## 1 Introduction  
Pulse position modulation (PPM) technology is a discrete pulse communication technology that is different from continuous wave communication technology. PPM transmits information by transmitting discrete pulse signals in the time domain. The position of pulse signals is determined by the data to be transmitted. The PPM signal has the advantages of a low duty cycle, high peak power and high energy efficiency. It has been widely studied in the field of optical communication [1, 2, 3], underwater communication [4, 5] and radio communication [6, 7, 8]. Early PPMs belong to analog modulation technology. The data to be transmitted are continuously variable analog data, and the position of the pulse signal is also continuously variable in the time domain. The current PPM technology is usually digital modulation, and the data to be transmitted are also digital signals. Each digital PPM symbol carries \(M\)-bit data, and the symbol duration \(T_{s}\) is evenly divided into \(Q=2^{M}\) time slots. One of the \(Q\) time slots is selected according to the \(M\)-bit data to be transmitted to transmit a pulse signal. Therefore, the pulse signal can only appear in a limited time slot position and is not continuously variable. A digital PPM in which each symbol duration is divided into \(Q\) uniform time slots is commonly referred to as \(Q\)-ary PPM [9, 10]. In particular, when the PPM symbol carries only one bit of data, the number of time slots \(Q\) is equal to 2. At this time, we call this binary PPM. The PPM technologies studied in this paper refer to digital modulation PPM technology. PPM modulation technology has been widely used in the field of optical communication, such as optical fiber communication and deep space communication, and its performance has also been studied, including the binary PPM modulation system and Q-ary PPM modulation system. The research shows that PPM modulation technology is an efficient optical communication technology with high energy efficiency and strong anti-interference ability. PPM modulation technology has also been widely used in the field of wireless communication. In particular, after the FCC licensed the 3-10 GHz band to ultra wideband (UWB) technology for free, UWB communication technology has been studied a great deal. PPM, as a technology to realize UWB communication, has also been studied to a great extent [11; 12], including regarding its bit error performance [13; 14; 15; 16; 17; 18]. However, most of the literature on wireless communication has studied the binary PPM system, and only a small amount of the literature has focused on the \(Q\)-ary PPM system [19; 20]. To the best of our knowledge, there is no accurate description of the bit error performance of the \(Q\)-ary PPM system under the AWGN channel. There are many studies on the bit error performance of the \(Q\)-ary PPM system in the field of optical communication, but most of them are based on the Poisson channel. These results are not applicable to the scenario of the AWGN channel [3; 21; 22]. As the noncoherent PPM system has the advantages of simple implementation, we mainly study the noncoherent PPM system and introduce the suboptimal receiver and the optimal receiver of the noncoherent \(Q\)-ary PPM system, respectively. The suboptimal receiver makes decisions through the threshold decision method. This method has low complexity, and the optimal receiver adopts the method of selecting the largest decision that has a better performance. We derive the accurate BER performance expression of the suboptimal receiver and the accurate BER performance expression of the optimal receiver for a noncoherent \(Q\)-ary PPM system and carry out a numerical simulation. The simulation results are in good agreement with the theoretical expression, which shows the correctness of the theoretical formula. We also analyze the change in the bit error performance of a noncoherent \(Q\)-ary PPM system with the increase in \(Q\).