# A Novel Non-Coherent \(Q\)-Ary TH-PPM Transceiver
## 3 Performance Analysis
### BER Performance Analysis under Non-Ideal Synchronization Conditions with Constant Time Offset  
In practical applications, the TH-PPM pulse is very narrow, which needs to achieve high accuracy of synchronization for the receiver. However, it is difficult to implement. Synchronization inaccuracy can be expressed by several factors: constant time offset and jitter. The following are the analyses of the BER performance under non-ideal synchronization conditions with constant time offset for the conventional TH-PPM receiver and the new TH-PPM receiver. Without loss of generality, a time synchronization error factor is defined as \(\mu\), representing the ratio of synchronization error to the width of the time slot \(\Delta T\). When \(\mu=0\), it indicates that the time synchronization is perfect. While \(\mu=0.1\), it indicates that there is a slight error in time synchronization. And then \(\mu=1\), it indicates that the time synchronization error has exceeded the width of the time slot \(\Delta T\).  
Due to the data decision flow of the conventional TH-PPM receiver, the BER performance is positively correlated with the sampling values of the output of the matched filter. Therefore, we can obtain the BER performance by analyzing the deterioration of the sampling values output by the matched filter under non-ideal synchronization conditions. Since the accuracy of time synchronization directly affects the envelope of energy within the sampling slot, the envelope of energy within the sampling slot can be expressed as,  
\[\left\{\begin{array}{l}A_{1}^{2\prime}=\int_{0}^{1-\mu}A_{1}^{2}dt=(1-\mu)A _{1}^{2},\;0\leq\mu\leq 1\\ A_{1}^{2\prime}=0,\;others\end{array}\right. \tag{16}\]  
where \(A_{1}^{2\prime}\) is the actual energy of the envelope within the sampling slot. And \(A_{1}^{2}\) representing the energy carried by a TH-PPM symbol. When \(\mu=0\), the time synchronization is perfect, and the actual energy of the envelope within the sampling slot is equal to the energy carried by the TH-PPM symbol. When \(0\leq\mu\leq 1\), the SNR of the output of the matched filter can be expressed as,  
\[\frac{S^{\prime}}{N^{\prime}}=\frac{N_{s}(1-\mu)A_{1}^{2}}{2\delta^{2}},\;0 \leq\mu\leq 1 \tag{17}\]  
For \(N_{s}\cdot{A_{1}}^{2}/{2\sigma^{2}}=\log_{2}^{Q}\cdot{E_{b}}/{N_{0}}\), there is  
\[\frac{S^{\prime}}{N^{\prime}}=(1-\mu)\cdot\log_{2}^{Q}\frac{E_{b}}{N_{0}},\;0 \leq\mu\leq 1 \tag{18}\]  
Substituting Equation (18) into Equation (14), we have  
\[P_{eb}=\frac{Q/2}{Q-1}\cdot\sum_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{k+1 }\exp\left(-\frac{k}{k+1}\cdot(1-\mu)\cdot\log_{2}^{Q}\frac{E_{b}}{N_{0}} \right),\;0\leq\mu\leq 1 \tag{19}\]  
The analysis is semi-quantitative, for time synchronization errors not only reduce the sampling value by lowering the envelope value of the TH-PPM pulse slot, but also raise the sampling value by raising the envelope value of the noise slot, which further reduces the SNR. Therefore, the analysis is the theoretical lower bound for BER when time synchronization errors exist. The actual BER performance is slightly worse, but it is sufficient for trend analysis.  
For the data decision flow of the new TH-PPM receiver, the data symbol is determined by which matched filter outputs a correlation peak. Thus, the start of the time slot is not required. In this way, the SNR of the decision flow will not be reduced by synchronization errors. The demodulation of the new TH-PPM relies on the relative position relationship between multiple pulses within a symbol, while a TH-PPM symbol's duration is typically about tens of microseconds or less. In such a short time range, it can be assumed that the timing synchronization is fixed and the relative position relationship between the pulses within the symbol remains constant. For parallel matched filters output by the pipeline, the amplitude of the correlation peak will not change. In ideal cases, the BER performance of the new TH-PPM receiver is not affected by the timing synchronization error, and it can still be described by Equation (15).