## 3 Low-Complexity Optimization of the Algorithm
### Reducing Matching Branches  
The proposed algorithm reduces the number of matching branches by averaging the local reference signals and leveraging the phase symmetry of the signal, so the following two aspects are explained in detail in this section. (a) Average local reference signal  
Suppose the observation interval \(N=5\). Due to the influence of the pre-modulation filter, inter-symbol interference exists between the signals. In the traditional MSD algorithm, the number of local reference signal waveforms is \(L=2^{N+2}=128\). However, since each sequence includes one uncertain bit in the head \(a_{0}\) and tail \(a_{6}\) caused by inter-symbol interference, its impact on subsequent matched filtering is minimal; therefore, the middle N-bit values can be grouped into different sets, with each group having four possible combinations, and the local reference signal waveform can be averaged for each group. Taking the five-symbol demodulation algorithm as an example, when the middle bit takes a value of 1 (corresponding to the top half of branches in Figure 2) and only real-part branches are computed (corresponding to \(\theta_{0}=0\) in Figure 2), the correlation operation details in Figure 2 are illustrated in Figure 3. Every four matching branches with the same middle bits [\(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\)] are optimized into a single branch. Image-part branches are also optimized according to the same method. In Figure 3, \(a_{3}=1\); \(k_{1}\) and \(k_{2}\) represent the average value of sampling symbols at the head and tail positions, respectively; and \(\widetilde{a}_{i}\) denotes the average sample value of symbol  
Figure 3: Average matched filter schematic diagram. after sampling and averaging. Under the assumption of no inter-symbol interference, one of the local reference signal sets for a particular code group can be expressed as follows:  
\[A_{L}^{\prime}=\{a_{1},a_{2},a_{3},a_{4},a_{5}\} \tag{9}\]  
This code group, in the presence of inter-symbol interference, can generate four distinct code groups, which can be represented as the following:  
\[A_{L}=\{a_{0},A_{L}^{\prime},a_{6}\} \tag{10}\]  
where \(a_{0}\) and \(a_{6}\) take values of 1 or 0. The modulation of the code group signals can be averaged as follows:  
\[\widetilde{L}(t)=\frac{1}{K}\sum_{i=0}^{K-1}L(t_{i}) \tag{11}\]  
where \(L(t)\) is the modulated signals corresponding to the code group \(A_{L}\). Based on Equation (10), we have \(K=4\). Thus, the number of local reference signals is decreased from \(2^{N+2}\) to \(2^{N}\), which is one-fourth of the original number of branches. Subsequently, only the optimized local reference signals need to be correlated, followed by selection and decision. (b) Leverage phase symmetry  
By leveraging signal phase symmetry, the proposed algorithm can further optimize the number of local reference signals without any performance loss. In Figure 2, it can be seen that the top half of the matching branches is different from the bottom half in phase, and this is because the middle bits take different values with 0 and 1. Figure 4 shows the phase tree of the PCM/FM signals, where the horizontal axis represents the increment in the number of symbols, and the vertical axis represents the phase increment. It is evident that the stored local reference signals exhibit the characteristic of phase symmetry. Specifically, among all local reference signals, the code groups \(\{a_{0},a_{1},a_{2},0,a_{4},a_{5},a_{6}\}\) and \(\{a_{0},a_{1},a_{2},1,a_{4},\)\(a_{5},a_{6}\}\) are symmetric, where \(\{a_{0},a_{1},a_{2},a_{4},\)\(a_{5},a_{6}\}\) represents any possible combination of symbols. According to Equation (5), in their corresponding complex forms, these two code groups are complex conjugates of each other. This can be described as follows.