# Cramér–Rao Lower Bound for Magnetic Field Localization around Elementary Structures

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contribution

- How does the achievable positioning performance depend on the magnetic sensor properties, i.e., the sensor noise variance?
- How does the positioning performance depend on the distance from ferromagnetic objects that cause magnetic field distortions?
- How does the ferromagnetic material properties, in particular the relative permeability, influence the achievable positioning performance?

## 2. The Static Magnetic Field

#### 2.1. Maxwell’s Equations for the Static Magnetic Field

#### 2.2. Boundary Values and Conditions

## 3. Ferromagnetic Structures in a Homogeneous Magnetic Field

#### 3.1. Sphere

#### 3.2. Cylinder

## 4. Positioning Performance—An Estimation–Theoretic Approach

#### 4.1. Measurement Model

#### 4.2. Fisher Information and the Cramér–Rao Lower Bound

#### 4.3. Discussion of Results

#### 4.3.1. Dependency on Magnetic Sensor Properties

#### 4.3.2. Dependency on Material Properties

#### 4.3.3. Dependency on Sensor Location

#### Sphere

#### Cylinder

## 5. Examples

#### 5.1. Cramér–Rao Lower Bound in the Presence of Quantization Noise

#### 5.2. Cramér–Rao Lower Bound in the Presence of Earth’s Magnetic Field Noise

#### 5.2.1. Characterization of the Earth’s Magnetic Field Noise

#### 5.2.2. Cramér–Rao Lower Bound

#### 5.2.3. Discussion of Results

**Σ**

_{∞}. Higher fluctuations lead to a higher trace, tr(

**Σ**

_{∞}), of that covariance matrix and consequently to a lower SNR and an increased Cramér–Rao lower bound.

#### 5.3. Cramér–Rao Lower Bound in the Presence of Quantization and Magnetic Field Noise

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Properties of the Jacobian Matrix for a Cylinder in a Homogeneous Magnetic Field

## Appendix B. A Sphere in a Homogenous Magnetic Field

#### Appendix B.1. Solution of Laplace’s Equation in Spherical Coordinates

#### Appendix B.1.1. Separation of the Laplace Equation

#### Appendix B.1.2. Solution of the Polar Part

#### Appendix B.1.3. Solution of the Azimuth Part

#### Appendix B.1.4. Solution of the Radial Part

#### Appendix B.1.5. The General Solution

#### Appendix B.2. Solution of the Boundary Value Problem

## Appendix C. A Cylinder in a Homogenous Magnetic Field

#### Appendix C.1. Solution of Laplace’s Equation in Cylinder Coordinates

#### Appendix C.1.1. Separation of the Laplace Equation

#### Appendix C.1.2. Solution of the Azimuth Part

#### Appendix C.1.3. Solution of the Height Part

#### Appendix C.1.4. Solution of the Radial Part

#### Appendix C.1.5. The General Solution

#### Appendix C.2. Solution of the Boundary Value Problem

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**Figure 1.**Solid sphere with permeability ${\mu}_{\mathrm{in}}$ in an outer medium with permeability ${\mu}_{\mathrm{out}}$.

**Figure 2.**Magnetic field strength and magnetic flux lines for a sphere with relative permeability ${\mu}_{\mathrm{r}}=100$ in an outer homogeneous magnetic field.

**Figure 3.**Solid cylinder with permeability ${\mu}_{\mathrm{in}}$ in an outer medium with permeability ${\mu}_{\mathrm{out}}$.

**Figure 4.**Magnetic field strength and magnetic flux lines for a cylinder with relative permeability ${\mu}_{\mathrm{r}}=100$ in an outer homogeneous magnetic field.

**Figure 5.**Dependency of position estimation standard deviation on the relative permeability ${\mu}_{\mathrm{r}}$.

**Figure 7.**Polar angle dependency of position estimation standard deviation for a sphere in a homogeneous magnetic field.

**Figure 8.**Location dependency ${\sigma}_{\mathrm{loc}}$ of position estimation standard deviation for a sphere in a homogeneous magnetic field.

**Figure 9.**Cramér–Rao lower bound for position estimation standard deviation versus the distance of the sensor to the center of a vertical cylindrical pillar in the Earth’s magnetic field according to (45) with ${\sigma}_{\mathrm{mat}}=1$.

**Figure 11.**Quantile–quantile plots for the Earth magnetic field fluctuation in Fuerstenfeldbruck, Germany during May 2016.

**Figure 13.**Cramér–Rao lower bound for position estimation standard deviation for a vertical cylindrical pillar in the Earth’s magnetic field according to (63) with ${\sigma}_{\mathrm{mat}}=1$.

**Figure 14.**Cramér–Rao lower bound for position estimation standard deviation for a vertical cylindrical pillar in the Earth’s magnetic field according to (64), with (66) and ${\sigma}_{\mathrm{mat}}=1$. Solid lines represent results for considering the Earth’s magnetic field noise and the sensors’ quantization noise. Dashed lines with markers (’+’) repeat results from Figure 9, where only quantization noise has been considered.

Sensor | Ref. | Range [$\mathrm{\mu}\mathbf{T}$] | Resolution ${\mathit{B}}_{\mathbf{res}}$ [nT] | Quant. Noise ${\mathit{\sigma}}_{\mathit{B}}^{2}$ [${\mathbf{nT}}^{2}$] | SNR $\frac{{\mathit{H}}_{\mathit{\infty}}^{2}}{2{\mathit{\sigma}}_{\mathit{H}}^{2}}=\frac{{\mathit{B}}_{\mathit{\infty}}^{2}}{2{\mathit{\sigma}}_{\mathit{B}}^{2}}$ |
---|---|---|---|---|---|

MMC246xMT | [24] | $\pm 600$ | 25 | $52.1$ | $4.2\times {10}^{6}$ |

HMC5983 | [25] | $\pm 88$ | 73 | 444 | $5.0\times {10}^{5}$ |

YAS532 | [26] | $\pm 1200$ | x,y: 150 | 1875 | $1.2\times {10}^{5}$ |

z: 250 | 5208 | ||||

AKM8975 | [27] | $\pm 1200$ | 300 | 7500 | $2.9\times {10}^{4}$ |

YAS529 | [28] | $\pm 300$ | x,y: 600 | 30,000 | $7.4\times {10}^{3}$ |

z: 1200 | 120,000 |

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**MDPI and ACS Style**

Dammann, A.; Siebler, B.; Sand, S.
Cramér–Rao Lower Bound for Magnetic Field Localization around Elementary Structures. *Sensors* **2024**, *24*, 2402.
https://doi.org/10.3390/s24082402

**AMA Style**

Dammann A, Siebler B, Sand S.
Cramér–Rao Lower Bound for Magnetic Field Localization around Elementary Structures. *Sensors*. 2024; 24(8):2402.
https://doi.org/10.3390/s24082402

**Chicago/Turabian Style**

Dammann, Armin, Benjamin Siebler, and Stephan Sand.
2024. "Cramér–Rao Lower Bound for Magnetic Field Localization around Elementary Structures" *Sensors* 24, no. 8: 2402.
https://doi.org/10.3390/s24082402