## 2 Traditional Multi-Symbol Detection Algorithm  
The PCM/FM modulated signals can be modeled as follows:  
\[\mathrm{S_{pcm/fm}}(t)=\cos(\omega_{c}t)\cos(k_{f}\!\int_{-\infty}^{t}f(\tau)d \tau)-\sin(\omega_{c}t)\sin(k_{f}\!\int_{-\infty}^{t}f(\tau)d\tau) \tag{1}\]  
where \(\omega_{c}\) represents the carrier frequency, \(k_{f}\) denotes the frequency offset, and \(f(t)\) is the signal after pre-filtering.  
The MSD technique exploits the correlation characteristics between signals. When the receiver receives a new symbol, it does not immediately make a decision; instead, it continues to observe the subsequent three to five symbols before making a decision [16]. The specific approach is as follows: the multi-symbol detector performs a correlation operation between the waveforms of the symbols to be decided and the local reference signal waveforms, selecting the waveform with the highest correlation coefficient as the decision for the symbols. Figure 1 shows the overall framework of the receiver system. The principle of the baseband non-coherent MSD algorithm is illustrated in Figure 2.  
In Figure 2, \(L\) (\(t\), \(a_{n}\), \(A_{i}\), \(\theta_{0}\)) represents the local reference signals, which corresponds to the PCM/FM signals of the symbols \([a_{n}\), \(A_{i}]\). \(a_{n}\) denotes the middle bit of the \(N\)-bit sequence to be observed and takes a value of 1 or 0. \(A_{i}\) is the combination of other bits excluding \(a_{n}\). Each pair of adjacent matching branches with \(\theta_{0}\) equaling 0 and \(pi/2\) represents the real and imaginary parts of a set of local reference signals.  
The received signals within the observation interval can be simplified as follows:  
\[s(t)=A\cos[\omega_{c}t+f(t)+\theta_{1}] \tag{2}\]  
where \(A\) is the amplitude, and \(\theta_{1}\) is the initial phase. After downconverting the signal \(s(t)\), the baseband quadrature signals are given by the following equations:  
Figure 1: System overall framework diagram.  
\[I(t)=\cos(f(t)+\theta_{1}) \tag{3}\] \[Q(t)=-\sin(f(t)+\theta_{1}) \tag{4}\]  
The complex form can be expressed as below:  
\[R(t)=I(t)+jQ(t)=\cos(f(t)+\theta_{1})-j\sin(f(t)+\theta_{1})=I_{R}-jQ_{R} \tag{5}\]  
The received signals within the observation interval, along with all possible combinations of local reference signals, can be expressed as follows:  
\[L(t)=\cos(f(t)+\theta_{2})-j\sin(f(t)+\theta_{2})=I_{L}-jQ_{L} \tag{6}\]  
where \(\theta_{2}\) is the initial phase of the local reference signals.  
Figure 2: Principle diagram of multi-symbol detection algorithm.  
The result of the conjugate complex correlation operation between the baseband signals in complex form within the observation interval and the local reference signals can be expressed as follows:  
\[R(t)\cdot L(t)=\cos(\theta_{2}-\theta_{1})+j\sin(\theta_{2}-\theta_{1})=I_{B}+jQ _{B} \tag{7}\]  
The square of the absolute value of the correlation result can be expressed with the following equation:  
\[S=N_{0}^{2}(I_{B}^{2}+Q_{B}^{2}) \tag{8}\]  
where \(N_{0}\) represents the number of samples per symbol. By comparing the magnitude of \(S\), the symbol of the signal within the observation interval can be determined.