## 3 System Model  
In our system model, a uniform linear array (ULA) with \(M\) elements are considered and \(D\) uncorrelated far-field signals are impinging on the ULA, where \(M>D\) is assumed. Figure 1 shows our considered ULA model. The received signals at time \(t\) can be expressed as  
\[\mathbf{x}(t)=\sum_{d=1}^{D}\mathbf{a}(\theta_{d})s_{d}(t)+\mathbf{n}(t), \tag{1}\]  
where \(\mathbf{x}(t)=[x_{1}(t),\dots,x_{M}(t)]^{\mathrm{T}}\) is the array output, \(\mathbf{a}(\theta_{d})\) is the steering vector for the signal \(d\) arriving at angle \(\theta_{d}\), \(s_{d}(t)\) is the impinging signal from the \(d\)th source at time \(t\), and \(\mathbf{n}(t)\) is the additive white Gaussian noise (AWGN). In the matrix form, (1) can be represented as  
\[\mathbf{X}=\mathbf{A}\mathbf{S}+\mathbf{N}, \tag{2}\]  
where \(\mathbf{X}\in\mathbb{C}^{M\times L}\), \(\mathbf{A}\in\mathbb{C}^{M\times D}\), \(\mathbf{S}\in\mathbb{C}^{D\times L}\) and \(\mathbf{N}\in\mathbb{C}^{M\times L}\), with \(L\) being the number of collected snapshots and \(\mathbb{C}\) represents the set of complex numbers. The steering matrix \(\mathbf{A}\) is  
\[\mathbf{A}=[\mathbf{a}(\theta_{1}),\mathbf{a}(\theta_{2}),\dots,\mathbf{a}(\theta_{D})], \tag{3}\]  
and the steering vector \(\mathbf{a}(\theta_{d})\) for the ULA can be written as  
\[\mathbf{a}(\theta_{d})=[1,e^{-j\frac{2\pi}{\eta}g\sin\theta_{d}},\dots,e^{-j(M-1) \frac{2\pi}{\eta}g\sin\theta_{d}}]^{\mathrm{T}}, \tag{4}\]  
where \(\eta\) is the wavelength of center frequency for signals, and \(\xi\) is the distance between the two adjacent elements of ULA. Figure 1: Our considered ULA model. If an infinitely large number of snapshots are collected, then \(\mathbf{n}(t)\) follows the AWGN perfectly, so the covariance matrix of the array output \(\mathbf{R}_{xx}\) can be described as  
\[ \begin{split}\mathbf{R}_{xx}&=\lim_{L\to\infty}\frac{1}{ L}\sum_{t=1}^{L}\mathbf{x}(t)\mathbf{x}^{\mathbf{H}}(t)\\ &=\mathbf{A}\mathbf{R}_{ss}\mathbf{A}^{\mathbf{H}}+\sigma_{\text{N}}^{2}\mathbf{I }_{M},\end{split}  \tag{5}\]  
where \(\mathbf{R}_{\text{ss}}\) is the covariance matrix of the impinging signals, \(\sigma_{\text{N}}^{2}\) is the variance of the noises and \(\mathbf{I}_{M}\) is the \(M\times M\) identity matrix. From [15], the eigenvalues of \(\mathbf{R}_{xx}\) can be represented in ascending order as  
\[\lambda_{1}=\cdots=\lambda_{M-D}<\lambda_{M-D+1}\leq\cdots\leq\lambda_{M}, \tag{6}\]  
where the noise-subspace eigenvalues are  
\[\lambda_{1}=\cdots=\lambda_{M-D}=\sigma_{\text{N}}^{2}, \tag{7}\]  
and the signal-subspace eigenvalues are  
\[\lambda_{M-D+1}\leq\cdots\leq\lambda_{M}. \tag{8}\]  
The gaps of eigenvalues \(\Delta\lambda_{i}\) are defined as  
\[\Delta\lambda_{i}\triangleq\lambda_{i+1}-\lambda_{i}\quad(i=1,\ldots,M-1). \tag{9}\]  
From (6) and (9), we have  
\[ \begin{cases}\Delta\lambda_{i}=0,&i=1,\ldots,M-D-1,\\ \Delta\lambda_{i}>0,&i=M-D,\\ \Delta\lambda_{i}\geq 0,&i=M-D+1,\ldots,M-1.\end{cases}  \tag{10}\]  
The first row of (10) represents the gaps between the noise-subspace eigenvalues; this paper calls these gaps \"NN gaps\" (noise-noise subspace eigenvalues gaps). The second row of (10) represents the gap between the greatest noise-subspace eigenvalue and the smallest signal-subspace eigenvalue; this paper call this gap \"NS gap\" (noise-signal subspace eigenvalues gap). The third row of (10) represents the gaps between the signal-subspace eigenvalues; this paper call these gaps \"SS gaps\" (signal-signal subspace eigenvalues gaps). In practice, the ideal covariance matrix \(\mathbf{R}_{xx}\) cannot be obtained.