New demodulation filter in digital phase rotation beamforming

Ultrasonics 44 (2006) 265–271 www.elsevier.com/locate/ultras

New demodulation filter in digital phase rotation beamforming Fabio Kurt Schneider a, Yang Mo Yoo a, Anup Agarwal a, Liang Mong Koh b, Yongmin Kim a,* a

Image Computing Systems Laboratory, Departments of Electrical Engineering and Bioengineering, University of Washington, Seattle, WA 98195-2500, United States b School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 27 December 2005; received in revised form 7 February 2006; accepted 8 February 2006 Available online 6 March 2006

Abstract In this paper, we present a new quadrature demodulation filter to reduce hardware complexity in digital phase rotation beamforming. Due to its low sensitivity to phase delay errors, digital quadrature demodulation is commonly used in ultrasound machines. However, since it requires two lowpass filters for each channel to remove harmonics, the direct use of conventional finite impulse response (FIR) filters in ultrasound machines is computationally expensive and burdensome. In our new method, an efficient multi-stage uniform coefficient (MSUC) filter is utilized to remove harmonic components in phase rotation beamforming. In comparison with the directly implemented FIR (DI-FIR) and the previously-proposed signed-power-of-two FIR (SPOT-FIR) lowpass filters, the proposed MSUC filter reduces the necessary hardware resources by 93.9% and 83.9%, respectively. In simulation, the MSUC filter shows a negligible degradation in image quality. The proposed method resulted in comparable spatial and contrast resolution to the DI-FIR approach in the phantom study. These preliminary results indicate that the proposed quadrature demodulation filtering method could significantly reduce the hardware complexity in phase rotation beamforming while maintaining comparable image quality.  2006 Elsevier B.V. All rights reserved. Keywords: Ultrasonic imaging; Beamforming; Phase rotation beamforming; Quadrature demodulation filter; Hardware complexity

1. Introduction The adoption of digital receive beamforming (DRBF) techniques based on dynamic focusing has greatly improved the ultrasound image quality in the last few decades [1]. In the DRBF, the enhanced time delay accuracy in digital processing provides higher signal-to-noise ratios (SNR) and better spatial resolution. In addition, the DRBF’s flexibility has enabled new imaging techniques (e.g., dynamic aperture and multi-beam) [2]. However, the DRBF significantly increases the computational complexity since it requires fast analog-to-digital converters (ADC) and front-end digital circuitries running at a high clock frequency. To alleviate the high-frequency requirement in ADCs and front-end cir-

*

Corresponding author. Tel.: +1 206 685 2271; fax: +1 206 543 3852. E-mail address: [email protected] (Y. Kim).

0041-624X/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.02.004

cuitries, interpolation beamforming (IBF) and phase rotation beamforming (PRBF) methods have been developed [2–4] and are commonly used in ultrasound machines [5]. However, IBF and PRBF methods require many computationally-expensive interpolation and demodulation filters, respectively. Thus, this high computational requirement makes the development of ultrasound machines with large channel counts (e.g., for 3D ultrasound systems [6]) or very low-end machines (e.g., handheld [7]) challenging. Various beamforming techniques, such as pipelinedsampled-delay-focusing (PSDF) [8], sigma-delta oversampled (SDO) [5] and direct I/Q [9], have been developed for further reducing the hardware burden in the DRBF. The PSDF technique relies on non-uniform sampling of the signals coming from the different receive channels to compensate for time delay differences in dynamic receive focusing [8]. While the PSDF succeeds in lowering the hardware complexity, the control of ADCs for non-uniform sampling

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Table 1 Acronyms used ADC CR DI-FIR DRBF FIR IBF LSB MSUC PRBF PSDF SDO SNR SPOT-FIR TGC

Analog-to-digital converter Contrast resolution Directly implemented FIR Digital receive beamforming Finite impulse response Interpolation beamforming Least significant bit Multi-stage uniform coefficient Phase rotation beamforming Pipelined-sampled-delay-focusing Sigma-delta oversampled Signal-to-noise ratio Signed-power-of-two FIR Time-gain compensation

