## 1 Introduction
However, the time-delay estimation accuracy of GCC in the wideband sampling system is decreased severely because the extremely sharp peak cannot be used to improve the accuracy of the estimated value by interpolation theory. For this situation, the sampling accuracy can be improved by zero-padding in the frequency domain, but the computational workload will be multiplied. Based on the aforementioned points, in this paper we focus on the optimization problem of the BCC algorithm--namely, \(\max_{\tau}R(\tau)\)--in the context of the research on phased-array antenna technology for satellite terminals. [PERSON] [11] applies the parabolic interpolation method to the peak calculation of \(R(\tau)\) to improve the accuracy of the time-delay estimation, but there is a misfit phenomenon. Therefore, [PERSON] [15] uses the interpolation theory to reconstruct \(R(\tau)\) so as to obtain a performance close to that of CRLB. However, the algorithm has two intrinsic limitations, the computational complexity and the very small grid size, which are key to improving the estimation precision. To cope with these limitations, [PERSON] [19] proposes a new two-step method to obtain coarse time-delay estimates based on [20] first, then obtains a fine solution via parabolic extremization and triangular interpolation. However, the algorithm requires matrix inversion, which makes it insufficient for large-scale sensor arrays. [PERSON] [21] designed an estimation method based on the envelope differences of correlation functions (EDCFs) while maintaining a high precision and lower computational complexity; however, the algorithm has a limited range of time-delay estimation. In summary, this paper proposes a polynomial inversion-based estimation algorithm that takes into account the accuracy and algorithm complexity while extending the range of time-delay estimation. The main contributions of this paper include:  
1. Taking the convex parabolic extremum equation determined by the three sets of cross-correlation values as TED and feeding the result of TED into the polynomial inversion equation, an accurate fractional delay estimate with the desired performance and a lower computational complexity can be obtained. 2. The CRLB is produced to adapt to more scenarios. The bias, variance, and MSE equations of the new algorithm are also derived. Combined with the CRLB and existing approaches, the simulations show that the proposed time-delay estimator is asymptotically unbiased, with its MSE performing on the CRLB. 3. Applying the algorithm to the RF multichannel broadband acquisition system, the synchronization error between the channels is less than 5 ps. The rest of the paper is organized as follows. In Section 2, we introduce the principle of fast delay estimation based on BCC. Section 3 describes the principle of polynomial inversion-based time-delay estimation. To evaluate the performance of the proposed algorithm, expressions for bias, variance, and MSE are derived, and the performance curves of the algorithm are depicted in conjunction with the produced CRLB in Sections 4 and 5. A design example is given in Section 6. Finally, we draw conclusions in Section 7. Notation: We use \(\tau\), \(\tilde{\tau}\), and \(\tilde{\tau}\) to represent the true time delay, the output of TED, and the optimized estimate of the polynomial, respectively. The subscripts \(()_{r}\) and \(()_{i}\) denote the imaginary and real parts of a complex variable. The symbol \(*\) means conjugate, \(\Re(\cdot)\) returns the real part of a complex number, and \(E\) stands for the mathematical expectation. \(R_{x}(\cdot)\) represents the autocorrelation function of signal \(x\), while \(R_{x_{1}x_{2}}(\cdot)\) denotes the correlation function of \(x_{1}\) and \(x_{2}\). The \(\hat{R}_{x_{1}x_{2}}(\cdot)\) stands for the estimate of \(R_{x_{1}x_{2}}(\cdot)\).