# An Improved CZT Algorithm for High-Precision Frequency Estimation
## 4 Experimental Analysis
### Simulation Studies  
In the first simulation, we compared the root mean square error (RMSE) and variation ranges of the frequency estimates of the improved CZT algorithm versus the SNR. Figure 2 shows the means and variation ranges of the frequency estimates of the various methods versus the SNR when the frequency is \(f_{c}=5070\) Hz; the compared algorithms include the conventional FFT, the Candan algorithm [19], the A&M algorithm [20], the QSE algorithm [21], the CZT Candan algorithm [22], the CZT-based algorithm [23], and the proposed improved CZT algorithm. Figure 2 shows the means and variation ranges of the frequency estimates of the various methods versus the SNR when \(f_{c}=5070\) Hz, where (a) is the signal disturbed by the Gaussian noise, and (b) is the signal disturbed by the uniform noise. As can be seen from Figure 2, the mean of the frequency estimates of the proposed algorithm is closest to the true frequency of all of the methods, which indicates that the proposed improved CZT algorithm has a higher estimation accuracy than the other methods. Moreover, the proposed algorithm is affected by noise less than other methods, and the fluctuation range of the frequency estimates of the proposed algorithm is less than that of the other methods when the SNR is relatively high. In addition, the mean of the frequency estimates for the CZT-based algorithm, the CZT Candan, and the proposed algorithm is closer to the true frequency when the SNR is relatively high, and it is affected by noise. The variation ranges of frequency estimates become larger when the SNR is relatively low. Figure 3 shows the RMSE of the frequency estimates of various methods and the Cramer-Rao lower bound (CRLB) versus the SNR, where (a) is the signal disturbed by the Gaussian noise, and (b) is the signal disturbed by the uniform noise. In Figure 3, the formula of the theoretical CRLB has the following expression:  
\[\text{CRLB}=\frac{\sigma^{2}}{{{a_{0}}^{2}}^{N-1}{4\pi^{2}}\frac{{\pi^{2}}}{ {f_{s}}^{2}}\left|\exp(2\pi j\frac{n}{f_{s}}f_{c})\right|^{2}} \tag{27}\]  
As can be observed from Figure 3, the larger the SNR, the smaller the RMSE of all algorithms. When \(\text{SNR}<-10\) dB in the signal disturbed by the Gaussian noise, the RMSE of the proposed improved CZT algorithm is less than that of the CZT-based algorithm and is slightly greater than that of the FFT algorithm. When \(\text{SNR}\geq-10\) dB in the signal  
Figure 1: The frequency estimates of the proposed method versus various values of \(q\). disturbed by the Gaussian noise and in the signal disturbed by the uniform noise, the RMSE of the proposed algorithm is smaller than that of the existing algorithms, which indicates that the proposed improved CZT algorithm has a better estimation performance than other methods. In the second simulation, the signal is disturbed by the Gaussian noise. Table 2 shows the mean of the frequency estimates of the various methods versus the frequency \(f_{c}\)  
Figure 3: The RMSEs of frequency estimates of various algorithms versus SNR. Figure 2: Means and variation ranges of frequency estimates of various methods versus SNR when \(f_{c}=5070\) Hz. when the SNR is 10 dB. As can be seen from Table 2, the mean of the frequency estimates of the proposed algorithm is closest to the true frequency of all of the methods, which indicates that the proposed algorithm has a better estimation performance than the other methods. Table 3 shows the RMSE of the frequency estimates of the various methods versus the frequency when the SNR is 10 dB. As can be observed from Table 3, the proposed algorithm has the smallest RMSE for most of the frequencies, with the exception that it has a higher RMSE than the other methods at individual frequencies, such as \(f_{c}=3800\) Hz and \(f_{c}=4400\) Hz, due to the fence effect of the CZT-based algorithm. From the results in Table 2 and Table 3, it can be seen that the frequency estimation accuracy of the proposed algorithm is better and more robust than for the other algorithms.