# An Improved CZT Algorithm for High-Precision Frequency Estimation
###### Abstract  
Estimating the frequencies of multiple superimposed exponentials in noise is an important problem due to its various applications in engineering. In order to obtain good inhibition of spectral leakage and improve the estimation accuracy, an improved Chirp-Z transform (CZT) algorithm is proposed for high-precision frequency estimation. Firstly, the proposed algorithm analyzes the characteristics of the CZT spectrum and utilizes the CZT spectrum to construct a bias correction factor for frequency bias estimation. Then, an expression between the bias correction factor and the frequency estimation error is derived to obtain a more accurate estimate of the frequency bias. Finally, the frequency estimate of the CZT is corrected by the estimated frequency bias so as to obtain a higher frequency estimation accuracy. Compared with the conventional CZT algorithm, the proposed improved CZT algorithm achieves a higher frequency estimation accuracy by correcting the frequency estimate of the CZT method using the estimated frequency bias. The proposed improved CZT algorithm is verified using simulation studies and experimental results, and the results show that it has a higher accuracy and better robustness than the existing methods. frequency detection; spectral leakage; Chirp-Z transform; correction factor +
Footnote †: journal: Applied Sciences  
to estimate the frequency by performing a linear regression on the phase of the received signal [16]. To cope with the problems of fence effects and spectral leakage in the classical FFT, the interpolated DFT (IpDFT) algorithm [17] is proposed to reduce these errors, and the transformed consecutive points of the DFT can be utilized to obtain a more accurate frequency estimation [18]. [PERSON] [19] estimated the frequency in two stages. A coarse frequency estimate is obtained by utilizing the amplitude spectrum of the DFT in the first stage, and a bias correction term is estimated and utilized to improve the frequency estimation accuracy in the second stage. [PERSON] and [PERSON] proposed the A&M algorithm in [20], which shifted the DFT coefficients by half and resulted in an improved frequency estimation performance. Based on [20], [PERSON] proposed the q-Shift estimator (QSE) algorithm in [21], whose coefficients are equivalent to interpolating the DFT of the signal by \(1/|q|\). The QSE algorithm can be considered a generalization of the A&M algorithm and has a better estimation performance than the A&M algorithm. [PERSON] [22] proposed combining the CZT method with the Candan algorithm [19] to improve the frequency resolution by deriving the bias correction term through the amplitude of the CZT spectral. In [23], [PERSON] proposed the CZT-based algorithm for the detection of multiple targets for the FMCW radar. As illustrated in [24], the CZT-based methods can improve the frequency resolution of the spectrum and reduce the measurement errors caused by spectral leakage by analyzing a narrow frequency band. Considering the advantages of the CZT method over other methods, we investigate the CZT method in depth in this paper. This paper proposes an improved CZT algorithm to further improve the frequency estimation accuracy of the CZT algorithm. The proposed algorithm first analyzes the characteristics of the CZT spectrum and utilizes the CZT spectrum to construct a bias correction factor for frequency bias estimation. Moreover, the relationship between the bias correction factor and the intrinsic frequency estimation error is derived, and from this, a more accurate estimate of the frequency bias is obtained. Then, the frequency bias estimate is used to modify the frequency estimate of the CZT algorithm to obtain a more accurate frequency estimate than the CZT algorithm. Finally, the proposed improved CZT algorithm is verified using simulation studies and experimental results, and the results show that it has a higher estimation accuracy and better robustness than the existing methods. The remainder of this paper is organized as follows.