is challenging [10]. In the SDO beamforming, one-bit sigma-delta modulators running at a high clock frequency are utilized to eliminate the complicated digital delay circuitries (e.g., interpolation and phase rotation) [11]. On the other hand, it suffers from artifacts caused by synchronous errors between the sigma-delta modulator at each channel and the post-beamforming demodulator [5,12]. In addition, sigma-delta modulators running at the required clock frequency for the typical wideband signals in medical ultrasound imaging are difficult to implement. In the direct I/Q method, the complex baseband signals utilized in the PRBF are directly sampled from the receive signals. Thus, there is no need for lowpass filtering, which is a major computational task in the PRBF, during demodulation to remove harmonic components [9]. However, the direct I/Q approach assumes narrow-band signals and no depthdependent attenuation, which is typically not satisfied in medical ultrasound imaging. Alternatively, the hardware complexity in the PRBF could be reduced by replacing the directly implemented finite impulse response (DI-FIR) filters with more efficient lowpass filters, such as signedpower-of-two (SPOT) FIR [13,14]. In the SPOT-FIR, filter coefficients are only represented by sums of a limited number of SPOT terms [15]. While the SPOT filter can substitute complex multipliers with adders and shifters, its computational complexity is still high.

In this paper, we propose a new quadrature demodulation filtering technique based on an efficient multi-stage uniform coefficient (MSUC) filter. The necessary hardware resources of the developed MSUC filter is compared with the conventional DI-FIR and the previously-proposed SPOT-FIR. Since the developed MSUC filter may degrade image quality, its effect on image quality is also analyzed. The acronyms used in this paper are listed in Table 1. 2. Methods 2.1. Digital phase rotation beamforming with digital quadrature demodulation Fig. 1 shows the block diagram of a digital phase rotation beamformer with quadrature demodulation. The receive ultrasound signals are amplified in proportion to depth in order to compensate for signal attenuation (i.e., time-gain compensation, TGC). After TGC, the RF signals are digitized by ADCs whose sampling frequency is typically 4f0 where f0 is the transducer center frequency. The digitized RF signal can be represented by x½n ¼ AI ½n  cos½2pf0 n  AQ ½n  sin½2pf0 n

ð1Þ

where AI[n] and AQ[n] are the baseband in-phase and quadrature signal, respectively. The baseband signal can be extracted by removing the carrier frequency through quadrature demodulation, which consists of mixing and lowpass filtering. The digitized RF signal (i.e., x[n]), which is originally centered around ±f0, is first multiplied with cosine (cos[2pf0n]) and sine (sin[2pf0n]) values. These multiplications generate not only the signals centered at 0, but also signal harmonics centered at ±2f0 in the in-phase and quadrature components of the mixed signal (MI[n] and MQ[n]). To remove these harmonics shown in Fig. 1, lowpass demodulation filtering after mixing is required. Afterwards, the receive beamforming based on dynamic focusing is performed on the extracted baseband signals (i.e., AI[n] and AQ[n]) for coherent summation consisting of delay, phase compensation and summation to improve SNR and spatial resolution. Table 2 lists the computational load of each processing function (in terms of multiplications and additions) in the

Fig. 1. Block diagram of a phase rotation beamformer with quadrature demodulation. Ffg represents the Fourier transform.

F.K. Schneider et al. / Ultrasonics 44 (2006) 265–271

Alternatively, the complexity of lowpass filtering in quadrature demodulation can be reduced by utilizing the SPOT-FIR filter. In this method, multiplications are substituted with shifts and additions so that the complexity mainly depends on the maximum number of sums of SPOT terms. For a symmetric m-tap FIR filter with b SPOT values per filter coefficient, (m + mb)/2  1 additions in Ba and Be and mb/2 shifts in Bd are required as shown in Fig. 2(b). We designed a 16-tap SPOT-FIR filter by applying a genetic algorithm [16] to determine the two 12-bit SPOT values for each filter coefficient.

Table 2 Number of operations in each PRBF function and its percentage to the total number of operations when the number of channels (C) is 32 and the filter tap size (m) is 16

Mixing Demodulation filter Phase compensation Sum Total

Multiplications

Additions

Number

%

Number

%

2C 2mC 4C – (2m + 6)C

5 84 11 – 100

– 2(m  1)C 2C C1 (2m + 1)C  1

– 91 6 3 100

267

2.3. New demodulation filter for phase rotation beamforming PRBF with C receive channels and m-tap demodulation filters. To identify the most computationally-demanding PRBF function, the percentage of each function to the total number of operations was also computed when C is 32 and m is 16. As shown in Table 2, the demodulation filtering accounts for 84% and 91% of the total multiplications and additions in the PRBF, respectively. Therefore, simplifying demodulation filtering can greatly help in reducing the overall PRBF computational load.

