## 3 Performance Analysis
### Performance of Suboptimal Receiver
Then, the probability that the noise slot envelope sampling value does not exceed the threshold is  
\[P_{c1}=\int_{0}^{U_{0}}p(u_{N})du_{N}\,. \tag{4}\]  
If the envelope value of each noise time slot is independent and identically distributed, the probability that the envelope value of the first \(k-1\) noise time slots does not exceed the threshold is  
\[{P_{c1}}^{\prime}(k)=\prod_{i=1}^{k-1}P_{c1}\,. \tag{5}\]  
When the \(k\)th time slot contains a signal, the probability that its envelope value exceeds the threshold is  
\[P_{c2}(k)=\int_{U_{0}}^{+\infty}p(u_{S})du_{S}\,. \tag{6}\]  
Therefore, when \(k\leqslant 2^{M-1}\), the probability of correct symbol judgment is  
\[P_{c3}(k\leq 2^{M-1})={P_{c1}}^{\prime}(k)\cdot P_{c2}(k)\,. \tag{7}\]  
When \(k=2^{M}\), the probability of correct symbol judgment is  
\[P_{c3}(k=2^{M})={P_{c1}}^{\prime}(2^{M})\cdot P_{c2}(2^{M})+{P_{c1}}^{\prime}( 2^{M})\cdot\left[1-P_{c2}(2^{M})\right]={P_{c1}}^{\prime}(2^{M})\,. \tag{8}\]Since \(k\) is uniformly distributed in \(2^{M}\) time slots, the probability of the correct symbol judgment of the suboptimal receiver of the noncoherent PPM system is  
\[P_{c}=\frac{1}{2^{M}}\Bigg{(}\sum_{k=1}^{2^{M}-1}P_{G}(k\leq 2^{M-1})+P_{c3}(k=2^{M} )\Bigg{)}\,. \tag{9}\]  
Thus, the symbol error rate is  
\[P_{ew}=1-P_{c}\,. \tag{10}\]  
Substituting (2)-(9) into (10) yields  
\[ \begin{split} P_{ew}=1-\frac{1}{2^{M}}\Bigg{\{}& \sum_{k=1}^{2^{M}-1}\Bigg{[}\prod_{i=1}^{k-1}\int_{0}^{U_{0}}\frac{u_{N}}{ \sigma^{2}}e^{-\frac{u_{N}2^{2}}{\sigma^{2}}}du_{N}\Bigg{]}\cdot\int_{U_{0}}^{ +\infty}\frac{u_{S}}{\sigma^{2}}e^{\left(-\frac{u_{S}2^{2}+A^{2}}{2\sigma^{2} }\right)}I_{0}\bigg{(}\frac{Au_{S}}{\sigma^{2}}\bigg{)}du_{S}\\ &\qquad\qquad+\prod_{i=1}^{2^{M}-1}\int_{0}^{U_{0}}\frac{u_{N}}{ \sigma^{2}}e^{-\frac{u_{N}2^{2}}{\sigma^{2}}}du_{N}\Bigg{\}}\\ =1-\frac{1}{2^{M}}\Bigg{\{}&\sum_{k=1}^{2^{M}-1} \Bigg{[}\left(1-e^{-\gamma^{2}\frac{A^{2}}{2\sigma^{2}}}\right)^{k-1}\cdot Q_ {1}\bigg{(}\frac{A}{\sigma},\gamma\frac{A}{\sigma}\bigg{)}\Bigg{]}+\left(1-e ^{-\gamma^{2}\frac{A^{2}}{2\sigma^{2}}}\right)^{2M-1}\Bigg{\}}\,.\end{split}  \tag{11}\]  
where \(Q_{1}(a,b)\) is the Marcum-\(Q\) function. As \(A^{2}/2\sigma^{2}=M\cdot E_{b}/N_{0}\), there are  
\[ \begin{split} P_{ew}=1-\frac{1}{2^{M}}\Bigg{\{}& \sum_{k=1}^{2^{M}-1}\Bigg{[}\left(1-e^{-\gamma^{2}M\cdot\frac{E_{b}}{N_{0}}} \right)^{k-1}\cdot Q_{1}\Bigg{(}\sqrt{2M\cdot\frac{E_{b}}{N_{0}}},\gamma\sqrt {2M\cdot\frac{E_{b}}{N_{0}}}\Bigg{)}\Bigg{]}\\ &\qquad\qquad+\left(1-e^{-\gamma^{2}M\cdot\frac{E_{b}}{N_{0}}} \right)^{2M-1}\Bigg{\}}\,.\end{split}  \tag{12}\]  
According to the assumption that the pulse signal is uniformly distributed in the possible time slots, the conversion relationship between the symbol error rate and bit error rate (BER) is [25]  
\[P_{eb}=\frac{2^{M-1}}{2^{M}-1}P_{ew}\,. \tag{13}\]  
Hence, the BER of the suboptimal receiver of noncoherent PPM system is  
\[ \begin{split} P_{eb}=\frac{2^{M-1}}{2^{M}-1}\Bigg{\{}& 1-\frac{1}{2^{M}}\Bigg{\{}& \sum_{k=1}^{2^{M}-1}\Bigg{[}\left(1-e^{-\gamma^{2}M\cdot\frac{E_{b}}{N_{0}}} \right)^{k-1}\cdot Q_{1}\Bigg{(}\sqrt{2M\cdot\frac{E_{b}}{N_{0}}},\gamma\sqrt {2M\cdot\frac{E_{b}}{N_{0}}}\Bigg{)}\Bigg{]}\\ &\qquad+\left(1-e^{-\gamma^{2}M\cdot\frac{E_{b}}{N_{0}}}\right)^{ 2M-1}\Bigg{\}}\,.\end{split}  \tag{14}\]  
As can be seen from the above formula, the threshold coefficient \(\gamma\) has a great impact on the performance of the noncoherent PPM with a sub-optimal receiver. The correct selection of the threshold coefficient can lead to better bit error performance. The impact of the threshold coefficient on the bit error performance is shown in Figure 4.  
Figure 4 gives the BER performance of the noncoherent suboptimal receiver of the 8-ary PPM versus the value of \(\gamma\), where the horizontal coordinate is the \(\gamma\), the vertical coordinate is the BER, the red line indicates the BER at different \(E_{b}/N_{0}\) and \(\gamma\) and the black dashed line indicates the trend of the optimal \(\gamma\) with \(E_{b}/N_{0}\).