## Appendix B Proof of Lemma 1
If \(\wp(\boldsymbol{\beta};\kappa)=0_{N_{\Omega}M}\) for some constant \(\kappa\) (\(\kappa\
eq 0\)), then \(\boldsymbol{\varphi}(\alpha_{i})\) and \(\boldsymbol{\varphi}\Big{(}\alpha_{i}^{{}^{\prime}}\Big{)}\) would be correlated, and \(\mathbf{G}(\boldsymbol{\alpha}_{\hbar})\) would not be full column rank. Based on this, \(\mathrm{P}\) is the probability of describing \(\mathbf{G}(\boldsymbol{\alpha}_{\hbar})\) as full column rank can be expressed as: \[\mathrm{P}=1-\mathrm{Pr}\{\wp(\boldsymbol{\beta};\kappa)=0\};\] (13)  
\[\mathrm{P}=1-\frac{\int\ A\ \mathbb{I}_{\wp(\boldsymbol{\beta};\kappa)=0}( \boldsymbol{\beta};\kappa)d(\boldsymbol{\beta},\kappa)}{\int\ A\ \mathbb{I}d( \boldsymbol{\beta},\kappa)}, \tag{14}\]  
here, if \(\wp(\boldsymbol{\beta};\kappa)=0\), for some \(\kappa\
eq 0\), \(\mathbb{I}_{\wp(\boldsymbol{\beta};\kappa)=0}(\boldsymbol{\beta};\kappa)=1\); otherwise, it is \(0\). Additionally, \(\mathbb{A}=\Big{\{}\big{[}\boldsymbol{\beta}^{\top},\kappa\big{]}^{\top}:s_{1}, \cdots s_{M}\in(s_{U},s_{D}],\omega_{1},\cdots,\omega_{N_{\Omega}}\in(0, \omega_{B}],\kappa\
eq 0\Big{\}}\), where \(\omega_{B}\) represents the utilized measurement bandwidth. Thus, \(\wp(\boldsymbol{\beta};\kappa)=0\) represents a set of points that have a finite solution. Hence, the Lebesgue integral \(\int\ A\ \mathbb{I}_{\wp(\boldsymbol{\beta};\kappa)=0}(\boldsymbol{\beta};\kappa)d( \boldsymbol{\beta},\kappa)=0\). Consequently, for the case of \(|\hbar|=2\), the probability of \(\mathbf{G}(\boldsymbol{\alpha}_{\hbar})\) being full column rank is \(1\). Therefore, for \(|\hbar|=1\) and \(2\), there exists a regularization constant \(C_{(2)}\) and a \(2\)-RIP constant \(\delta_{(2)}\) (where \(\delta_{(2)}\in(0,1)\)), such that \(\mathbf{G}(\boldsymbol{\alpha}_{\hbar})\) is full column rank. Thus, inequality in Lemma 1 is proven. Next, we specify the minimum possible values for the RIP constant \(\delta_{(2)}\) and the regularization constant \(C_{(2)}\) to satisfy the equality in Lemma 1. Let \(\lambda_{max}\) and \(\lambda_{min}\) represent the maximum and minimum possible eigenvalues of \((\mathbf{G}(\boldsymbol{\alpha}_{\hbar}))^{\mathrm{H}}\mathbf{G}( \boldsymbol{\alpha}_{\hbar})\) for \(\forall\hbar\subset\{1,\cdots,I\}\)and \(\left|\hbar\right|=1,2\). Furthermore, let \(\lambda\) denote a general eigenvalue of \((\mathbf{G}(\mathbf{\alpha}_{\hbar}))^{\mathbf{H}}\mathbf{G}(\mathbf{\alpha}_{\hbar})\). Then we have:  
\[\frac{\lambda}{C_{(2)}}\in\left[\frac{\lambda_{min}}{C_{(2)}},\frac{\lambda_{ max}}{C_{(2)}}\right], \tag{10}\]  
we can impose the constraint \(\delta_{(2)}\in(0,1)\)[35] to minimize \(\delta_{(2)}\). The optimal solution for \(C_{(2)}^{\star}\) can be expressed as:  
\[C_{(2)}^{\star}=\underset{C_{(2)}}{min}\ max\Bigg{\{}1-\frac{\lambda_{min}}{C_ {(2)}},\frac{\lambda_{max}}{C_{(2)}}-1\Bigg{\}}; \tag{11}\]  
\[\text{s.t.}\ 0<1-\frac{\lambda_{min}}{C_{(2)}}<1; \tag{12}\]  
\[0<\frac{\lambda_{max}}{C_{(2)}}-1<1, \tag{13}\]  
according to Equations (12) and (13), the range of \(C_{(2)}\) allowed by the RIP condition in Lemma 1 is \((max\{\lambda_{max}/2,\lambda_{min}\},\lambda_{max})\). Additionally, the optimal regularization constant is given by \(C_{(2)}^{\star}=\frac{\lambda_{max}+\lambda_{min}}{2}\), and the corresponding minimum possible RIP constant is \(\delta_{(2)}^{min}=\frac{\lambda_{max}-\lambda_{min}}{\lambda_{max}+\lambda_{ min}}\). Therefore, Lemma 1 is proven. \(\square\)