Magnetic background noise cancellation in real-world environmentsP. J. M. Wöltgens and R. H. Koch Citation: Review of Scientific Instruments 71, 1529 (2000); doi: 10.1063/1.1150490 View online: http://dx.doi.org/10.1063/1.1150490 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/71/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Unambiguous position and orientation tracking using a rotating magnet J. Appl. Phys. 114, 114502 (2013); 10.1063/1.4821506 Evaluation of cancellation coil for precision magnetic measurements with strong prepolarization field insideshielded environment J. Appl. Phys. 111, 083916 (2012); 10.1063/1.4706564 Magnetorelaxometry of magnetic nanoparticles in magnetically unshielded environment utilizing a differentialfluxgate arrangement Rev. Sci. Instrum. 76, 106102 (2005); 10.1063/1.2069776 Noise properties of magnetic and nonmagnetic tunnel junctions J. Appl. Phys. 93, 7020 (2003); 10.1063/1.1558655 Low-noise flux-gate magnetic-field sensors using ring- and rod-core geometries Appl. Phys. Lett. 78, 1897 (2001); 10.1063/1.1358852

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17

Magnetic background noise cancellation in real-world environmentsP. J. M. Woltgensa) and R. H. KochIBM T. J. Watson Research Center, Yorktown Heights, New York 10598

~Received 23 February 1999; accepted for publication 5 November 1999!

Measurements of the magnetic field noise and the spatial correlation of the noise were made in atypical laboratory environment using two three-axis fluxgate magnetometers. The magnitude of themagnetic field noise was found to be approximately 100 pT/AHz at 10 Hz with a correlation of 90%at a separation of 1 m between the two sensors. The correlation was significantly reduced from noiseinduced eddy currents near large metallic surfaces. ©2000 American Institute of Physics.@S0034-6748~00!02103-1#

I. INTRODUCTION

Measurements of small magnetic fields at low frequen-cies ~,100 Hz! have promising applications in the medicalfield, such as magnetoencephalography and magneto-cardiography.1–4 So far, magnetically shielded rooms havebeen required for most such biomagnetic measurements inorder to shield the magnetic background noise of a typicallaboratory environment from the measurement. The highcosts of these shielded rooms have limited the use of thesesystems to a few laboratories. Biomagnetic applicationswould potentially gain in popularity if the necessity of mag-netic shielding could be reduced or completely eliminated. Inorder to achieve this, the magnetic background noise needsto be cancelledinstead ofshieldedfrom the measurement.Thus, noise cancellation is an important issue for biomag-netism. In this article a systematic investigation will be pre-sented on magnetic background noise and its cancellation ina laboratory environment.

Background noise cancellation is based on the use of asignal sensor and one or more reference sensors. The signalsensor is positioned in such a way that it measures as muchas possible of the signal of interest, superimposed on theunavoidable environmental background noise. The referencesensors are arranged such that they measure as little as pos-sible of the signal of interest while measuring a backgroundresembling as closely as possible the background noise at thesignal sensor. The background noise in the signal channel isthen estimated on the basis of the reference signalsBref . Theestimated signal of interestBcorr is then obtained by remov-ing the estimated background noiseBback from the measuredsignal Bsig. This subtraction or cancellation can be per-formed actively or passively: actively, by cancelling thephysical quantityBback at the input of the signal sensor5 andpassively, by subtractingBback after the output of the signalsensor.6,7

In a real-world environment there are many differentbackground sources at different locations and therefore themagnetic background at each point in space consists of adifferent mix of fields from these background sources. Thisis a basic dilemma in noise cancellation: how to locate the

reference sensor with respect to the signal sensor. On onehand, the reference sensor should be as close as possible tothe signal sensor in order to measure the same background asthe signal sensor; on the other hand, the reference sensorshould be far away from the signal sensor, so that it measuresas little as possible of the signal of interest.

This is the major question this article tries to answer: towhat extent is the background constant as a function of po-sition, i.e., how far away from each other can the signal andreference sensors be in order to obtain a certain backgroundrejection?

In this article, first a comparison will be made of a fewmethods of cancellation. The coherence is identified as thequantity ruling the degree of cancellation, and based on thisthe feasible amount of cancellation is investigated as a func-tion of the distance between a signal sensor and its reference.Finally, the influence of close-by conductors on magneticbackground cancellation will be discussed.

II. CANCELLATION METHODS

The methods which were investigated and will be dis-cussed below are~in order of increasing complexity!: ~i! sub-traction,~ii ! balancing in the time domain, and~iii ! balancingin the frequency domain.

