## 3 Performance Analysis
### Performance of Suboptimal 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 the demodulation of a suboptimal PPM system is as follows: first, set a threshold value, start from the first time slot of each symbol and successively compare the envelope sampling value of each time slot with the threshold  
Figure 3: Symbol decision flow of optimal receiver. value. When the sampling value of a time slot envelope exceeds the threshold for the first time, it can be considered that the corresponding pulse signal is transmitted in the slot at this time, and it can be determined as the corresponding \(M\)-bit information; in particular, if the sampling values from the first time slot to the \(2^{M}\) time slot do not exceed the threshold, we agree to determine that the \(2^{M}\) time slot has sent a signal. Based on the above idea, assuming that the transmitted signal pulse is located in the \(k\)th time slot, if the sampling values of the first \(k-1\) time slots are less than the threshold value but the sampling value of the \(k\)-th time slot is also less than the threshold value, the PPM symbol judgment is incorrect. We call this error leakage error \(p_{1}\). If at least one sampling value in the first \(k-1\) time slot exceeds the threshold, a judgment error also occurs, and this corresponds to a false alarm error \(p_{2}\). Adding \(p_{1}\) and \(p_{2}\) allows for the bit error performance expression of the suboptimal PPM receiver to be obtained. However, when calculating the false alarm error \(p_{2}\), it is necessary to consider the case that one sampling value exceeds the threshold value in the first \(k-1\) time slots and that two sampling values exceed the threshold value. Until \(k-1\) sampling values exceed the threshold value, the calculation is quite complex. Therefore, we first calculate the correct probability of the symbol judgment of the suboptimal PPM receiver and obtain the error probability by taking its complement to obtain the bit error performance expression of the suboptimal PPM receiver. Based on the above considerations, assuming that the signal pulse transmitted by the PPM system is located in the \(k\)th time slot, the sampling envelope value of the first \(k-1\) time slots does not exceed the threshold, and when the envelope sampling value of the \(k\)th time slot exceeds the threshold, the transmitted symbol decision is correct. It should be noted that, when the transmitted signal is the \(2^{M}\) time slot, as long as the envelope sampling value of the first \(2^{M-1}\) time slots does not exceed the threshold, the decision is correct regardless of whether the envelope sampling value of the \(2^{M}\) time slot exceeds the threshold. The above is the case in which a symbolic decision is correct, and the case where the decision is incorrect can be obtained by taking its complement. We first calculate the probability of a correct decision and then obtain the probability of an incorrect decision by taking its complement. Defining the threshold coefficient as \(\gamma\), the decision threshold is \(U_{0}=\gamma A\).