $<1,2) 0≤g<1 and [b+a(1-b)]/$ \eta \,$≥1,3)$ \gamma \,$≥1 and 0≤[b+a(1-b)]/$ \eta $<1,and 4)$ \gamma \,$≥1 and [b+a(1-b)]/$ \eta \,$≥1. However,since the above equation does not admit a close-form solution in general,the analytical PD and PFA are intractable except for some special cases. Hence,we resort to the Monte Carlo simulations to evaluate the detection performance of the MEAWK,both for matched and mismatched signals. For simplicity,only the deterministic signal is considered. 5    Numerical Examples

In this section,we first compare the detection performance of the AED with other relative detectors. Then we evaluate the detection performance of the MEAWK.

In this subsection,we compare the detection performance of the AED with some well-known detectors,namely,the KGLRT,AMF,ACE,and DMRao. Precisely,we compare their PDs for the deterministic,random,matched,and mismatched signals. As for the random signal,only the Swerling I fluctuation model is considered.

The noise is modeled as an exponentially correlated random vector with one-lag correlation coefficient. That is to say,the (i,j)-th element of the covariance matrix R is given by s²e^│i-j│,i,j=1,2,···,N,with s²=1 and e=0.95. Moreover,to reduce the computational complexity,the PFA is set to PFA=10^-3,and we choose N=12 and L=2N in all figures. For the Monte Carlo simulations,to evaluate the PD and the threshold necessary to ensure a prescribed PFA,we resort to 2×10⁴ and 200/PFA independent data realizations,respectively.

Fig. 1 shows the PD of the AED for matched signals,also in comparison with the KGLRT,AMF,ACE,and DMRao. The signal is assumed to be known up to an unknown scaling factor,i.e.,s_t=as₀,where the actual signal steering vector s₀ is known a priori and the signal amplitude a is unknown. The subscripts “TH” and “MC” in the legend stand for theoretical results and Monte Carlo simulation result,respectively. It is seen that the theoretical results match the Monte Carlo simulation results pretty well. Moreover,when the signature of the signal is known a priori,the PD of the AED is lower than those of the other detectors. This seems reasonable,since the knowledge of the signature of the signal is not taken into account by the AED. Particularly,the KGLRT has the highest PD. Furthermore,compared with the case that the deterministic signal model,it needs additional SCNR to achieve the same PD for a certain detector for random signals.

图 1(Fig. 1)

Fig. 1 PDs of the KGLRT,AMF,DMRao,ACE,and AED for matched signals

Fig. 2 depicts the detection performance of the detectors for mismatched signals. In Fig. 2,the signal mismatch quantity cos²f,defined in Eq. (31),is set to be 0.7. It is shown that under this system parameters,the AED exhibits the highest PD both for random and deterministic signal. Note that in Fig. 2(b) only the Monte Carlo results are provided.

图 2(Fig. 2)

Fig. 2 PDs of the KGLRT,AMF,DMRao,ACE,and AED for mismatched signals

In this subsection,we compare the detection performance of the MEAWK with its natural competitors,both for matched and mismatched signals. The operational parameters are the same as those in the above subsection.

We use Eq. (40) rather than Eq. (39) to perform Monte Carlo simulations. A remarkable advantage of Eq. (40) is that it does not need any matrix multiplication or matrix inversion operation. Compared to Eq. (39),this reduces the computational complexity significantly.

Fig. 3 shows the PD of the MEAKW with different tunable parameters,also in comparison with the KGLRT (i.e.,the plan shown in Fig. 3). For convenience,the values of the PD of the MEAWK less than 0.8 are set to be 0.8. It is shown that with many pairs of tunable parameters a and g,the MEAWK can provide a higher PD than the KGLRT.

图 3(Fig. 3)

Fig. 3 PDs of the MEAWK with different tunable parameters for matched signals. SCNR=16 dB

The detection performance of the MEAWK,with specific g and a,under different SCNRs is displayed in Fig. 4. The results highlight that the MEAWK can provide a higher PD than the other detectors for all the values of the SCNR in the interval between 8 dB and 16 dB.

图 4(Fig. 4)

Fig. 4 PD versus SCNR for mismatched signals. a=0.6 and g=0.8

Fig. 5 demonstrates the detection perfor-mance of the detectors for mismatched signals. The amount of the signal mismatch is quantified by Eq. (31). The results in Fig. 5 indicate that the MEAWK,with appropriate tunable parameters,can be either more robust or more selective than the existing detectors,depending on the system requirement.

图 5(Fig. 5)

Fig. 5 PD versus cos2f for mismatched signals. SCNR=20 dB

In the first part of the paper,we have considered the problem of detecting a target,with completely unknown signature. We derive the Rao and Wald tests,which coincide with the GLRT,i.e.,the AED. Moreover,we derive the exact statistical distribution of the AED. According to this distribution,one can easily calculate

the PD and PFA. Moreover,it is shown that the AED is more robust to the signal mismatch than other detectors. At the second part of the paper,we propose a two-parameter tunable detector,i.e.,the MEAWK. This idea is inspired by the relationship between the AED and other detectors. The MEAWK is of considerable flexibility in governing the level to which the mismatched signal is rejected. With proper tunable parameters,it can exhibit improved detection performance w.r.t. its natural competitors for matched signals. In addition,it can also become more robust or more selective than the existing detectors for mismatched signals. As for how to select the tunable parameters,a rule of thumb is that if a selective detector is needed to reject mismatched signals,the values of the tunable parameters should be set relatively large^[23]. An instance is that a radar workswin tracking mode. On the other hand,if a robust detector is needed,the values of the tunable parameters should be set as small as possible. An example is that a radar operates in searching mode.