A. Subtraction

The simplest way of cancellation is subtraction of thereference channelBref from the signal channelBsig, as isdone in the measurement of a gradient, i.e., using the refer-enceBref as the estimate for the noise in the signal channel

Bback~ t !5Bref~ t !. ~1!

B. Linear regression fit in the time domain

A more sophisticated way to estimate the noise in thesignal channel is to make a linear regression fit in the timedomain of the signal channelBsig with respect to the refer-ence channelsBref,j . This will result in an estimate for thenoise in the signal channelBback:

Bback~ t !5(j

ajBref,j~ t !, ~2!

where the coefficientsaj are the~time independent! fit coef-ficients of the regression fit.a!Electronic mail: woelt@us.ibm.com

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 71, NUMBER 3 MARCH 2000

15290034-6748/2000/71(3)/1529/5/$17.00 © 2000 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17

The values of the coefficients are chosen such that thestandard deviation ofBsig(t) with respect toBback(t) is mini-mized, by setting

]

]ai(

t@Bsig~ t !2Bback~ t !#250 ~3!

for all i. In matrix notation this can be expressed as

b5Ga, ~4!

whereb is a vector with components

bi5(t

@Bsig~ t !Bref,i~ t !#, ~5!

G is the symmetric correlation matrix with components

G i j 5(t

@Bref,i~ t !Bref,j~ t !#, ~6!

and a is a vector the componentsaj of which are the fitparameters.

The multiple-regression equation is then solved by com-puting the inverse matrixG21 of G and evaluating

a5G21b. ~7!

C. Linear regression fit in the frequency domain

A more refined way of cancellation is to do a linearregression fit in the frequency domain.~See Ref. 8.! Such afit results in an estimate for the noise in the signal channelBback( f ) according to

Bback~ f !5(j

H j~ f !Bref,j~ f !, ~8!

where the transfer functionsH j ( f ) are the optimum fre-quency response functions, the results of the linear regres-sion fit. These functionsH j ( f ) are chosen such that the spec-tral density ofBback( f )2Bsig( f )5Bc( f ) is minimized, i.e.,

]Scc~ f !

]Hi~ f !50 and

]Scc~ f !

]Hi* ~ f !50 ~9!

for all i, with Scc( f ) the power-spectral density ofBc andHi* ( f ) the complex conjugate ofHi( f ). Again, this minimi-zation can be expressed in terms of a matrix equation

s5Gfh, ~10!

wheres is a vector with components

sj5Sj ,sig~ f !. ~11!

Here, Sj ,sig( f ) is the cross-spectral density ofBref,j (t) andBsig(t). Further,Gf is a Hermitian matrix with components

G f ,i j 5Si j ~ f ! ~12!

with Si j ( f ) the cross-spectral density ofBref,i(t) andBref,j (t). Finally, h is a vector with components

hj5H j~ f !. ~13!

Analogously to the time-balanced case, the solution forh canbe found by inversion ofGf .

III. EXPERIMENTAL SETUP

To investigate the magnetic background noise in a labo-ratory and its cancellation, we measured the noise simulta-neously at two locations in our laboratory, using a setup asdepicted in Fig. 1. For magnetic sensors we used two Bill-ingsley TFM100-RLN three-axis fluxgate magnetometerswith a noise level of 10– 15 pT/AHz in the frequency rangeof 0.1–100 Hz. The sensors were placed in our lab on ahorizontal plate of dimensions 48 in.348 in. ~1.22 m31.22m!. One sensor~sensor 1! remained in a fixed position, whilethe position of the other sensor~sensor 2! was varied alongtwo perpendicular directions parallel to the edges of the platedesignated thex axis andy axis. Both sensors were located 2in. ~0.051 m! above the platform surface. The sensors wereoriented parallel to each other to within less than 0.03 rad,and thex, y, andz channels of each three-axis fluxgate wereorthogonal to better than 0.017 rad. We varied the height ofthe plate above the floor and the material of the plate, com-paring nonconductive~wooden! plates and conductive plates@1/8 in. ~3.18 mm! thick, made of aluminum and nonstainlesssteel#. Further, we moved the whole setup to an adjacent labin order to test reproducibility of our results. Thex, y, andzsignals of both sensors were filtered via 120 Hz low-passfilters with a roll-off of 224 dB/octave. The filter outputswere subsequently digitized and processed on a personalcomputer. The time traces were taken during 14 s with asampling rate of 600 Hz. For spectral analysis we used theaverage of the spectral estimates of 15 half overlapping, 1.7s long subdivisions of the main time trace.

