Precompensated Excitation Waveforms to

1564 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 51, no. 11, november 2004

Precompensated Excitation Waveforms toSuppress Harmonic Generation in MEMS

Electrostatic TransducersShiwei Zhou, Paul Reynolds, Member, IEEE, and John Hossack, Member, IEEE

Abstract—Microelectromechanical systems (MEMS)electrostatic-based transducers inherently produce harmon-ics as the electrostatic force generated in the transmit modeis approximately proportional to the square of the appliedvoltage signal. This characteristic precludes them from be-ing effectively used for harmonic imaging (either with orwithout the addition of microbubble-based contrast agents).The harmonic signal that is nonlinearly generated by tissue(or contrast agent) cannot be distinguished from the in-herent transmitted harmonic signal. We investigated twoprecompensation methods to cancel this inherent harmonicgeneration in electrostatic transducers. A combination offinite element analysis (FEA) and experimental results arepresented. The first approach relies on a calculation, ormeasurement, of the transducer’s linear transfer function,which is valid for small signal levels. Using this transferfunction and a measurement of the undesired harmonic sig-nal, a predistorted transmit signal was calculated to cancelthe harmonic inherently generated by the transducer. Dueto the lack of perfect linearity, the approach does not workcompletely in a single iteration. However, with subsequentiterations, the problem becomes more linear and convergestoward a very satisfactory result (a 18.6 dB harmonic re-duction was achieved in FEA simulations and a 20.7 dBreduction was measured in a prototype experiment). Thesecond approach tested involves defining a desired function[including a direct current (DC) offset], then taking thesquare root of this function to determine the shape of therequired input function. A 5.5 dB reduction of transmittedharmonic was obtained in both FEA simulation and exper-imental prototypes test.

I. Introduction

There has been growing interest in the potential of mi-croelectromechanical systems (MEMS) processed elec-

trostatically operated transducer as a potential replace-ment for piezoelectric transducer (PZT)-based ultrasoundtransducers or as an enabling technology for more com-plex transducer configurations [e.g., two-dimensional (2-D)catheter arrays]. This type of electrostatic transducer hasbeen studied for several decades. However, the applicationof integrated circuit (IC) semiconductor manufacturing re-lated microlithographic techniques to these devices overthe past decade has resulted in significant progress [1]–

Manuscript received January 30, 2004; accepted June 23, 2004.This work was funded in part by the Whitaker Foundation and NIHgrant EB2349.

The authors are with the Department of Biomedical Engineer-ing, University of Virginia, Charlottesville, VA 22908-0759 (e-mail:[email protected]).

[15]. These devices are frequently referred to as capacitivemicromachined ultrasonic transducers (cMUT).

The MEMS transducers possess a number of inherentadvantages with respect to conventional PZT transduc-ers. These advantages include: The potential to integrateassociated transmit/receive electronics adjacent to theelement—signal integrity is maintained and an expensivebulky cabling may be avoided. Low cost for high volume—IC processes are capable of very high volumes with mod-est cost. Transducer cell-to-cell, element-to-element, andtransducer array to transducer array consistency due tothe inherently high precision, accuracy, and repeatabilityof microlithographic processes. Well matched to fluids—the membrane possesses a very low acoustic impedanceand, hence, is practically nonresonant when in contactwith water-like loads (including tissue).

One limitation of cMUT is the fact that these devicesare inherently nonlinear because the force output is pro-portional to the square of the applied voltage [3]. In normalpractice, a large DC potential in comparison to the appliedalternating current (AC) signal is used to bias and thuslinearize the response of the device. However, there clearlyare practical limits to this approach. In diagnostic ultra-sound, there is a need for relatively high acoustic outputs,but there are clear safety, cost, and practical limits to thelevel of DC voltage that can be used. More specifically,the transducer becomes more efficient as the membraneis drawn down under increasing bias levels [16]. However,once the deflection exceeds more than one third of the gapdimension, the device becomes unstable and the membranecollapses downward onto the base substrate. It is also im-portant to note that, in a recent comparison of cMUT andPZT technology for diagnostic ultrasound, Mills and Smith[17] observed that it was necessary to further improve theefficiency of cMUT technology to make it truly competi-tive. This underlines the need to operate the transducer asaggressively as possible.

