# Simplified Derivative-Based Carrierless PPM Using VCO and Monostable Multivibrator
## 2 Principle of Signal Component Analysis for PWM and PPM
### Analysis of Components in PWM Signals  
PWM signal is represented by the \(S_{PWM}(t)\). PWM involves encoding information by varying the duration of the positive pulses within a square wave. The pulse width is directly proportional to the amplitude of the information signal. When the information signal exceeds the voltage of the sawtooth waveform, positive pulses are generated. Conversely, when the information signal is lower than the sawtooth waveform, negative pulses are produced. The relationship between the PWM signal and the sawtooth waveform is illustrated in Figure 1. It should also be noted that \(S_{PWM}\left(t\right)\) can interchangeably be expressed by \(\phi_{PWM}\left(t\right)\).  
From Figure 1, the signal \(S_{PWM}\left(t\right)\) over a single period can be expressed by Equation (1), where \(A\) denotes the amplitude of the PWM signal. The variables \(t_{0}\) and \(T_{p}\) represent the initial time reference (usually set to 0) and the pulse width duration, respectively, while \(T\) determines the period of the PWM signal. This expression is then analyzed to determine the components of the signal using the Fourier series, where \(F_{0}\) and \(F_{n}\) denote the fundamental frequency and \(n\)-th harmonic frequency. Both are calculated using Equations (2) and (3), respectively, where the latter is expressed in a complex form, with \(j\) indicating the imaginary part. Later, for the complex form, \(j\) indicates the imaginary part.  
\[S_{PWM}(t)=\left\{\begin{array}{ll}A;&t_{0}\leq t\leq t_{0}+T_{p}\\ 0;&t_{0}+T_{p}\leq t\leq t_{0}+T\end{array}\right. \tag{1}\]  
\[F_{0}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}S_{PWM}(t)dt=\frac{1}{T}\left(\int_{0}^{T_{p}} Adt+\int_{T_{p}}^{T}0 dt\right)=A\frac{T_{p}}{T}\text{, where }t_{0}=0 \tag{2}\]  
\[F_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}S_{PWM}(t)\cdot e^{-jm\omega_{0}}dt= \frac{1}{T}\left(\int_{0}^{T_{p}}A\cdot e^{-jn\omega_{0}t}dt+\int_{T_{p}}^{T} 0\cdot e^{-jn\omega_{0}t}dt\right) \tag{3}\]  
Subsequently, \(S_{PWM}\left(t\right)\) is re-expressed in terms of the Fourier series using \(F_{0}\) and \(F_{n}\), yielding the PWM signal as shown in Equation (4). By substituting \(\omega_{0}=\frac{2\pi}{T}\), where \(\omega_{0}\) represents the angular fundamental frequency, into Equation (4) and rearranging, the original \(S_{PWM}\left(t\right)\) from Equation (1) is re-expressed as Equation (5). This re-expression, written in Equation (5), represents a summation of sinusoidal components.  
\[\begin{array}{l}S_{PWM}(t)=F_{0}+\sum_{n=-\infty}^{\infty}F_{n}e^{jn \omega_{0}t}\\ =A\frac{T_{p}}{T}+\sum_{n=1}^{\infty}\left(-\frac{2A}{jn\omega_{0}}\text{sin} \left(n\omega_{0}\big{(}t-T_{p}\big{)}\right)+\frac{2A}{Thr\omega_{0}}\text{ sin}\left(n\omega_{0}t\right)\right)\end{array} \tag{4}\]  
Figure 1: Comparison between the information signal and the sawtooth waveform (carrier) yields the output \(S_{PWM}\left(t\right)\).  
\[S_{PWM}(t)=A\frac{T_{p}}{T}+\sum_{n=-\infty}^{\infty}\biggl{(}- \frac{2A}{Tn\bigl{(}\frac{2\pi}{T}\bigr{)}}\text{sin}\bigl{(}n\omega_{0}\bigl{(}t -T_{p}\bigr{)}\bigr{)}+\frac{2A}{Tn\bigl{(}\frac{2\pi}{T}\bigr{)}}\text{sin} \bigl{(}n\omega_{0}t\bigr{)}\biggr{)} \tag{5}\] \[=A\frac{k_{p}m(t)}{T}+\sum_{n=1}^{\infty}\Bigl{(}-\frac{A}{n\pi} \text{sin}\bigl{(}n\omega_{0}\bigl{(}t-k_{p}m(t)\bigr{)}\bigr{)}+\frac{A}{n\pi} \text{sin}\bigl{(}n\omega_{0}t\bigr{)}\Bigr{)}\]  
From Equation (5), the pulse width \(T_{p}\) is proportional to the information signal \(m(t)\), given by \(T_{p}=k_{p}m(t)\leq T\). The proportionality factor \(k_{p}\) defines modulation depth in PWM, determining how \(m(t)\) influences pulse duration. It is observed that the PWM signal, \(S_{PWM}\left(t\right)\), consists of a lowest frequency term, which is the information signal. In addition, it includes higher frequency terms, corresponding to phase modulation with a sine wave at various frequencies.