# Anti-Multipath Performance Improvement of an M-ary Position Phase Shift Keying Modulation System
## 3 The Scheme of Multipath Delay Estimation and Normalized Symbol Joint Decision
### Normalized Symbol Joint Decision  
The premise of symbol joint decision is to estimate the multipath delay and then the separation of multipath. The key is to make the symbol information contained in the sub-path participate in the decision, that is, the sampling integral decision value of the current symbol and the sampling integral decision value including the current symbol in the multipath are used as the decision basis [17]. The symbol joint decision also introduces the noise of the sub-path when using the sub-path symbol information to participate in the decision, which causes the bit error rate not to reach the minimum level. Therefore, this paper proposes a normalized symbol joint decision method. After the signal is transmitted through multipath channel, the receiving expression is given by Equation (3). After obtaining the relative delay of each path, the delay value of the i-th path is eliminated and the waveform with no delay for each path is obtained. Then, each path signal is normalized and added to the main path, and the noise introduced by the sub-path is reduced while using the symbol information included in the sub-path, thereby improving the signal-to-noise ratio and reducing the bit error rate. The added signal is given by  
\[r(t)=\sum\limits_{i=1}^{n}\frac{a_{i}}{a_{1}}a_{i}g(t) \tag{6}\]  
where \(a_{1}\) is the path gain of the main path and \(\frac{a_{i}}{a_{1}}\) is the value obtained by normalizing the path gain of the i-th path with the main path gain. It is assumed that when the signal is transmitted, the energy of each symbol is \(E_{s}\), and the amplitude gain of the i-th path is \(a_{i}\), so the amplitude of the equivalent baseband signal on the i-th path is \(a_{i}\sqrt{E_{s}}\), and the symbol energy is \({a_{i}}^{2}E_{s}\). Assuming that the noise power spectral density is \(N_{0}\), the signal-to-noise ratio on the \(i\)-th path is given by  
\[\gamma_{i}=\frac{{a_{i}}^{2}E_{s}}{N_{0}} \tag{7}\]  
when the received signals are combined, the weighting coefficient of each path is \(x_{i}\), then the amplitude of the equivalent baseband signal of the i-th path after weighting is \(x_{i}a_{i}\sqrt{E_{s}}\), and the amplitude of the combined signal is given by  
\[A=\sum\limits_{i=1}^{n}x_{i}a_{i}\sqrt{E_{s}} \tag{8}\]  
The signal-to-noise ratio of the combined signal is given by  
\[\begin{split}\gamma(x_{1},x_{2},\cdots,x_{n})&= \frac{(\sum\limits_{i=1}^{n}x_{i}a_{i}\sqrt{E_{s}})^{2}}{\sum\limits_{i=1}^{n }{x_{i}}^{2}N_{0}}\\ &=\frac{E_{s}}{N_{0}}\frac{(\sum\limits_{i=1}^{n}x_{i}a_{i})^{2} }{\sum\limits_{i=1}^{n}{x_{i}}^{2}}\end{split} \tag{9}\]  
1. When the receiver adopts the symbol joint decision scheme, the weighting coefficient of each path is 1, that is, \(x_{i}=1(i=1,2,\cdots,n)\). The signal-to-noise ratio of the combined signal is given by \[\gamma_{S}(x_{1},x_{2},\cdots,x_{n})=\frac{E_{s}}{N_{0}}\frac{(\sum\limits_{i=1}^{n }a_{i})^{2}}{\sum\limits_{i=1}^{n}1^{2}} \tag{10}\]
2. When the receiver adopts the normalized symbol joint decision scheme, the weighting coefficients of the i-th path is \(\frac{a_{i}}{a_{1}}\), that is, \(x_{i}=\frac{a_{i}}{a_{1}}\). The signal-to-noise ratio of the combined signal is given by \[\gamma_{N}(x_{1},x_{2},\cdots,x_{n})=\frac{E_{s}}{N_{0}}\frac{(\sum\limits_{i= 1}^{n}\frac{a_{i}^{2}}{a_{1}})^{2}}{\sum\limits_{i=1}^{n}(\frac{a_{i}}{a_{1}}) ^{2}}\] (11)  
The calculation shows that \(\gamma_{N}(x_{1},x_{2},\cdots,x_{n})\) is larger than \(\gamma_{S}(x_{1},x_{2},\cdots,x_{n})\), that is, the normalized symbol joint decision scheme improves the signal-to-noise ratio and reduces the bit error rate compared with the symbol joint decision scheme. The proof is shown in Appendix A.