Very recently, Bayram et al. [18] described a new ap-proach for operating cMUT in a more linear manner. Thisinvolves using them with the center of the membrane col-lapsed down while the membrane adjacent is free to vi-brate. A 10 dB reduction in harmonic was reported. How-ever, these results were based on FEA simulation resultsrather than relying on experimental confirmation. Anotherlimitation of MEMS transducers, not discussed here, isthat, because they are surface machined onto the surfaces

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zhou et al.: comparison of precompensation approaches and electrostatic transducers 1565

of low acoustic loss silicon wafers, there is a relatively highlevel of acoustic crosstalk [19], [20].

At the same time as cMUT are evolving to the pointat which they are being considered for diagnostic imagingapplications [6], [21], [22], tissue harmonic imaging (THI)[23], wherein the acoustic signal generated nonlinearly bythe tissue is isolated, beamformed and scan-converted toform a B-mode image, has displaced fundamental-based,B-mode imaging for a large number of clinical applications.Significant improvements in spatial resolution and imagecontrast have been widely observed [24]. It is evident thatthe improvement in image quality outweighs the reducedpenetration depth for most clinical applications. Neverthe-less, because THI’s penetration is inherently limited andthe harmonic signal is a result of nonlinear propagation,there is clearly interest in using relatively high, yet stillsafe, acoustic intensities in order to maximize its effective-ness. However, the improved resolution of THI is mostlylost if the transducer array emits harmonic energy becauseit then becomes impossible to distinguish harmonic signalsgenerated via nonlinear effects in tissue from componentsgenerated nonlinearly by the transducer and propagatedlinearly in tissue. The arguments relating to the need tosuppress harmonic generation in the transducer also applyto imaging modes optimized for contrast agent imaging.For example, the pulse inversion approaches [25]–[27] inwhich linear components cancel and nonlinear harmoniccomponents sum, is based on the assumption that thetransducer does not itself generate significant nonlinearcomponents.

In this paper, we compare two precompensation ap-proaches to achieve a reduction of the inherent harmoniccomponent obtained with cMUT. The approaches wereconsidered using experimental analysis and a simple 1-D model for electrostatic force generation. Additionally,finite-element analysis (FEA) was used to simulate the ge-ometry and materials of the actual MEMS transducer thatwe tested experimentally. The reduction in the transmit-ted harmonic level that we observed can be used in one oftwo ways (or in partial combination). For preselected levelsof DC bias and AC signal level, it achieves a reduced har-monic signal. Alternatively, for the same absolute amountof acceptable harmonic generation, it enables higher fun-damental signal levels to be achieved.

II. Theory and Method

Fig. 1 illustrates schematically the typical structure ofan electrostatic transducer. It consists of a capacitor with arigid silicon substrate (lower capacitor plate) and a movingsilicon nitride membrane (upper capacitor plate). Gener-ally, the silicon nitride membrane has a metal electrodeformed on the top surface, and the substrate is inherentlyconductive due to heavy doping. When a voltage, V, is ap-plied to this capacitor, the silicon membrane will deformand generate an electrostatic force [3]:

F =12εA

V 2

d2 , (1)

Fig. 1. Schematic of an electrostatic transducer. 1, Electrode; 2, sili-con nitride membrane; 3, silicon wall support; 4, heavily doped siliconsubstrate; 5, vacuum; 6, DC bias voltage; 7, AC excitation voltage.

where ε is the permittivity of the layer between the twoplates (typically a vacuum), A is the capacitor effectiveelectrode surface area, V is the source voltage, and d is thedistance between the two electrodes. This electrostaticallygenerated plate force generates membrane displacementsthat in turn produce the output acoustic pressure.

The driving signal generally comprises a DC voltagebias (VDC) and a desired ultrasonic AC voltage pulse(VAC). When these are applied together, we obtain:

V 2 = V 2DC + 2VDCVAC + V 2

AC . (2)

Thus, according to (2), we obtain a DC component(which produces no ultrasound output whatsoever), ascaled replica of the desired AC pulse and an undesiredtransmitted harmonic component in the acoustic result.By inspection, the ratio of the desired AC pulse to the un-desired harmonic quantity can be increased by increasingthe relative level of the applied DC bias. However, this biaslevel is limited by practical limits, including, in particular,the threshold beyond which the membrane collapses and‘sticks’ to the lower plate [3]. Therefore, for a given max-imum permitted bias voltage and maximum permissibledegree of harmonic generation, there exists a strict limitto the amplitude for VAC—the desired ultrasonic signal.