Our new lowpass filtering technique is based on an efficient multi-stage uniform coefficient (MSUC) filter. Since the harmonics introduced during mixing only occur at ±2f0, the MSUC filter that is used in image processing and computer vision for efficiently implementing Gaussian-like filters with large tap size [17–19] can be used for removing the harmonics with reduced hardware complexity. Reduction in the complexity of the MSUC filter is due to two factors: the use of uniform coefficients and the recursive implementation of each stage. In the MSUC filter, uniform coefficients are utilized at each stage in order to replace multiplications with additions. The impulse response of each uniform coefficient filter stage can be represented by  1=ai 0 6 k < Li  1 hi ðkÞ ¼ ð2Þ 0 otherwise

2.2. DI-FIR and SPOT-FIR filters Multipliers and adders are utilized in the DI-FIR filter to convolve the input signal with filter coefficients. As shown in Fig. 2(a), the complexity in DI-FIR filtering can be reduced by taking advantage of the symmetry in the filter coefficients, resulting in m/2 multiplications in Bb and a total of m  1 additions in Ba and Bc per output value when m is the filter tap size. In this paper, we designed a symmetric 16-tap FIR filter quantized at 12 bits based on the least squares method.

where Li is the number of taps in the ith stage and 1/ai is the uniform coefficient determined by the number of least significant bits (i.e., LSBi) discarded in the ith cascaded

Coefficients

+

Coefficients

×

m/2

+

m/2-1

×

Ba

Bb

mb/2

m/2

+

Input data

Input data

m/2

+

Bc

Ba

Bd

Input

×

×

Input

×

data +

L2

L1 +

c

Ls +

+

shift accumulator registers

data

Be

b

Li coefficients

×

+

+

a

1 1 1 1

mb/2-1

LSB1

+

+

+

LSB2

d

Fig. 2. (a) DI-FIR filter with symmetric coefficients, (b) SPOT-FIR filter with symmetric coefficients, (c) direct implementation of a uniform coefficient filter stage and (d) s-stage MSUC filter using recursive implementation.

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stage output (i.e., ai = LSBi/Li). The frequency response of the s-stage MSUC filter is given by X ð1  zL1 Þ  ð1  zL2 Þ    ð1  zLs Þ Ai ð3Þ H ðzÞ ¼ s ð1  z1 Þ i where Ai is the gain of the ith stage (i.e., Ai ¼ Li =2LSBi ). Thus, this frequency response can be controlled by changing parameters, such as s, Li, and LSBi. From Eq. (3), the zero(s) for each stage is positioned at ±bi Æ fs/Li, where fs is the sampling frequency and bi is an array of integer values in the range of 1 to Li/2 (e.g., zeros at ±fs/5 and ±2fs/5 for Li = 5). In this paper, the filter parameters were determined by minimizing the least squares error in the MSUC filter frequency response with respect to the same ideal filter used when designing the reference DI-FIR filter. An exhaustive search method is used over a limited parameter space where the number of taps of each stage is smaller than 5 to provide zeros at frequencies larger than the typical cutoff frequencies and the maximum number of cascaded stages is set to make the number of total additions in the MSUC filter smaller than that in the efficient SPOT-FIR filter. In our MSUC filter, each uniform coefficient filter stage can be directly implemented using multiple adders as shown in Fig. 2(c). Alternatively, for further hardware reduction, it can be recursively implemented using one accumulator and as many shift registers as the number of taps of the ith filter stage (i.e., Li). These recursively-implemented stages can be cascaded to compose the s-stage MSUC filter shown in Fig. 2(d). The ith-stage recursive implementation of the MSUC filter can be represented by y i ðk þ 1Þ ¼ y i ðkÞ þ

1  ½xi ðk þ 1Þ  xi ðk þ 1  Li Þ ai

ð4Þ

where a new output value, yi(k + 1), is generated by adding the scaled difference between a new input value, xi(k + 1), and a Li-delayed input value, xi(k + 1  Li), to the current output value, yi(k). According to Eq. (4) and Fig. 2(d), the MSUC filter requires only two additions per cascaded stage, and the number of cascaded stages is typically smaller than a half of the tap size of the DI-FIR and SPOT-FIR filters. Thus, the MSUC filter can significantly reduce the complexity by eliminating the need of multiplications and reducing the number of additions compared to both DI-FIR and SPOT-FIR filters. Fig. 3 shows the frequency responses of the 16-tap DI-FIR and SPOT-FIR filters with the proposed MSUC filter when a minimum stopband attenuation of 40 dB and the normalized cutoff and stopband frequencies of 0.35 and 0.5 are specified. The actual filter parameters are listed in Table 3. The MSUC filter presents higher stopband attenuation than the DI-FIR and SPOT-FIR filters. However, the MSUC filter has a wider transition band compared to both filters, which could potentially attenuate the baseband signal that is located close to the transition band. The DI-FIR and SPOT-FIR filters show similar passband response, but the DI-FIR presents higher stopband attenu-

Fig. 3. Frequency responses of the DI-FIR, SPOT-FIR and MSUC filters.

