## 1 Introduction  
Obtaining accurate and fast frequency estimates of complex exponential signals is a classic problem in signal processing. It has extensive applications in navigation [1], power systems [2; 3], metabolomics [4], and aerospace communication [5; 6]. The frequency offsets in these systems may lead to demodulation failure, especially for highly dynamic applications, such as satellite communication, aviation communication control systems, and missile control systems. Thus, an efficient and simple frequency estimation method is necessary to ensure real-time communication for the above applications. Frequency estimation can be seen as a search process. A coarse frequency estimate can be obtained by searching the spectral maximum. However, the estimation accuracy of algorithms based on the spectrum calculated by a DFT is limited by the frequency resolution. The obtained estimate must be an integral multiple of the frequency resolution, which deviates from the fact that the true frequency usually lies between spectral lines. Thus, it is necessary to determine a fine frequency estimation to further reduce the residual frequency offset. Many estimators have been proposed based on the interpolation of DFT and DTFT samples, which is an efficient way to perform fine frequency estimation [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. The interpolations of [PERSON] estimator [7], Candan estimator [8; 9], Quinn estimator [10], parabolic interpolation estimator [11; 12], and Fang estimator [13] are based on DFT samples. The residual frequency offset can be reduced with only a few additional multiplication operations after obtaining a coarse estimation. However, the RCSTL estimator [14], which uses three DTFT samples to assist with fine frequency estimation, obtains a smaller estimation error than other approaches. Its estimation error still deviates from the CRLB. Therefore, A&M in [16] first introduced the iteration operation into interpolation algorithms and further reduced the offset. IpDFT/e-lpDFT [19] were proposed to reduce the spectral leakage of coarse estimation by iteratively interpolating the weighted signal. The weighted signal is generated by applying different windows on the received signal [20; 21; 22; 23]. Moreover,iterative algorithms can also obtain precise frequency estimates for various signals, such as the real sinusoid signal [26], the damped complex exponential signal [28], and the two-dimensional (2-D) complex exponential signal [29]. The asymptotic variances of some iterative estimators are very close to the CRLB, but an edge effect problem is encountered in which the variance may increase obviously at the edge of the frequency offset range. At the end, we summarize the performance of the key estimators in Table 1:  
In fact, even a slight improvement in the estimation accuracy is important for signals in highly dynamic applications. Thus, to further explore the performance potential of estimators based on interpolation and obtain robust estimates, we present a new iterative algorithm in this paper. It includes two stages. In the coarse estimation stage, we first pad \(M-N\) zeros after the \(N\)-point signal samples. Then, a coarse estimate can be obtained by maximizing the magnitude of the \(M\)-point FFT, which is an efficient computation method of the DFT. In the fine estimation stage, every iteration, three DTFT samples are calculated to obtain a fine estimate. The general form of the estimator and a mean square error (MSE) analysis are also presented so that the estimator parameters can be adjusted according to the given design requirements. Simulation results reveal that the RMSE of the proposed estimator is closest to the CRLB in most cases. At the same time, this estimator has a relatively simple form and low computational complexity, so it can satisfy the high accuracy and fast estimation speed requirements of the given application environments.