# A Local Search Maximum Likelihood Parameter Estimator of Chirp Signal
## 4 Simulation and Analysis
### Numerical Results and Analysis  
For simplicity, the proposed method is called LsML, TF coarse estimator is called TFCE. The estimator that uses denoised signal \(y^{\prime}(n)\), as shown in Figure 7, is called DnCNN-XXX. We compare LsML with six other methods including RWT [8] and FrFT [15], which directly use the observation \(y(n)\), TFCE, DnCNN-FrFT, DnCNN-RWT and DnCNN-LsML. Figure 8 compare the estimation performance of the above methods in estimating chirp rate \(\hat{k}\) and initial frequency \(\widehat{f}_{0}\). Here, the rotation angle of RWT is from 85 to 91 degrees with an interval of 0.01, the fractional power of FrFT is from 0.96 to 1.08 with an interval of 0.0001, the performance metric is the natural logarithm of Root Mean Squared Error, as described in Equations (10) and (11). \[\ln\text{RMSE}_{\hat{t}_{0}}=\ln\left(\sqrt{\frac{1}{\text{M}}\sum_{i=1}^{ \text{M}}\left(f_{0i}-\widehat{f}_{0i}\right)^{2}}\right) \tag{10}\]  
\[\ln\text{RMSE}_{\hat{k}}=\ln\left(\sqrt{\frac{1}{\text{M}}\sum_{i=1}^{\text{M} }\left(k_{i}-\hat{k}_{i}\right)^{2}}\right) \tag{11}\]  
where M = 500 is the number of signals in the test set, \(f_{0i}\) and \(k_{i}\) are the true value, \(\widehat{f}_{0i}\) and \(\hat{k}_{i}\) are estimation results. The parameter search step of LsML is 0.02, we set the limitation width \(\sigma\) as \(f\in\left[f_{0}{}^{\prime}-0.15,f_{0}{}^{\prime}+0.15\right]\) and \(k\in\left[k^{\prime}-0.35,\ k^{\prime}+0.35\right]\). Table 2 lists the average runtime of 100 signals conducted by these methods. Figure 8: _Cont_. Figure 7: Schematic diagram of DnCNN-XXX. It can be concluded from these results that the proposed LsML has an excellent performance for chirp parameter estimation. By comparing the results of FrFT, RWT, DnCNN-FrFT, and DnCNN-RWT in Figure 8a, we can see that after DnCNN denoising, the accuracy of \(\hat{\kappa}\) is significantly improved. This means that \(y^{\prime}(n)\) has a higher SNR than \(y(n)\). But by comparing the results of LsML and DnCNN-LsML in Figure 8a,b, we can see that observation \(y(n)\) is more suitable as input to the ML fine estimator. This means that after denoising of DnCNN, although the SNR of \(y^{\prime}(n)\) has improved a lot, as shown in Figure 9, the signal parameters have changed slightly. As analyzed before, the accuracy of TFCE determines the value of limitation width \(\sigma\), which will further affect the performance of LsML. The impacts of different \(\sigma\) on estimation accuracy of LsML are shown in Figure 10a,b. We can see that when the SNR is very low, we need a high value of limitation width \(\sigma\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **RWT** & **DnCNN-RWT** & **FrFT** & **DnCNN-FrFT** & **TFCE** & **LsML** \\ \hline Time consumed & 0.8925 s & 0.9271 s & 0.6826 s & 0.7464 s & 0.0819 s & 0.2559 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average runtime of 100 signals. Figure 8: lnRMSE of chirp rate and initial frequency estimation versus Signal-to-Noise (SNR). (**a**) is lnRMSE of chirp rake, (**b**) is lnRMSE of initial frequency. Figure 9: The SNR of observations and denoised signals. For generality and comparability, we conducted simulation experiments in the background of additive Gaussian noise. Generally, as long as the spectrum width of noise is much larger than the bandwidth of the system, and the spectral density in this bandwidth can basically be considered as a constant, then we can treat it as white noise. For example, thermal noise and shot noise have uniform power spectral density in a wide frequency range; they can usually be considered as white noise. In specific application scenarios, there are different noise environments. As for the special interference under certain conditions, it needs to be studied separately and explored in-depth, e.g., the Doppler shift and multipath in the field of underwater acoustic engineering.