The significance of the need to suppress the transmit-ted harmonic from MEMS electrostatic transducers, andapproaches for achieving this suppression, was apparentlyrealized by several individuals working independently [28]–[30]. Each individual developed slightly different proposedsolutions as briefly described below. In this paper, we im-plemented the iterative linear approach [28] and the squareroot approach [29]

A. Iterative Linear Approach

In this approach [28] one attempts to find the approx-imately linear transfer function valid for frequencies inthe vicinity of the unwanted harmonic. The observed har-monic from an initial driving signal is measured. There-after, one estimates, using the transfer function, a compen-sating waveform that, when applied, will largely cancel the



undesired harmonic. Some degree of iteration is requiredbecause the transducer is inherently nonlinear. However,because the harmonic error signal gets smaller with eachiteration, the procedure becomes successively more linearand is convergent.

B. Square Root Approach

Savord and Ossman [29] recognized that the output isapproximately determined by the square of the appliedinput voltage. Therefore, a preferred waveform is defined(with no harmonic content), a DC bias is superimposed,and the square root of this function is taken to determinethe required DC and AC quantities that should be appliedto obtain the desired waveform.

C. Slope Precompensation Approach

Fraser [30] identified that, as a result of the nonlin-ear generation, positive and negative going slopes in thewaveform have different absolute rates of change (slope).Therefore, the applied waveform is artificially modified tocompensate for this waveform slope distortion. Where theslope is expected to steepen, a shallower slope is appliedto the input waveform as a compensatory procedure.

1. Approach I—Iterative Linear Approach: Althoughthe electrostatic transducer operates in a nonlinear manner[28], we can assume approximate linearity when the ACsignal level is small compared to the DC bias. In this firstapproach, a DC bias voltage of 50 V was used. This isabout half of the absolute maximum rating of the device weused. (The device we used was a prototype single elementdevice donated by Sensant Corp., San Leandro, CA.)

The approach is described via numbered steps in Fig. 2.In Step 1, a 40 V peak, 2.5 MHz, 30% −6 dB fractionalbandwidth Gaussian pulse was used as the AC excitation.This AC excitation is relatively large compared with DCbias (about 80%) and was expected to generate significantharmonics around 5 MHz. The corresponding pressure out-put signal was converted to the frequency domain in Step 2and filtered in Step 3 to isolate the harmonic signal thatwas to be suppressed.

The transducer’s approximately linear transfer functionwithin the vicinity of the second harmonic then was eval-uated using Steps 4 to 6. A 7 MHz, 90% −6 dB fractionalbandwidth Gaussian pulse with sufficiently low amplitudeso as to obtain an approximately linear operation (5 V, i.e.,10% of DC bias voltage level) was used. This amplitudelevel was selected experimentally based on it being the low-est level (i.e., the amplitude that would give the most accu-rate estimate of the transducers linear transfer function)without having excessive electronic noise corrupting themeasurement. In Step 6, the transfer function was calcu-lated by dividing the measured output by the input (bothof them were converted to the frequency domain).

Subsequently, in Step 7, the harmonic signal to be sup-pressed was divided by the transducer’s transfer functionin the frequency domain. The result was converted to the

Fig. 2. Schematic of iterative linear approach.

time domain via an inverse fast Fourier transform (IFFT)and inverted so as to produce a compensation signal an-tiphase with respect to the existing (undesired) harmonicsignal. In Step 8, the required compensation signal wassummed with the original large amplitude 2.5 MHz exci-tation to arrive at the final required excitation.

After executing this process through one iteration, thesecond harmonic component at 5 MHz vicinity will be re-duced, but there is a possibility that higher harmonic com-ponents will result from the nonlinear response to the can-cellation signal we have added in the vicinity of 5 MHz.However, the procedure can be used iteratively, extract-ing the higher harmonics from new results and dividingby the same transfer function again to obtain new can-cellation signals. This procedure can be applied multipletimes until a satisfactory result is achieved. Because therequired excitation functions to be superimposed becomesmaller with successive iterations, the approach converges.In this paper, two iterations were found to be sufficient toprovide very satisfactory results.

