## Appendix A Deduction of the MSE Expression  
The deduction process of the final expression of the MSE is given in this section. To obtain \(E\Big{[}\big{(}\delta-\delta\big{)}^{2}\Big{]}\), we need to compute the numerator and denominator of the MSE:  
\[ \begin{split}(\delta-\delta)^{2}_{nomi}=&(p- \delta)^{2}U_{p}^{2}+(p+\delta)^{2}U_{-p}^{2}+4U_{0}^{2}\delta^{2}\text{cos}^{ 2}\bigg{(}\frac{\pi N}{M}p\bigg{)}\\ &-2(p^{2}-\delta^{2})U_{p}U_{-p}+4U_{p}U_{+p}(p-\delta)\delta \text{cos}\bigg{(}\frac{\pi N}{M}p\bigg{)}\\ &+4U_{p}U_{-p}(p+\delta)\delta\text{cos}\bigg{(}\frac{\pi N}{M}p \bigg{)},\end{split}  \tag{1}\]  
and  
\[(\delta-\delta)^{2}_{\text{denomi}}=\bigg{(}A_{p}+A_{-p}-2A_{0}\text{cos} \bigg{(}\frac{\pi N}{M}p\bigg{)}\bigg{)}^{2}. \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Estimators** & **Complex Multiplications** & **Complex Additions** \\ \hline [PERSON] (N points) & \(\frac{N}{2}log_{2}N+1\) & \(Nlog_{2}N+3\) \\ [PERSON] (\(N\) points) & \(\frac{N}{2}log_{2}N+1\) & \(Nlog_{2}N+3\) \\ A\&M (\(N\) points, 2 iterations) & \(\frac{N}{2}log_{2}N+4N+2\) & \(Nlog_{2}N+4N\) \\ Fan (\(M\) points, 2 iterations) & \(\frac{M}{2}log_{2}M+5N+8\) & \(Mlog_{2}M+5N+1\) \\ Fang (\(M\) points) & \(\frac{M}{2}log_{2}M\) & \(Mlog_{2}M\) \\ RCSTL (\(M\) points) & \(\frac{M}{2}log_{2}M+1\) & \(Mlog_{2}M\) \\ Proposed (\(M\) points, 2 iterations) & \(\frac{M}{2}log_{2}M+5N\) & \(Mlog_{2}M+5(N-1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational complexities of the estimators used in the simulation. Because \(U_{p}\) is the real part of \(W_{p}\) at all frequencies and \(w(n)\sim N(0,\sigma^{2})\), \(E(U_{p}^{2})=E(U_{-p}^{2})=E(U_{0}^{2})=N\sigma^{2}/2\). To compute \(E(U_{p}U_{+p})\), we go back to the DTFT expression:  
\[U_{p}=\text{Re}(W_{p}e^{-j\varphi_{p}})=\text{Re}(\sum_{n=0}^{N-1}w(n)e^{-j(2 \pi f_{p}n+\varphi_{p})}). \tag{11}\]  
The expectation of \(U_{p}U_{-p}\) is  
\[ \begin{split} E\big{[}U_{p}U_{-p}\big{]}&=E\bigg{[} \sum_{n=0}^{N-1}\big{(}w_{r}(n)\cos(2\pi f_{p}n+\varphi_{p})+w_{i}(n)\sin(2\pi f _{p}n+\varphi_{p})\big{)}\cdot\\ &\qquad\sum_{m=0}^{N-1}\big{(}w_{r}(m)\cos(2\pi f_{-p}m+\varphi_ {-p})+w_{i}(m)\sin(2\pi f_{-p}m+\varphi_{-p})\big{)}\,\bigg{]},\end{split}  \tag{12}\]  
where \(\varphi_{p}=\varphi+\pi\frac{N-1}{M}(\delta-p)\) and \(\varphi_{-p}=\varphi+\pi\frac{N-1}{M}(\delta+p)\) are the phases of \(X_{p}\) and \(X_{-p}\), respectively. \(f_{p}=\frac{N}{M}(\delta-p)\) and \(f_{-p}=\frac{N}{M}(\delta+p)\) are the frequencies of \(X_{p}\) and \(X_{-p}\), respectively. According to the independence characteristic of Gaussian white noise, \(E(w_{r}(n)w_{i}(m))=0\), and \(E(w_{i}(n)w_{r}(m))=0\). If \(n\
eq m\), \(E(w_{r}(n)w_{r}(m))=0\), and \(E(w_{i}(n)w_{i}(m))=0\). Therefore, only autocorrelation items (\(n=m\)) are retained in the expectation, which can be written as  
\[ \begin{split} E\big{[}U_{p}U_{-p}\big{]}=& E\bigg{[}\sum_{n=0}^{N-1}\Big{(}w_{r}^{2}(n)\cos(2\pi f_{p}n+ \varphi_{p})\cos(2\pi f_{-p}n+\varphi_{-p})+\\ &\quad w_{i}^{2}(n)\sin(2\pi f_{p}n+\varphi_{p})\sin(2\pi f_{-p} n+\varphi_{-p})\Big{)}\bigg{]}\\ =& N\sigma^{2}/2\sum_{n=0}^{N-1}\big{(}\cos(2\pi f_ {p}n+\varphi_{p})\cos(2\pi f_{-p}n+\varphi_{-p}) \tag{13}\] \[+\sin(2\pi f_{p}n+\varphi_{p})\sin(2\pi f_{-p}n+\varphi_{-p}) \big{)}\\ =& N\sigma^{2}/2\sum_{n=0}^{N-1}\big{(}\cos(2\pi(f_{p }-f_{-p})n+(\varphi_{p}-\varphi_{-p}))\big{)}.\end{split}  \tag{14}\]  
After some algebra, we have  
\[ \begin{split} E\big{[}U_{p}U_{-p}\big{]}=&\frac{N \sigma^{2}}{4}\bigg{[}e^{j(\varphi_{p}-\varphi_{-p})}\frac{1-e^{j(2\pi(f_{p}- f_{-p})n)}}{1-e^{j(2\pi(f_{p}-f_{-p})n/M)}}+\\ &\quad e^{-j(\varphi_{p}-\varphi_{-p})}\frac{1-e^{-j(2\pi(f_{p}- f_{-p})n)}}{1-e^{-j(2\pi(f_{p}-f_{-p})n/M)}}\bigg{]}.\end{split}