A new beamforming technique for ultrasonic imaging systems

Ultrasonics 38 (2000) 156–160 www.elsevier.nl/locate/ultras

A new beamforming technique for ultrasonic imaging systems G. Cincotti *, G. Cardone, P. Gori, M. Pappalardo Dipartimento di Ingegneria Elettronica, Universita` di Roma Tre, Via della Vasca Navale, 84, 00146 Rome, Italy

Abstract We propose a simple, versatile and inexpensive beamforming method that performs the aperture windowing of an ultrasonic transducer array in the transmit mode, without modifying the driver voltage, but simply controlling the length of the electric pulse driving the array elements. A conversion formula has been determined that permits us to compute, for a desired emitted pulse amplitude, the corresponding driving pulse length to be applied. Any shading function can be implemented over any type of transducer array, using very low-cost hardware. Computer simulations and experimental measurements, with a 3.8 MHz convex array, confirm the effectiveness of this approach in enhancing the contrast resolution, since the off-axis intensity in the radiated beam pattern is largely reduced. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Apodization; Beamforming; Driving pulse; Sidelobes

1. Introduction Aperture apodization is a technique widely used in ultrasonic imaging systems [1–4], employed in order to lower sidelobes of the produced beam at the expense of a broadening of the main lobe. Although apodization in reception, where low voltage signals are handled, is usually performed without problems, apodization in transmission requires high voltage, high-speed linear amplifiers, i.e. additional, expensive electronics, in order to reach a satisfactory signal-to-noise ratio and penetration depth. In this work we propose an alternative beamforming technique that allows us to achieve results comparable to those obtained with conventional apodization, but with a simpler, less expensive system. The energy reduction needed for the apodization is imposed with a shortening of the length of the electric driving pulse, instead of decreasing its amplitude. What is necessary is then to analyze which modifications result for the emitted ultrasonic pulse with respect to that obtained with conventional apodization. A design curve is presented that, starting from an amplitude apodization, permits us to compute the distribution of time lengths to be applied in order to obtain, with the proposed method, a comparable result. It is then shown that, for * Corresponding author. Fax: +39-655-79-078. E-mail address: [email protected] (G. Cincotti)

a proper operation, this technique has to be used in conjunction with a focusing delay correction, necessary because the reduction of the driving pulse length modifies the shape of the resulting ultrasonic pulse and then the instant in which its maximum occurs. The effectiveness of the proposed beamforming technique is demonstrated through numerical simulations and experimental measurements of the beam patterns of a 3.8 MHz convex array.

2. Influence of the driving pulse length on the emitted ultrasonic pulse In this section we investigate how changing the duration of the electric pulse driving a single piezoelectric transducer affects the emitted ultrasonic pulse. As a first case, we consider a Gaussian pulse as temporal driving waveform

C A

g(t)=exp −2

BD

t−T/2 2 T

.

(1)

In Eq. (1) T is the double of the standard deviation of the Gaussian pulse and is the parameter we change. We intend to compare the effect, obtained by varying T, on the energy of the ultrasonic pulse with that obtained with conventional apodization where only the amplitude level, say A, of the electric pulse is varied.

0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 08 0 - 3

G. Cincotti et al. / Ultrasonics 38 (2000) 156–160

157

Let us assume that the transducer can be modeled as having a Gaussian frequency response of the kind

C A BD

f−f 2 0 U( f )=U exp −a , (2) 0 Bf 0 where f is the center frequency, B is the −6 dB relative 0 bandwidth, U is a constant and a=1.2 ln 10. If the 0 system constituted by the driving electronics and the transducer can be described as a linear system, the emitted ultrasonic pulse p(t) is given by the convolution of the driving pulse g(t) and of the transmit impulse response u(t) (3)

p(t)=u(t)1g(t),

where u(t) is the anti-Fourier transform of the expression in Eq. (2) and 1 denotes convolution. For the proposed method, the emitted energy is computed, in the frequency domain, as

P

e(T )=

2

p |U( f )G( f )|2 df=U2 T2 0 2

−2 f−f 2 2 0 × exp[−(pfT )2] df, (4) exp −2a Bf −2 0 where G( f ) denotes the Fourier transform of g(t). Eq. (4) is simply evaluated as

