# Optimization of the Processing Time of Cross-Correlation Spectra for Frequency Measurements of Noisy Signals

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals

#### 2.1. Cross-Correlation

#### 2.2. Cross-Correlation Spectrum

- 1.
- The CCS’s frequency range $\left[{f}_{a};{f}_{e}\right]$ and the frequency resolution $\Delta f$ are set by the user. $w$ is set to 0;
- 2.
- ${f}_{w}$ is initialized to ${f}_{a}$ (${f}_{w}={f}_{a}$);
- 3.
- The sine function ${x}_{{f}_{W}}\left(n\right)$ is generated according to Equation (17);
- 4.
- The cross-correlation function ${\varphi}_{{x}_{{f}_{w}}y}\left(k\right)$ between the ${x}_{{f}_{W}}\left(n\right)$ and the measuring signal $y\left(n\right)$ is calculated;
- 5.
- The cross-correlation spectrum’s value $K\left({f}_{w}\right)$ for the current frequency ${f}_{w}$ is determined by identifying the amplitude of the cross-correlation function ${\varphi}_{{x}_{{f}_{w}}y}\left(k\right)$. In [13], it is accomplished by finding the function’s maximum value;
- 6.
- The frequency ${f}_{w}$ is increased by $\Delta f$, i.e., $w=w+1$;
- 7.
- The steps from 3–6 are repeated, until $K\left({f}_{w}={f}_{e}\right)$ 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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**Figure 1.**Calculation process of cross-correlation spectrum [9].

**Figure 2.**Cross-correlation spectrum of a sine function [9].

**Figure 4.**Cross-correlation spectra of the sinus signals and their sum [9].

**Figure 17.**Typical SMI-signal for rotational speed measurement (Reprinted with permission from [17] © The Optical Society).

**Figure 20.**Linearity between Doppler frequency and rotational speed (Reprinted with permission from [16] © Dr. Hui Sun).

Phase Difference | Behavior of Accuracy |
---|---|

$\left[0;0.5\pi \right]$ | Increasing phase difference leads to accuracy deterioration |

$\left[0.5\pi ;\pi \right]$ | Increasing phase difference leads to accuracy improvement |

$\left[\pi ;1.5\pi \right]$ | Increasing phase difference leads to accuracy deterioration |

$\left[1.5\pi ;2\pi \right]$ | Increasing phase difference leads to accuracy improvement |

Original CCS | New CCS Method |
---|---|

286.93 ms | 30.82 ms |

**Table 3.**Linearity of method in [17], original CCS and new CCS method.

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% |

**Table 4.**NRMSE of results using the method of [17], original CCS and new CCS method.

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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Liu, 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