# A Novel Non-Coherent \(Q\)-Ary TH-PPM Transceiver
## 2 System Model
### New TH-PPM Transceiver  
The conventional TH-PPM, which is a combination of time-hopping and PPM directly, has the advantage of simplicity and intuition. However, the receiver of conventional TH-PPM requires good synchronization, which is difficult to implement due to the pulse width of the TH-PPM signal being extremely narrow. A minimal synchronization error will lead to significant BER performance degradation. To address this problem, this paper proposed a waveform design method for coupling time-hopping and PPM, which enables certain orthogonality between different TH-PPM symbols, allowing demodulation with parallel matched filter for TH-PPM receiver, which can reduce the time synchronization accuracy requirements and facilitate engineering implementation. The new TH-PPM signal can be represented by Equation (3). \[s^{(k)}(t)=\sum_{j}p\Big{(}t-jT_{f}-c_{j}^{(k)}T_{c}-c_{j}^{(k)}\Theta PPK_{[j/N _{k}]}^{(k)}\Big{)} \tag{3}\]  
where, \(s(t)\) represents the TH-PPM signal. \(k\) represents the user ID. \(p(t)\) represents the pulse waveform of the TH-PPM signal. \(T_{f}\) represents the pulse repetition period. \(c_{j}\) represents the time-hopping sequence, which is limited from 1 to \(N_{h}\). And \(T_{c}\) represents the width of the time-hopping chip. \(PPK_{[j/N_{k}]}^{(k)}\) indicates the PPM symbol which is determined by the data to be transmitted, and the data to be transmitted changes only every \(N_{s}\) hops. This represents the coupling encoding of the PPM symbol and the time-hopping sequence, which is different from the conventional TH-PPM signal. The block diagram of the new TH-PPM transmitter is shown in Figure 5.  
As same as the conventional TH-PPM signal, the symbol duration \(T_{s}\) of the new TH-PPM signal is evenly divided into individual time slots \(T_{f}\), while each \(T_{f}\) is evenly divided into \(N_{h}\) time slots \(T_{c}\), and then each \(T_{c}\) is evenly divided into \(Q\) time slots \(\Delta T\). The new TH-PPM signal structure is shown in Figure 6.  
Figure 5: Block diagram of the new TH-PPM transmitter. Figure 6: Structure diagram of the new TH-PPM signal. As shown in Figure 6, compared to the conventional TH-PPM signal in Figure 2, the position of the pulses in the third layer is not only determined by the data to be transmitted but also determined by the time-hopping sequences. In Figure 2, when the data symbol to be transmitted is 4, all the \(N_{\text{s}}\) pulses in the third layer are located at the fourth \(\Delta T\). However, in Figure 6, when the data symbol to be transmitted is 4, the \(N_{\text{s}}\) pulses in the third layer are located in different \(\Delta T\) from each other. For example, the first pulse of the new TH-PPM is located in the fourth \(\Delta T\) of the \(Q\) slots, while the second pulse is located in the thirteenth \(\Delta T\) of the \(Q\) slots. For the new TH-PPM signal, the time-hopping sequence and the PPM modulated symbol are no longer independent. In addition, the new TH-PPM signal doesn't change the parameters of the pulse width, the number of pulses, and the pulse repetition period. That is, only the position of the pseudo-random occurrence of the signal is changed. Therefore, the time-domain waveform and spectrum of the new TH-PPM signal are the same as those of the conventional TH-PPM signal. For the first layer and the second layer, the new TH-PPM has the same signal structure as the conventional TH-PPM. Due to the orthogonality between different symbols in the new TH-PPM signal, parallel-matched filters can be used for the receiver, as shown in Figure 7. From Figure 7, it can be seen that the new TH-PPM receiver coupled with the time-hopping sequence with the PPM obtains \(Q\) group-matched filter coefficients, which are used for the \(Q\)-matched filter. When one of the \(Q\)-matched filters outputs a correlation peak, we can determine the transmitted data symbol by the corresponding matched filter. In this way, we can reduce the requirement for time synchronization accuracy.