# Simplified Derivative-Based Carrierless PPM Using VCO and Monostable Multivibrator
## 4 Performance Analysis
### Error Analysis of Demodulation Using Integration  
For the analysis of error values in demodulation using integration, we begin by comparing the PPM signal with added noise to recover the original PPM signal. When the PPM signal, combined with noise, is subjected to comparison through zero crossing detection, the output from the comparison is obtained, as depicted in Figure 12.  
Figure 11: Novel derivative-based PPM modulation using VCO and monostable multivibrator.  
To illustrate the transmission of noise in the spatial domain to the time-domain noise, we consider the probability density function (PDF) of additive white Gaussian noise (AWGN), as shown in Equation (32). In Equation (32), the variance of AWGN denotes \(\sigma_{n}^{2}\) and \(n\) represents noise. When a noise signal is added to the PPM signal and subjected to signal comparison at any reference level, or even at a reference level of 0 (zero crossing), the result will be similar. The magnitude-domain interference that is converted to time-domain interference varies with the signal slope, where a higher slope results in less interference, as illustrated in Figure 12.  
\[f(n)=\left(1/\sqrt{2\pi\sigma_{n}^{2}}\right)e^{-\frac{\pi^{2}}{2\sigma_{n}^{2 }}} \tag{32}\]  
Following this, let \(s\) be the signal slope of the PPM signal, expressed by \(\frac{\Delta\tau}{\Delta t}\), where \(v\) denotes the PPM voltage signal and \(t\) represents time. Given this, the relationship of the interference transitioning from magnitude-domain interference to time-domain interference can be expressed by Equation (33). After that, Equation (32) is recast using the relationship in Equation (33), producing Equation (34).  
\[\Delta t=\Delta v/s\ \rightarrow\ \Delta v=n=s\Delta t \tag{33}\]  
\[f(\Delta t)=\left(1/\sqrt{2\pi\sigma_{n}^{2}}\right)e^{-\frac{(s\Delta\tau)^{ 2}}{2\sigma_{n}^{2}}} \tag{34}\]  
When the time-shifted PPM signal without a carrier is used to control the integrator with a reference voltage input of \(V_{r}\), it influences the initial condition of \(n_{o}\). Consequently, the expression for \(n_{o}\), representing the magnitude of AWGN output, is formulated in Equation (35). With this initial condition, the PDF of \(n_{o}\), which is a noise output, is acquired and mathematically stated in Equation (36).  
\[\int_{0}^{\Delta t}V_{r}dt=V_{r}\Delta t=n_{o}\rightarrow\ \Delta t=n_{o}/V_{r} \tag{35}\]  
\[f(n_{o})=\left(1/\sqrt{2\pi\sigma_{n}^{2}}\right)e^{-((\frac{\Delta\tau}{V_{r }})^{2}/2\sigma_{n}^{2})}=\left(1/\sqrt{2\pi\sigma_{n}^{2}}\right)e^{-((n_{o} )^{2}/2\sigma_{n}^{2}(\frac{V_{r}^{2}}{2}))} \tag{36}\]  
Figure 12: Comparison of the noise signal added to the PPM signals, where \(v\) is the PPM voltage signal, and \(t\) represents time. (**a**) Low slope input, and (**b**) High slope input.  
By calculating the power output from Equation (36), the power of the noise signal \(n_{o}\) at the output of the integrator during demodulation is thus equal to \(\frac{N_{o}V_{r}^{2}}{S^{2}}\). From the perspective of PPM as a narrow rectangular positive pulse phase modulation, the maximum variation angle is less than \(2\pi\geq k_{p}m(t)\), where \(k_{p}\) is the modulation index and \(m(t)\) is the information signal. If the information signal is fully demodulated, the message signal power \(S_{m}\) is obtained, as expressed by Equation (37), where a low-pass bandwidth \(B_{L}\) is used along with the low-pass bandwidth \(B_{L}\), and the noise signal is attenuated to \(\frac{B_{L}N_{o}V_{r}^{2}}{S^{2}}\)  
\[S_{m}=k_{p}^{2}\overline{m^{2}(t)} \tag{37}\]  
Once the power of both the information and filter noise signals is acquired, the signal-to-noise ratio is then calculated and presented in Equation (38).  
\[\frac{S_{m}}{N}=\left(k_{p}^{2}\overline{m^{2}(t)}\right)/\left(\frac{B_{L}N_ {o}V_{r}^{2}}{S^{2}}\right)=\frac{k_{p}^{2}S^{2}}{V_{r}^{2}}\overline{\frac{m ^{2}(t)}{B_{L}N_{o}}} \tag{38}\]