# Quantitative Evaluation for Magnetoelectric Sensor Systems in Biomagnetic Diagnostics

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## Abstract

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## 1. Introduction

## 2. Evaluation Metrics for Magnetoelectric Sensor Systems

#### 2.1. Input–Output–Amplitude–Relation

#### 2.1.1. Limit-of-Detection

#### 2.1.2. Limit-of-Quantification

#### 2.1.3. Linear Region

#### 2.1.4. 1-dB-Compression-Point and 3-dB-Compression-Point

#### 2.1.5. Dynamic Range

#### 2.1.6. Determination of the Input–Output–Amplitude–Relation

#### 2.2. Frequency Response (Magnitude and Phase Response)

- Mean Passband Amplitude$$\overline{a}=\frac{1}{M}\sum _{\mu =0}^{M-1}\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mathrm{p}1}\le {\mathsf{\Omega}}_{\mu}\le {\mathsf{\Omega}}_{\mathrm{p}2}$$
- Passband Ripple$$\begin{array}{c}\hfill {\delta}_{\mathrm{p}}=20\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{log}_{10}\left(\right)open="("\; close=")">\frac{\overline{a}+{\delta}_{\mathrm{p},\mathrm{max}}}{\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\\ \mathrm{with}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\delta}_{\mathrm{p},\mathrm{max}}=\mathrm{max}\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\hfill \end{array}\hfill \mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\delta}_{\mathrm{p},\mathrm{min}}=\mathrm{min}\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\\ \mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mathrm{p}1}\le {\mathsf{\Omega}}_{\mu}\le {\mathsf{\Omega}}_{\mathrm{p}2}\hfill $$
- Passband Edge Frequencies$${\mathsf{\Omega}}_{\mathrm{p}1}=arg\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}$$$${\mathsf{\Omega}}_{\mathrm{p}2}=arg\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}$$
- Stopband Edge Frequencies$${\mathsf{\Omega}}_{\mathrm{s}1}=arg\left(\right)open="\{"\; close="\}">\left|H\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\mathrm{max}\left(\right)open="("\; close=")">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}{\mathsf{\Omega}}_{\mathrm{s}1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}\ll {\mathsf{\Omega}}_{\mathrm{p}1}$$$${\mathsf{\Omega}}_{\mathrm{s}2}=arg\left(\right)open="\{"\; close="\}">\left|H\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\mathrm{max}\left(\right)open="("\; close=")">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}{\mathsf{\Omega}}_{\mathrm{s}2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}\gg {\mathsf{\Omega}}_{\mathrm{p}2}$$
- Transition Bands$$\Delta {\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{\mathrm{p}1}-{\mathsf{\Omega}}_{\mathrm{s}1}$$$$\Delta {\mathsf{\Omega}}_{2}={\mathsf{\Omega}}_{\mathrm{s}2}-{\mathsf{\Omega}}_{\mathrm{p}2}$$
- −3 dB Angular Frequencies, Bandwidth$${\mathsf{\Omega}}_{-3\mathrm{dB},1}=arg\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\frac{1}{\sqrt{2}}\phantom{\rule{0.166667em}{0ex}}\overline{a}$$$${\mathsf{\Omega}}_{-3\mathrm{dB},2}=arg\left(\right)open="\{"\; close="\}">\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\frac{1}{\sqrt{2}}\phantom{\rule{0.166667em}{0ex}}\overline{a}$$$$w={\mathsf{\Omega}}_{-3\mathrm{dB},2}-{\mathsf{\Omega}}_{-3\mathrm{dB},1}.$$