P

C A BD

AB S C

p 3/2 e(T )=U2 T2 0 2

2B2 f2 0 2a+(pTf B)2 0 2a(pTf )2 0 . (5) ×exp − 2a+(pTf B)2 0 When conventional apodization is used, the duration T is fixed, say T9 , and there is an amplitude factor A, variable over the aperture, that multiplies p(t). The emitted energy is then found for this case as

D

AB S p 3/2

2B2 f2 0 2 2a+(pT 9 f B)2 0 2a(pT f )2 9 0 ×exp − . (6) 2a+(pT 9 f B)2 0 By equating Eqs. (5) and (6), we derive a relationship between A and T

e: (A)=U2 A2T9 2 0

C

Fig. 1. Equivalent amplitude apodization level as a function of the driving pulse length T, for Gaussian excitation pulse.

increasing and this gives a limit on the maximum length T9 that can be taken by the driving pulse in order to obtain an equivalent apodization effect. In other words, the useful part of the curve is obtained for the range T=0 to T , T being the value in which the maximum M M occurs. So far we have considered a Gaussian pulse. This has permitted us to obtain closed-form results. In many practical cases, however, the electric driving pulse is best modeled by a half-wave square pulse of duration T g(t)=

G

1

0≤t≤T

0

elsewhere.

(8)

The ultrasonic pulses emitted by a transducer driven by three electric pulses of different lengths T are shown in Fig. 2 as a function of the normalized time t∞=t/T9 , where T9 is the maximum used length of the electric pulse. The emitted pulses are normalized to the peak amplitude obtained when the driving pulse length is T=T9 .

D

A

B

T dT2 exp − A=A , 0 (a+bT2)1/4 a+bT2

(7)

where a=2a, b=(pBf )2, d=a(pf )2, and A = 0 0 0 (a+bT9 2)1/4/T 9 exp[dT9 2/(a+bT9 2)]. Eq. (7) is plotted in Fig. 1 where a relative bandwidth B=60% has been considered. This curve can be used for converting a desired amplitude apodization into a corresponding ‘time’ apodization, giving the lengths of the driving pulses to be applied. However, it is not monotonically

Fig. 2. Ultrasonic emitted pulses versus the normalized time t∞=t/T 9, for three different electric pulse lengths T. Full line refers to T=T9 , dashed line to T=T9 /2 and dotted line to T=T 9 /10.

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In the preceding developments, we used the equality of the emitted energy as a criterion for establishing an equivalence between ‘time’ and conventional apodization. Another possible point of view is to require that the ultrasonic emitted pulses have the same peak amplitude. We verified by numerical methods that the two criteria lead to comparable results. For the present example, we shall refer to the second one since it better evidences the need for a temporal shift of the pulses emitted by different elements of the transducer. It is seen from Fig. 2 that both the emitted pulse peak amplitude p and its position t depend on the 0 0 electric pulse length T and that a reduction of the driving pulse length produces a corresponding reduction of the emitted pulse amplitude. Therefore, the implementation of the present beamforming method requires a time delay compensation to obtain a correct focusing. This can be simply achieved by aligning the centers of the driving pulses and applying to each array element a pulse given by

g (t)= 1

G

1

T9 −T 2

≤t≤

T9 +T 2

(9)

0 elsewhere.

In Fig. 3 we plot the peak amplitude p , normalized to 0 the value corresponding to a pulse length T=T 9 , as a function of the normalized parameter q=T/T9 , for a transducer with relative bandwidth B=80%. The maximum exploitable length T is found, for square wave M excitation, to be equal to 1/(2f ). The curve in Fig. 3 0 has been numerically obtained and it has been found that it is rather well fitted by the function p (q)=2q−q2, (10) I which is also plotted in Fig. 3 as a full line. This function can be used as a design curve in the sense that, choosing

Fig. 3. Peak pressure p , normalized with respect to the case T=T 9, 0 versus the parameter q. The full line refers to the fitting function p , I whereas the dashed line refers to the exact curve.

an amplitude windowing function, the lengths of the driving pulses can be determined by inverting Eq. (10).

