## 3 Performance Analysis
### Performance of Optimal Receiver  
Each pulse of the PPM system carries \(M\)-bit information, and each symbol is divided into \(Q=2^{M}\) time slots. The basic idea of optimal PPM system demodulation is to successively sample the envelope sampling value of the first time slot to the envelope value of the \(2^{M}\) time slot, to compare the size and to judge the time slot with the largest sampling value as the time slot sent by the pulse signal to judge it as the corresponding \(M\)-bit information. The optimal receiver does not need to set the threshold value, so the difficulty of the dynamic adjustment of the threshold is avoided.  
Based on the above idea, assuming that the transmitted pulse signal is in the \(k\)th time slot, the decision error occurs when the envelope sampling value of one and only one noise time slot exceeds the envelope sampling value of the \(k\)th time slot. When the envelope sampling value of only two noise time slots exceeds the envelope sampling value of the \(k\)th time slot, a decision error is also made. By analogy, when the envelope sampling value of all noise time slots exceeds the envelope sampling value of the \(k\)th time slot, an error is also determined. Only by adding the probabilities of all the above decision error cases can the BER of the optimal noncoherent PPM receiver be obtained, but the calculation is very complex. Therefore, we also calculate the BER of the optimal noncoherent PPM receiver by calculating the probability of a correct decision first and then by finding its complement.  
According to the above considerations, it is assumed that the signal pulse transmitted by the PPM system is located in the \(k\)th time slot. As long as the envelope sampling value of all noise time slots does not exceed the envelope sampling value of the \(k\)th time slot, the symbol decision is correct. First, the probability that a noise slot envelope sampling value does not exceed the signal pulse slot envelope sampling value is  
\[P_{c1}=\int_{0}^{u_{S}}p(u_{N})du_{N}\,. \tag{15}\]  
Then, the probability that all noise slot envelope sampling values are less than the signal pulse slot envelope sampling values is  
\[P_{c2}=\prod_{i=1}^{2^{M}-1}P_{c1}\,. \tag{16}\]  
Taking the average of all possible sampling values of the signal yields  
\[P_{c3}=\int_{0}^{\infty}P_{c2}\cdot p(u_{S})du_{S}\,. \tag{17}\]  
Figure 4: BER of 8-ary noncoherent PPM suboptimal receiver vs. \(\gamma\).  
Since the signal pulses are evenly distributed in all possible time slots, the probability of a correct decision for any symbol is  
\[P_{c}=\frac{1}{2^{M}}\sum_{i=1}^{2^{M}}P_{c3}=P_{c3}\,. \tag{18}\]  
By taking its complement, it can be obtained that the optimal symbol error rate of the noncoherent PPM receiver is  
\[P_{ew}=1-P_{c}\,. \tag{19}\]  
Substituting (2), (3) and (15)-(18) to (19) yields  
\[P_{ew}=\sum_{k=1}^{2^{M}-1}\binom{2^{M}-1}{k}\frac{(-1)^{k+1}}{k+1}\exp\left(- \frac{k}{k+1}\frac{A^{2}}{2\sigma^{2}}\right) \tag{20}\]  
where  
\[\binom{2^{M}-1}{k}=\frac{(2^{M}-1)!}{k!(2^{M}-1-k)!}\,. \tag{21}\]  
Considering \(A^{2}/2\sigma^{2}=M\cdot E_{b}/N_{0}\), then (20) is simplified as  
\[P_{ew}=\sum_{k=1}^{2^{M}-1}\binom{2^{M}-1}{k}\frac{(-1)^{k+1}}{k+1}\exp\left( -\frac{k}{k+1}\cdot M\cdot\frac{E_{b}}{N_{0}}\right). \tag{22}\]  
Similarly, from the conversion relationship between the symbol error rate and BER (13), the optimal BER of the noncoherent PPM receiver is  
\[\begin{split} P_{eb}&=\frac{2^{M-1}}{2^{M}-1}P_{ew} \\ &=\frac{2^{M-1}}{2^{M}-1}\cdot\sum_{k=1}^{2^{M}-1}\binom{2^{M}-1 }{k}\frac{(-1)^{k+1}}{k+1}\exp\left(-\frac{k}{k+1}\cdot M\cdot\frac{E_{b}}{N_ {0}}\right).\end{split} \tag{23}\]  
The above BER is consistent with the BER of the noncoherent demodulation of the \(M\)-ary orthogonal signal because the PPM signal is also an orthogonal signal in essence.