# A Parallel-Phase Demodulation-Based Distance-Measurement Method Using Dual-Frequency Modulation
## 2 The Principles of AMCW-Based ToF Measurement  
General AMCW-based ToF measurement scheme is illustrated in Figure 1. The illumination source is modulated within the frequency range of 20-30 MHz usually. This modulated light is emitted onto the object, which reflects the modulated light back to the detector. Detected reflected light signal is demodulated by a demodulation pixel or by an optical shutter [12; 13; 14].  
The harmonic emitted light signal \(E(t)\), reflected light signal \(R(t)\), and the \(n\)-th demodulation signal \(M_{n}(t)\) are expressed below  
\[E(t)=E_{A}\sin(2\pi ft)+E_{DC} \tag{1}\]  
\[R(t)=R_{A}\sin(2\pi ft-\varphi_{ToF})+R_{DC} \tag{2}\]  
\[M_{n}(t)=M_{A}\sin(2\pi ft+\delta_{n})+M_{DC} \tag{3}\]  
\[\delta_{n}=\frac{2\pi(n-1)}{N},\ n=1,2,3,\ldots,N, \tag{4}\]  
where \(E_{A}\), \(R_{A}\), and \(M_{A}\) represent the amplitudes of the emitted light signal, the reflected light signal, and the demodulation signal, respectively. \(f\) denotes the modulation frequency of all signals. \(E_{DC}\), \(R_{DC}\), and \(M_{DC}\) represent the DC components of the emitted light signal, the reflected light signal, and the demodulation signal, respectively. \(\varphi_{ToF}\) is the phase delay of the reflected light signal. In the demodulation signal, \(\delta_{n}\) represents the \(n\)th phase shift for \(n\) = 1, 2, \(\ldots\), \(N\). The number of phase shifts, identical to the number of samples, is \(N\).  
During the demodulation process, the sample \(I_{m}(\delta_{n})\), which is the cross-correlation function between the reflected light signal and demodulation signal for one integration time, is calculated for \(n\) = 1, 2, \(\ldots\), \(N\) in sequence [26]. The integration in the demodulation process can be expressed as  
\[\begin{array}{ll}I_{m}(\delta_{n})&=\int\limits_{0}^{T_{\text{int}}}R(t)M_{n }(t)dt\\ &=\int\limits_{0}^{T_{\text{int}}}[R_{A}\sin(2\pi ft-\varphi_{ToF})+R_{DC}] \cdot[M_{A}\sin(2\pi ft+\delta_{n})+M_{DC}]dt\\ &=\frac{R_{A}M_{A}T_{\text{int}}}{2}\cos(\varphi_{ToF}+\delta_{n})\ +\ R_{DC}M_{DC}T_{\text{int}},\ n=1,2,3,\ldots N \end{array} \tag{5}\]  
In Equation (5), \(T_{\text{int}}\) is the integration time required to calculate one sample. During the sample integration, sinusoidal terms which contain \(f\) are generally negligible under the assumption that the inverse of \(T_{\text{int}}\) is much smaller than \(f\). With multiple samples, the phase delay between the emitted light signal and reflected light signal is calculated through a simple trigonometric calculation [26; 27]. The distance is also determined using this phased delay accordingly. The results are given below  
\[\varphi_{ToF}=\tan^{-1}\!\left(\!\!\!\!\!\!  
\[d=\frac{c}{4\pi f}\varphi_{T\circ F}, \tag{7}\]  
where \(c\) is the velocity of light and \(d\) is the distance from the sensor to the object. More details can be found in the literature [18; 22; 24; 27; 28]. To measure a single distance, \(N\) integration times are required for \(N\) integration processes in Equation (5). By processing the phase demodulation in parallel, the total time for one-point distance measurement can be reduced. Details regarding the reduction of the total time for one-point distance measurement are presented in Section 4.