# A Parallel-Phase Demodulation-Based Distance-Measurement Method Using Dual-Frequency Modulation
## 4 Proposed Parallel Phase Demodulation Using Dual Frequencies
### Dual-Frequency Modulation Based on Parallel Phase Demodulation  
In order to solve the \(2\pi\) ambiguity problem, the conventional dual-frequency modulation [24; 25] method is combined with parallel phase demodulation. By combining these two methods, the demodulation process is especially reformed, which is different with sequential demodulation process mentioned in Section 3.3. The mathematical description is as follows. First, the emitted light signal \(E(t)\) is modulated simultaneously using two frequencies. In this case, the emitted light signal \(E(t)\) and the reflected light signal \(R(t)\) can be expressed as follows:  
\[E(t)\ =\ E_{1}\sin(2\pi f_{1}t)+E_{2}\sin(2\pi f_{2}t)+E_{DC} \tag{16}\]  
\[R(t)=R_{1}\sin(2\pi f_{1}t-\varphi_{1})+R_{2}\sin(2\pi f_{2}t-\varphi_{2})+R_ {DC}, \tag{17}\]  
where \(E_{1}\), \(E_{2}\), \(R_{1}\), and \(R_{2}\) represent the amplitude of the \(f_{1}\) component of the emitted light signal, the amplitude of the \(f_{2}\) component of the emitted light signal, the amplitude of the \(f_{1}\) component of the reflected light signal, and the amplitude of the \(f_{2}\) component of the reflected light signal, respectively. Above \(\varphi_{1}\) and \(\varphi_{2}\) are, correspondingly, the phase shifts of the reflected light signal for each frequency component. Likewise, the demodulation signal is also generated using two frequencies simultaneously. The demodulation signal can be expressed as  
\[M_{n}(t)=M_{1}\sin(2\pi f_{1}t+m\delta_{n})+M_{2}\sin(2\pi f_{2}t+l\delta_{n}) \tag{18}\]\[\delta_{n}=\frac{2\pi(n-1)}{N},\ n=1,2,3,\ldots,N \tag{19}\]  
\[m=\frac{f_{1}}{f_{e}},\ l=\frac{f_{2}}{f_{e}},\ f_{e}=\gcd(f_{1},\ f_{2}), \tag{20}\]  
where \(M_{1}\) and \(M_{2}\) represent the amplitude of the \(f_{1}\) component of the demodulation signal and the amplitude of the \(f_{2}\) component of the demodulation signal, respectively. \(f_{e}\) is the effective frequency, which is identical to the greatest common division of \(f_{1}\) and \(f_{2}\). The demodulation signal does not have a DC component because demodulation is done using software, as opposed to other methods in the literature [22; 24; 25]. Moreover, as expressed by Equation (18), the demodulation signal is also generated as the sum of the dual-frequency signal, which is also one of the major differences compared to earlier approaches [22; 24; 25]. Using the signals above, the \(2\pi\) ambiguity problem is revisited by referring to previous methods [22; 24; 25]. When the phase shift of the demodulation signal is 0, sample \(I_{m}(0)\) can be expressed as follows  
\[ \begin{array}{rl}I_{m}(0)&=\int\limits_{0}^{T_{\rm int}}R(t)M_{0}(t)dt\\ &=\int\limits_{0}^{T}\big{\{}R_{1}\sin(2\pi f_{1}t-\varphi_{1})+R_{2}\sin(2\pi f _{2}t-\varphi_{2})+R_{DC}\big{\}}|M_{1}\sin(2\pi f_{1}t)+M_{2}\sin(2\pi f_{2}t )|dt\\ &=\int\limits_{0}^{T_{\rm int}}\big{\{}\frac{R_{1}M_{1}}{2}\cos(\varphi_{1})- \frac{R_{1}M_{1}}{2}\cos(4\pi f_{1}t-\varphi_{1})+\frac{R_{2}M_{2}}{2}\cos( \varphi_{2})-\frac{R_{2}M_{2}}{2}\cos(4\pi f_{2}t-\varphi_{2})\big{\}}dt\\ &+\int\limits_{0}^{T_{\rm int}}\big{\{}\frac{R_{2}M_{1}}{2}\cos(2\pi(f_{2}-f_{ 1})t+\varphi_{1})-\frac{R_{2}M_{1}}{2}\cos(2\pi(f_{2}+f_{1})t-\varphi_{1}) \big{\}}dt\\ &+\int\limits_{0}^{T_{\rm int}}\big{\{}\frac{R_{1}M_{2}}{2}\cos(2\pi(f_{1}-f_{ 2})t+\varphi_{2})-\frac{R_{1}M_{2}}{2}\cos(2\pi(f_{1}+f_{2})t-\varphi_{2}) \big{\}}dt\end{array}  \tag{21}\]  
Because the inverse of \(T_{\rm int}\) is much smaller than \(f_{1}\), \(f_{2}\), \(\big{|}f_{1}-f_{2}\big{|}\) and \(f_{1}+f_{2}\), the terms which contain the \(f_{1}\), \(f_{2}\), \(\big{|}f_{1}-f_{2}\big{|}\) and \(f_{1}+f_{2}\) can be neglected.