# Simplified Derivative-Based Carrierless PPM Using VCO and Monostable Multivibrator
## 2 Principle of Signal Component Analysis for PWM and PPM
### Analysis of Components in PPM Signals  
For the analysis of PPM, we present an indirect approach, demonstrating that PPM arises from PWM with and without delay. The PPM signal is obtained by subtracting the delayed PWM signal, where the delay equals the width of the positive pulse, from the original PWM signal. The equation representing the components of the delayed PWM signal, where \(\varepsilon\) denotes the time delay, is derived from Equation (5). The delayed signal, \(S_{PWM}\left(t\right)\) or \(S_{PWM}\left(t-\varepsilon\right)\), is expressed in Equation (6), where \(k_{p}\) retains the same definition as in Equation (5). \[S_{PWM}(t)=A\frac{k_{p}m(t-\varepsilon)}{T}+\sum_{n=1}^{\infty}\biggl{(}\frac {A}{n\pi}\text{sin}\bigl{(}n\omega_{0}(t-\varepsilon)\bigr{)}-\frac{A}{n\pi} \text{sin}\bigl{(}n\omega_{0}\bigl{(}t-\varepsilon-k_{p}m(t-\varepsilon) \bigr{)}\bigr{)}\biggr{)} \tag{6}\]  
After obtaining the delayed PWM signal shown in Equation (6), the PWM signal without delay, as expressed in Equation (5), is first subtracted by the delayed PWM signal. The resulting expression is then rearranged, giving rise to Equation (7). As can be observed in Equation (7), the third term represents the pulse carrier signal, with the positive pulse width equal to \(\varepsilon\). \[ \begin{array}{c}\phi_{PWM}\bigl{(}t-\varepsilon\bigr{)}-\phi_{PWM}(t)=-A \frac{k_{p}(m(t)-m(t-\varepsilon))}{T}\\ +\sum_{n=1}^{\infty}\frac{A}{n\pi}\bigl{(}\text{sin}\bigl{(}n\omega_{0}\bigl{(} t-k_{p}m(t)\bigr{)}\bigr{)}-\text{sin}\bigl{(}n\omega_{0}\bigl{(}t-\varepsilon-k_{p}m(t- \varepsilon)\bigr{)}\bigr{)}\bigr{)}\\ -\sum_{n=1}^{\infty}\frac{A}{n\pi}\bigl{(}\text{sin}\bigl{(}n\omega_{0}t\bigr{)} -\text{sin}\bigl{(}n\omega_{0}(t-\varepsilon)\bigr{)}\bigr{)}\end{array}  \tag{7}\]  
The subtraction process of the PWM signals with and without delay is illustrated in Figure 2. The first row of signal displays \(\phi_{PWM}\left(t\right)\), while the second row shows \(\phi_{PWM}\left(t-\varepsilon\right)\). The third row illustrates the subtraction of the two upper signals, where the positive portion represents the PPM signal, \(\phi_{PWM}\left(t\right)\), and the negative portion corresponds to the pulse carrier signal. Finally, the last row shows the resulting output, which produces the PPM signal, \(\phi_{PWM}(t)\). Therefore, to isolate only the PPM modulation terms, the pulse carrier component is discarded. This pulse carrier term, illustrated in Figure 3, corresponds to a DC term that represents its duty cycle. In this context, the DC term is defined as \(D\). To ensure that the PPM signal maintains a low voltage level of 0 volts and a high pulse level of 1 volt (or any positive value), the DC component \(D\) of the pulse carrier term must be compensated. The resulting equation for the PPM signal is given in Equation (8). \[ \begin{array}{c}\phi_{PPM}(t)=-A\frac{k_{p}(m(t)-m(t-\varepsilon))}{T}\\ +\sum_{n=1}^{\infty}\frac{A}{n\pi}\bigl{(}\text{sin}\bigl{(}n\omega_{0}\bigl{(} t-k_{p}m(t)\bigr{)}\bigr{)}-\text{sin}\bigl{(}n\omega_{0}\bigl{(}t-\varepsilon-k_{p}m(t- \varepsilon)\bigr{)}\bigr{)}\bigr{)}+D\end{array}  \tag{8}\]  
The term \(\frac{m(t)-m(t-\varepsilon)}{T}\) in Equation (8), when compared with the period \(T\), can be approximated as the derivative of the information signal (\(\frac{dn(t)}{dt}\)). Based on this approximation,the equation for the components of the PPM signal, excluding the rectangular pulse carrier, is ultimately expressed in Equation (9). \[ \begin{array}{c}\phi_{PPM}(t)=-Ak_{p}\frac{dm(t)}{dt}\\ +\sum_{t=1}^{\infty}\frac{\Lambda}{\pi t}\left(\sin\left(n\omega_{0}\left(t-k_{p }m(t)\right)\right)-\sin\left(n\omega_{0}\left(t-\varepsilon-k_{p}m(t- \varepsilon)\right)\right)\right)+D\end{array}  \tag{9}\]  
From Equation (9), it is shown that the components of the PPM signal consist of a low-frequency term, which is the direct current, and the information that has been differentiated.