2. Approach II—Square Root Approach: This methodis based on the observation that the output is approxi-mately related to the square of the applied input voltage[29]. This approach is described schematically in Fig. 3. InStep 1, a desired ultrasound pulse function is first defined(2.5 MHz, 30% −6 dB fractional bandwidth Gaussian ACsignal). In Step 2, a DC quantity is added to it to providebiasing and so that real numbers can be assured when thesquare root operation is applied. Thereafter, the compos-ite function was scaled (multiplied) by the DC quantity,and the required excitation was determined by taking thesquare root of it. This substep was used so that the re-sulting excitation waveforms being used in this approach



Fig. 3. Schematic of square root approach.

and the iterative linear approach are comparable in am-plitude. In Step 5, the DC quantity is subtracted so thatthe new precompensated AC signal (the square root input)can be isolated. The DC level is generated by a suitablehigh-voltage DC source, and the AC signal is producedby programming the waveform into an arbitrary functiongenerator and amplifying to obtain the correct signal level.

3. Approach III—Slope Precompensation Approach:This approach [30] is based on using the observation thatthe pressure output waveform of a cMUT is distorted bythe nonlinearity of the device so as to evolve from be-ing symmetric (broadly triangular in form) to asymmetric(broadly saw-tooth in form). The positive going portionof the waveform exhibits a steeper slope than that foundin the original waveform, and the negative going portionexhibits a shallower slope. Therefore, to compensate forthis nonlinear effect, a predistorted waveform is used sothat, after the slope distortion induced by the nonlineareffects, it will revert back to a symmetric waveform withreduced harmonic content. Fig. 4 is a schematic illustra-tion of this approach. Where the slope is expected to besteepen, a shallower slope is applied to the input waveformas a compensatory procedure [30]. Because this approachhas been described only in a heuristic manner, we have notattempted to implement it systematically in this paper. Itis thought to be unlikely that it will yield a result betterthan Approaches I or II above.

III. FEA Simulation Results

The PZFlex FEA software (Weidlinger Associates, Inc.,Los Altos, CA) with a nonlinear electrostatic solver option[31] was used for the simulation of cMUT transducers tak-ing account of device geometry and material properties.PZFlex is well suited as it uses a time-domain solver that

Fig. 4. Slope precompensation approach. The solid line is the originalinput, the dotted line is the slope-distorted output, and the dashedline illustrates the predistorted waveform, which will revert back toa semblance of the original waveform after nonlinear distortion.

Fig. 5. Finite-element simulation model (only central region of themodel is shown).

is particularly efficient for relatively large-scale transientproblems. A time-domain approach also is more amenableto exact nonlinear modeling than is possible in frequencydomain solvers because the actual output level on a time-instant-by-time-instant basis is known and included in themodel.

Our FEA model is illustrated in Fig. 5 and correspondsto a prototype device donated by Sensant Corp. (San Le-andro, CA). It is a single element cMUT device with 1.9-mm diameter circular piston shape. There are 1380 cMUTcells in the actual prototype device, but only one cell wassimulated in FEA. Therefore, to ensure a match betweenthe electrical loading conditions in the FEA simulationand in the experiment, the series connected source resis-tance in the simulation was scaled by multiplying the ac-tual impedance (50 Ω) by the number of cells. Parameters



TABLE ICritical Dimensions and Material Properties.

Silicon membrane diameter 50 µm Silicon membrane thickness 2 µmVacuum gap diameter 50 µm Vacuum gap thickness 0.16 µm

Silicon density 2340 kgm−3 Silicon nitride (Si3N4) density 3270 kgm−3

Silicon longitudinal velocity 9000 ms−1 Si3N4 longitudinal velocity 11000 ms−1

Silicon shear velocity 5840 ms−1 Si3N4 shear velocity 6250 ms−1

Silicon relative permittivity 11.5 Si3N4 relative permittivity 7.5

Fig. 6. Original AC input signal in time (left) and frequency (right) domains.

Fig. 7. Finite-element simulation of pressure output produced by the original input in time (left) and frequency (right) domains.

for each cell in the Sensant cMUT transducer are describedin Table I.

PZFlex permits the use of different finite-element meshsizes in one model. In this simulation, we used a finelysampled mesh with varying element sizes in different re-gions. There are a total of 6688 elements in the model,and the smallest of them in the vacuum gap is 0.02 µm ineach dimension. The simulation was executed on IBM In-telliStation Z Pro PC (IBM, White Plains, NY) with 3 GBRAM. Each simulation run took about 24 hours. We arecurrently looking into more efficient implementations.


