# Quantitative Evaluation for Magnetoelectric Sensor Systems in Biomagnetic Diagnostics

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## Abstract

**:**

## 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$$\begin{array}{cc}\hfill {\delta}_{\mathrm{p}}=20\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{log}_{10}\left(\frac{\overline{a}+{\delta}_{\mathrm{p},\mathrm{max}}}{\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}& \mathrm{with}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\delta}_{\mathrm{p},\mathrm{max}}=\mathrm{max}\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\right\}\hfill \\ \hfill \mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\delta}_{\mathrm{p},\mathrm{min}}=\mathrm{min}\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}& \mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mathrm{p}1}\le {\mathsf{\Omega}}_{\mu}\le {\mathsf{\Omega}}_{\mathrm{p}2}\hfill \end{array}$$
- Passband Edge Frequencies$${\mathsf{\Omega}}_{\mathrm{p}1}=arg\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}<{\mathsf{\Omega}}_{\mathrm{z}}$$$${\mathsf{\Omega}}_{\mathrm{p}2}=arg\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\overline{a}-{\delta}_{\mathrm{p},\mathrm{min}}\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}>{\mathsf{\Omega}}_{\mathrm{z}}$$
- Stopband Edge Frequencies$${\mathsf{\Omega}}_{\mathrm{s}1}=arg\left\{\left|H\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\mathrm{max}\left(\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}<{\mathsf{\Omega}}_{\mathrm{s}1}\right)\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}\ll {\mathsf{\Omega}}_{\mathrm{p}1}$$$${\mathsf{\Omega}}_{\mathrm{s}2}=arg\left\{\left|H\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\mathrm{max}\left(\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}>{\mathsf{\Omega}}_{\mathrm{s}2}\right)\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mu}\gg {\mathsf{\Omega}}_{\mathrm{p}2}$$
- 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$${\mathsf{\Omega}}_{-3\mathrm{dB},1}=arg\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\frac{1}{\sqrt{2}}\phantom{\rule{0.166667em}{0ex}}\overline{a}\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mathrm{s}1}\le {\mathsf{\Omega}}_{\mu}\le {\mathsf{\Omega}}_{\mathrm{p}1}$$$${\mathsf{\Omega}}_{-3\mathrm{dB},2}=arg\left\{\left|\widehat{H}\left({e}^{j{\mathsf{\Omega}}_{\mu}}\right)\right|\stackrel{!}{=}\frac{1}{\sqrt{2}}\phantom{\rule{0.166667em}{0ex}}\overline{a}\right\}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathsf{\Omega}}_{\mathrm{p}2}\le {\mathsf{\Omega}}_{\mu}\le {\mathsf{\Omega}}_{\mathrm{s}2}$$$$w={\mathsf{\Omega}}_{-3\mathrm{dB},2}-{\mathsf{\Omega}}_{-3\mathrm{dB},1}.$$

#### 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 |

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**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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## Share and Cite

**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