Optimization of the Processing Time of Cross-Correlation Spectra for Frequency Measurements of Noisy Signals
Abstract
:1. Introduction
2. Fundamentals
2.1. Cross-Correlation
2.2. Cross-Correlation Spectrum
- 1.
- The CCS’s frequency range and the frequency resolution are set by the user. is set to 0;
- 2.
- is initialized to ();
- 3.
- The sine function is generated according to Equation (17);
- 4.
- The cross-correlation function between the and the measuring signal is calculated;
- 5.
- The cross-correlation spectrum’s value for the current frequency is determined by identifying the amplitude of the cross-correlation function . In [13], it is accomplished by finding the function’s maximum value;
- 6.
- The frequency is increased by , i.e., ;
- 7.
- The steps from 3–6 are repeated, until has been determined.
3. New Version of Cross-Correlation Spectrum
3.1. Theory
3.2. Simulations
4. Phase Measurement Using Cross Correlation
5. Frequency and Phase Measurement Method
6. Application Example
6.1. Self-Mixing Interferometry
6.2. Experiments on SMI-Signals
7. Discussions
- Creating reference signals with different frequencies can require a lot of processing time, which could be reduced by improved algorithms;
- Depending on the application’s requirement, the spectrum’s resolution could vary in different frequency ranges. While a lower resolution is initially used, critical frequency ranges could be assigned higher resolutions. This way, an optimum between accuracy and computational effort could be achieved.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Phase Difference | Behavior of Accuracy |
---|---|
Increasing phase difference leads to accuracy deterioration | |
Increasing phase difference leads to accuracy improvement | |
Increasing phase difference leads to accuracy deterioration | |
Increasing phase difference leads to accuracy improvement |
Original CCS | New CCS Method |
---|---|
286.93 ms | 30.82 ms |
Rotational Speed in RPM | Linearity of Method in [19] | Linearity with Original CCS | Linearity with New CCS Method |
---|---|---|---|
5 | −0.21 | −0.24% | +0.23% |
10 | −0.18 | −0.20% | +0.20% |
20 | −0.10 | −0.06% | +0.08% |
30 | +0.01 | 0.03% | −0.00% |
40 | +0.10 | 0.09% | −0.10% |
50 | +0.14 | 0.11% | −0.10% |
75 | +0.31 | 0.35% | −0.27% |
100 | +0.21 | 0.40% | +0.32% |
125 | −0.26 | −0.21% | +0.14% |
150 | −0.12 | −0.16% | +0.10% |
175 | +0.18 | 0.03% | +0.02% |
200 | −0.21 | −0.12% | +0.18% |
Rotational Speed (RPM) | NRMSE of Method in [19] | NRMSE with Original CCS (%) | NRMSE with New CCS (%) |
---|---|---|---|
5 | 0.16 | 0.16 | 0.18 |
10 | 0.06 | 0.06 | 0.07 |
20 | 0.05 | 0.08 | 0.09 |
30 | 0.08 | 0.14 | 0.13 |
40 | 0.15 | 0.18 | 0.21 |
50 | 0.13 | 0.13 | 0.12 |
75 | 0.07 | 0.04 | 0.05 |
100 | 0.06 | 0.06 | 0.07 |
125 | 0.06 | 0.06 | 0.09 |
150 | 0.07 | 0.08 | 0.08 |
175 | 0.05 | 0.06 | 0.07 |
200 | 0.06 | 0.03 | 0.03 |
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Liu, Y.; Liu, J.; Kennel, R. Optimization of the Processing Time of Cross-Correlation Spectra for Frequency Measurements of Noisy Signals. Metrology 2022, 2, 293-310. https://doi.org/10.3390/metrology2020018
Liu Y, Liu J, Kennel R. Optimization of the Processing Time of Cross-Correlation Spectra for Frequency Measurements of Noisy Signals. Metrology. 2022; 2(2):293-310. https://doi.org/10.3390/metrology2020018
Chicago/Turabian StyleLiu, Yang, Jigou Liu, and Ralph Kennel. 2022. "Optimization of the Processing Time of Cross-Correlation Spectra for Frequency Measurements of Noisy Signals" Metrology 2, no. 2: 293-310. https://doi.org/10.3390/metrology2020018