## 4 MSE Analysis of the Proposed Algorithm  
To perform a MSE analysis for the developed iterative algorithm, we give the expression of the spectral magnitude of a complex exponential signal in noise:  
\[|Y(k)|=|X(k)+W(k)|, \tag{18}\]where \(W(k)\) is the DFT of additive Gaussian white noise \(w(n)\). The spectral peak in the noise is  
\[|Y(k_{m})|=|X(k_{m})+W(k_{m})|=\Bigg{|}A\frac{\sin\left(\frac{\pi N}{M}\delta \right)}{\sin\left(\frac{\pi}{M}\delta\right)}+W(k_{m})e^{-j\varphi}e^{-j\pi \frac{N-1}{M}\delta}\Bigg{|}. \tag{19}\]  
Because \(|W(k)|\ll|X(k)|\) at a high \(SNR\), \(|Y_{k_{m}}|=A_{k_{m}}+Re(e^{-j\varphi_{k_{m}}}W_{k_{m}})\), \(A_{k_{m}}\) is the magnitude of \(X_{k_{m}}\), and \(\varphi_{k_{m}}\) is the phase of \(X_{k_{m}}\). Therefore, an approximation of (19) can be expressed as  
\[|Y(k_{m})|\approx|X(k_{m})|+\text{Re}(W(k_{m})e^{-j\varphi_{k_{m}}})\approx A_ {k_{m}}+U_{k_{m}}, \tag{20}\]  
where \(A_{k_{m}}=|X(k_{m})|=A\cdot\sin\left(\frac{\pi N}{M}\delta\right)/\sin\left( \frac{\pi}{M}\delta\right)\), \(U_{k_{m}}=Re(W(k_{m})e^{-j\varphi_{k_{m}}})\), \(\varphi_{k_{m}}=\phi+\pi\frac{N-1}{M}\delta\). Similarly, the DTFT samples can be expressed as \(|Y(k_{m}+p)|\approx A_{k_{m}+p}+U_{k_{m}+p}\), \(|Y(k_{m}-p)|\approx A_{k_{m}-p}+U_{k_{m}-p}\). To simplify the analysis, we denote \(Y_{0}=Y(k_{m})\), \(Y_{p}=Y(k_{m}+p)\), \(Y_{-p}=Y(k_{m}-p)\), \(A_{0}=A_{k_{m}}\), \(A_{p}=A_{k_{m}+p}\), \(A_{-p}=A_{k_{m}-p}\), \(U_{0}=U_{k_{m}}\), \(U_{p}=U_{k_{m}+p}\), and \(U_{-p}=U_{k_{m}-p}\). Consequently, (16) becomes  
\[ \begin{split}\hat{\delta}=&\frac{p(A_{p}+U_{p})-p(A _{-p}+U_{-p})}{(A_{p}+U_{p})+(A_{-p}+U_{-p})-2(A_{0}+U_{0})\cos\left(\frac{\pi N }{M}p\right)}\\ =&\frac{p(A_{p}-A_{-p}+U_{p}-U_{-p})}{A_{p}+A_{-p}-2A _{0}\cos\left(\frac{\pi N}{M}p\right)}\cdot\frac{1}{1+\frac{U_{p}+U_{-p}-2U_{ 0}\cos\left(\frac{\pi N}{M}p\right)}{A_{p}+A_{-p}-2A_{0}\cos\left(\frac{\pi N }{M}p\right)}}.\end{split}  \tag{21}\]  
Because \(\frac{U_{p}+U_{-p}-2U_{0}\cos\left(\frac{\pi N}{M}p\right)}{A_{p}+A_{-p}-2A_{ 0}\cos\left(\frac{\pi N}{M}p\right)}\ll 1\), the first-order Taylor expansion of the second item on the right side of (21) is  
\[\frac{1}{1+\frac{U_{p}+U_{-p}-2U_{0}\cos\left(\frac{\pi N}{M}p\right)}{A_{p}+ A_{-p}-2A_{0}\cos\left(\frac{\pi N}{M}p\right)}}\approx 1-\frac{U_{p}+U_{-p}-2U_{0} \cos\left(\frac{\pi N}{M}p\right)}{A_{p}+A_{-p}-2A_{0}\cos\left(\frac{\pi N }{M}p\right)}. \tag{22}\]  
By introducing (22) into (21), the frequency offset is  
\[\begin{split}\hat{\delta}=&\frac{p(A_{p}-A_{-p})}{A _{p}+A_{-p}-2A_{0}\cos\left(\frac{\pi N}{M}p\right)}+\frac{p(U_{p}-U_{-p})}{A _{p}+A_{-p}-2A_{0}\cos\left(\frac{\pi N}{M}p\right)}\\ &-\frac{p(A_{p}-A_{-p})\left(U_{p}+U_{-p}-2U_{0}\cos\left(\frac{ \pi N}{M}p\right)\right)}{\left(A_{p}+A_{-p}-2A_{0}\cos\left(\frac{\pi N}{M}p \right)\right)^{2}}\\ &-\frac{p(U_{p}-U_{-p})\left(U_{p}+U_{-p}-2U_{0}\cos\left(\frac{ \pi N}{M}p\right)\right)}{\left(A_{p}  
Then, we compute the expectation of the estimation error and the MSE of the frequency estimate:  
\[ \begin{split} E(\hat{\delta}-\delta)&=\frac{p(U_{p}-U_{ -p})}{A_{p}\!+\!A_{-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p\right)}-\delta\cdot \frac{U_{p}\!+\!U_{-p}-2U_{0}\cos\!\left(\frac{\pi N}{M}p\right)}{A_{p}\!+\!A_ {-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p\right)}\\ &=\!\frac{(p-\delta)U_{p}-(p+\delta)U_{-p})-2U_{0}\delta\cos\! \left(\frac{\pi N}{M}p\right)}{A_{p}\!+\!A_{-p}-2A_{0}\cos\!\left(\frac{\pi N} {M}p\right)},\end{split}  \tag{25}\]  
and  
\[E\!\left[\left(\delta-\delta\right)^{2}\right]=E\!\left[\frac{(p-\delta)U_{p} -(p+\delta)U_{-p})}{A_{p}\!+\!A_{-p}-2A_{0}\cos\!\left(\frac{\pi N}{M}p\right) }-\frac{2U_{0}\delta\cos\!\left(\frac{\pi N}{M}p\right)}{A_{p}\!+\!A_{-p}-2A_ {0}\cos\!\left(\frac{\pi N}{M}p\right)}\right]^{2}. \tag{26}\]  
According to deduction in Appendix A, the final MSE expression is obtained as follows:  
\[E\!\left[\left(\delta-\delta\right)^{2}\right]\approx\left\{