We first tested the iterative linear approach in an FEAmodel. We selected a DC bias of 50 V, and the originalAC driving signal was a 2.5 MHz, 30% −6 dB fractional

bandwidth Gaussian pulse as shown in the Fig. 6. Thepressure output just above the transducer element in thewater is illustrated in Fig. 7. The second harmonic com-ponent about 5 MHz is readily evident and is only 8.2 dBbelow the level of the fundamental.

Another simulation was made using a much smaller ACsignal (5 V peak, 10% of the DC bias) with 7 MHz centralfrequency and 90% −6 dB fractional bandwidth. The ap-proximate transfer function was calculated across the en-tire frequency range, and the harmonic component in theprevious simulation was isolated by using a finite impulseresponse (FIR) band-pass filter with cut-off frequencies at4 MHz and 10 MHz. Using the calculated transfer func-tion, the required cancellation signal was calculated usingthe procedure discussed in the previous section. The newAC input signal and its pressure output are illustrated inFigs. 8 and 9. We observed that the second harmonic was



Fig. 8. Finite-element simulation. The first iteration precompensated AC input signal (left) and resulting pressure output (right).

Fig. 9. Finite-element simulation of pressure output. Comparison infrequency domain of original output and after using a first iterationstep of the iterative linear approach.

reduced from −8.2 dB down to −23.7 dB, but a higherorder harmonic component occurred near 7.5 MHz due tothe nonlinear operation of the transducer operating on thenewly applied 5 MHz signals. This harmonic is significant,(−10 dB magnitude compared to the fundamental compo-nent). Therefore, a second iteration of the procedure wasused.

After performing the second iteration, the second har-monic at 5 MHz was reduced significantly to −26.8 dB,and the higher harmonic generated in the first iterationstep was reduced from −10 dB to −30.9 dB (Figs. 10 and11).

The procedure could be repeated to eliminate more ofthe higher harmonics. However, those high-frequency com-ponents are not of interest in most harmonic-imaging sit-uations (which are primarily concerned with the secondharmonic). In this paper, we applied the procedure onlytwice and considered the frequency range below 8 MHz.This frequency range is well within the useful bandwidthof the hydrophone used in our experiments.


The same FEA model was used for testing the squareroot approach. The original AC driving function was thesame 2.5 MHz, 30% −6 dB fractional bandwidth Gaussianpulse as illustrated in Fig. 6, left. The DC bias level was50 V. Fig. 12 illustrates the calculated square-root inputfunction and its associated pressure output. In the fre-quency domain result in Fig. 13 (in which the square rootapproach was used), the second harmonic was reduced by5.5 dB with respect to the level at which no precompen-sation was used. However, due to the VDCVAC term in (2)and the higher harmonic components in the square-root in-put function, some second and higher harmonics are stillpresent in the output, even when using this square rootapproach.

IV. Experimental Results

Experiments were performed using the donated Sen-sant single-element transducer. We applied the same ACdriving functions and 50 V DC bias that we used in theFEA simulations. The AC signal was generated on a com-puter and loaded into a Sony-Tektronix 2021 ArbitraryWaveform Generator (AWG) (Tektronix, Beaverton, OR).The AWG outputs were amplified by a radio frequencyamplifier (ENI 50 dB, Electronic Navigation Industries,Rochester, NY, 325LA) and connected to the transducervia a series DC blocking capacitor. The DC bias was gener-ated by a DC power supply (HP 6515A, Agilent, Palo Alto,CA). The hydrophone [SEA (Onda Corp, Sunnyvale, CA),GL-0085] was placed 4 mm away from the transducer (inthe far-field) and was used to monitor the output acousticsignal. Thereafter, the result was amplified, then recordedon an oscilloscope. Fig. 14 describes the configuration ofour experiment.


We programmed the original AC signal (2.5 MHz, 30%Gaussian pulse shown in Fig. 6, left) as used in the FEAsimulation. The experimentally measured pressure output



Fig. 10. The second precompensated AC input signal (left) and its pressure output (right) (finite-element simulation).

Fig. 11. Comparison of pressure output in frequency domain afterthe second iteration of the iterative linear approach (finite-elementsimulation).

is shown in Fig. 15. The second harmonic was at a −8.3 dBlevel. Thereafter, we applied a 5 V, 7 MHz, 90% Gaussianpulse (small amplitude—10% of DC bias), and used the ex-perimental measured results to determine the transducer’sapproximate transfer function and, hence, to calculate therequired compensation input signal. After the first itera-tion, the second harmonic was reduced from −8.3 dB to−17.6 dB, but there is a −14 dB third harmonic due tothe nonlinear operation of the transducer acting on the 4–5 MHz components in the compensation signals (Fig. 16).Following the same procedure as illustrated in Fig. 2, werepeated the process to complete a second iteration. Thefinal result is illustrated in Fig. 17. It is observed that thesecond harmonic is further reduced to −28.9 dB, and thehigher order harmonic was reduced to −26.8 dB.

