# Adaptive Modulation Tracking for High-Precision Time-Delay Estimation in Multipath HF Channels
## 3 MATE Algorithm
### TOA Tracking  
After completing parameter initialization, the TOA tracking stage begins. The tracking loop employs closed-loop feedback mechanisms to achieve signal synchronization, and a good tracking loop design can accurately strip the carrier of the signal to obtain the phase deviation of the modulating signal [19], so as to realize high-precision delay estimation, so the tracking loop is the key link in realizing HF TOA. The structure diagram of the MATE tracking algorithm is shown in Figure 4.  
TOA tracking consists of two parts: the modulating signal phase tracking loop and the carrier tracking loop. Among them, the carrier tracking loop can be divided into the phase-locked loop (PLL) and frequency-locked loop (FLL) [20], and the purpose of carrier tracking is to try its best to make the carrier signal it replicates consistent with the received HF carrier signal, so as to completely strip the carrier in the HF signal through frequency mixing [21]. The modulating signal phase tracking loop is mainly a delay-locked loop (DLL) [22], the main purpose of which is to calculate the phase difference between the received signal and the transmitted signal to accomplish TOA estimation. The main components of both loops are as follows: correlator, discriminator, loop filter, and local Numerically Controlled  
Figure 4: The structure of the TOA tracking. The fundamental principle is outlined as follows: The received signal is first subjected to a mixing correlation operation with the in-phase (\(I\)) and quadrature (\(Q\)) carrier reference signals generated by the local carrier NCO. This critical step achieves both carrier stripping and signal down-conversion. Subsequently, the processed baseband signal is directed to three parallel correlation channels, where it undergoes precise correlation operations with the advanced (\(E\)), prompt (\(P\)), and delayed (\(L\)) modulation reference signals generated by the modulated signal generator. Through a meticulously designed integrate-and-dump operation (accumulating signal energy over a fixed time interval \(T\) followed by reset), six key correlation values are ultimately obtained: \(I_{E}\), \(Q_{E}\), \(I_{P}\), \(Q_{P}\), \(I_{L}\), and \(Q_{L}\). These correlation values can be uniformly expressed by the following mathematical expression:  
\[I =A\mathrm{sinc}(\pi\delta fT)*R(\delta\tau+kd)*\cos(\delta\tau)+n_{I} \tag{9}\] \[Q =A\mathrm{sinc}(\pi\delta fT)*R(\delta\tau+kd)*\sin(\delta\tau)+n _{Q} \tag{10}\]  
where \(A\) is the signal amplitude; \(sinc(\pi\delta fT)=sin(\pi\delta fT)/(\pi\delta fT)\); \(\delta f\) is the carrier frequency error; \(T\) is the coherent integration time; \(R(\cdot)\) is the modulating signal autocorrelation function; \(\delta\tau\) is the modulating signal phase error; \(d\) is the neighbor correlator spacing; for the early, prompt, and late branches, k takes the values of \(-1\), \(0\), and \(+1\), respectively; and \(n_{I}\) and \(n_{Q}\) are the zero-mean Gaussian white noises of the two independently uncorrelated branches of \(I/Q\), respectively. Subsequently, the PLL, FLL, and DLL loops utilize the carrier phase discriminator, carrier frequency discriminator, and modulating signal phase discriminator to discriminate the tracking error of these six \(I/Q\) messages to obtain the carrier phase error, carrier frequency error, and modulating signal phase error, respectively. The algorithms used for each discriminator and their characterization are given in Table 1.  
The discriminator output discrimination error will be sent to the loop filter for low-pass filtering process to reduce the noise of the discrimination error in the loop. This helps avoid the high-frequency noise caused by over-regulation of the voltage-controlled oscillator and is conducive to the stability of the loop [25; 26; 27]. Figure 5 shows a block diagram of the second-order digital loop filter, and its filtering transfer function is as follows:  
\[F(z)=\frac{1}{K}\bigg{(}a_{2}\omega_{n}+\frac{T_{s}}{2}\frac{1+z^{-1}}{1-z^{-1 }}\omega_{n}^{2}\bigg{)} \tag{11}\]