## Appendix A Deduction of the MSE Expression
 \tag{15}\]  
By substituting \(\varphi_{p}-\varphi_{-p}=-\pi\frac{N-1}{M}p,f_{p}-f_{-p}=-\frac{2N}{M}p\) into (15), we obtain  
\[E\big{[}U_{p}U_{-p}\big{]}=\frac{N\sigma^{2}}{2}\frac{\sin(\frac{2\pi Np}{M})} {\sin(\frac{2\pi p}{M})}\approx\frac{N\sigma^{2}}{2}\frac{\sin(\frac{2\pi Np}{M })}{\frac{2\pi p}{M}}. \tag{16}\]  
In the same way, \(E\big{[}U_{0}U_{p}\big{]}\)=\(E\big{[}U_{0}U_{-p}\big{]}=N\sigma^{2}/2\frac{\sin(\pi Np/M)}{\sin(\pi p/M)} \approx N\sigma^{2}/2\frac{\sin(\pi Np/M)}{\pi p/M}\). Then, we introduce the above results into (12) and obtain  
\[E\Big{[}(\delta-\delta)^{2}\Big{]}_{nomi}=N\sigma^{2}\bigg{[}p^{2}+\delta^{2}+2 \delta^{2}\text{cos}^{2}\bigg{(}\frac{\pi N}{M}p\bigg{)}-(3\delta^{2}+p^{2}) \frac{\sin\left(\frac{2\pi Np}{M}p\right)}{\frac{2\pi}{M}p}\bigg{]}. \tag{17}\]To compute the denominator of \(E\Big{[}(\delta-\delta)^{2}\Big{]}\), we return to the expression of \(A_{p}\)  
\[A_{p}{=}|X(k_{m}+p)|{=}A\cdot\frac{\sin\!\left(\frac{\pi N}{M}(\delta-p)\right)} {\sin\!\left(\frac{\pi}{M}(\delta-p)\right)}\approx A\cdot\frac{\sin\!\left( \frac{\pi N}{M}(\delta-p)\right)}{\frac{\pi}{M}(\delta-p)}. \tag{11}\]  
Similarly, \(A_{-p}\approx A\cdot\frac{\sin\!\left(\frac{\pi N}{M}(\delta+p)\right)}{\frac{ \pi}{M}(\delta+p)}\), \(A_{0}\approx A\cdot\frac{\sin\!\left(\frac{\pi N}{M}\delta\right)}{\frac{\pi}{ M}\delta}\). By introducing \(A_{p}\), \(A_{-p}\), and \(A_{0}\) into (10), we obtain  
\[ \begin{split} A_{p}{+}A_{-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p \right)\approx& A\cdot\Bigg{[}\frac{\sin\!\left(\frac{\pi N}{M}( \delta-p)\right)}{\frac{\pi}{M}(\delta-p)}{+}\frac{\sin\!\left(\frac{\pi N}{M }(\delta{+}p)\right)}{\frac{\pi}{M}(\delta{+}p)}\\ &-\frac{2\sin\!\left(\frac{\pi N}{M}\delta\right)\cos\!\left( \frac{\pi N}{M}p\right)}{\frac{\pi}{M}\delta}\Bigg{]}\\ \approx&\frac{2 AM}{\pi}\cdot\Bigg{(}\frac{p^{2}\sin \!\left(\frac{\pi N}{M}\delta\right)\cos\!\left(\frac{\pi N}{M}p\right)}{ \delta(\delta^{2}-p^{2})}\\ &-\frac{p\delta\cos\!\left(\frac{\pi N}{M}\delta\right)\sin\! \left(\frac{\pi N}{M}p\right)}{\delta(\delta^{2}-p^{2})}\Bigg{)}.\end{split}  \tag{12}\]  
Then,  
\[ \begin{split} E\Big{[}(\delta-\delta)^{2}\Big{]}_{\text{demomi}}=& \Big{(}A_{p}{+}A_{-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p\right) \Big{)}^{2}\\ =&\Big{(}\frac{2 AMp}{\pi}\Big{)}^{2}\cdot\frac{ \Big{[}p\sin\!\left(\frac{\pi N}{M}\delta\right)\cos\!\left(\frac{\pi N}{M}p \right)}{\delta^{2}(\delta^{2}-p^{2})^{2}}\\ &-\frac{\delta\cos\!\left(\frac{\pi N}{M}\delta\right)\sin\! \left(\frac{\pi N}{M}p\right)\Big{]}^{2}}{\delta^{2}(\delta^{2}-p^{2})^{2}}. \end{split}  \tag{13}\]  
Since there are two cases in which the denominator equals zero, which leads to a meaningless result, we further analyze these two cases. When \(|\delta|=p\),  
\[ \begin{split}& A_{p}{+}A_{-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p \right)\\ &\approx\frac{2 AMp}{\pi}\cdot\frac{p\sin\!\left(\frac{\pi N}{M} \delta\right)\cos\!\left(\frac{\pi N}{M}p\right)-\delta\cos\!\left(\frac{\pi N }{M}\delta\right)\sin\!\left(\frac{\pi N}{M}p\right)}{\delta(\delta^{2}-p^{2}) }\\ &\approx\frac{2 AMp}{\pi}\cdot\frac{\delta\sin\!\left(\frac{\pi N}{ M}(\delta-p)\right)-(\delta-p)\sin\!\left(\frac{\pi N}{M}\delta\right)\cos\! \left(\frac{\pi N}{M}p\right)}{\delta(\delta^{2}-p^{2})}\\ &\approx\frac{2 ANp}{\delta+p}\cdot\left(\sin c\!\left(\frac{\pi N }{M}(\delta-p)\right)-\sin c\!\left(\frac{\pi N}{M}\delta\right)\cos\!\left( \frac{\pi N}{M}p\right)\right)\\ &\approx AN\cdot\left(1-\sin c\!\left(\frac{\pi N}{M}\delta \right)\cos\!\left(\frac{\pi N}{M}p\right)\right);\end{split}  \tag{14}\]when \(\delta=0\),  
\[