Laser Digital Phase Rangefinder Uses Vector Inner Product Method to Determine Phase

Digital phase rangefinder, especially involving the calculation method of vector inner product (Vector InnerProduct) digital phase. Quoted from linear algebra, the inner product of N-dimensional vectors is a generalization of quantity product. The N-dimensional vector can be obtained on the basis that the inner product of N-dimensional vectors X and Y and the norms of vectors X and Y satisfy Schwartz's inequality. The angle between X and Y, ∅=arccos(X,Y)/(ⅡXⅡ·Ⅱ YⅡ). When the sinusoidal signal sampling satisfies Nyquist's law and the sinusoidal digital sequence satisfies the requirements of the real number domain, the vector inner product method can be used to calculate the phase difference of the two sinusoidal signals. It is much simpler than the fast Fourier transform phase difference method. It has a small amount of calculations, short calculation time, and simple requirements for the processor. It is the product of a combination of pure mathematical principles and signal processing. It is used in the fields of phase measurement and optical-mechanical distance measurement. It has broad application prospects.

The laserdigital phase distance meter uses the vector inner product phase method, including an optical system, a circuit system and a data processing method. It is characterized in that one channel is a reference mixed signal, the other is a return measurement mixed signal, and the two channels are sinusoidal with the same frequency. The signals undergo analog/digital conversion respectively and become two N-point sinusoidal digital signal sequences X and Y, which are sent to the data processor. The X and Y sequences are used as two N-dimensional vectors to perform vector inner product operations and obtain two N The angle between the two dimensional vectors is the phase difference between the two sinusoidal signals of the same frequency.

Main technical areas:

1. Principle of phase ranging

The phase laser rangefinder uses a fixed-frequency high-frequency sinusoidal signal to continuously modulate the luminous intensity of the light source to measure distance. Since the phase delay Δφ caused by the modulated optical signal propagating back and forth over the distance to be measured is proportional to the optical path to be measured, the distance to be measured can be obtained by measuring the phase difference between the returned optical signal and the emitted optical signal. L. Since △φ=o●△, the relationship between the distance to be measured L and the phase delay △ is:

Therefore, a key part of phase ranging is the detection of the phase difference between the returned optical signal and the emitted optical signal.

2. Detection of phase difference

Phase difference detection methods can be divided into two categories: analog and digital. The analog method is to directly process two sinusoidal analog signals, such as delay phase measurement, pulse technology phase measurement, etc. The digital method is to use digital signal processing methods to process the two-way sinusoidal sequences obtained by sampling after sampling the two-way signals. The analog method has a long history, and currently more and more digital methods are used. I will not introduce the analog method too much here, but only give a brief introduction to the fast Fourier transform (FFT) method that is currently widely used.

The so-called FFT phase measurement method is to perform fast discrete Fourier transform on the sampled reference signal and echo signal digital sequence respectively, and find the phase according to the real and imaginary parts of the respective spectral peaks, which can be used as the phase of the respective corresponding signals, and then Find the phase difference between the two. The following is a brief description of the principle of FFT.
The N-point discrete Fourier transform (DFT) can be expressed as:

In the formula, W₄=e~2πlN, called the butterfly factor.

The FFT algorithm decomposes a long sequence of DFT into a short sequence of DFT²]. Time decimation FFT (DIT) is to decompose the input sequence x(n) of N points into two parts according to even numbers and odd numbers into even sequence y(n)=x(2n) and odd sequence z(n)=x(2n+1) , so the N-point FFT of x(n) can be expressed as:

Further extraction is done in the same way, and finally a set of 2-point DFT can be obtained.

Figure 1 is the signal flow diagram of the 8-point radix-2 DIT FFT. It can be seen that the output sequence (left column) is arranged sequentially, while the input sequence (right column) is scrambled. But there are certain rules for scrambling, namely "bit reversal". In actual work, the input data x(n) is generally a real sequence. For the operation of real number sequence FFT, the real sequence x(n) can be considered to be a complex sequence with an imaginary part of zero. In this way, the complex sequence FFT can be performed exactly as described above. Perform calculations. But you can also use N/2-point complex FFT to calculate the DFT of an N-point real number sequence, set the even numbers of the sequence as the real part, and the odd numbers as the imaginary part, and also separate them at the end. Theoretically, this cuts the amount of computation in half and therefore the amount of storage in half.