#### Frequency Response Determination

## 3. Figures of Merit for Sensor Signal Evaluation

#### 3.1. System Noise

^{2}/Hz can be represented as amplitude spectral density,

#### Measurement

#### 3.2. Signal-to-Noise Ratio

#### 3.3. Application Specific Capacity

## 4. Exemplary Evaluation of Magnetoelectric Sensor Systems

#### 4.1. Exchange Bias Magnetoelectric Sensor

#### 4.2. Electrically Modulated ME Sensor

#### 4.3. Overview of the ME Evaluation Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A/D | Analog/Digital |

ACF | Autocorrelation function |

AS | Amplitude spectrum |

ASC | Application specific capacity |

ASD | Amplitude spectral density |

DFT | Discrete Fourier transform |

DR | Dynamic range |

ECG | Electrocardiography |

EEG | Electroencephalography |

EMC | Electromagnetic compatibility |

ENBW | Equivalent noise bandwidth |

HP | Highpass |

JFET | Junction-gate-field-effect transistor |

LOD | Limit-of-Detection |

LOQ | Limit-of-Quantification |

LTI | Linear time-invariant |

ME | Magnetoelectric |

MCG | Magnetocardiography |

OPM | Optically Pumped Magnetometer |

PCB | Printed circuit board |

PS | Power spectrum |

PSD | Power spectral density |

PTB | Physikalisch Technische Bundesanstalt |

RMS | Root mean square |

SNNR | Signal-plus-noise to noise ratio |

SNR | Signal-to-noise ratio |

SQUID | Super conducting quantum interference device |

## Appendix A. MCG Prototype Signal

t [s] | 0 | $0.25$ | $0.3$ | $0.35$ | $0.44$ | $0.47$ | $0.5$ | $0.52$ | $0.56$ | $0.6$ | $0.75$ | $0.85$ | 1 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$s\left(t\right)$ [pT] | 0 | 0 | $2.1$ | 0 | 0 | $-10.5$ | 70 | $-7$ | 0 | 0 | $12.6$ | 0 | 0 |

$\frac{ds\left(t\right)}{dt}$ [$\frac{\mathrm{pT}}{\mathrm{s}}$] | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