However, discrepancies can be found between the FEAsimulated output waveforms and experimental result. Webelieve that the explanation for the discrepancy is that it ischallenging to accurately characterize the materials prop-erties and device geometry in the actual cMUT transducer.

A better understanding and characterization of the devicewill improve the accuracy of the FEA prediction.

The very high level of the harmonic signal for the case inwhich no precompensation was used is quite striking in thisexperiment. Clearly, this device could not be used for har-monic imaging without some form of compensation. Con-ventional practice would be to reduce the level of the ACsignal relative to DC but the consequent impact on signal-to-noise ratio (SNR) resulting from such an approach is ob-vious. One additional factor partially explaining the highharmonic level in this case is that we excited the trans-ducer at less than its internal resonant ‘center’ frequency,which we estimate to lie in the range of 7 to 10 MHz. Wechose to operate at lower frequencies because experimentalmeasurement of high bandwidth, multiharmonic signals atthese lower frequency ranges was considerably simpler, andmore accurate, and eliminated any hydrophone bandwidthconcerns.


Using the same experiment apparatus, we tested thesquare root approach using the same approach as in thesimulation above. The square root driving function wasthe same as that used in the simulation (Fig. 12), and itscorresponding pressure output is shown in Fig. 18. Com-pared with the original output spectrum, we observe thatthe second harmonic is reduced by 5.5 dB from −8.3 dBto −13.8 dB. Clearly, the iterative linear approach gives amore satisfactory reduction of the harmonic generation.

V. Discussion and Conclusions

Two precompensation approaches were evaluated in thispaper. Both of these methods enable us to use AC driv-ing voltages that are a larger fraction of the amplitudeof the DC bias while reducing the level of undesired har-monic signal generation. Using the first, more complex,iterative linear approach, we achieved 18.6 dB suppressionin the transmitted second harmonic in FEA simulation,and 20.7 dB in an experiment. It is believed that the fact



Fig. 12. The square root AC input signal (left) and its pressure output (right) (finite-element simulation).

Fig. 13. Finite-element simulation of output pressure spectra in thefrequency domain resulting from the original input and using squareroot-based compensation approach.

that the experimental result demonstrated a reduction ap-proximately 2 dB better than that obtained in the FEAsimulation is coincidental. This may have been caused, forexample, by imperfect hydrophone alignment in the ex-periment resulting in part of the harmonic signal beingmissed. (The harmonic signal is more directional than thefundamental signal.)

Using the square root approach, a 5.5 dB harmonic re-duction in generated harmonic was obtained in both FEAsimulation and experiment.

These methods can be used in practice because the ar-bitrary function generators that are required for the trans-mitter circuit are becoming more common among state-of-the-art premium ultrasound scanners [32], [33]. Ultimately,the approaches will result in improved sensitivity in har-monic imaging modes when cMUT are used for diagnos-tic THI. Similar improvements will be obtained when themethod is used to image contrast agents using nonlineardetection methods.

Fig. 14. Experimental configuration.

Acknowledgment

We are grateful to Sensant Corp., San Leandro, CA, forthe donation of a prototype single element cMUT.

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Fig. 15. Experimental measured pressure output from the original input (2.5 MHz, 30% Gaussian pulse, peak amplitude 80% of DC biasvoltage) in time (left) and frequency (right) domains.

Fig. 16. Experimental measured pressure output from the precompensated input signal after the first iteration of the iterative linear approachin time (left) and frequency (right) domains.

Fig. 17. Experimental measured pressure output from the pre-compensated input signal after the second iteration of the ‘Iterative Linear’approach in time (Left) and frequency (Right) domains



Fig. 18. Experimentally measured pressure output using the square-root input waveform shown in time (left) and frequency (tight) domains.

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[23] T. Christopher, “Finite amplitude distortion-based inhomoge-neous pulse echo ultrasonic imaging,” IEEE Trans. Ultrason.,Ferroelect., Freq. Contr., vol. 44, pp. 125–139, 1997.

