# Simplified Derivative-Based Carrierless PPM Using VCO and Monostable Multivibrator
## 3 Proposed Techniques of Demodulation and Modulation
### Proposed Carrierless PPM Techniques
#### 3.2.1 Demodulation via Integration: Derivative Recovery from PPM  
PPM demodulation offers two options: converting PPM to PWM or converting PPM to PAM and then passing it through an LPF. Based on the analysis of the PPM signal without a carrier shown in Figure 5, the time interval (\(\Delta t_{n}\)) between the \(n\)-th pulse and the previous pulse (\(n-1\)) is analyzed. The relationship between these time intervals is then expressed in Equation (10), where \(\Delta m\)\((t_{n})\) equals to \(m\)\((t_{n})-m\)\((t_{n-1})\) after rearranging.  
\[\Delta t_{n}=(nT+m(t_{n}))-((n-1)T+m(t_{n-1}))=T+\Delta m(t_{n}) \tag{10}\]  
Hence, when the PPM signal controls the switch in an integrator circuit shown in Figure 6, the switch opens when the signal is LOW, causing the integrator to operate. When the signal is HIGH, the output of the integrator is set to 0. In this circuit, \(-V_{r}\) serves as a constant reference voltage, providing a fixed baseline for comparison. The output voltage, \(V_{o}\), is obtained through the mathematical method of integration and is given by Equation (11). The integration process is governed by the resistor \(R\) and capacitor \(C\), where their values determine the rate of voltage accumulation over time.  
\[\begin{array}{l}V_{o}=\frac{1}{\text{RC}}\int\limits_{(n-1)T+m(t_{n-1})}^{ nT+m(t_{n})}V_{r}dt=\frac{V_{r}}{\text{RC}}\Big{(}t|_{(n-1)T+m(t_{n-1})}^{ nT+m(t_{n})}\Big{)}\\ \quad\quad\quad\quad\quad=\frac{V_{r}}{\text{RC}}(nT+m(t_{n})-((n-1)T+m(t_{n- 1})))\end{array} \tag{11}\]  
As shown in Equation (11), the terms in parentheses are like the ones presented in Equation (10) so Equation (11) is rewritten as Equation (12). Considering the term \(\Delta m\)\((t_{n})\) in relation to time \(T\) in Equation (12), the relationship for \(V_{o}\) is expressed as in Equation (13)  
Figure 5: Analysis of the carrierless PPM signal, focusing on the time interval (\(\Delta t_{n}\)) between consecutive pulses.  
Figure 6: Integrator circuit controlled by the PPM signal through the switch.  
where the term \(\frac{\Delta m(t_{n})}{T}\) appears. This term can be approximated as the derivative of the information signal \(m(t)\) with respect to time \(t\).  
\[V_{o}=\frac{V_{r}}{RC}(T+\Delta m(t_{n})) \tag{12}\]  
\[V_{o}=\frac{V_{r}\times T}{RC}\bigg{(}1+\frac{\Delta m(t_{n})}{T}\bigg{)}\cong \frac{V_{r}\times T}{RC}+\frac{V_{r}\times T}{RC}\cdot\frac{dm(t)}{t} \tag{13}\]  
From Equation (13), \(V_{o}\) represents the peak value of the sawtooth signal generated by the integration, as shown in Figure 7. When considering Equation (13), it can be observed that the result of the integration varies with the derivative of the information signal. Therefore, the demodulation process described above produces an output that is the derivative of the information signal, \(m(t)\). To recover the original information signal, \(\widetilde{m}\)\((t)\), the value of \(V_{o}\) from Equation (13) must be passed through a High-Pass Filter (HPF), and integration is required to recover the original information signal. This is because the DC term, when integrated, increases linearly. Thus, we can design the HPF in conjunction with the integrator to form a Band-Pass Filter (BPF). The low-frequency response of the BPF behaves like an HPF, while the high-frequency response behaves like an LPF, which is analogous to the behavior of the integrator.  
This analysis demonstrates that the integration-based demodulation of carrierless PPM fundamentally produces the derivative of the original message signal. This recurring outcome suggests that differentiating the message signal prior to modulation could simplify the overall system.