Table 3 Parameters for the DI-FIR, SPOT-FIR and MSUC filters DI-FIR

SPOT-FIR

MSUC

Tap size

16

16

FCP fcutoff fstopband Attenstopband

12 bits 0.35 0.5 38 dB

12 bits 0.35 0.5 38 dB

L1 = L2 = 2, L3 = L4 = L5 = 3 – 0.2 0.5 38 dB

FCP, Li, fcutoff, fstopband, and Attenstopband are the filter coefficient precision, tap size of the ith-stage of the MSUC filter, normalized cutoff frequency, normalized stopband frequency, and minimum stopband attenuation, respectively.

ation than the SPOT-FIR filter so that it is used as a reference filter for image quality evaluation. 3. Results and discussion 3.1. Image quality evaluation To analyze the effect of the proposed filter on image quality, the simulation and phantom studies were performed. A simulation model was generated by utilizing the commonlyused Field II program [20]. For the phantom study, a tissue mimicking phantom (Model 539 Multipurpose Phantom, ATS Laboratories Inc., Bridgeport, CT, USA) was used. A commercial ultrasound machine (i.e., SA-9900, Medison Corp., Korea) was modified to acquire pre-beamformed data. The parameters for the simulation and phantom studies are summarized in Table 4. For quantitative comparison, the mean square error (MSE) and lateral and axial resolution were computed. The MSE is given by MSE ¼

N 1 X 2 ðX DI-FIR  X MSUC Þ N i¼1

ð5Þ

where XDI-FIR and XMSUC are the normalized envelope signals obtained by applying the DI-FIR and MSUC filters, respectively, and N is the number of samples.

F.K. Schneider et al. / Ultrasonics 44 (2006) 265–271 Table 4 Parameters used in simulation and phantom studies Parameter

Simulation

Phantom

Speed of sound [m/s] Transducer type Transducer center frequency (f0) [MHz] Receive signal bandwidth at 45 dB Transmited pulse Element height/pitch [mm] Elevational focus [mm] Transmit focus [mm] Receive focus Number of transducer elements Number of receive channels Simulation frequency [MHz] Sampling frequency [MHz] Apodization Number of scanlines ADC quantization [bits]

1540 Linear 3.5 1.4f0 1 square 14/0.44 40 40 Dynamic 160 32 112 14 Uniform 129 8

1450 Convex 3.5 1.4f0 1 square 12/0.317 50 50 Dynamic 192 32 — 15.4 Uniform 192 8

lateral and axial resolution of the point spread function, respectively. No significant degradation in spatial resolution was observed when applying the MSUC filter compared to both the DI-FIR and SPOT-FIR. Fig. 5(a)–(c) shows the tissue mimicking phantom images with the dynamic range of 60 dB obtained by applying the DI-FIR, SPOT-FIR and MSUC filters, respectively. The different filters provide similar image quality. For quantitative comparison, the MSE was computed by considering all the points in the images. The MSE between the DI-FIR filter and the developed MSUC filter was 1.1 · 105, which is negligible. Fig. 5(d) and (e) are the lateral and axial resolution for the different filters at the focal region and show that the resolution difference is insignificant. Additionally, the contrast resolution (CR) was computed for the region marked in Fig. 5(a) by CR ¼ 1 

Fig. 4(a) shows the point spread function of the simulation model. The computed MSE at the center scanline in Fig. 4(a) was less than 0.001. Fig. 4(b) and (c) show the

269

Ic Is

ð6Þ

where Ic and Is are the average intensity inside cyst and speckle regions at the same depth, respectively [21]. The

Fig. 4. Simulation results: (a) point spread function, (b) lateral resolution and (c) axial resolution with the DI-FIR, SPOT-FIR and MSUC filters.

Fig. 5. Phantom study results with 60 dB dynamic range: ultrasound images with (a) DI-FIR filter, (b) SPOT-FIR filter, (c) MSUC filter, (d) lateral resolution, and (e) axial resolution.