IV. MAGNETIC BACKGROUND

We tested the previously described three ways of cancel-lation using our setup with a large separation~0.60 m! in thex direction between the two fluxgate sensors on a woodenplate. The results are presented in Fig. 2.

The background spectrum in Fig. 2 is typical for a labo-ratory: at frequencies below 1 Hz the power spectral densityhas a 1/f 2 frequency dependence, which is associated withrandom switching noise with a characteristic frequency be-low our frequency window. Between 1 and 60 Hz the spec-

FIG. 1. The setup for coherence/cancellation measurements in a laboratoryenvironment.

1530 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 P. J. M. Woltgens and R. H. Koch

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17

trum is roughly flat, with an average level of about100 pT/AHz. This white noise below 60 Hz has been ob-served by other groups1,4 in laboratory environments, but itsorigin is as yet unknown. At 60 Hz, the magnetic field fromthe power line has a level of 60 000 pT/AHz or 46 000 pTrms.As a reference, Fig. 2 also shows a magnetic backgroundspectrum taken outside the laboratory building using a super-conducting quantum interference device~SQUID! magneto-meter. A comparison with the noise spectrum taken in thelaboratory shows that the white magnetic laboratory noisebelow 60 Hz is associated with the laboratory building andnot with geomagnetic noise.

V. CANCELLATION RESULTS

Figure 2 further shows the cancellation spectra resultingfrom the cancellation methods discussed in the previous sub-section. Taking the gradient reduces the overall backgroundnoise by a factor of four and the 60 Hz line peak by a factorof 7.

At the expense of a small increase in the noise at fre-quencies below 5 Hz, time balancing reduces the 60 Hz peakby a factor of 22. The balancing ofB1x with respect not onlyto B2x , but also to the orthogonal componentsB2y andB2z

eliminates noise due to a misalignment of the sensors withrespect to each other. Apparently, the noise decrease at 60Hz in the balanced spectrum outweighs the noise increase atlower frequencies.

Finally, B1x was balanced in the frequency domain withrespect toB2x , B2y , and B2z . This form of cancellationprovided the best background noise rejection: at frequenciesbelow 60 Hz a reduction of about 143 and at 60 Hz a re-duction of 3003. For frequencies other than 60 Hz, the re-sulting cancelled noise spectrum is very close to the limit set

by the fluxgate noise of 10 pT/AHz. The better performanceof frequency balancing as compared to the previously de-scribed methods can be understood if one realizes that in areal-world environment multiple magnetic source arepresent, emitting magnetic fields at different frequencies.Thus, at different frequencies the outputs of signal and ref-erence sensors may very well reflect different combinationsof magnetic sources and therefore display different correla-tions.

VI. LIMITS ON CANCELLATION

In the real world, multiple magnetic sources are present,limiting the feasible amount of cancellation between twosensors to finite values. We investigated how these limitsdepend on the distance between the signal sensor and itsreference sensor, using the previously described setup withmetallic and nonmetallic plates as measurement platforms.The relevant parameter to characterize the maximum noisereduction is the coherencegsig,ref

2 of a signal Bsig and itsreferenceBref , and is defined as follows:

gsig,ref2 ~ f !5

uSsig,ref~ f !u2

Ssig~ f !Sref~ f !, ~14!

with Ssig,ref( f ) the cross-spectral density between the refer-ence and the signal sensor. Here,Ssig( f ) andSref( f ) are thepower-spectral densities of the signal sensor and the refer-ence sensor, respectively. The coherence is thus a normalizedcross-spectral density between the two signals, having realvalues between 0 and 1. The power-spectral densitySuncorrofthe part ofBsig uncorrelated toBref is then given by

Suncorr5@12gsig,ref2 ~ f !]Ssig. ~15!

The averaged coherence between 0 and 50 Hz as a func-tion of the sensor–sensor distance is plotted in part~b! ofFig. 3. Qualitatively similar behavior was observed for trans-lations of sensor 2 with respect to sensor 1 in they directionand for the coherence betweeny andz channels of the sen-sors. The exact dropoff of the coherence with distance de-pends on the circumstances, however, the qualitative behav-ior of the dropoff remains the same. The upper limit of thecoherence, at small sensor–sensor distances, is 0.99: this

FIG. 2. Magnetic background spectrum and cancellation spectra of thisspectrum in an unshielded laboratory environment. Fluxgate sensors 1 and 2are placed on a wooden platform, separated by a distance of 24.0 in.~0.61m! in the x direction. The thick solid line is the magnetic background spec-trum of B1x . The dotted line is the gradient signalB1x–B2x , the dashed lineis the time-balanced spectrum ofB1x with respect toB2x , B2y , andB2z .The thin solid line is the frequency balanced spectrum ofB1x with respect toB2x , B2y , andB2z . The solid, dashed, and dotted horizontal lines indicatethe levels of the 60 Hz peaks in the corresponding spectra. The thinnestsolid line denotes the magnetic background spectrum outside the laboratorybuilding, as measured with a SQUID magnetometer.

