## 4 Resolving the SHANM with SDP
### SDP of SHANM  
It can be noted that \(P_{n}^{m}(cos\theta)\) is an order \(n\) trigonometric polynomial, \(P_{n}^{m}(cos\theta)=\sum_{\upsilon=-n}^{n}\gamma_{n,m,l}e^{i\upsilon\theta}\) with unique coefficients \(\{\gamma_{n,m,l}\}\), and the SH in Equation (2) is described using Fourier series as [58]  
\[Y_{n}^{m}(\theta,\varphi)=\sum_{l=-n}^{n}A_{n,m}\gamma_{n,m,l}e^{i\mu\theta}e ^{i\mu\varphi}, \tag{21}\]  
where \(A_{n,m}=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}\). From Equation (21), Equation (8) can be rewritten as  
\[\boldsymbol{P}(t)=\boldsymbol{BOD}\big{(}\overrightarrow{\psi}\big{)} \boldsymbol{s}(t)+\boldsymbol{c}(t) \tag{22}\]where \(\mathbf{D}(\overrightarrow{\psi})=[f(\psi_{1}),\ldots,f(\psi_{l}),\) and \(\mathbf{f}(\psi_{l})=f_{\theta}(\theta_{i})\mathbf{\otimes}\mathbf{f}_{\varphi}(\varphi_{i} ).f_{\theta}(\theta_{i})=[e^{-iN\theta_{i}},\ldots,1,\ldots,\)\(e^{iN\theta_{i}}]^{T}\),  
\[\mathbf{f}_{\varphi}(\varphi_{i})=\Big{[}e^{-iN\varphi_{i}},\ldots,1,\ldots,e^{iN \varphi_{i}}\Big{]}^{T}, \tag{23}\]  
\(\mathbf{O}\) denotes a constant matrix generated by \(A_{n,m}\gamma_{n,mJ}\). Hence, putting Equation (22) into Equation (12), we have  
\[\mathbf{R}=E\Big{\{}\mathbf{P}(t)\mathbf{P}^{H}(t)\Big{\}}=\widetilde{\mathbf{O}}\mathbf{D}( \overrightarrow{\psi})\mathbf{R}_{s}\mathbf{D}^{H}(\overrightarrow{\psi})\widetilde {\mathbf{O}}^{H}+\mathbf{C} \tag{24}\]  
where \(\widetilde{\mathbf{O}}=\mathbf{B}\mathbf{O}\).  
Using the relationship shown in Equation (24) regarding the spherical harmonic steering vector and Vandermonde matrix, starting with a noiseless scenario where \(C=0\) we arrive at the low-dimensional SDP implementation for the SHANM method.  
**Proposition 1**.: _For certain location sources \(\overrightarrow{\psi}\), assuming the objective value of the subsequent optimization puzzle gives the norm \(\|\mathbf{R}\|_{T}\), implying_  
\[\|\mathbf{R}\|_{T}= \underset{\mathbf{Q}\mathbf{M}}{\text{min}}\frac{1}{2\varkappa^{2}}Tr( \widetilde{\mathbf{O}}S(\mathbf{Z})\widetilde{\mathbf{O}}^{H})+\frac{1}{2}Tr(\mathbf{M}) \tag{25}\] \[s.t.\ \begin{bmatrix}\widetilde{\mathbf{O}}S(\mathbf{Z})\widetilde{\mathbf{O}}^{H }&\mathbf{R}\\ \mathbf{R}^{H}&\mathbf{M}\end{bmatrix}\geq 0,\]  
_if the highest \(SH\) order \(N\) is high enough and a strong duality condition is ensured, then \(\|\mathbf{R}\|_{T}=\|\mathbf{R}\|_{T}\), where \(\|\mathbf{R}\|_{T}\) is the \(SH\) atomic norm described in Equation (20) with \(\mathbf{R}\in\mathbb{C}^{(N+1)^{2}\times(N+1)^{2}}\)._  
\[\varkappa=\|\widetilde{\mathbf{O}}\|_{F}=\sqrt{\sum_{i=0}^{N}\frac{(4I+1)}{4\pi}} \|\mathbf{B}\|_{F}, \tag{26}\]  
_and \(S(\mathbf{Z})\) is a Hermitian matrix, which is a function of \(\mathbf{Z}\) as [35]_  
\[S(\mathbf{Z})=\begin{bmatrix}\mathbf{Z}_{0}&\mathbf{Z}_{-1}&0&\mathbf{Z}_{-2N}\\ \mathbf{Z}_{1}&\mathbf{Z}_{0}&0&\mathbf{Z}_{-2N+1}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{Z}_{2N}&\mathbf{Z}_{2N-1}&\ldots&\mathbf{Z}_{0}\end{bmatrix} \tag{27}\]  
_where \(\mathbf{Z}_{v}\) is a Toeplitz matrix described by lth row of \(\mathbf{Z}\) to be_  
\[\mathbf{Z}_{l}=\begin{bmatrix}\mathbf{Z}_{l,0}&\mathbf{Z}_{l,-1}&\ldots&\mathbf{Z}_{l,-2N}\\ \mathbf{Z}_{l,1}&\mathbf{Z}_{l,0}&\ldots&\mathbf{Z}_{l,-(2N-1)}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{Z}_{l,2N}&\mathbf{Z}_{l,2N-1}&\ldots&\mathbf{Z}_{l,0}\end{bmatrix} \tag{28}\]