# A Novel Non-Coherent \(Q\)-Ary TH-PPM Transceiver
## 2 System Model
### Conventional TH-PPM Transceiver  
According to [5], the conventional TH-PPM signal model can be expressed as (1). \[s^{(k)}(t)=\sum_{j}p\Big{(}t-jT_{f}-c_{j}^{(k)}T_{c}-PPM_{[j/N_{k}]}^{(k)} \Big{)} \tag{1}\]  
where \(s(t)\) represents the output signal of the TH-PPM transmitter, \(k\) 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 1 to \(N_{h}\), and \(T_{c}\) represents the period of the time-hopping chip, \(PPM_{[j/N_{k}]}^{(k)}\) 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. The block diagram of the conventional TH-PPM transmitter is shown in Figure 1.  
For \(Q\)-ary TH-PPM, the symbol duration \(T_{s}\) 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 \(\Delta T\). Figure 1: Block diagram of the conventional TH-PPM transmitter. Without loss of generality, taking 16-ary TH-PPM as an example, take \(N_{s}=N_{h}=10\), take the TH sequence listed as {1,4,9,5,3,3,5,9,4,1}, assuming \(PPM_{[j/N_{s}]}^{(k)}=4\) is the data to be transmitted, a 16-ary TH-PPM signal is shown in Figure 2.  
As shown in Figure 2, the TH-PPM signal can be divided into three layers. The third layer is the same as the conventional \(Q\)-ary PPM signal structure, with each PPM symbol duration divided into individual time slots, each of which is \(\Delta T\), one of the time slots chosen to transmit the PPM pulse modulated by the data symbol. There will be a pulse in the \(uth\)\(\Delta T\) slot when the data symbol which will be transmitted is \(n\in(1,Q)\). For example, when the data symbol will be transmitted is 1, there will be a pulse in the first \(\Delta T\) slot, and when the data symbol will be transmitted is 2, there will be a pulse in the second \(\Delta T\) slot, etc. In Figure 2, the data symbol for transmitting is 4, so there is a pulse in the forth \(\Delta T\) slot. The middle layer is a layer that embodies the pseudo-randomness of the time-hopping, where \(T_{f}\) is the pulse repetition period. Each pulse repetition period \(T_{f}\) is divided into \(N_{s}\) time-hopping chips. The width of each TH chip is \(T_{c}\). In each pulse repetition cycle, the TH sequence pseudo-randomly selects a chip to transmit a PPM symbol. From the perspective of the entire mediate layer, the signal is pseudo-randomly transmitted in the time domain. The top layer represents the pulse repetition frequency. If the pulse repetition time is \(T_{f}\) and the pulse repetition frequency is \(N_{s}\), then \(T_{s}=N_{s}*T_{f}\) is the duration of a complete TH-PPM symbol. Within each pulse repetition period \(T_{f}\), one TH-PPM pulse is transmitted. \(N_{s}\) TH-PPM pulses constitute a TH-PPM symbol, and the duration of the TH-PPM symbol is the same as the duration of the \(Q\)-ary PPM symbol with the same information rate. For \(Q\)-ary TH-PPM, we can derive the following relationship from Figure 2.  
\[ \begin{array}{rcl}\Delta T&=&T_{c}/Q\\ &=&\left(T_{f}/N_{h}\right)/Q\\ &=&\left(\left(T_{s}/N_{s}\right)/N_{h}\right)/Q\\ &=&\frac{T_{c}}{N_{s}-N_{h}-Q}\end{array}  \tag{2}\]  
The block diagram of a conventional TH-PPM receiver can be shown in Figure 3.  
Figure 2: Schematic diagram of 16-ary TH-PPM signal. For conventional TH-PPM receivers, we sample the output of the matched filter at every time slot from 1 to \(Q\), which synchronization is required to determine the start of the time slot. Then, if the maximum sampling value is \(U_{k}\), we can determine the data symbol as \(k\), where \(k\) is limited from 1 to \(Q\). The decision flow of the conventional TH-PPM receiver is shown in Figure 4.  
1. Start data decision;
2. Sample the values of the matched filter output at time slot 1, 2,..., \(Q\);
3.