FIG. 3. ~a! The way the averaged coherence between 0 and 50 Hz is deter-mined.~b! The averaged coherenceg1x2x,av

2 betweenB1x andB2x as a func-tion of the distance between sensor 2 and sensor 1 in thex direction. Thelabels indicate the different plate materials for which the results were ob-tained.

1531Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Magnetic background noise cancellation

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17

limit is set by the sensor noise power of the fluxgate sensorswhich is about 1/100 of the average background noise powerbetween 0 and 50 Hz. In this two channel analysis, the mis-alignment between the two sensors can decrease the maxi-mum value of the coherence if they are not aligned to within10% of each other. The alignment errors in these measure-ments were less than 10% in the relative alignment. If allthree axis were used as references to determine the maxi-mum coherence, any alignment errors would not decrease themaximum value.

In these measurements, as the spacing of the two sensorswas increased the coherence decreased. The magnitude ofthe noise powerSuncorrvaried as the spacing between the twosensorsDx as Suncorr}Dx1.660.2. There is no simple way tounderstand this particular power law from basic consider-ations of magnetic sources yet, the general falloff with dis-tance can be understood. The total field noise impinging onboth sensors, the main and the reference sensors, consists ofcontributions from a large number,N, of independentsources. At any one frequency, the complex amplitude ratioof the field on the main sensor and the reference sensor fromeach of theN sources,Rj ( f ) ~where j 51,2,3,...,N) will de-pend the detailed nature of the source, including its positionrelative to the main and reference sensors. If there is onlyone noise source, i.e.,N51, than Eq.~8! and a single refer-ence sensor, as used in these measurements, would be able tocorrect completely for the noise source by adjusting the cor-relation coefficientH1( f ) to be equal toR1( f ). However,with two or more independent noise sources, the choice of asingle value ofH1( f ) can only be a compromise. Only in thelimit where Dx→0 will all the Rj ( f ) approach the samevalue, one, and perfect cancellation be achieved.

Further, a loss of coherence can be observed in the pres-ence of a conductive plate instead of a nonconductive one.This loss occurs in the field directions parallel to the plate,i.e., thex and they directions, but is absent in thez directionperpendicular to the plate. Using Eq.~15!, we can computethe maximum achievable noise amplitude reduction from thecoherence betweenBsig andBref :

maximum reduction51/A12g1x2x,av2 . ~16!

The resulting maximum achievable background noise reduc-tion computed from the data presented in Fig. 3 is plotted inFig. 4.

The results indicate that in a two-sensor configurationthe maximum achievable noise reduction is better than a fac-tor of 10 for separations of less than 0.20 m for frequenciesbetween 0 and 50 Hz. To achieve higher noise rejection fac-tors at these distances, more reference sensors with lowernoise levels are necessary. Some of these sensors should beorthogonal to the direction of interest such as to compensatefor the misalignment between the sensors.

VII. DEGRADATION OF COHERENCE NEAR METAL

The observed loss of coherence in the magnetic fieldparallel to a conductive plate can be explained by taking intoaccount electromagnetic induction.

To get a first-order estimate of these inductive effects weperform the following analysis: consider a circular conduc-tive plate radiusR and thicknessu in an homogeneous ap-plied magnetic fieldBa . The direction perpendicular to theplate is designated thez axis, thex andy axes are orthogonaldirections in the plane of the plate.Ba has az componentBa,z5B0eivt, wherev is the angular frequency. We considerthe limit in which the induced magnetic fieldBind is small incomparison to the applied alternating current fieldBa . Themagnitude of the induced current densityJind at a distancerfrom the center of the plate is then given by

Jind~r !}ivBa,zr

2r, ~17!

with r the resistivity of the plate.At a distanced!R from the surface of the plate such

local eddy currentsJind resemble infinitely long line currentsinducing a magnetic field which drops off as 1/d. UsingAmpere’s law, it can be shown that thez component of the

FIG. 5. Illustration of the changes in direction of the in-plane component ofthe localBind on the topside of the plate as a function of the position on thex axis. The circles indicate the local eddy current paths, and the vectorsBind,1 , Bind,2 , andBind,3 indicate the magnetic field induced on the topside ofthe plate by these local eddy currents at positions 1, 2, and 3. Note theopposite values ofx component ofBind in positions 1 and 3, whereas theycomponent ofBind changes only by a factor of 21/2 from position 2 topositions 1 and 3.