## References

- Sanchez-Reyes, L.M.; Rodriguez-Resendiz, J.; Avecilla-Ramirez, G.N.; Garcia-Gomar, M.L.; Robles-Ocampo, J.B. Impact of EEG Parameters Detecting Dementia Diseases: A Systematic Review. IEEE Access
**2021**, 9, 78060–78074. [Google Scholar] [CrossRef] - Williamson, S.J. Advances in Biomagnetism: Proceedings of the Seventh International Conference on Biomagnetism, New York, NY, USA, 13–18 August 1989; Plenum Press: New York, NY, USA, 1989. [Google Scholar]
- Boto, E.; Hill, R.M.; Rea, M.; Holmes, N.; Seedat, Z.A.; Leggett, J.; Shah, V.; Osborne, J.; Bowtell, R.; Brookes, M.J. Measuring functional connectivity with wearable MEG. NeuroImage
**2021**, 230, 117815. [Google Scholar] [CrossRef] [PubMed] - Elzenheimer, E.; Laufs, H.; Schulte-Mattler, W.; Schmidt, G. Magnetic Measurement of Electrically Evoked Muscle Responses with Optically Pumped Magnetometers. IEEE Trans. Neural Syst. Rehabil. Eng.
**2020**, 28, 756–765. [Google Scholar] [CrossRef] [PubMed] - Shirai, Y.; Hirao, K.; Shibuya, T.; Okawa, S.; Hasegawa, Y.; Adachi, Y.; Sekihara, K.; Kawabata, S. Magnetocardiography Using a Magnetoresistive Sensor Array. Int. Heart J.
**2019**, 60, 50–54. [Google Scholar] [CrossRef] [Green Version] - Janosek, M.; Butta, M.; Dressler, M.; Saunderson, E.; Novotny, D.; Fourie, C. 1-pT Noise Fluxgate Magnetometer for Geomagnetic Measurements and Unshielded Magnetocardiography. IEEE Trans. Instrum. Meas.
**2020**, 69, 2552–2560. [Google Scholar] [CrossRef] - Macfarlane, P.W.; van Oosterom, A.; Pahlm, O.; Kligfield, P.; Janse, M.; Camm, A.J. Comprehensive Electrocardiology, 2nd ed.; Springer: London, UK, 2010. [Google Scholar] [CrossRef]
- Koch, H. Recent advances in magnetocardiography. J. Electrocardiol.
**2004**, 37, 117–122. [Google Scholar] [CrossRef] - Hayes, P. Converse Magnetoelectric Resonators for Biomagnetic Field Sensing. Ph.D. Thesis, Kiel University, Kiel, Germany, 2020. [Google Scholar]
- Hayes, P.; Jovičević Klug, M.; Toxværd, S.; Durdaut, P.; Schell, V.; Teplyuk, A.; Burdin, D.; Winkler, A.; Weser, R.; Fetisov, Y.; et al. Converse Magnetoelectric Composite Resonator for Sensing Small Magnetic Fields. Sci. Rep.
**2019**, 9, 16355. [Google Scholar] [CrossRef] - Jahns, R.; Knöchel, R.; Greve, H.; Woltermann, E.; Lage, E.; Quandt, E. Magnetoelectric sensors for biomagnetic measurements. In Proceedings of the 2011 IEEE International Symposium on Medical Measurements and Applications, Bari, Italy, 30–31 May 2011; pp. 107–110. [Google Scholar] [CrossRef]
- Bald, C.; Schmidt, G. Processing Chain for Localization of Magnetoelectric Sensors in Real Time. Sensors
**2021**, 21, 5675. [Google Scholar] [CrossRef] - Hoffmann, J.; Elzenheimer, E.; Bald, C.; Hansen, C.; Maetzler, W.; Schmidt, G. Active Magnetoelectric Motion Sensing: Examining Performance Metrics with an Experimental Setup. Sensors
**2021**, 21, 8000. [Google Scholar] [CrossRef] - Sander, T.; Jodko-Władzińska, A.; Hartwig, S.; Brühl, R.; Middelmann, T. Optically pumped magnetometers enable a new level of biomagnetic measurements. Adv. Opt. Technol.
**2020**, 9, 247–251. [Google Scholar] [CrossRef] - Ripka, P. Magnetic Sensors and Magnetometers; Artech House: Boston, MA, USA, 2001. [Google Scholar]
- Hayes, P.; Schell, V.; Salzer, S.; Burdin, D.; Yarar, E.; Piorra, A.; Knöchel, R.; Fetisov, Y.K.; Quandt, E. Electrically modulated magnetoelectric AlN/FeCoSiB film composites for DC magnetic field sensing. J. Phys. D Appl. Phys.
**2018**, 51, 354002. [Google Scholar] [CrossRef] - Ohm, J.R.; Lüke, H.D. Signalübertragung: Grundlagen der digitalen und analogen Nachrichtenübertragungssysteme, 11th ed.; Springer-Lehrbuch; Springer: Berlin, Germany, 2010. [Google Scholar] [CrossRef]
- Hänsler, E. Statistische Signale: Grundlagen und Anwendungen, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar] [CrossRef]
