## 2 Communication System Model and Theory
### Correlation Cost Factor-Based Updates to Matched Filters Reception Method for M-SS (CCF-MSS)  
When the channel exhibits strong time variability and low SNR, many velocity prediction methods often yield suboptimal results, which is attributed to the significant reduction in the overall correlation of the time-varying channel. It has been demonstrated that even with rapid time variation in the channel, the channel correlation of consecutive symbols remains relatively high [25; 28], which also indicates that the frequency shifts of consecutive symbols are similarly close; we can utilize the accurate estimation results of preceding symbols to guide the frequency shift estimation direction of subsequent symbols. Here, we consider two methods for selecting frequency shift estimation results: The first method involves selecting the frequency shift estimation results based on the matrix peaks of the two-dimensional correlation matrix mentioned in Section 2.2. The second method involves selecting the frequency shift estimation results based on the correlation cost factors calculated from the two-dimensional correlation matrix. This second method represents an innovative approach that we propose, which will be introduced and theoretically derived in the following sections. The signal processing flow of the proposed method is illustrated in the reception section of Figure 1. The received signal is first synchronized and then input into the \"Doppler Coarse Estimation\" module to preliminarily estimate the Doppler value of the first symbol. Subsequently, the estimation result is passed to the \"Matched Filter\" module for fine Doppler estimation and computation of the correlation cost factor. Based on the outcome of the correlation cost factor, the final decoding result is output, and the current Carrier Frequency Offset (CFO) value is saved to feed back into the \"Matched Filter\" module as an initial reference point for the Doppler search of the next symbol. As illustrated in Figure 1, the computational efficiency of CCF-MSS does not show a marked improvement over WDE-MSS or CDE-MSS, primarily because WDE-MSS and CDE-MSS do not involve precise Doppler estimation, a process that demands substantial computational resources. When juxtaposed with FDE-MSS under identical search iterations, CCF-MSS merely necessitates an additional division operation, which accounts for a negligible computational load. Consequently, it can be inferred that FDE-MSS and CCF-MSS exhibit similar computational demands given the same number of searches. Nevertheless, CCF-MSS distinguishes itself by calculating a correlation cost factor during the search process; should this factor fall below a predetermined threshold, the search can be prematurely terminated. This capability allows for significant reductions in computational effort compared to the conventional fixed-iteration FDE-MSS, and it also enhances the accuracy of the decoding results. We first designed a specialized frame structure, as illustrated in Figure 3, in which the first two symbols of each frame are assigned the same spre  
Figure 2: Doppler estimation process. as the repeated code group, to facilitate Doppler estimation. Although this may slightly reduce the transmission rate, it allows for a more accurate initial Doppler estimation. After performing preprocessing operations such as synchronization and filtering on the received signal, the first repeated code group in the signal is subjected to coarse Doppler estimation on a frame-by-frame basis, yielding the initially estimated carrier frequency \(f_{c}^{*}\) followed by searching for frequency shifts with a precision of \(df\) and a range of \(2k\) for the carrier frequency, which generates a two-dimensional correlation matrix, and after calculating the correlation cost factors at each point in the matrix, outputs the index positions and despreading results. Subsequently, the matching filters are updated based on the correlation cost factors.