FIG. 4. The averaged reduction computed from the average coherenceg1x2x,av

2 of Fig. 3 between thex channels of the two fluxgate sensors as afunction of the separation of sensors 1 and 2 in thex direction. The labelsindicate the different plate materials for which the results were obtained.

1532 Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 P. J. M. Woltgens and R. H. Koch

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17

induced field in the center of the plate should scale as: weexpect the induced magnetic fieldsBind to scale with theinduced current density, thus giving

Bind~r !}ivBa,zr

2r. ~18!

In the rectangular plates used in this experiment, theseeddy currents will in first approximation run in concentriccircles around the center of the plate as depicted in Fig. 5.Moving along thex axis, the local eddy-current directionrotates, and similarly, the in-plane component of the locallyinduced magnetic field is rotated. Thus, at different positionsalong thex axis a sensor measures, in addition to the appliedfield Ba , a differently directed induced fieldBind . The big-gest change occurs in thex component ofBind , which re-verses in direction from position 1 to position 3. The totalfield at sensor 2 can then, in first approximation, be ex-pressed as:B2x5B1x1(Bind,2x2Bind,1x), which means thatthe coherence loss scales with the magnitude of the localBind .

To corroborate this picture, we performed a frequency-dependent multiple regression analysis ofB1x vs B2x , B2y ,and B2z and computed transfer functionsH2x , H2y , andH2z , with sensor 1 located at the origin and sensor 2 posi-tioned somewhere along thex axis. For the case of a non-conductive plate, we trivially find thatH2x51, and H2y

5H2z50. With a conductive plate, however, the transferfunctionH2z increases linearly with frequency, while havinga frequency-independent phase. Further, a comparison ofH2z for different plate materials shows that the magnitude ofH2z at a certain frequency drops with increasing resistance ofthe platform, as is shown in Fig. 6. More quantitatively, atany given frequencyH2z is about 43larger for the aluminumplate than for the steel plate, this factor is the sameas the conductivity ratio for aluminum and steel (rAl

52.6731028 V m, rFe510.0031028 V m).In general, metallic objects in applied ac magnetic fields

will generate induced fields, which effectively mix orthogo-nal components into each component of the applied magneticfield. As this mix may vary significantly over space the co-herence between the magnetic field at two points in space isdegraded.

1J. Vrbaet al., in Advances in Biomagnetisrn Research: Biomag96, editedby C. Aine, Y. Okada, G. Stroink, S. Swithenby, and C. Wood~Springer,New York, in press!.

2J. Vrba, inSQUID Sensors: Fundamentals, Fabrication, and Applications,edited by H. Weinstock~Kluwer Academic, Dordrecht, 1996!, p. 117.

3J. Vrba and J. McKay, Appl. Supercond.5, 431 ~1997!.4H. J. M. ter Brakeet al., Inst. Phys. Conf. Ser.148, 1503~1995!.5R. H. Koch, J. R. Rozen, J. Z. Sun, and W. J. Gallagher, Appl. Phys. Lett.63, 403 ~1993!.

6Y. Tavrin, Y. Zhang, M. Muck, A. I. Braginski, and C. Heiden, IEEETrans. Appl. Supercond.3, 2477~1993!.

7J. Borgmann, P. David, G. Ockenfuß, R. Otto, J. Schuber, W. Zander, andA. I. Braginski, Rev. Sci. Instrum.68, 2730~1997!.

8G. M. Jenkins and D. G. Watts,Spectral Analysis and its Applications~Holden-Day, San Francisco, 1968!, p. 485ff.

FIG. 6. The frequency dependence of the transfer functionH2z of B2z to B1x

as obtained from the frequency-dependent balancing ofB1x with respect toB2x , B2y , andB2z . Sensor 2 is positioned on thex axis 0.305 m away fromsensor 1. Both sensors are located 50.8 mm above the surface of the plate.The thin solid line indicatesH2z for a nonconductive plate, the thin dashedline for a steel plate, and the thick solid line indicatesH2z for an aluminumplate.

1533Rev. Sci. Instrum., Vol. 71, No. 3, March 2000 Magnetic background noise cancellation

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.105.215.146 On: Mon, 22 Dec 2014 05:18:17