- Tumański, S. Handbook of Magnetic Measurements; Sensors Series; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Brinkmann, B. Internationales Wörterbuch der Metrologie: Grundlegende und Allgemeine Begriffe und Zugeordnete Benennungen (VIM) Deutsch-Englische Fassung ISO/IEC-Leitfaden 99:2007, 4th ed.; Beuth Wissen; Beuth Verlag GmbH: Berlin, Germany, 2012. [Google Scholar]
- DIN 32645:2008-11. Chemische Analytik: Nachweis-, Erfassungs- und Bestimmungsgrenze unter Wiederholbedingungen-Begriffe, Verfahren, Auswertung. Available online: https://www.beuth.de/de/norm/din-32645/110729574 (accessed on 21 January 2021). [CrossRef]
- Wenclawiak, B.W.; Koch, M.; Hadjicostas, E. Quality Assurance in Analytical Chemistry; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Mitra, S.K. Digital Signal Processing: A Computer-Based Approach, 4th ed.; McGraw-Hill: New York, NY, USA, 2011. [Google Scholar]
- Proakis, J.G.; Manolakis, D.G. Digital signal processing, 4th ed.; Pearson New International Edition ed.; Always Learning; Pearson: Harlow, UK, 2014. [Google Scholar]
- Papula, L. Mathematik für Ingenieure und Naturwissenschaftler: Band 3: Vektoranalysis, Wahrscheinlichkeitsrechnung, Mathematische Statistik, Fehler- und Ausgleichsrechnung, 7th ed.; Springer Vieweg: Wiesbaden, Germany, 2016. [Google Scholar] [CrossRef]
- Tietze, U.; Schenk, C.; Gamm, E. Electronic Circuits: Handbook for Design and Application, 2nd ed.; First Indian Reprint ed.; Springer: New Delhi, India, 2012. [Google Scholar] [CrossRef]
- Norton, H.N. Handbook of Transducers; Prentice-Hall: Englewood Cliffs, NJ, USA, 1989. [Google Scholar]
- Madisetti, V.K.; Williams, D.B. The Digital Signal Processing Handbook; The Electrical Engineering Handbook Series; CRC Press: Boca Raton, FL, USA, 1997. [Google Scholar]
- Rohde & Schwarz GmbH & Co. KG. R&S UPV Audio Analyzer Operating Manual; Rohde & Schwarz GmbH & Co. KG: München, Germany, 2015. [Google Scholar]
- Keithley Instruments Inc. Model 6220 DC Current Source Model 6221 AC and DC Current Source Users Manual; Keithley Instruments Inc.: Cleveland, OH, USA, 2008. [Google Scholar]
- Sternickel, K.; Braginski, A.I. Biomagnetism using SQUIDs: Status and perspectives. Supercond. Sci. Technol.
**2006**, 19, S160–S171. [Google Scholar] [CrossRef] - Elzenheimer, E.; Laufs, H.; Sander-Thömmes, T.; Schmidt, G. Magnetoneurograhy of an Electrically Stimulated Arm Nerve. Curr. Dir. Biomed. Eng.
**2018**, 4, 363–366. [Google Scholar] [CrossRef] [Green Version] - Reermann, J.; Durdaut, P.; Salzer, S.; Demming, T.; Piorra, A.; Quandt, E.; Frey, N.; Höft, M.; Schmidt, G. Evaluation of magnetoelectric sensor systems for cardiological applications. Measurement
**2018**, 116, 230–238. [Google Scholar] [CrossRef] - Welch, P. The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust.
**1967**, 15, 70–73. [Google Scholar] [CrossRef] [Green Version] - Weik, M.H. Signal-plus-noise to noise ratio. In Computer Science and Communications Dictionary; Springer: Boston, MA, USA, 2001; p. 1583. [Google Scholar] [CrossRef]
- Bald, C.; Elzenheimer, E.; Sander-Thömmes, T.; Schmidt, G. Amplitudenverlauf des Herzmagnetfeldes als Funktion des Abstandes. In Proceedings of the Workshop Biosignal Processing 2018—Innovative Processing of Bioelectric and Biomagnetic Signals, Erfurt, Germany, 21–23 March 2018. [Google Scholar]
- Scher, A.M.; Young, A.C. Frequency Analysis of the Electrocardiogram. Circ. Res.
**1960**, 8, 344–346. [Google Scholar] [CrossRef] [Green Version] - Schlichthärle, D. Digital Filters: Basics and Design; Springer eBook Collection; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar] [CrossRef]
- Shannon, C. Communication in the Presence of Noise. Proc. IRE
**1949**, 37, 10–21. [Google Scholar] [CrossRef] - Salzer, S.D. Readout Methods for Magnetoelectric Sensors. Ph.D. Thesis, Kiel University, Kiel, Germany, 2018. [Google Scholar]
- Stanford Research Systems. Operating Manual and Programming Reference, Model SR785 Dynamic Signal Analyzer; Stanford Research Systems Inc.: Sunnyvale, CA, USA, 2017. [Google Scholar]
