# A Novel Non-Coherent \(Q\)-Ary TH-PPM Transceiver
## 3 Performance Analysis
### BER Performance Analysis under Ideal Synchronization Conditions  
In this part, it is assumed that the receiver has achieved perfect synchronization, both for the conventional TH-PPM receiver and the new TH-PPM receiver. For the conventional TH-PPM receiver, the TH-PPM pulses are recovered through envelope detection. Then the sampling values are achieved every \(\Delta T\) slot from the output of the matched filter. The noise envelope sampling without a signal follows the Rayleigh distribution as shown in Equation (4), while the envelope sampling with a signal follows the Rician distribution as shown in Equation (5),  
\[p(u_{1})=\frac{u_{1}}{\sigma^{2}}\exp\left(-\frac{{u_{1}}^{2}}{2\sigma^{2}}\right) \tag{4}\]  
\[p(u_{2})=\frac{u_{2}}{\sigma^{2}}\exp\left[-\frac{{u_{2}}^{2}+{N_{\rm s}}\cdot {A_{1}}^{2}}{2\sigma^{2}}\right]{I_{0}}\left[\frac{{u_{2}}\cdot\left(\sqrt{{N _{\rm s}}}\cdot{A_{1}}\right)}{2\sigma^{2}}\right] \tag{5}\]  
where \(u_{1}\) represents the sampling value of the matched filter when there is no TH-PPM pulse in the \(\Delta T\) slot, \(\sigma^{2}\) represents the variance of Gaussian white noise. \(u_{2}\) represents the sampling value of the matched filter when there is a TH-PPM pulse in the \(\Delta T\) slot, \(A_{1}\) is the amplitude of the output of the matched filter with a TH-PPM pulse, and \(I_{0}(\cdot)\) is the zero-order Bessel function. For the conventional TH-PPM receiver, if the sampling value at the time slot without the TH-PPM pulse does not exceed the sampling value at the time slot with the TH-PPM pulse, then the sample is judged to be correct, and the probability is shown in Equation (6). \[P_{1}=\int_{0}^{u_{2}}p(u_{1})du_{1} \tag{6}\]  
For a TH-PPM symbol, there are \(Q-1\) time slots without a TH-PPM pulse. Thus, when the \(Q-1\) sampling values do not exceed the sampling value at the time slot with the TH-PPM pulse, the TH-PPM symbol can be determined correctly. The probability is  
\[P_{2}=\prod_{i=1}^{Q-1}P_{1} \tag{7}\]  
The average of all possible values can be obtained by  
\[P_{3}=\int_{0}^{\infty}P_{2}\cdot p(u_{2})du_{2} \tag{8}\]  
Since the transmitted signal is evenly distributed across the time slots, the probability of obtaining a correct symbol decision is  
\[P_{c}=\frac{1}{Q}\sum_{i=1}^{Q}P_{3}=P_{3} \tag{9}\]Then we can obtain the symbol error rate of the conventional TH-PPM receiver by  
\[P_{ew}=1-P_{c} \tag{10}\]  
Substituting Equations (4)-(9) into Equation (10) yields  
\[P_{ew}=\sum_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{k+1}\exp\biggl{(}-\frac{k }{k+1}\frac{N_{s}\cdot{A_{1}}^{2}}{2\sigma^{2}}\biggr{)} \tag{11}\]  
For \(N_{s}\cdot{A_{1}}^{2}/2\sigma^{2}=\log_{2}^{Q}\cdot E_{b}/N_{0}\), we have  
\[P_{ew}=\sum_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{k+1}\exp\biggl{(}-\frac{ k}{k+1}\cdot\log_{2}^{Q}\cdot\frac{E_{b}}{N_{0}}\biggr{)} \tag{12}\]  
From [20], we have the conversion relationship between the symbol error rate and bit error rate (BER) is  
\[P_{eb}=\frac{Q/2}{Q-1}P_{ew} \tag{13}\]  
Hence, the BER of the conventional TH-PPM receiver under ideal synchronization conditions is  
\[ \begin{array}{ll}P_{eb}&=\frac{Q/2}{Q-1}P_{ew}\\ &=\frac{Q/2}{Q-1}\cdot\sum\limits_{k=1}^{Q-1}\binom{Q-1}{k}\frac{(-1)^{k+1}}{ k+1}\exp\biggl{(}-\frac{k}{k+1}\cdot\log_{2}^{Q}\cdot\frac{E_{b}}{N_{0}} \biggr{)}\end{array}  \tag{14}\]  
For the new \(Q\)-ary TH-PPM receiver, there are \(Q\) parallel matched filters. And every matched filter outputs a sampling value. If the data symbol transmitted is not correlated with one matched filter, the output of the matched filter will be noise, and which sampling value follows the Rayleigh distribution as shown in Equation (4). While the data symbol transmitted is correlated with one matched filter, the output of the matched filter will be a correlation peak, whose sampling value follows the Rician distribution as shown in Equation (5).