3. Problems with the Fast Fourier Transform (FFT) phase measurement method

From the above introduction to the principle of the FFT algorithm, it can be seen that the FFT algorithm is very complicated. It is more used in time-frequency transformation. When used for phase measurement, the biggest disadvantage is the large amount of calculation, followed by the long calculation time. The requirements for the processor are high, so the size and power consumption of the system will be subject to certain restrictions.

Technical indicators:

The laserdigital phase rangefinderwe introduced uses the vector inner product phase method to find a distance measurement calculation method that has a small amount of calculations, a short calculation time, low requirements on the processor, and reduces both volume and power consumption.

1. Theoretical basis of vector inner product (VIP) digital phase calculation method

In elementary linear algebra, we know the concept of vector space V on the real number field R, and we also know the concept of n-dimensional vector space. The concept and properties of n-dimensional vector inner product are introduced below. Refer to Engineering Linear Algebra (Second Edition), Advanced Mathematics Press, 1981, 1 (reprint), Chapter 5, Section 1 (P108~109). In analytic geometry, for two- or three-dimensional vectors, we know the quantity product of vectors, that is, for X,
Y two vectors, there are:
X·Y=|X|·|Y|cosφ
φ is the angle between the two vectors. In the rectangular coordinate system, there are
X·Y={x₁x₂,x₃}~{y₁,y₂,y₃}=x₁J₁+x₂Y₂+xJ₃

The inner product of n-dimensional vectors is a generalization of quantity product. n-dimensional vectors do not have the intuitive concepts of length and angle like three-dimensional vectors. Therefore, it is only promoted based on the rectangular coordinate calculation formula of quantity product. Define the inner product of n-dimensional vectors:

The length (or norm) of n-dimensional vector X:

The inner product of vectors satisfies:

(x,Y)²≤(x,x)(r,r)

The above formula is called Schwartz inequality, and it can be obtained from Schwartz inequality:

So there is the following definition:

is called the angle between n-dimensional vectors X and Y.

Although time series analysis and signal processing can also be studied without using vector space, the expression of vector space can bring a lot of convenience. Here, we use the concepts of vector spaces and vectors to process sequences of sinusoidal numbers. On the one hand, the sinusoidal digital sequence obtained by A/D sampling is a real number sequence, which fully conforms to the properties and requirements of n-dimensional vectors in the vector space on the real number domain R. On the other hand, according to Nyquist's law, if the frequency band of the signal itself is limited and the sampling frequency is greater than or equal to twice the highest frequency contained in the signal, the original signal can theoretically be completely restored based on its discrete sampling values. , so the sampled digital sequence completely contains the information of the sine wave. To sum up the above two aspects, we can completely use the vector inner product method to detect the phase of the digital sequence.
In the digital phase ranging system, the two demodulated sine signals are sampled by A/D. Assuming n bits are sampled, two n-point sine digital sequences are obtained:

The above two digital sequences should satisfy: The sampling of the two digital sequences 1, X, and Y should satisfy Nyquist's law. 2.X.
Y satisfies the properties and requirements of n-dimensional vectors in the vector space on the real number field R. Then the angle φ between X and Y is:

In

(1) |X| is the norm of the digital sequence X
(2)|Y| is the norm of the digital sequence Y
(3)(X,Y) is the inner product of the digital sequences X and Y
(4)|X|≠0,|Y|≠0
Taking X and Y as two n-dimensional vectors, according to the above vector inner product phase detection principle, the angle between the two n-dimensional vectors can be obtained, that is, the phase of the two sinusoidal signals. From the phase difference, the flight time of light between these two points can be calculated, and from this the distance can be obtained.
2. Simulation experiment of vector inner product (VIP) digital phase calculation method
Mathematical Experimental Language (MATLAB) is the most influential and dynamic software in the international scientific community (especially in the field of automatic control). It originated from matrix operations and has developed into a highly integrated computer language. It provides powerful scientific operations, flexible programming flow, and system simulation functions. Next, we use Matlab language to conduct a simulation experiment on a digital phase meter implemented using the vector inner product method. The experiment content is divided into three major steps, as follows:
(1) Simulate the generated signal source, that is, generate two sinusoidal digital sequences with phase differences.
(2) Simulate the calculation of phase difference.
(3) Convert the resulting cosine value into an angle.
The specific simulation program will not be described in detail here. The simulation experiment can prove that the digital phase meter implemented using the vector inner product method is not only simple and feasible, but also has high accuracy.

