# 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
Thus, sample \(I_{m}(0)\) can be expressed as  
\[I_{m}(0)=\frac{R_{1}M_{1}T_{\rm int}}{2}\cos(\varphi_{1})+\frac{R_{2}M_{2}T_{ \rm int}}{2}\cos(\varphi_{2}). \tag{22}\]  
Likewise, if the phase shift \(\delta_{n}\) is applied to the demodulation signal, sample \(I_{m}(\delta_{n})\) can be expressed as  
\[I_{m}(\delta_{n})=\frac{R_{1}M_{1}T_{\rm int}}{2}\cos(\varphi_{1}+m\delta_{n} )+\frac{R_{2}M_{2}T_{\rm int}}{2}\cos(\varphi_{2}+l\delta_{n}). \tag{23}\]  
Using angle sum identities and the orthogonality of trigonometric functions, the following equations are satisfied. \[\begin{array}{rl}\sum\limits_{n=1}^{N}I_{m}(\delta_{n})\cos(m\delta_{n})= \frac{R_{1}M_{1}T_{\rm int}N}{2}\cos(\varphi_{1})\\ \sum\limits_{n=1}^{N}I_{m}(\delta_{n})\sin(m\delta_{n})=-\frac{R_{1}M_{1}T_{\rm int }N}{2}\sin(\varphi_{1})\\ \sum\limits_{n=1}^{N}I_{m}(\delta_{n})\cos(l\delta_{n})  
The M9-852-0100 laser diode by Thorlabs Inc., which generates infrared light at a wavelength of 852 nm, is used as the illumination source. The APD 120 by Thorlabs Inc. is used as the detector, which converts the reflected light signal into a voltage signal. The PXI system by National Instruments Inc. is used to control the measurement system, collect the measurement signal, and process the signals. The PXI system consists of a PXIe-5160 digitizer module, a PXIe-5423 arbitrary function generator module, and a PXI-8880 controller module. The PXIe-5160 fast digitizer module collects the measured voltage signal from the avalanche photodiode with a sampling frequency of 625 MHz. The PXIe-5423 module generates a waveform for modulating the laser intensity and for generating the demodulation signal. The PXI-8880 controller module controls the entire measurement system and processes the samples. All modules are combined in a PXI-1082 chassis, which provides the power, cooling, and communication buses for the PXI modules. Figure 6 shows the overall signal flows of the measurement system. The laser controller generates a DC component for the signal, and the function generator generates a modulation signal. Part of the modulation signal is connected to one channel of the digitizer to be used as the demodulation signal. The remaining generated modulation signal is combined with the DC component of the laser controller and used to modulate the laser diode, which emits the optical signal onto the object. The emitted light signal is then reflected and detected by the avalanche photodiode, after which the avalanche photodiode transmits the measured voltage signal to the digitizer through another channel. The digitizer collects input signals and transmits them to the controller. The controller calculates the distance using the signals received. One distinct advantage of this system setup is that the modulation signal and demodulation signal can be generated using only one function generator. Figure 5: (**a**) The whole distance measurement system; (**b**) optical components of the system.