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Table 5 MSE and CR values from the phantom study Parameter

Value

MSE CR for DI-FIR CR for MSUC

1.1 · 105 0.732 0.730

computed CR values for the DI-FIR and MSUC filters are listed in Table 5. No significant difference (i.e., <0.3%) in the contrast resolution between the DIR-FIR and MSUC filters exists. Even though no perceptible degradation in image quality was observed from the simulation and phantom studies with the developed MSUC filter, more systematic image quality assessment may be needed before clinical use. 3.2. Hardware resources Before evaluating the amount of necessary hardware resources, the DI-FIR and SPOT-FIR filters were optimized by utilizing the symmetric filter coefficients as shown in Fig. 2(a) and (b), respectively. In the DI-FIR and SPOTFIR filters, every two symmetric input data are first added together in Ba and convolved with the filter coefficient. The convolution in the DI-FIR filter is performed by using eight 8 · 12-bit multipliers and a 21-bit 8-input adder in Bb and Bc, respectively. In the SPOT-FIR, the convolution is carried out through two shifters per input based on predetermined 12-bit SPOT terms in Bd, followed by a 21-bit 16-input adder in Be. On the other hand, for the MSUC filter shown in Fig. 2(d), a 5-stage filter (i.e., tap sizes of 2, 2, 3, 3, and 3 for stages S1 through S5, respectively) was implemented with full precision adders (i.e., no truncation error) except for the last two stages where division by two is performed. The efficient ripple-block carry look-ahead and Wallace tree-based architectures were used for the adders and multipliers, respectively. We used the gate counts for various circuits reported by Omondi [22]. Table 6 summarizes the estimated hardware complexity along with the register counts for all three filters. Due to the recursive implementation of the MSUC filter, its output Table 6 Output data rate (ODR), register count (RC) in bits, estimated gate count per block (EGCB) with IB bits for the block input data, and the estimated gate count total (EGCT) for the DI-FIR, SPOT-FIR and MSUC filters Technique

ODR

RC

Block

IB

EGCB

EGCT

DI-FIR

fb

160

Ba Bb Bc

8 9 21

1344 16048 3380

20772

SPOT-FIR

fb

336

Ba Be

8 21

1344 6568

7931

MSUC

fs

131

S1 S2 S3 S4 S5

8 9 10 12 13

182 200 252 320 320

1274

Fig. 6. Gate counts for demodulation filtering as a function of the number of receive channels with the DI-FIR, SPOT-FIR and MSUC filters.

data rate, fs, is usually four times higher than that in the DI-FIR and SPOT-FIR filters, which use the beamforming frequency, fb. However, the MSUC filter can generate four outputs in a period of 1/fb, which leads to the same beamforming throughput as the DI-FIR and SPOT filters. As shown in Table 6, the proposed MSUC filter can reduce the necessary gate count by 93.9% and 83.9% compared to the DI-FIR and SPOT-FIR filters, respectively. Additionally, the MSUC filter reduces the required number of registers in bits (131 for data delay) compared to the DIFIR (160 for data and coefficients storage) and SPOTFIR (336 for data shift) filters. Therefore, the proposed MSUC filter significantly cuts down the hardware complexity while providing similar image quality, in terms of MSE, spatial resolution and CR, compared to the DI-FIR and SPOT-FIR filters. The advantage of the developed MSUC filter is more clearly shown in Fig. 6 where the gate counts for various system configurations (i.e., different number of receive channels) are shown. The gate count savings provided by the MSUC filter are significant in low-end ultrasound machines (e.g., 16 or 32 channels) where reduction in cost, size and power consumption is critical. These low-end systems could be used in newly emerging environments, e.g., distributed diagnosis and home healthcare [23], for routine screening and monitoring in addition to traditional diagnosis. The savings offered by the MSUC filter would be also beneficial to the development of high channel count systems where the computational burden for beamforming is enormously high (e.g., 3D beamforming for a 2D array transducer with thousands of elements). 4. Conclusion In this paper, the new quadrature demodulation filtering method based on the multi-stage uniform coefficient (MSUC) filter is proposed. From both simulation and phantom studies, the proposed method has shown comparable image quality to the commonly-used directly implemented FIR filter in terms of the MSE, spatial resolution and CR while providing significant reduction in the

