# An Improved CZT Algorithm for High-Precision Frequency Estimation
## 4 Experimental Analysis
### Simulation Studies
The third simulation is utilized to verify the performance of the proposed algorithm for the multiple-tone signal. Figure 4 shows the means of the frequency estimates of the multiple-tone signal versus the SNR when the frequencies are (a) \(f_{c}=3050\) Hz, (b) \(f_{c}=\)14,000 Hz, and (c) \(f_{c}=\)18,000 Hz. The compared algorithms include the CZT Candan algorithm [22], the CZT-based algorithm [23], and the proposed improved CZT algorithm. As can be observed in Figure 4, three methods have the ability to recognize multiple frequencies. As the SNR increases, the estimated frequency results tend to be stable. The CZT-based algorithm and the CZT Candan algorithm perform better in (a), while the proposed algorithm performs better in (b) and (c). Therefore, the proposed algorithm outperforms the other algorithms for the multi-tone signal. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Preset** & \multicolumn{6}{c}{**Mean of Frequency Estimates/Hz**} \\ \cline{2-7}
**Frequency/Hz** & **FFT** & **Candan** & **A\&M** & **CZT-based** & **QSE** & **CZT Candan** & **Proposed Algorithm** \\ \hline
3000 & 2990.0933 & 2990.7082 & 2995.0410 & 3001.4171 & 3001.3411 & 3000.7109 & 3000.4748 \\
3200 & 3171.3110 & 3179.0949 & 3185.6476 & 3199.6263 & 3199.6462 & 3199.8110 & 3199.8726 \\
3400 & 3443.1377 & 3408.5396 & 3421.5705 & 3398.0031 & 3398.1102 & 3398.9988 & 3399.3300 \\
3600 & 3624.3555 & 3619.3810 & 3612.1630 & 3601.7010 & 3601.6092 & 3600.8477 & 3600.5631 \\
3800 & 3805.5732 & 3805.3871 & 3802.7203 & 3799.9102 & 3799.9149 & 3799.9543 & 3799.9689 \\
4000 & 3986.7910 & 3987.9414 & 3993.3664 & 3998.1386 & 3998.2887 & 3999.0689 & 3999.3789 \\
4200 & 4168.0088 & 4178.8213 & 4183.9950 & 4201.9565 & 4201.8515 & 4200.9798 & 4200.6542 \\
4400 & 4439.8354 & 4415.9718 & 4419.9163 & 4400.1941 & 4400.1838 & 4400.0991 & 4400.0672 \\
4600 & 4621.0532 & 4617.6298 & 4610.5368 & 4598.4021 & 4598.4880 & 4599.1998 & 4599.4658 \\
4800 & 4802.2710 & 4802.2373 & 4800.9280 & 4801.9527 & 4801.8480 & 4800.9787 & 4800.6545 \\
5000 & 4983.4888 & 4985.4033 & 4991.7410 & 5000.4779 & 5000.4520 & 5000.2367 & 5000.1557 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Means of the frequency estimates versus the frequency when the SNR is 10 dB. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Preset** & \multicolumn{6}{c}{**RMSE of Frequency Estimates/Hz**} \\ \cline{2-7}
**Frequency/Hz** & **FFT** & **Candan** & **A\&M** & **CZT-Based** & **QSE** & **QSE** & **CZT Candan** & **Proposed Algorithm** \\ \hline
3000 & 21.8910 & 20.5400 & 11.8729 & 3.1414 & 2.9734 & 1.6241 & 1.1728 \\
3200 & 63.3943 & 46.2041 & 31.7640 & 0.8257 & 0.7829 & 0.5627 & 0.5766 \\
3400 & 95.3217 & 19.1503 & 47.6832 & 4.9017 & 4.6463 & 2.5550 & 1.8097 \\
3600 & 53.8185 & 42.8342 & 26.9534 & 3.7670 & 3.5642 & 1.9191 & 1.3526 \\
3800 & 12.3152 & 11.9157 & 10.1294 & 0.1985 & 0.1925 & 0.3969 & 0.5168 \\
4000 & 29.1880 & 26.6527 & 15.0581 & 4.1845 & 3.9612 & 2.1415 & 1.5042 \\
4200 & 70.6913 & 46.8121 & 35.4033 & 4.4196 & 4.1842 & 2.2624 & 1.5841 \\
4400 & 88.0247 & 35.3550 & 44.0314 & 0.4288 & 0.4084 & 0.4441 & 0.5366 \\
4600 & 46.5215 & 38.9633 & 23.3930 & 3.5353 & 3.3457 & 1.8123 & 1.2881 \\
4800 & 5.0182 & 4.9683 & 20.4068 & 5.1889 & 4.9222 & 2.7312 & 1.9435 \\
5000 & 36.4850 & 32.2607 & 18.4582 & 1.0561 & 0.9995 & 0.6479 & 0.6154 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSEs of the frequency estimates versus the frequency when the SNR is 10 dB.