3. Simulated and experimental beam patterns We present now computer simulations and experimental measurements performed with a 3.8 MHz convex array with radius of curvature R=40 mm and 70 active elements of width w=0.382 mm and height H=12 mm. The array has an acoustical lens with focal length F= 80 mm (in elevation) and is electronically focused at a distance Z=90 mm. The transducer has a frequency spectrum with relative bandwidth B=53%. The experiments have been performed with an ESAOTE ultrasonic system, measuring the radiated field with a Sonic 804-097 hydrophone in a water tank. This medical imaging system does not support conventional apodization. The pressure field is computed, at every point, by using the Rayleigh–Sommerfeld diffraction formula, simplified as suggested in Ref. [5]. The beam pattern can be determined either by finding the maximum over the time of the transient acoustic field [6 ] or by evaluating the time integral of the instantaneous intensity distribution [7]. In this work the latter method was used for a more accurate comparison between simulated and measured beam patterns, because the hydrophone used is sensitive to the integrated intensity of the acoustic pressure. A comparison between conventional apodization and the proposed beamforming technique is given in Fig. 4 where the beam profile of the unshaded aperture is also plotted for reference. The beam patterns are evaluated in the focal plane Z=90 mm and a Gaussian shading function with a −20 dB truncation at the border of the

Fig. 4. Simulated beam profile on the focal plane Z=90 mm for a 3.8 MHz convex array. The dashed line refers to the case of no apodization, the dotted line to conventional apodization and the full line to the proposed beamforming technique. A −20 dB truncated Gaussian shading function has been used.

G. Cincotti et al. / Ultrasonics 38 (2000) 156–160

aperture is imposed. For the proposed technique, the relative amplitude of each array element has been converted into the corresponding driving pulse length by using Eq. (10). It can be seen that the two methods affect the beam profile in a similar way; in fact, in both cases, the off-axis intensity is largely reduced. These favorable effects, of course, must be paid for by an increase of the main-lobe width, as in any apodization process. Experimental and numerical results for a −6 dB truncated Gaussian shading function are presented in Fig. 5(a) and (b) respectively. From Fig. 5, the importance of correcting the focusing time delays is pointed out; in fact, without correction, the off-axis intensity is reduced but the beam width increases remarkably. This effect can also be observed in Fig. 6(a) and (b), where measured and simulated on-axis beam profiles are plotted. It is apparent that the peak intensity moves towards lower z if the focusing delays are not properly applied. As far as the hardware implementation is concerned,

159

(a)

(b)

Fig. 6. On-axis beam profiles for a 3.8 MHz convex array. The dotted line refers to the case of no apodization, the full line and dashed line refer to the proposed beamforming technique, with and without focusing correction respectively. A −6 dB truncated Gaussian shading function has been chosen. (a) Experimental data obtained in a water tank. (b) Computer simulations. (a)

a limit on the application of the proposed method is given by the fact that the duration of the electric pulse cannot be arbitrarily varied. A lower limit exists for the length practically obtainable with a pulser, whereas an upper bound value is related, as has been seen, to the center frequency of the transducer.

4. Conclusions (b)

Fig. 5. Beam profile on the focal plane Z=90 mm for a 3.8 MHz convex array. The dotted line refers to the case of no apodization, the full line and dashed line refer to the proposed beamforming technique, with and without focusing correction respectively. A −6 dB truncated Gaussian shading function has been chosen. (a) Experimental data obtained in a water tank. (b) Computer simulations.

In this work, an efficient, flexible and low-cost beamforming technique that performs aperture windowing, in the transmit mode, by varying the electric pulse length from one element to another is proposed. Computer simulations and experimental measurements show that the contrast resolution of the proposed technique is very close to that obtained with conventional apodization.

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Acknowledgement The authors are grateful to Francesco Pomata of Esaote Biomedica S.p.A., for his constructive suggestions and useful discussions.

References [1] P.J. ’t Hoen, Aperture apodization to reduce the off-axis intensity of the pulsed-mode directivity function of linear arrays, Ultrasonics (1982) 231.

[2] J.N. Wright, Resolution issues in medical ultrasound, Proc. IEEE Ultrason. Symp. (1985) 793. [3] C.M.W. Daft, W.E. Engeler, Windowing of wide-band ultrasound transducer, Proc. IEEE Ultrason. Symp. (1996) 1541. [4] K.E. Thomenius, Evolution of ultrasound beamformers, Proc. IEEE Ultrason. Symp (1996) 1615. [5] J. Lu, J.F. Greenleaf, A study of two-dimensional array transducers for limited diffraction beams, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 41 (1994) 724. [6 ] V. Murino, A. Trucco, A. Tesei, Beam pattern formulation and analysis for wide-band beamforming systems using sparse arrays, Signal Processing 56 (1997) 177. [7] K. Raum, W.D. O’Brien Jr., Pulse-echo field distribution measurement technique for high frequency ultrasound sources, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 44 (1997) 810.