## 3 Proposed Algorithm  
To ensure the estimation accuracy of our model, we focus on the fine frequency estimation process and propose a new estimator. The steps of the proposed estimator are shown in Algorithm 1. First, we give the expressions of the two neighboring DTFT samples of the spectral peak:  
\[X(k_{m}\pm p)=Ae^{i\phi}e^{j2\pi\frac{N-1}{M}(\delta\mp p)}\frac{\sin(\frac{ \pi N}{M}(\delta\mp p))}{\sin(\frac{\pi}{M}(\delta\mp p))}. \tag{7}\]  
Then, we obtain their magnitudes:  
\[|X(k_{m}\pm p)|=A\left|\frac{\sin(\frac{\pi N}{M}(\delta\mp p))}{\sin(\frac{ \pi}{M}(\delta\mp p))}\right|. \tag{8}\]  
From the definitions of \(\delta\) and \(p\), it can be deduced that \(|\delta\mp p|<3/2\). Considering that \(0<\frac{N}{M}<1/2\), we can obtain \(-3\pi/4<\frac{\pi N}{M}(\delta\mp p)<3\pi/4\). Thus, \(\sin(\frac{\pi N}{M}(\delta\mp p))\) and \(\sin(\frac{\pi}{M}(\delta\mp p))\) have the same sign in this range, namely, \(\frac{\sin(\frac{\pi N}{M}(\delta\mp p))}{\sin(\frac{\pi}{M}(\delta\mp p))}>0\). In this way, (8) can be changed to  
\[|X(k_{m}\pm p)|=A\frac{\sin(\frac{\pi N}{M}(\delta\mp p))}{\sin(\frac{\pi}{M} (\delta\mp p))}. \tag{9}\]  
The ratio of \(|X(k_{m}+p)|\) to \(|X(k_{m})|\) is  
\[\frac{|X(k_{m}+p)|}{|X(k_{m})|}=\frac{\sin(\frac{\pi N}{M}(\delta-p))}{\sin( \frac{\pi}{M}(\delta-p))}/\frac{\sin(\frac{\pi N}{M}(\delta))}{\sin(\frac{ \pi}{M}(\delta))}. \tag{10}\]  
Since \(M\gg\pi(\delta-p)\), we obtain  
\[\frac{|X(k_{m}+p)|}{|X(k_{m})|}\approx\frac{\sin(\frac{\pi N}{M}(\delta-p))} {\sin(\frac{\pi N}{M}(\delta))}/\frac{\delta-p}{\delta}. \tag{11}\]  
Figure 1: Magnitudes of the DTFT and DFT of a complex exponential signal.  
(11) can also be expressed as:  
\[\frac{|X(k_{m}+p)|}{|X(k_{m})|}\frac{\delta-p}{\delta}\approx\frac{\sin(\frac{ \pi N}{M}(\delta-p))}{\sin(\frac{\pi N}{M}(\delta))}. \tag{12}\]  
According to the transformation formula of a trigonometric function, (12) can be decomposed as  
\[\frac{|X(k_{m}+p)|}{|X(k_{m})|}\frac{\delta-p}{\delta}\approx\frac{\sin(\frac{ \pi N}{M}\delta)\cos(\frac{\pi N}{M}p)-\cos(\frac{\pi N}{M}\delta)\sin(\frac{ \pi N}{M}p)}{\sin(\frac{\pi N}{M}\delta)}. \tag{13}\]  
Similarly,  
\[\frac{|X(k_{m}-p)|}{|X(k_{m})|}\frac{\delta+p}{\delta}\approx\frac{\sin(\frac{ \pi N}{M}\delta)\cos(\frac{\pi N}{M}p)+\cos(\frac{\pi N}{M}\delta)\sin(\frac{ \pi N}{M}p)}{\sin(\frac{\pi N}{M}\delta)}. \tag{14}\]  
After adding (13) to (14), we obtain  
\[\frac{|X(k_{m}+p)|}{|X(k_{m})|}\frac{\delta-p}{\delta}+\frac{|X(k_{m}-p)|}{|X (k_{m})|}\frac{\delta+p}{\delta}\approx 2\cos(\frac{\pi N}{M}p). \tag{15}\]  
Finally, the estimate of \(\delta\) can be obtained by  
\[\delta=\frac{p|X(k_{m}+p)|-p|X(k_{m}-p)|}{|X(k_{m}+p)|+|X(k_{m}-p)|-2|X(k_{m})| \cos(\frac{\pi N}{M}p)}. \tag{16}\]  
The frequency estimate is  
\[\hat{f}=(k_{m}+\delta)\frac{f_{s}}{M}. \tag{17}\]  
From (17), we can obtain a precise estimate of the frequency offset, but the above analysis is based on a signal without noise and some approximations. In fact, the frequency estimation derived from (17) usually deviates from the true frequency due to the influence of noise, so we introduce an iteration operation into the fine estimation process. The procedures of the proposed estimator are shown in Algorithm 1.  
```
0: Signal samples \(y\)
0: Frequency estimate \(\hat{f}\)
1: Coarse estimation: calculate the \(M\)-point DFT, and obtain the location of the spectral peak \(k_{m}\) as the coarse frequency estimate.
2: Fine estimation: initialize the iteration variables; \(\delta=0,q=1\)
3:for\(q<Q\)do
4: Compute the DTFT samples \(|X(k_{m}+p)|\), \(|X(k_{m})|\), and \(|X(k_{m}-p)|\) according to (8)
5: Compute \(\hat{\delta}\) according to (16)
6: Update the iteration variables: \(k_{m}=k_{m}+\delta,q=q+1\)
7: Compute \(\hat{f}\) according to (17)
8:return\(\hat{f}\)
```  
**Algorithm 1** The proposed iterative estimator.