3.1 Algorithm model establishment
Because of nonideal channel model or sample number for estimation, the original Doppler estimation makes Eqs. (5) and (6) untenable. To enhance the authenticity and reliability of the analysis in this section, an estimation deviation \(\Delta\) is considered existing between the values on both sides of the equal sign in (5) and (6). According to our simulations, the following remarks have been concluded.

When the actual Doppler shift \(f_{d}\) and SNR \(\gamma_{s}\) are increasing, the estimation deviation \(\Delta\) is generally decreasing. Besides, the maximum of the observed \(\Delta\) in simulation can nearly be 23% of \(f_{d}\).

The larger sampling interval leads to smaller deviation, i.e., \(T_{s1} < T_{s2} \Rightarrow \Delta_{1} > \Delta_{2}\). Because the noise bandwidth becomes smaller when larger sampling interval is applied, thus SNR is equivalently increased, which results in smaller deviation.

The deviation \(\Delta_{1} (\Delta_{2} )\) is approximately a linear decreasing function of \(m(n)\).
Based on above remarks, Eqs. (5) and (6) for joint Doppler shift and SNR estimations in (7) are reformulated as
$$\hat{f}_{d1} = \frac{{\sqrt {2  \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2  (m\pi f_{d} T_{s} )^{2} } \right)} }}{{m\pi T_{s} }} + \frac{\Delta }{m}$$
(9)
$$\hat{f}_{d2} = \frac{{\sqrt {2  \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2  (n\pi f_{d} T_{s} )^{2} } \right)} }}{{n\pi T_{s} }} + \frac{\Delta }{n}$$
(10)
3.2 Choice of sampling intervals
In this part, above two Eqs. (9) and (10) are utilized to analyze the effect of two sampling intervals, i.e., the values of \((m,n)\). In this paper, the maximum value of the normalized Doppler shift \(f_{m}\) is prespecified 0.1 for simulation and analysis. Due to the limit in (8) and the assumption as \(m < n\), the maximum value for n is 5. Thereby, there are two types of value choice for \((m,n)\):

\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is integer: \(m = 1\), and \(n \in \{ 2,3,4,5\}\).

\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is fraction: m and n are relatively prime and \(2 \le m < n \le 5\).
In this paper, the estimation deviation \(\Delta\) is prespecified when doing simulation for analysis. For reliability, it is relaxed to 30% from the observed maximum 23%. The analysis for different \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is conducted by two cases according to whether \(\Delta T_{s}\) is fixed or changing. In addition, the mean square error (MSE) of the final Doppler shift estimation \(\hat{f}_{d}\) is applied for comparison, which is defined by
$$E_{f} = E\left[ {\left {{{\hat{f}_{d} } \mathord{\left/ {\vphantom {{\hat{f}_{d} } {f_{d} }}} \right. \kern\nulldelimiterspace} {f_{d} }}  1} \right^{2} } \right]$$
(11)
where \(E\left[ \cdot \right]\) denotes the expectation operation.
3.2.1 Case I (\(\Delta T_{s}\) is fixed)
In this case, \(f_{m} \in \{ 0.01:0.01:0.1\}\), \(SNR \in \{ 0:3:30\}\) and \(\Delta T_{s}\) is fixed 0.03, which corresponds to the worst case (estimation deviation reaches 30% of \(f_{d}\)). After computer simulation, the MSE of Doppler shift estimation is shown in Fig. 1, in which different values of m and n make the resulted MSE fluctuates. From Fig. 1a, it can be found that larger n can result in smaller MSE. When using the same n, the fraction n/m outperforms integer n/m, and larger m is better for fraction n/m. Obviously, the originally used \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\) in [23] is the worst choice.
In Fig. 1b, the SNR increasing makes the MSE changing like a concave curve. Relatively, the turning point that the MSE changes from decreasing to increasing is latter for lager n. Though in previous remarks, larger \(f_{d}\) and \(\gamma_{s}\) leading to smaller deviation \(\Delta\) is derived, the final estimation \(\hat{f}_{d}\) obtained by equations solving is nonlinearly and comprehensively affected by \(f_{d}\), \(\gamma_{s}\) and \(\Delta\). When negative effects outweigh positive effects, the MSE curve rises. Therefore, in Fig. 2, we can also find concave curves of MSE changing for different \(f_{m}\) (\({{f_{d} = f_{m} } \mathord{\left/ {\vphantom {{f_{d} = f_{m} } {T_{s} }}} \right. \kern\nulldelimiterspace} {T_{s} }}\)), in which the negative effects is more evident when \(f_{m}\) is smaller.
3.2.2 Case II (\(\Delta T_{s}\) is changing)
In this case, \(\Delta T_{s}\) is changing from 0.005 to 0.03 by step 0.005. Figure 3 displays the MSE of Doppler shift estimation for different \(\Delta T_{s}\), which increases as \(\Delta T_{s}\) becomes larger. For small \(\Delta T_{s}\), the increase in MSE is rapid and the performances of n/m’s are not consistent with previous performance presented in Fig. 1. Such deviation suggests us to choose the best sampling intervals (namely m and n) carefully. For further analysis, the MSEs are averaged over \(\Delta T_{s}\)’s, results are shown in Fig. 4, and nearly the same conclusion found in Figs. 1 and 2 can be obtained. Hence, the following analysis is carried out with the same parameter setting as that in Case I.
According to all above presentations and analysis, it seems that \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {5 \mathord{\left/ {\vphantom {5 4}} \right. \kern\nulldelimiterspace} 4}\) is the best choice, but it is hard to confirm because the deviation is modeled approximately, and the estimator can be sensitive for two very close sampling intervals. To find a better n/m for practical use, another performance called mismatch MSE (mMSE) is applied.
$$E_{f} (n,m) = \frac{1}{M  1}\sum\limits_{{1 \le m_{t} < n_{t} \le 5}} \quad {(\hat{f}_{d} (n,m)  \hat{f}_{d} (n_{t} ,m_{t} ))^{2} }$$
(12)
where M is the total number of \((n,m)\). Actually, \(E_{f} (n,m)\) can be viewed as a measurement for estimation stability under the principle of LeastSquare (LS). The most stable n/m, i.e., the one results in smallest \(E_{f} (n,m)\), can also be robust against nonideal factors.
The mMSE results are presented in Fig. 5. It is obvious that \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\) is the worst choice. Moreover, a larger gap between n and m or a smaller n for fraction n/m can lead to higher stability of mMSE. Based on all figures and above comprehensive analysis, \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern\nulldelimiterspace} 3}\) and \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) are two better choices for the scenario in this paper. On the other side, the fraction \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern\nulldelimiterspace} 3}\) has an advantage of computation reduction over \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) for its larger m.