# Simplified Derivative-Based Carrierless PPM Using VCO and Monostable Multivibrator
## 3 Proposed Techniques of Demodulation and Modulation
### Proposed Carrierless PPM Techniques
#### 3.2.2 Demodulation via PLL: Phase-to-Derivative Relationship  
For PPM demodulation using a PLL approach, the PPM signal is used as an input to the PLL circuit. The block diagram of the demodulation process is shown in Figure 8. In this approach, an RS flip-flop and a monostable multivibrator are used as a phase detector and the rest of the structure is abided by the conventional PLL structure. Figure 7: An illustration of the relationship between the integration process and the resulting \(V_{o}\), where the black dashed line represents the approximated signal derived from Equation (13), the red line denotes the integrator circuit output, and the navy-blue line corresponds to the PPM signal. To recover the information signal through the PPM demodulation process without the carrier using PLL, a conventional structure of PLL illustrated in Figure 9 is analyzed. To elaborate the traditional block diagram of the PLL approach, key variables shown in Figure 9 are detailed as follows:  
\(\phi_{i}(S)\) represents the Laplace transform of the phase input \(\phi_{i}\) (\(t\)),  
\(\phi_{o}(S)\) represents the Laplace transform of the phase output \(\phi_{o}\) (\(t\)),  
\(V_{o}(S)\) represents the Laplace transform of the LPF output \(V_{o}\) (\(t\)),  
\(\omega_{r}\) represents the Laplace transformation of a constant of free response,  
\(\omega_{L}\) represents the cutoff frequency of the LPF,  
\(F(S)\) refers to the transfer function of the LPF,  
\(k_{b}\) is a constant of a phase detection circuit,  
\(k_{o}\) is a constant of VCO sensitivity, and  
\(B\) is a gain of an integrator circuit. Referring to Figure 9, let us consider the points \(V_{o}\) (\(S\)) and \(\phi_{o}\) (\(S\)). The corresponding relationships are expressed in Equations (14) and (15), respectively. \[V_{o}(S)=F(S)k_{b}(\phi_{i}(S)-\phi_{o}(S)) \tag{14}\]  
\[\phi_{o}(S)=\frac{B}{S}(\omega_{r}+k_{o}V_{o}(S)) \tag{15}\]  
Substituting Equation (15) into Equation (14) yields Equation (16) and later Equation (16) is adjusted, giving rise to Equation (17). \[V_{o}(S)=F(S)k_{b}\bigg{(}\phi_{i}(S)-\frac{B}{S}(\omega_{r}+k_{o}V_{o}(S)) \bigg{)} \tag{16}\]  
\[V_{o}(S)=F(S)k_{b}\phi_{i}(S)-\frac{B}{S}\omega_{r}F(S)k_{b}-\frac{B}{S}k_{o}V _{o}(S)F(S)k_{b} \tag{17}\]  
Figure 8: Block diagram of the PLL used for PPM demodulation. Figure 9: Conventional block diagram of the PLL approach. Translating the term \(\frac{B}{S}k_{0}V_{o}(S)F(S)k_{b}\) to the left-hand side (LHS) produces Equation (18). In the same equation, \(F(S)\) is defined as a first-order passive LHF, where its transfer function is expressed by \(\frac{\omega_{L}}{S+\omega_{L}}\) and plugged into Equation (18), producing Equation (19). \[V_{o}(S)+\frac{B}{S}k_{0}V_{o}(S)F(S)k_{b}=F(S)k_{b}\phi_{i}(S)-\frac{B}{S} \omega_{r}F(S)k_{b} \tag{18}\]  
\[V_{o}(S)+\frac{B}{S}k_{0}V_{o}(S)\bigg{(}\frac{\omega_{L}}{S+\omega_{L}}\bigg{)} k_{b}=\bigg{(}\frac{\omega_{L}}{S+\omega_{L}}\bigg{)}k_{b}\phi_{i}(S)-\frac{B}{S} \omega_{r}\bigg{(}\frac{\omega_{L}}{S+\omega_{L}}\bigg{)}k_{b} \tag{19}\]  
After that, Equation (19) is rearranged, as shown in Equation (20). Once Equation (20) is obtained, the denominator, \(S(S+\omega_{L})\), is multiplied throughout Equation (20), resulting in Equation (21) with a slight rearrangement. Figure 7:0  
Figure 7:1  
To facilitate the solution for the response, the inverse Laplace transform of Equation (21) is applied, resulting in Equation (22). Figure 7:2  
It is assumed that the PLL is a stable system that has reached a steady state. Consequently, the solution for the forced response is determined by setting \(\phi_{i}(t)=\omega_{i}t+\theta_{i}\) and substituting this expression into Equation (22), resulting in Equation (23). Figure 7:3  
When the forcing function on the right-hand side of the equation is constant, the first and second derivatives of the assumed solution are equal to zero (according to the undetermined coefficient method). Thus, Equation (23) can be rewritten as Equation (24) and Equation (24) is reformulated as Equation (25).