3. Algorithm content

The laserdigital phase distance meter uses the vector inner product phase method, including an optical system, a circuit system and a data processing method. It is characterized in that one channel is a reference mixed signal, the other is a return measurement mixed signal, and the two channels are sinusoidal signals of the same frequency. The signals undergo analog/digital conversion respectively and become two N-point sinusoidal digital signal sequences X and Y, which are sent to the data processor. The X and Y sequences are used as two N-dimensional vectors to perform vector inner product operations and obtain two N The angle between the two dimensional vectors is the phase difference between the two sinusoidal signals of the same frequency.
This phase measurement method is used for optical, mechanical, and electrical integrated distance measurement. It uses digital sequences of reference signals and measurement signals for processing. This method is used without using fast Fourier transform. The hardware part is the same as the existing technology, except that the micro Ranging can be achieved by changing the software part of the processor. The flow chart of the two-way sinusoidal signal and the phase difference algorithm implemented on the microprocessor is as follows

(1) Enter the sinusoidal digital sequences X and Y;

(2) Calculate the norm ∥X∥ of the digital sequence X;
(3) Calculate the norm ∥Y∥ of the digital sequence Y;
(4) Calculate the inner product (X, Y) of the digital sequences X and Y;
(5) Calculation

Advantages of this technology:

The vector inner product digital phase calculation method is a digital phase measurement method. It processes the sampled reference signal and echo signal digital sequence, but it is much simpler than the fast Fourier transform phase measurement method. In comparison, it has the following advantages: 1. The amount of calculation is small and the idea is simple; 2. The requirements for the processor are low and the hardware circuit is simplified; 3. The calculation speed is fast and it can achieve real-time processing. The following is a comparison of the computational complexity of the two phase measurement methods. Now we use a computer as the processor. Since a digital computer takes much more time to perform multiplication than addition, we only consider the multiplication among the two algorithms and only compare the multiplications. For two N-point sinusoidal digital sequences, the ratio of the number of multiplications of fast Fourier phase measurement (FFT) and vector inner product phase measurement (VIP) is:

When N=8,

Description of the drawings

Figure 1 is the 8-point base 2DIT FFT signal flow chart.

Figure 1

Figure 2 is a flow chart of the vector inner product phase calculation algorithm.

Figure 2

Figure 3 is the hardware schematic block diagram.

Figure 3

Figure 4 is the principle block diagram of the laserranging system.

Detailed ways

In Figure 2,
1. Enter the sine number sequence X and Y;
2. Calculate the norm ∥X∥ of the digital sequence X;
3. Calculate the norm ∥P∥ of the digital sequence Y;
4. Calculate the inner product (X, Y) of the number sequences X and Y;
5. Calculation

Figure 3 is the hardware principle block diagram of the digital phase meter. 8. Analog/digital (A/D) converter, which samples two analog sinusoidal signals respectively. 9. Data processor, which contains a vector inner product algorithm software package, processes digital signals.

Figure 4 is a schematic block diagram of a laserranging system based on the vector inner product phasing method. 10. Direct digital synthesizer; 11. Microprocessor; 12. Measurement signal mixing; 13. Reference signal mixing; 14. Modulation emission; 15. Photoelectric reception; 16. Light reflector.
Based on the principle of phase ranging and the theory of digital phase meter implemented by vector inner product method, the phase ranging system we designed uses 8051 microcontroller as the core processing module. The system principle block diagram is shown in Figure 4. To generate the system signal source, we use precise direct digital synthesis DDS (10) technology, which is controlled by the control word issued by the microcontroller (11) to generate two signals. One channel is mixed with another channel before and after transmission. The reference signal is obtained by mixing (13) before transmission. The measurement signal is obtained by mixing (14) after modulation, transmission and reception. The reference signal and the measurement signal are passed through A /D(8) After sampling, two digital signal sequences are obtained, which are processed by the microcontroller. The microcontroller calls the vector inner product (VIP) digital phase calculation method to calculate the phase, thereby obtaining the measurement distance.
Without changing all the hardware of the original phase-type digital rangefinder, only the software is replaced by a vector inner product digital phase algorithm to calculate the phase. Finally, based on the relationship between phase and distance, the distance can be obtained.

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