Comments on “Gradient-Induced Acoustic and MagneticField Fluctuations in a 4T Whole-Body MR Imager”

Alan Barnett*

The results published in the article Gradient-Induced Acous-tic and Magnetic Field Fluctuations in a 4T Whole-Body MRImager by Wu et al. (Magn Reson Med 2000;44:532–536) ap-pear to be consistent with the response of a time-stationarylinear system. Since a linear system is more simply describedthan a nonlinear system, the authors are urged to reanalyzetheir data to test the linear-system hypothesis. Magn ResonMed 46:207, 2001. Published 2001 Wiley-Liss, Inc.†

Key words: MRI; acoustic noise; magnetic field fluctuations;time stationary linear system

In a recent article, Wu et al. (1) measured the acoustic andmagnetic noise generated by the gradients in a Siemens 4Twhole-body MRI machine. Their results appear to be con-sistent with the response of a time-stationary linear sys-tem. This observation can simplify the characterization ofthe system, since the response y(t) of a time-stationarylinear system to an arbitrary excitation x(t) is the convo-lution of the excitation x(t) with the impulse responsefunction h(t) of the system (2):

y~t! 5 x~t!*h~t!. [1]

Taking the Fourier transform of Eq. [1] and squaring it yields:

P~v! 5 uX~v!H~v!u2, [2]

where P(v) is the power spectrum, X(v) is the Fouriertransform of x(t), and the frequency response H(v) is theFourier transform of h(t).

Wu et al. measured the sound pressure level and mag-netic field variations present in a 4T whole body scannerwhen they applied two different waveforms to the gradientcoils: a “single edge” pulse consisting of a slow rampfollowed by a rapid decay, and a trapezoidal pulse. In addi-tion, they varied the duration of the plateau of the trapezoidalpulse. Among other things, they discovered that:

1) the power spectrum of the acoustic noise generatedby the x- and y-gradients has peaks at 432 Hz and950 Hz for all pulse shapes, and

2) the amplitude of the peaks depends on the pulse shapeand the duration of the plateau of the trapezoidal pulse.

From Eq. [2] we see that a time-stationary linear systemwould exhibit these characteristics if:

1) the frequency response H(v) has resonances at432 Hz and 950 Hz, and

2) the amplitude of each peak is proportional the squareof the Fourier transform of the excitation functionevaluated at the frequency of the peaks.

That the system is linear is strongly suggested by Fig.3 of the article, which shows that the height of the 950 Hzpeak in the acoustic and magnetic signals is a periodicfunction of plateau duration for the trapezoidal excitation.There are about 7.5 cycles in 8 ms, corresponding to aperiod of about 1.06 ms.

The trapezoidal excitation, shown in Fig. 1b of the article is:

x~t! 5 5 I0

T2

1 t 1 t

t2 ST

21 tD , t , 2

T2

I0 2T2

, t ,T2

I0

T2

1 t 2 t

t

T2

, t ,T2

1 t6 , [3]

where x(t) is the current waveform.Taking the Fourier transform of Eq. [3] plugging into Eq.

[2], the power spectrum can be written as:

P~v; t, T! 5 2I01sinSvt2 D

tv2 22

~1 2 cos~vT 1 vt!!~X~v!!2. [4]

Viewed as a function of plateau duration T, P oscillateswith period 2p/v 5 1/f. For f 5 950 Hz, the period is1.05 ms, in excellent agreement with the data shown inFig. 3 of Wu et al.

I suggest that Wu et al. reanalyze their data to test thehypothesis that the noise generation can be modeled as atime-stationary linear system.

Alan BarnettClinical Brain Disorders BranchNational Institute of Mental HealthNational Institutes of HealthBethesda, Maryland

REFERENCES

1. Wu Y, Chronik BA, Bowen C, Mechefske CK, Rutt BK. Gradient-inducedacoustic and magnetic field fluctuations in a 4T whole-body MR imager.Magn Reson Med 2000;44:532–536.

2. Oppenheim AV, Willsky AS, Young IT. Signals and systems. EnglewoodCliffs, NJ: Prentice Hall; 1983.

*Correspondence to: Alan Barnett, Ph.D., Building 10 Rm. 4D20, NationalInstitutes of Health, Bethesda, MD 20892.A response to this letter will be published in a forthcoming issue.Received 9 January 2001; accepted 28 March 2001.

207Published 2001 Wiley-Liss, Inc. † This article is a US Governmentwork and, as such, is in the public domain in the United States of America.

Magnetic Resonance in Medicine 46:207 (2001)

LETTER TOTHE EDITOR