# A Novel Non-Coherent \(Q\)-Ary TH-PPM Transceiver
## 3 Performance Analysis
### BER Performance Analysis under Non-Ideal Synchronization Conditions with Jitter  
Jitter is another factor that leads to a deterioration in synchronization accuracy. In this part, the BER performance under non-ideal synchronization conditions with jitter is analyzed. We define \(\varepsilon\) as the factor of jitter by Equation (20),  
\[\varepsilon=\frac{\Delta t}{\Delta T} \tag{20}\]Jitter usually changes randomly over time. For convenience, we define it as the max synchronization error caused by a jitter during a statistical period of time and \(\Delta T\) is the time slot of the TH-PPM signal model. So, the real jitter changes randomly from 0 to \(\varepsilon\) in time. When \(\varepsilon=0\), it indicates that the jitter is zero and the synchronization is perfect.  
For conventional TH-PPM receivers, jitter affects the envelope of energy within the sampling slot, the envelope of energy within the sampling slot can be expressed as,  
\[\left\{\begin{array}{l}A_{2}^{2}\prime=\int_{0}^{1-\varepsilon}A_{1}^{2}dt=(1 -\varepsilon)A_{1}^{2},\;0\leq\varepsilon\leq 1\\ A_{2}^{2}\prime=0,\;others\end{array}\right. \tag{21}\]  
where Equation (21) is the same as (16) by replacing \(\mu\) with \(\varepsilon\). According to the analysis from (16) to (19) in Section 3.2, we can have the BER performance with jitter as,  
\[P_{eb}=\frac{Q/2}{Q-1}\cdot\sum_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{k+ 1}\exp\left(-\frac{k}{k+1}\cdot(1-\varepsilon)\cdot\log_{2}^{Q}\frac{E_{b}}{N_ {0}}\right),\;0\leq\varepsilon\leq 1 \tag{22}\]  
Compare (22) with (19), it is easy to know that, for conventional TH-PPM, the BER performance with jitter is the same as the BER performance with constant time offset when \(\varepsilon=\mu\). In other words, the BER performance of the conventional TH-PPM is quite sensitive to jitter. The analysis is semi-quantitative, for jitter not only reduces the sampling value by lowering the envelope value of the TH-PPM pulse slot, but also raises the sampling value by raising the envelope value of the noise slot, which further reduces the SNR. Therefore, the actual BER performance is slightly worse with jitter, but it is sufficient for trend analysis.  
For the new TH-PPM receiver, thanks to the reduced requirement for synchronization accuracy, we can expand the integration time width of matched filters from \(\Delta T\) to \((1+\varepsilon)\cdot\Delta T\), shown in Figure 10. In this way, the matched filter can output a correlation peak without energy loss of the TH-PPM pulse.  
Although the matched filter can output a correlation peak without energy loss by expanding the integration width, the noise output of the matched filter will rise to \(N^{\prime}=(1+\varepsilon)\cdot N\). Then the SNR will deteriorate as,  
\[\frac{S^{\prime}}{N^{\prime}}=\frac{S}{(1+\varepsilon)\cdot N} \tag{23}\]  
Then, for the new TH-PPM, we can have the BER performance with jitter as (24),  
\[P_{eb}=\frac{Q/2}{Q-1}\cdot\sum_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{k+ 1}\exp\left(-\frac{k}{k+1}\cdot\frac{1}{(1+\varepsilon)}\cdot\log_{2}^{Q} \cdot\frac{E_{b}}{N_{0}}\right) \tag{24}\]  
Figure 10: Expand the integration width of matched filters to \((1+\varepsilon)\cdot\Delta T\).