[24] K. Spencer, L. Weinert, and R. Lang, “The role of echocardio-graphic harmonic imaging and contrast enhancement for im-provement of endocardial border delineation,” J. Amer. Soc.Echocardiogr., vol. 13, pp. 131–138, 2000.

[25] C. Chapman and J. Lazenby, “Ultrasound imaging system em-ploying phase inversion subtraction to enhance the image,” U.S.patent No. 5,632,277, 1997.

[26] J. Hwang and D. Simpson, “Two pulse technique for ultrasonicharmonic imaging,” U.S. patent No. 5,951,578, 1999.

[27] P. Burns, “Pulse inversion Doppler ultrasonic diagnostic imag-ing,” U.S. patent No. 6,095,980, 2000.

[28] J. Hossack, “Medical diagnostic ultrasound system and methodfor harmonic imaging with an electrostatic transducer,” U.S.patent No. 6,461,299, 2002.

[29] B. Savord and W. Ossman, “Circuit and method for exciting amicro-machined transducer to have low second order harmonictransmit energy,” U.S. patent No. 6,292,435, 2001.

[30] J. Fraser, “Capacitive micromachined ultrasonic transduc-ers,” U.S. patent No. 6,443,901, 2002.

[31] D. Vaughn and J. Mould, PZFlex Time Domain Finite ElementAnalysis Package. Los Altos, CA: Weidlinger Associates, Inc.,2003.

[32] C. Cole, A. Gee, and T. Lui, “Method and apparatus for trans-mit beamformer,” U.S. patent No. 5,675,554, 1996.

[33] J. A. Hossack, J.-H. Mo, and C. Cole, “Transmit beamformerwith frequency dependent focus,” U.S. patent No. 5,608,690,1997.

Shiwei Zhou was born in Beijing, China in1974. He received the B.S. and M.S. degreesin optical-electrical engineering from the Bei-jing Institute of Technology, Beijing, China, in1996 and 1999, respectively. He is currentlyworking towards the Ph.D. degree in medi-cal ultrasound imaging at the Department ofBiomedical Engineering of the University ofVirginia, Charlottesville, VA. His research in-terests are finite element analysis (FEA) mod-eling for various ultrasound transducers in-cluding CMUTs, multi-layer transducers, and

2-D array transducers; applications of digital signal processing tech-niques in ultrasound; new transducer techniques and optimization.

Paul Reynolds (M’98) was born in Dundee, Scotland in 1972. Heearned his B.Eng. in electrical and Mechanical Engineering from theUniversity of Strathclyde in Glasgow, Scotland in 1994. He receivedhis Ph.D. in 1998 from the Electrical Engineering Department at theUniversity of Strathclyde, for research into the finite element mod-elling of piezoelectric ultrasonic transducers and their applications.



In 1999 he was a Visiting Professor at the University of Coloradoin Boulder, CO, before joining Weidlinger Associates Inc., (WAI) inCalifornia as a Senior Research Engineer. At WAI, his primary focusis the development of their finite element modelling package, PZFlex,for use in industry and academia for ultrasound related problems,support of customers in the use of PZFlex, and general engineeringconsultancy.

John A. Hossack (S’90–M’92–SM’02) wasborn in Glasgow, Scotland, in 1964. He earnedhis B.Eng. Hons(I) degree in electrical elec-tronic engineering from Strathclyde Univer-sity, Glasgow, in 1986 and his Ph.D. de-gree in the same department in 1990. From1990 to 1992, Dr. Hossack was a post doc-toral researcher in the E. L. Ginzton Labo-ratory of Stanford University working underB. A. Auld’s guidance. His research was onmodeling of 0:3 and 1:3 piezoelectric compos-ite transducers. In 1992, he joined Acuson,

Mountain View, CA, initially working on transducer design. Duringhis time at Acuson his interests diversified into beamforming and 3-Dimaging. Dr. Hossack was made a Fellow of Acuson for ‘excellence intechnical contribution’ in 1999. In 2000 he joined the Biomedical En-gineering Department at the University of Virginia, Charlottesville,VA.

His current interests are in improved 3-D ultrasound imaging andhigh bandwidth transducers/signal processing. Dr. Hossack is a mem-ber of the IEEE and serves on both the Administrative Committeeand the Technical Program Committee of the Ultrasonics Section. Healso is an Associate Editor of the IEEE Transactions on Ultrasonics,Ferroelectrics, and Frequency Control.


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