- Stanford Research Systems. MODEL SR830 DSP Lock-In Amplifier; Stanford Research Systems Inc.: Sunnyvale, CA, USA, 2011. [Google Scholar]
- Zabel, S.; Reermann, J.; Fichtner, S.; Kirchhof, C.; Quandt, E.; Wagner, B.; Schmidt, G.; Faupel, F. Multimode delta-E effect magnetic field sensors with adapted electrodes. Appl. Phys. Lett.
**2016**, 108, 222401. [Google Scholar] [CrossRef] - Reermann, J.; Zabel, S.; Kirchhof, C.; Quandt, E.; Faupel, F.; Schmidt, G. Adaptive Readout Schemes for Thin-Film Magnetoelectric Sensors Based on the delta-E Effect. IEEE Sens. J.
**2016**, 16, 4891–4900. [Google Scholar] [CrossRef] - Ludwig, A.; Quandt, E. Optimization of the ΔE-effect in thin films and multilayers by magnetic field annealing. IEEE Trans. Magn.
**2002**, 38, 2829–2831. [Google Scholar] [CrossRef] - Durdaut, P.; Salzer, S.; Reermann, J.; Röbisch, V.; McCord, J.; Meyners, D.; Quandt, E.; Schmidt, G.; Knöchel, R.; Höft, M. Improved Magnetic Frequency Conversion Approach for Magnetoelectric Sensors. IEEE Sens. Lett.
**2017**, 1, 1–4. [Google Scholar] [CrossRef] - Salzer, S.; Durdaut, P.; Röbisch, V.; Meyners, D.; Quandt, E.; Höft, M.; Knöchel, R. Generalized Magnetic Frequency Conversion for Thin-Film Laminate Magnetoelectric Sensors. IEEE Sens. J.
**2017**, 17, 1373–1383. [Google Scholar] [CrossRef] - Durdaut, P.; Penner, V.; Kirchhof, C.; Quandt, E.; Knöchel, R.; Höft, M. Noise of a JFET Charge Amplifier for Piezoelectric Sensors. IEEE Sens. J.
**2017**, 17, 7364–7371. [Google Scholar] [CrossRef] - Lage, E.; Kirchhof, C.; Hrkac, V.; Kienle, L.; Jahns, R.; Knöchel, R.; Quandt, E.; Meyners, D. Exchange biasing of magnetoelectric composites. Nat. Mater
**2012**, 11, 523–529. [Google Scholar] [CrossRef] - Spetzler, B.; Bald, C.; Durdaut, P.; Reermann, J.; Kirchhof, C.; Teplyuk, A.; Meyners, D.; Quandt, E.; Höft, M.; Schmidt, G.; et al. Exchange biased delta-E effect enables the detection of low frequency pT magnetic fields with simultaneous localization. Sci. Rep.
**2021**, 11, 5269. [Google Scholar] [CrossRef] [PubMed] - Durdaut, P.; Salzer, S.; Reermann, J.; Röbisch, V.; Hayes, P.; Piorra, A.; Meyners, D.; Quandt, E.; Schmidt, G.; Knöchel, R.; et al. Thermal-Mechanical Noise in Resonant Thin-Film Magnetoelectric Sensors. IEEE Sens. J.
**2017**, 17, 2338–2348. [Google Scholar] [CrossRef] - Reermann, J.; Elzenheimer, E.; Schmidt, G. Real-Time Biomagnetic Signal Processing for Uncooled Magnetometers in Cardiology. IEEE Sens. J.
**2019**, 19, 4237–4249. [Google Scholar] [CrossRef] - Urs, N.O.; Golubeva, E.; Röbisch, V.; Toxvaerd, S.; Deldar, S.; Knöchel, R.; Höft, M.; Quandt, E.; Meyners, D.; McCord, J. Direct Link between Specific Magnetic Domain Activities and Magnetic Noise in Modulated Magnetoelectric Sensors. Phys. Rev. Appl.
**2020**, 13. [Google Scholar] [CrossRef] - Gussak, I.; Antzelevitch, C.; Wilde, A.A.M.; Friedman, P.A.; Ackerman, M.J. Electrical Diseases of the Heart: Genetics, Mechanisms, Treatment, Prevention, 1st ed.; Springer: London, UK, 2008. [Google Scholar]
- Elzenheimer, E. Analyse Stimulationsevozierter Muskel- und Nervensignale Mithilfe Elektrischer und Magnetischer Sensorik. Ph.D. Thesis, Chair of Digital Signal Processing and System Theory. Kiel University, Kiel, Germany, 2022. [Google Scholar]
- QuSpin. Specification QZFM Gen-3. Available online: https://quspin.com/products-qzfm/ (accessed on 9 December 2021).
- Bertrand, F.; Jager, T.; Boness, A.; Fourcault, W.; Le Gal, G.; Palacios-Laloy, A.; Paulet, J.; Léger, J.M. A 4He vector zero-field optically pumped magnetometer operated in the Earth-field. Rev. Sci. Instrum.
**2021**, 92, 105005. [Google Scholar] [CrossRef] - TDK Corporation. Ultrasensitive Magnetic Sensor-Nivio xMR. Available online: https://product.tdk.com/system/files/dam/doc/content/event/techfro2020/tech20_17.pdf (accessed on 9 December 2021).

