# A Parallel-Phase Demodulation-Based Distance-Measurement Method Using Dual-Frequency Modulation
## 4 Proposed Parallel Phase Demodulation Using Dual Frequencies
### Parallel Phase Demodulation Method  
The conventional AMCW method demodulates the optical signal reflected from an object using hardware such as an optical shutter, demodulation pixel, or an image intensifier [12; 14; 24]. Specifically, only one demodulation signal can be used per integration time; therefore, demodulation must be done sequentially. The parallel phase demodulation method proposed in this paper undertakes computational demodulation using software rather than hardware. The overall schematics of the sequential and parallel phase demodulations are shown in Figure 4.  
Figure 4a shows the sequential demodulation process of conventional AMCW method. In the conventional method, the samples \(l_{m}(\delta_{n})\), \(n=1\), \(2\),..., \(N\), are sequentially obtained for distance calculations. The sequential processing is inevitable in this case because it adopts a demodulation process in its hardware [12; 14; 24]. Figure 4b shows the proposed parallel phase demodulation process. Reflected light signal \(R(t)\) is measured and converted to digital format; then, the samples \(l_{m}(\delta_{n})\), \(n\)=1, 2,..., \(N\), are calculated using \(N\) number of demodulation signals in parallel computation. The proposed method requires 1/\(N\) total processing time per single distance measurement compared to that of sequential demodulation. In addition, the proposed method provides ideally zero DC components of demodulation \(M_{DC}\), which improves distance accuracy compared to that of the conventional method, as described in Equation (10). The proposed method demodulates digitized reflected light; and in software, positive and negative values of the demodulation signal, for example, ideal sine wave functions with zero DC components, can be used. Whereas the conventional method based on hardware demodulation uses a physical demodulation signal [12; 14; 24], which has a positive value for all time, a sine wave form with positive, non-zero DC component value can be used. Another advantage of parallel phase demodulation is that it can reduce unwanted noise effects during the calculation process. To verify this, the overall procedure of the parallel phase demodulation  
Figure 4: Comparison of sequential phase demodulation and parallel phase demodulation methods. (a) Sequential demodulation process of the conventional AMCW method; (b) parallel demodulation process of the proposed method. method is described mathematically. First, the reflected light signal \(R(t)\) and the demodulation signal \(M_{n}(t)\) can be expressed as follows  
\[R(t)=R_{A}\sin(2\pi ft-\varphi_{ToF})+R_{DC} \tag{12}\]  
\[M_{n}(t)=M_{A}\sin(2\pi ft+\delta_{n}), \tag{13}\]  
where all related parameters are identical to those in Section 2, and as mentioned above, the demodulation signal in Equation (13) has zero DC components. The \(n\)th sample \(I_{m}(\delta_{n})\), which is calculated by demodulating the reflected light signal, can then be expressed as  
\[I_{m}(\delta_{n})=\int\limits_{0}^{T_{\text{int}}}R(t)M_{n}(t)dt. \tag{14}\]  
This equation can be rewritten as shown below  
\[ \begin{array}{ll}I_{m}(\delta_{n})&=\int\limits_{0}^{T_{\text{int}}}[R_{A} \sin(2\pi ft-\varphi_{ToF})+R_{DC}][|M_{A}\sin(2\pi ft+\delta_{n})]dt\\ &=\int\limits_{0}^{T_{\text{int}}}[\frac{R_{A}M_{n}}{2}\cos(\varphi_{ToF}+ \delta_{n})-\frac{R_{A}M_{n}}{2}\cos(4\pi ft-\varphi_{ToF}+\delta_{n})+R_{DC} M_{A}\sin(2\pi ft+\delta_{n})]dt\\ &=\frac{R_{A}M_{n}T_{\text{int}}}{2}\cos(\varphi_{ToF}+\delta_{n})\end{array}  \tag{15}\]  
Hence, there is no DC component on \(M_{n}(t)\), which is related to photon shot noise [12]. With the parallel phase demodulation method, it is shown that such DC terms on \(M_{n}(t)\) can be excluded because all demodulation signals are implemented in the software program, as mentioned above. Moreover, in the sample \(I_{m}(\delta_{n})\), there is no DC term related to \(R_{DC}\), which includes the external light. This indicates that the proposed method can reduce the effects of external light on the accuracy of the measurement.