F.K. Schneider et al. / Ultrasonics 44 (2006) 265–271

amount of hardware needed. We believe that the MSUCbased demodulation filtering method would be useful in lowering the computational cost from low-end (e.g., portable) to high channel count (e.g., 3D) ultrasound machines without any perceptible degradation in image quality. Acknowledgements This project is supported by the Singapore Government’s A*STAR, through the Singapore–University of Washington Alliance (SUWA). F.K. Schneider gratefully acknowledges the support of the Brazilian Government through CAPES and UTFPR. References [1] S. Stergiopoulos, Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real Time Systems, CRC Press, Boca Raton, FL, 2000. [2] B.D. Steinberg, Digital beamforming in ultrasound, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (1992) 716–721. [3] J.N. Wright, C.R. Cole, A. Gee, Method and apparatus for a baseband processor for a receive beamformer system, US Patent No. 5,928,152, 1999. [4] M. O’Donnell, W.E. Engeler, J.J. Bloomer, J.T. Pedicone, Method and apparatus for digital phase array imaging, US Patent No. 4,983,970, 1991. [5] S.R. Freeman, M.K. Quick, M.A. Morin, R.C. Anderson, C.S. Desilets, T.E. Linnenbrink, M. O’Donnell, Delta-sigma oversampled ultrasound beamformer with dynamic delays, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46 (1999) 320–332. [6] A. Fenster, D.B. Downey, 3D ultrasound imaging: a review, IEEE Eng. Med. Biol. 15 (1996) 41–51. [7] J. Kellett, Ultrasound in the palm of your hand: the dawn of a new golden age of bedside medicine, Eur. J. Int. Med. 15 (2004) 335–336. [8] J.H. Kim, T.K. Song, S.B. Park, A pipelined sampled delay focusing in ultrasound imaging systems, Ultrason. Imaging 9 (1987) 75–91. [9] K. Ranganathan, M.K. Santy, T.N. Blalock, J.A. Hossack, W.F. Walker, Direct sampled I/Q beamforming for compact and very low-

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18] [19] [20] [21]

[22]

[23]

271

cost ultrasound imaging, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51 (2004) 1082–1094. S.H. Chang, K.I. Jeon, M.H. Bae, S.B. Park, Sampling-point-wise digital dynamic focusing in the receive mode of an ultrasonic linear array imaging system, IEEE Ultrason. Symp. (1993) 1163–1165. S.E. Noujaim, S.L. Garverick, M. O’Donnell, Phased array ultrasonic beam forming using oversampled A/D converters, US Patent No. 5,203,335, 1993. M. Kozak, M. Karaman, Digital phased array beamforming using single-bit delta–sigma conversion with non-uniform oversampling, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48 (2001) 922–931. M. Karaman, A. Atalar, H. Koymeh, VLSI circuits for adaptive digital beamforming in ultrasound imaging, IEEE Trans. Med. Imaging 12 (1993) 711–720. C.R. Hazard, G.R. Lockwood, Theoretical assessment of a synthetic aperture beamformer for real-time 3-D imaging, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46 (1999) 972–980. Y.C. Lim, S.R. Parker, FIR filter design over a discrete powers-oftwo coefficient space, IEEE Trans. Acoust. Speech Signal Process. 31 (1983) 583–591. Y.M. Yoo, W.Y. Lee, T.K. Song, Correlator design using genetic algorithms for medical imaging, in: Int. Conf. on Artif. Intell., 2000, pp. 1149–1152. W.M. Wells III, Efficient synthesis of Gaussian filters by cascaded uniform filters, IEEE Trans. Pattern Anal. Mach. Intell. 8 (1986) 234– 239. J. Shen, W. Shen, S. Castan, T. Zhang, Sum-box technique for fast linear filtering, Signal Process. 82 (2002) 1109–1126. R. Andonie, E. Carai, Gaussian smoothing by optimal iterated uniform convolutions, Comput. Artif. Intell. 11 (1992) 363–373. J.A. Jensen, Field: a program for simulating ultrasound systems, Med. Biol. Eng. Comput. 34 (1996) 351–353. K.L. Gammelmark, J.A. Jensen, Multielement synthetic transmit aperture imaging using temporal encoding, IEEE Trans. Med. Imaging 22 (2003) 552–563. A.R. Omondi, Computer Arithmetic System—Algorithms, Architecture and Implementations, Prentice Hall International, Hertfordshire, UK, 1994. E.H. Kim, J.J. Kim, F.A. Matsen III, Y. Kim, Distributed diagnosis and home healthcare (D2H2) and patient-centered electronic medical record, in: Proc. of Int. Bioeng. Conf., Singapore, 2004, pp. 461– 468.