## 4 Performance Evaluation
### Simulation Experiment  
First, we set the number of system channels to 8, the number of training samples to 16, the dimension of signal subspace to 3, and the expected PFA to \(10^{-3}\). A total of \(100/\text{PFA}\) Monte Carlo simulations are run to derive the detection threshold, and \(10^{4}\) independent repeated tests are used to calculate the PD. In the simulation experiment, the speckle component follows a zero-mean complex Gaussian random process, and its covariance matrix is [31]  
\[\left[\mathbf{R}\right]_{j,k}=\delta^{|j-k|},1\leq j,k\leq N \tag{38}\]  
where \(\delta\) denotes the one-lag correlation coefficient, set to 0.95. In this paper, the SCR is defined as [32]  
\[\text{SCR}_{\text{out}}{=}\xi^{\text{H}}\mathbf{H}^{\text{H}}\mathbf{R}^{-1} \mathbf{H}\xi \tag{39}\]  
In the experiment, we set \(\xi=\alpha\xi_{0}\), where \(\mathbf{H}\) and \(\xi_{0}\) are randomly generated, but fixed after generation. \(\alpha\) is the set regulatory factors to meet the given SCR as  
\[\alpha=\sqrt{\frac{\text{SCR}_{\text{out}}}{\xi_{0}^{\text{H}}\mathbf{H}^{ \text{H}}\mathbf{R}^{-1}\mathbf{H}\xi_{0}}} \tag{40}\]  
In addition, the detection performance when the signal mismatch occurs is also discussed in this paper. For subspace signals, signal mismatch refers to the fact that the actual signal \(\mathbf{s}_{0}\) is not entirely located in the assumed subspace \(\mathbf{H}\). For the purpose of quantifying the degree of signal mismatch, we employ the generalized cosine square [33]  
\[\text{cos}^{2}\phi=\frac{\mathbf{s}_{0}^{\text{H}}\mathbf{R}^{-1}\mathbf{H}( \mathbf{H}^{\text{H}}\mathbf{R}^{-1}\mathbf{H})^{-1}\mathbf{H}^{\text{H}} \mathbf{R}^{-1}\mathbf{s}_{0}}{\mathbf{s}_{0}^{\text{H}}\mathbf{R}^{-1} \mathbf{s}_{0}} \tag{41}\]  
We consider K distribution clutter, whose texture component follows a gamma distribution with the PDF as [18]  
\[p_{\tau}(\tau)=\frac{1}{\Gamma(\varrho)}\Big{(}\frac{\varrho}{u}\Big{)}^{ \varrho}\tau^{v-1}\exp\Big{(}-\frac{\varrho}{u}\tau\Big{)} \tag{42}\]  
where \(v\) is the shape parameter, representing the degree of the clutter trailing, and \(u\) is the scale parameter, representing the size of clutter power. Note that the smaller the shape parameter, the heavier the clutter trailing degree and the stronger the non-Gaussian characteristics. In the experiment, we set \(v\) to 1. Under the condition of a given texture component \(\tau\), the condition PDF of the clutter amplitude is  
\[f(r|\tau)=\frac{2r}{\tau}\exp\left(-\frac{r^{2}}{\tau}\right) \tag{43}\]  
From the full probability formula, the PDF of the clutter amplitude is  
\[f(r)=\int_{0}^{\infty}p(\tau)f(r|\tau)d\tau \tag{44}\]  
Substituting (42) and (43) into (44) results in  
\[ \begin{split} f(r)=&\int_{0}^{\infty}p(\tau)f(r| \tau)d\tau=\int_{0}^{\infty}\frac{v^{v}}{\Gamma(v)u^{v}}\tau^{v-1}\frac{2r}{ \tau}\exp\left(-\frac{v}{u}\tau-\frac{r^{2}}{\tau}\right)d\tau\\ =&\frac{2}{2^{v-1}\Gamma(v)\sqrt{u}}\bigg{(}\frac{2 r}{\sqrt{u}}\bigg{)}^{v}K_{v-1}\bigg{(}\frac{2r}{\sqrt{u}}\bigg{)}\end{split}  \tag{45}\]where \(K_{\texttt{P}}(\cdot)\) represents the modified Bessel function of the second kind and \(v\)-th order. The cumulative distribution function (CDF) of the K distribution can be obtained by integrating (45) as  
\[F(r)=1-\frac{1}{2^{v-1}\Gamma(v)}\bigg{(}\frac{2r}{\sqrt{u}}\bigg{)}^{v}K_{v} \bigg{(}\frac{2r}{\sqrt{u}}\bigg{)} \tag{46}\]  
In Figure 2, we use simulated data to compare the variation in PDs of the proposed detectors with respect to SCR; we can find that AE-GLRT, AE-Rao, AE-Wald, and AE-Gradient have similar PDs, but the performance of AE-Rao is slightly better than the other three proposed detectors. Specifically, when the SCR = 27 dB, AE-Rao can detect almost exactly for the given PFA. However, the PD of AE-Durbin is relatively low, when the SCR=27 dB, PD is close to 0.91; when the SCR=35 dB, PD is close to 1. Figure 3 depicts the variation in PDs when signal mismatch occurs at SCR = 35 dB.