**Figure 2.**Input–Output–Amplitude–Relation with labeling of the typical regions (noise region (I), linear region (II) and compression/saturation region (III)). Transition areas are marked in cyan. Furthermore, characteristic quantities, mean value within noise region, Limit-of-Detection (LOD), Limit-of-Quantification (LOQ), 1-dB-compression-point, 3-dB-compression-point and maximum value are marked in different colors.

**Figure 3.**Different methods for LOD computation for the same signal. The simulated sample by sample computation (

**a**) of the standard deviation and the mean value yields the same results as the stochastic parameters of the applied random process (${\mu}_{\mathrm{n}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0$, ${\sigma}_{\mathrm{n}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}\mathrm{pT}$). The averaging window applied in (

**b**) results in a reduced spread and therefore an SNR gain at the cost of a reduction in bandwidth (${\mu}_{\mathrm{n}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}\mathrm{pT}$, ${\sigma}_{\mathrm{n}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}70\phantom{\rule{0.166667em}{0ex}}\mathrm{fT}$).

**Figure 5.**Amplitude response (asymmetrical passband) with predominant bandpass characteristic including metrics labeling. For a response with lowpass characteristic, only the right abscissa axis is required.

**Figure 6.**Frequency response measurement of a sensor system in a magnetically shielded environment by lock-in-amplifier, current source and cylinder coil.

**Figure 7.**MCG prototype signal based on SQUID-MCG-Data. (

**a**) SQUID Measurement—time domain. (

**b**) Prototype MCG—time domain. (

**c**) SQUID Measurement—power spectral density. (

**d**) Prototype MCG—power spectral density.

**Figure 8.**Summation of noise amplitude densities. Two exemplary noise amplitude densities (constant and arbitrary shape) are provided (

**a**). Summation is performed from DC up to an increasing upper cutoff frequency to obtain the corresponding RMS value (

**b**) for both densities. Assuming a sensor −3 dB cutoff frequency of 500 Hz, the colored areas under curve (

**a**) yield RMS amplitudes of 25 pT and 23 pT, which will vary if a different upper frequency limit is applied (

**b**).

**Figure 9.**MCG-Prototype signals (cf. Figure 7b) superimposed with white noise (left column) and high-pass filtered noise (right column). The second row shows the low-pass filtered sum of the signal and noise, while the third row shows application specific capacity and the power spectral densities of the signal, noise and weighted noise (cf. Equation (39)). (

**a**) MCG Signal plus white noise—time signal. (

**b**) MCG Signal plus HP noise—time signal. (

**c**) MCG Signal plus white noise—time signal, filtered. (

**d**) MCG Signal plus HP noise—time signal, filtered. (

**e**) MCG Signal and white noise—PSD. (

**f**) MCG Signal and HP noise—PSD.

**Figure 10.**Sensors systems used in this study: In (

**a**) an exchange bias magnetoelectric sensor is shown with integrated readout electronics. In (

**b**) an electrically modulated ME sensor is presented with integrated preamplifier and external shielded battery supply (gray box; $\pm 9$ V). (

**a**) Exchange bias magnetoelectric sensor (cantilever) with integrated readout. (

**b**) Electrically modulated ME sensor with integrated preamplifier and external battery.

**Figure 11.**Measurements for the evaluation of the exchange bias magnetoelectric sensor system. In (

**a**) the amplitude response and in (

**c**) the phase response of the sensor system near the mechanical resonance are depicted. The noise measurement equalized with amplitude response is shown in (

**b**). The Input–Output–Amplitude–Relation of the sensor, with an external magnetic field at f = ${f}_{\mathrm{res}}$, is depicted in (

**d**).

**Figure 12.**Measurements for the evaluation of the electrically modulated ME sensor. In (

**a**) the amplitude response and in (

**c**) the phase response of the sensor system are depicted. The noise measurement equalized with amplitude response is shown in (

**b**). The Input–Output–Amplitude–Relation of the sensor, with an external magnetic field at f = 10 Hz, is depicted in (

**d**).

**Figure 13.**Application-specific noise requirements for MCG (Heart-Rate-Variability analyses by detection of the R-Peak; Disease localization by solving the inverse problem) specified by the amplitude density. Sensor positioned directly over the chest [55].

