## 1 Introduction  
Estimating the delay error between the channels of a sensor array system to realize the angle-of-arrival estimation and beam tracking of the incoming signal is an important task in modern signal processing. This estimation is used extensively in acoustic imaging [1; 2], channel sounding [3; 4], wireless sensor networks [5], and satellite navigation and communication [6; 7; 8; 9]. Efficient and high-precision TDE is critical to prevent the beam'squint' phenomena of large-scale and ultra-wideband-array antennas Figure 1a, which is an essential component of broadband communication systems such as LEO satellite communication and THz communication systems. By high-precision time-delay estimation and difference compensation (Variable Fractional Delay Filter), the array can operate over a large signal bandwidth without beam squint. Moreover, it is key to solving the problem frequency selective fading in broadband large-scale antenna systems. To obtain a flat pattern over the desired frequency range, the antenna's coherent bandwidth should be greater than the bandwidth of the signal. According to this theory, the system must have the ability to perform delay compensation to coherently combine signals, which is an intrinsic property of TTD beamforming, as shown in Figure 1b. Thus, high-performance delay estimators are extensively highlighted in future ultra-wideband communication system scenarios. In recent years, the main algorithms used in TDE are the cross-correlation method [1; 2; 11], generalized phase delay estimation [12], adaptive delay estimation [13; 14; 15], wavelet estimation [16], and compressive sensing [17; 18]. Generalized phase delay estimation captures a set of frequency points and the associated phase information in the band, then estimates the time delay based on their linear relationship. However, this algorithm is only suitable for the case of noise-free or uncorrelated Gaussian noise interference. The adaptive time-delay estimation algorithm overcomes the deficiencies of the generalized phase delay estimation algorithm, which requires a priori knowledge of the signal and noise; however, in order to achieve unbiased estimation, the algorithm needs to increase the order of the fractional time-delay FIR filter, leading to an increase in the computational complexity, while the improvement is not significant under low signal-to-noise ratio (SNR) conditions. [PERSON] [16] combines wavelet transform with the time-delay estimation approach to obtain more accurate time-delay information; yet, the algorithm requires complex calculations and its response time is limited. In an attempt to reduce the burden of data transmission bandwidth, scientists such as [PERSON] [17] have proposed to carry out time-delay error estimation based on compressed sensing ideas. Additionally, [PERSON] [18] gave the CRLB performance bound, but the algorithm was only effective for presenting sparsity after changes. Compared with the mentioned algorithms, the cross-correlation algorithm (CC) is currently the most widely used. The CC algorithm can be divided into Basic Cross-Correlation (BCC) and Generalized Cross-Correlation (GCC) based on the presence or absence of frequency-domain whitening filters. The BCC algorithm, by calculating the  
Figure 1: (**a**) Architecture of typical TTD-based wideband large-scale antenna systems (TTD—True Time Delay; ADC—Analog-to-Digital Converter; DDC—Digital Down-Converter; VFD—Variable Fractional Delay); (**b**) maximal signal bandwidth vs. number of elements [10]. cross-correlation value \(R(\tau)\) of the two signals, solves \(\max_{\tau}R(\tau)\) to yield \(\tilde{\tau}\), which is the relative time-delay estimate of the two signals. The GCC algorithm first performs whitening filtering before calculating the cross-correlation function to eliminate the correlation between channels, thereby reducing the width of the main lobe of the correlation function. It is mostly used in fields such as microphone arrays and ultrasonic positioning.