## 4 Experiment and Analysis  
In order to validate this proposed method, the DLIA with multiple frequencies automatic frequency estimation ability is implemented in a PC, and different types of to-be-test signals are used. The frequency deviation \(\Delta f\) between the measured signal and reference is the key factor affecting the amplitude and phase measurement accuracy of the DLIA. Thus, the frequency estimation accuracy is verified first. Assuming the test is \(s[n]=\sin(2\pi f/f_{s}n)+w[n]\), where the sampling frequency \(f_{s}=10\) kSPS; and the length of the signal sequence is 4096. The \(w[n]\) is the Gaussian white noise which can be generated by Equation (27). In different SNR test conditions, the amplitude of the to-be-tested signal and the SNR are defined first. Then, according to Equation (27a), the required variance of the Gaussian white noise \(w[n]\) can be calculated. After that, the additive Gaussian white noise \(w[n]\) can be obtained by multiplying the Gaussian white noise \(\tilde{w}[n]\) with zero mean value and variance of 1 by the desired standard deviation \(D\). \[SNR=10\text{log}_{10}(\frac{A^{2}}{2D^{2}}) \tag{27a}\] \[w[n]=D\times\tilde{w}[n] \tag{27b}\]  
The parameters of the PSO-based frequency estimation method are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline N & signal sequence length & 4096 \\ \(f_{s}\) & sampling rate & 10 kSPS \\ M & population size & 20 \\ T & iteration times & 20 \\ \(\omega_{max}\) & max inertial weight & 0.9 \\ \(\omega_{min}\) & min inertial weight & 0.4 \\ \(c_{1}\) & self-cognition factor & 0.6 \\ \(c_{2}\) & group cognition factor & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters setting of PSO-based frequency estimation method. Figure 5: Amplitude-frequency response of IAF with different filter length N; (**a**) N = 100; (**b**) N = 500; (**c**) N = 1000; (**d**) N = 2000. In Figure 6, the signal which satisfies the Nyquist sampling theorem has been tested in different SNR situations. The largest relative frequency estimation error calculated by Equation (28) is \(0.11\%\) when the SNR is \(-3\) dB. In this condition, according to Equation (10), the relative error of amplitude estimation caused by frequency deviation between the reference and the measured signal is \(0.06\%\). Moreover, when the SNR of the measured signal is greater than \(-3\) dB, the frequency estimation error is less than \(0.1\%\), which means the amplitude estimation error of the DLIA caused by frequency deviation can be neglected. \[\text{Frequency Relative error}=\frac{|f_{estimate}-f|}{f} \tag{28}\]  
From Figure 6, it can be seen that the frequency estimation error increased with the decrease in the SNR of the measured signal. The principle of the frequency estimation method is to find an optimal frequency which makes Equation (15) reach its maximum as shown in Figure 3b. The Gaussian white noise is uniformly distributed in the frequency domain. When the SNR of the measured signal is decreased, the optimum of the refined spectrum shown in Figure 3b deviates from the real optimal frequency points due to the noise energy at the specific frequency line. After verifying the frequency estimating accuracy of the proposed method, the amplitude accuracy is subsequently tested. Assume that the to-be-tested signal is \(s[n]=A\sin(2\pi f/f_{s}n)+w[n]\) where \(A\) is the amplitude of the signal-needed estimate. Moreover, for the proposed method in this verification situation, the accuracy frequency value \(f\) of the measured signal is unknown, and it will be automatically estimated by the proposed method. The simulation results are shown in Figure 7.  
Figure 6: Frequency estimation results under different SNR: (**a**) SNR = 10 dB, (**b**) SNR = 0 dB, (**c**) SNR = 1 dB, (**d**) SNR = \(-3\) dB. As shown in Figure 7, the to-be-tested signal under different frequencies and different SNR conditions have been verified.