**Figure 14.**Exemplary prototype design for a clear presentation of application-related metrics [55].

Metrics | Exchange Bias ME Sensor [13] | Electrically Modulated ME Sensor [10,16] |
---|---|---|

Operation | Room | Room |

Temperature | temperature | temperature |

Inherent | ≈4 pT/$\sqrt{\mathrm{Hz}}$ | ≈70 pT/$\sqrt{\mathrm{Hz}}$ |

Noise | at 7.684 kHz | at 10 Hz |

Bandwidth | ≈12.5 Hz (−6 dB) | unknown |

Sensitivity | ≈98 kV/T | ≈40 kV/T |

Availability | under development | under development |

Parameters | Exchange Bias ME Sensor | Electrically Modulated ME Sensor |
---|---|---|

Amplitude Response | ||

${f}_{\mathrm{res}}$ | 7684 Hz | |

${f}_{-3\mathrm{dB}}$ | ${f}_{-3\mathrm{dB},1}$ = 7680 Hz (low)${f}_{-3\mathrm{dB},2}$ = 7689 Hz (high) | 11.3 kHz |

${f}_{-6\mathrm{dB}}$ | ${f}_{-6\mathrm{dB},1}$ = 7677 Hz (low) ${f}_{-6\mathrm{dB},2}$ = 7692 Hz (high) | 15 kHz |

Q | 854 | |

$\left|{\mathrm{Slope}}_{-3\mathrm{dB}/-6\mathrm{dB}}\right|$ | 0.94 dB/Hz (low) 0.92 dB/Hz (high) | 0.805 dB/kHz |

${B}_{3\mathrm{dB}}$ | 9 Hz (bandpass) | 11.3 kHz (lowpass) |

${B}_{6\mathrm{dB}}$ | 15 Hz (bandpass) | 15 kHz (lowpass) |

Sensitivity | ||

${\u03f5}_{\mathrm{sys}}$ | 63 kV/T at ${f}_{\mathrm{res}}$ | 5.76 kV/T at 10 Hz |

Noise | ||

$\sqrt{{\widehat{S}}_{\mathrm{B}}\left(f\right)}$ | 6 pT/$\sqrt{\mathrm{Hz}}$ at ${f}_{\mathrm{res}}$ | 66 pT/$\sqrt{\mathrm{Hz}}$ at 10 Hz |

${B}_{\mathrm{n}}^{\mathrm{rms}}$ | 20 pT (${f}_{-3\mathrm{dB},\mathrm{low}}$ to ${f}_{-3\mathrm{dB},\mathrm{high}}$) | 11.7 nT (1 Hz to ${f}_{-3\mathrm{dB}}$) |

Input-Output-Relation | ||

${\overline{b}}_{\mathrm{out}}$ | 11 pT | 55 pT |

$LOD$ | 22 pT | 102 pT |

$LOQ$ | 42 pT | 210 pT |

${b}_{1\mathrm{dB}}$ | 6 µT | 18 µT |

${b}_{3\mathrm{dB}}$ | 18 µT | 23 µT |

${b}_{\mathrm{max}}$ | 27 µT | |

$DR$ | 103 dB | 98 dB |

Application (MCG) Specific Quantities | ||

$SNR$ | −90 dB | −11 dB |

$ASC$ | 9.8 $\times {10}^{-7}$ dB Hz | 23 dB Hz |

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**MDPI and ACS Style**

Elzenheimer, E.; Bald, C.; Engelhardt, E.; Hoffmann, J.; Hayes, P.; Arbustini, J.; Bahr, A.; Quandt, E.; Höft, M.; Schmidt, G.
Quantitative Evaluation for Magnetoelectric Sensor Systems in Biomagnetic Diagnostics. *Sensors* **2022**, *22*, 1018.
https://doi.org/10.3390/s22031018

**AMA Style**

Elzenheimer E, Bald C, Engelhardt E, Hoffmann J, Hayes P, Arbustini J, Bahr A, Quandt E, Höft M, Schmidt G.
Quantitative Evaluation for Magnetoelectric Sensor Systems in Biomagnetic Diagnostics. *Sensors*. 2022; 22(3):1018.
https://doi.org/10.3390/s22031018

**Chicago/Turabian Style**

Elzenheimer, Eric, Christin Bald, Erik Engelhardt, Johannes Hoffmann, Patrick Hayes, Johan Arbustini, Andreas Bahr, Eckhard Quandt, Michael Höft, and Gerhard Schmidt.
2022. "Quantitative Evaluation for Magnetoelectric Sensor Systems in Biomagnetic Diagnostics" *Sensors* 22, no. 3: 1018.
https://doi.org/10.3390/s22031018