# 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
Figure 7:4  
Figure 7:5  
For the sake of convenience in analyzing, let B be 1, and Equation (25) is simplified as shown in Equation (26). When considering the PPM signal equation as phase modulation, where \(\phi_{i}(t)\) is proportional to \(m(t)\), the frequency term \(\omega_{i}\) in Equation (26) can be expressed as the derivative of \(\omega_{i}=\frac{d\phi_{i}(t)}{dt}=\frac{dm(t)}{dt}\). Consequently, Equation (26) can be rewritten as Equation (27). Figure 7:6  
Figure 7:7  
Finally, it is evident that the output voltage of the LPF in Equation (27), which is a component of the PLL, is the derivative of the transmitted message signal. Thus, the PLL-based demodulation approach also confirms that the demodulated output is the derivative of the message signal. This further supports the idea of applying differentiation before modulation to streamline the process. 3.2.3. Demodulation via Quasi-FM-PWM: Frequency Modulation Perspective  
The analysis of demodulating PPM signals without a carrier using a PLL reveals that, by nature, the PLL circuit is an FM demodulator or a phase demodulator where the information signal is differentiated. Consequently, we conclude that carrierless PPM can be considered a form of frequency modulation, where the information signal is differentiated. This offers an alternative perspective on the matter. Furthermore, in the research conducted by [PERSON] et al. [19], they proposed an FM demodulation technique by converting the information signal into a quasi-FM-PWM signal. Their analysis demonstrated that a quasi-FM-PWM signal contains low-frequency components that vary with frequency, which in turn also varies with the information signal, as expressed in Equation (28). In the same equation, \(A\) represents the amplitude and \(T_{i}\) is the period of the information signal, \(m(t)\). Additionally, \(k\) is a proportionality constant that influences the transformation of FM signals into quasi-FM-PWM signals. Based on their analysis, the quasi-FM-PWM signal is expressed as Equation (29), where the component of PWM signal is converted from FM signals by a monostable multivibrator circuit. Figure 7:8  
\[\phi_{pwFM}(t)=\frac{AT_{p}}{T_{i}}+\sum_{n=1}^{\infty}\left\{ \begin{array}{ l}-Asin\big{(}n\omega_{i}(t-T_{p})\big{)}/n\pi\\ +Asin(n\omega_{i}t)/n\pi\end{array} \right\} \tag{29}\]  
Plugging the expression in Equation (28) into Equation (29) leads to Equation (30), in which the information signal, \(m(t)\), appears within the quasi-FM-PWM waveform. \[\phi_{pwFM}(t)=AT_{p}km(t)+\sum_{n=1}^{\infty}\left\{ \begin{array}{l}-Asin \big{(}n\omega_{i}(t-T_{p})\big{)}/n\pi\\ +Asin(n\omega_{i}t)/n\pi\end{array} \right\} \tag{30}\]  
Therefore, in the case of a carrierless PPM signal, which is a form of phase modulation, if considered from the perspective of frequency modulation, it can be realized as frequency modulation where the information signal is differentiated. When it is converted into a quasi-FM-PWM signal using a monostable multivibrator, it yields its low-frequency term, which varies with the derivative of the information signal, as demonstrated by Equation (31) [19; 23]. \[\frac{AT_{p}}{T_{i}}=AT_{p}k\frac{dm(t)}{dt} \tag{31}\]  
As a result, the PPM signal without a carrier can be demodulated by utilizing a monostable multivibrator, followed by passing its output through an LPF. This process produces an output that corresponds to the derivative of the information signal, with the rate of amplification being dependent on the pulse width of the monostable multivibrator. To retrieve the original message, the differentiated signal must be integrated. The block diagram of the system is shown in Figure 10.  
The quasi-FM-PWM demodulation method likewise yields the derivative of the message signal as its output. This consistent finding across different demodulation techniques motivates the proposed approach of differentiating the message signal prior to PPM generation. Figure 10. Demodulating PPM from the perspective of FM modulation, in conjunction with its conversion to PWM.