# An Improved CZT Algorithm for High-Precision Frequency Estimation
## 4 Experimental Analysis
### Field Test Results  
In this section, a test system for the FMCW radar was built to verify the performance of the proposed algorithm in real-world engineering applications, as shown in Figure 5, where the resolution of the AD converter is 12-bit in the FMCW radar. The test system includes an FMCW radar, a laser rangefinder, a reflector plate mounted on the wall, and a rail for mounting the FMCW radar. In this test system, the rail is used to adjust the distance between the reflector plate on the wall and the FMCW radar, and the laser rangefinder is used to measure the reference distance between the FMCW radar and the reflector plate. According to the principle of the FMCW radar in [9], various distances between the reflector plate and the FMCW radar mean that the frequency value of the beat signal received by the radar varies, and the distance information can be obtained from the frequency estimates of the beat signal. In this test system, the accuracy of the frequency estimation determines the distance estimation accuracy, so this system is utilized to test the estimation performance of the proposed improved CZT algorithm in practical applications. The parameters of the test system are shown in Table 4.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameter** & **Value** \\ \hline Initial frequency & 23.6 GHz \\ Chirp bandwidth & 999.807 MHz \\ Sampling points & 1024 \\ Sampling rate & 123.2 KHz \\ Electromagnetic wave speed & 299,709 Km/s \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of the FMCW test system.  
Figure 4: Means of the frequency estimates for the multiple-tone signal versus the SNR.  
Figure 5: Test system for FMCW radar.  
Utilizing the test system above, the distance between the FMCW radar and the reflector is adjusted between 1000 mm and 2000 mm. Table 5 gives the frequency estimates versus the various distances as well as the frequencies of the various algorithms, including the 1024-point 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 algorithm. Here, the means of the errors are defined as the difference between the frequency estimates and the true frequency. As can be seen in Table 5, the means of the proposed improved CZT algorithm are closer to the true values than those of other algorithms. Therefore, the proposed algorithm has a higher estimation accuracy than the other methods. Figure 6 shows the estimation errors of the frequency estimates under the different distances of the various algorithms. As can be seen in Figure 6, the maximum relative error of the proposed algorithm is 5.20 Hz, and the minimum error is 0.25 Hz while the maximum error of the CZT-based algorithm is 8.14 Hz, and the maximum error of the CZT Candan algorithm is 5.85 Hz. Moreover, the frequency estimates of the proposed algorithm are closer to the reference frequency than those of other algorithms. Therefore, the proposed improved CZT algorithm can effectively improve the frequency estimation capability of the FMCW radar and thus can improve the ranging accuracy of the FMCW radar system.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Distances** & **True** & \multicolumn{6}{c}{**Frequency Estimates/Hz**} \\ \cline{3-10} \multicolumn{1}{c}{/**mm**} & **Frequency/Hz** & **FFT** & **Candan** & **A\&M** & **CZT-Based** & **QSE** & **CZT** & **Proposed** \\  & & & & & **Algorithm** & **Algorithm** & **Algorithm** & **Candan** & **Algorithm** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 5: Frequency estimates versus various distances and frequencies of the various algorithms.