## 4 Experiment and Analysis
In Figure 7a, the signal frequency \(f\) is 50 Hz, and the estimation value of the signal frequency calculated by the proposed method is 50.0085 Hz, which is very close to the setting value. Moreover, in the experiments, the varying measured signal amplitude does not affect the frequency estimation accuracy of the proposed method. When the SNR is \(-1\) dB, the proposed automatic frequency estimation-based DLIA structure has the largest relative amplitude error which is 1.7%. With the SNR greater than 0 dB, for example, SNR = 10 dB and SNR = 15 dB, the relative amplitude error is less than 1%. When the SNR of the measured signal is decreased, the noise has more power in the view of both the time domain and the frequency domain. The frequency estimation error will increase as the increasement of noise power. According to Equation (10), the part of the amplitude measurement error caused by frequency deviation is increased when the SNR is low. The practical lowpass filter IAF used in this proposed method has the limited passband and attenuation of the stopband. Therefore, when the SNR of the measured signal is low, the energy of the residual noise filter by the IAF will be large. The amplitude measurement error is a combination of the amplitude error caused by the frequency estimation error and the residual noise, and it is increased when the SNR is low. Then, in the test condition where the SNR is \(-1\) dB, 0 dB, and 5 dB, respectively, the 100 independent frequency estimations and amplitude estimations of the proposed method are processed. This verification has two test signals of which the frequencies are 50 Hz and 100 Hz, respectively. The amplitude of the 50 Hz test signal is 1, and that of the 100 Hz test signal is 0.5. The distributions of the relative frequency measurement error are shown in Figure 8a-c. When the SNR is \(-1\) dB, most of the relative frequency measurement error is less than 0.03%, especially in the test signal where the frequency is 100 Hz. With the SNR of the to-be-tested signal increasing, the relative frequency measurement errors are gradually concentrated at the range [0,0.01%]. Similar to the relative frequency measurement error, the relative errors of the amplitude measurement shown in Figure 8d-f decrease with the increasement of the SNR. Moreover, in all 100 independent tests, the relative errors of both frequency measurements and amplitude measurements are not larger than the errors  
Figure 7: Amplitude estimation results under different SNR conditions and different signal frequencies: (**a**) signal frequency 50 Hz, (**b**) signal frequency 200 Hz, (**c**) signal frequency 500 Hz, (**d**) signal frequency 2000 Hz. shown in Figures 6 and 7; that means this proposed method has good calculation stability in different SNR conditions. The test results for the amplitude measurements of multiple frequency components under different frequency values and SNR are shown in Table 2. Due to the multiple frequency components, the local SNR [31] is used to describe the effect of noise on the useful signals, and the local SNR is:  
\[LSNR_{i}=10\text{log}_{10}(\frac{A_{i}^{2}}{2D^{2}}) \tag{29}\]  
where \(A_{i}\) is the amplitude of the i-th frequency component, and \(D^{2}\) is the variance of noise. In this multi-frequencies test, the variance \(D^{2}\) of Gaussian white noise is 0.16, according to Equation (29), and the local SNR of each signal frequency can be calculated as shown in Table 2.  
Figure 8: The frequency and amplitude estimation result distributions of the proposed DLIA method: (**a**) frequency error distribution (SNR = \(-1\) dB), (**b**) frequency error distribution (SNR = 0 dB), (**c**) frequency error distribution (SNR = 5 dB), (**d**) amplitude error distribution (SNR = \(-1\) dB), (**e**) amplitude error distribution (SNR = 0 dB), (**f**) amplitude error distribution (SNR = 5 dB). The largest relative error of the amplitude estimation is 1.40% when the signal frequency and amplitude are 50 Hz and 0.5, respectively, under the condition that the local SNR is \(-1.07\) dB.