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On the stability of wideband beam patterns with respect to weighting and envelope fluctuations
Ultrasonics 42 (2004) 997–1003 www.elsevier.com/locate/ultras
On the stability of wideband beam patterns with respect to weighting and envelope fluctuations Simone Curletto a, Caterina Parodi b, Andrea Trucco a
a,*
Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, Via Opera Pia 11A, Genova 16145, Italy b Esaote Biomedica S.p.A., Genoa, Italy
Abstract This paper presents a study on the worsening caused by a random perturbation of the weighting window (modeling, in this way, the uncertainty on the channel sensitivity), or by an alteration of the shape of the transmitted pulse, for a wideband array. In the stability evaluation, main-lobe width and signal to noise ratio evaluated at different dB levels of the beam pattern play the role of quality parameters. Two different profiles of the weighting window have been used for this analysis: a typical raised cosine, and a window optimized by a Simulated Annealing procedure. Also for the pulse envelope, two distinct shapes have been chosen: perfectly Gaussian and experimentally measured. The analysis of the obtained results provides useful hints about the amount and the kind of the beam pattern worsening, when realistic fluctuations occur. Moreover, the results show that the weighting windows provided by the simulated annealing procedure are particularly robust to such fluctuations. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Wideband beam pattern; Weighting window; Pulse envelope; Sensitivity fluctuations
1. Introduction This paper refers to wideband linear beamforming [1,2]. Among all of the applications which the beamforming method is applicable to, here we consider the one referring to active acoustical imaging systems [2,3]. In wideband conditions the beam pattern (BP) depends on both the weighting window and the shape of the pulse envelope adopted [3–5], and pulses of short time duration (i.e., wideband signals) are required. Images obtained by using wideband pulses are generally of a higher quality (e.g., the range resolution improves) and potentially contain more information than those obtained when narrow-band signals are employed [2,3]. However, in wideband conditions there are more than one definition of BP [4–6]. Among all of its possible definitions, the one chosen here considers the BP as the maximum over time of the beam signals’ absolute value. The shape of the BP is strongly related to the performances of the imaging system. In particular, the main *
Corresponding author. Fax: +39-010-353-2134. E-mail address: [email protected] (A. Trucco).
0041-624X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2003.12.006
lobe width and the side lobes level are related respectively to the angular resolution and to the presence of artifacts in the acoustic images generated. This paper presents a study on the worsening caused by a random perturbation of the weighting window (modeling, in this way, the uncertainty on the channel sensitivity), or by an alteration of the shape of the transmitted pulse, for a wideband array. In the stability evaluation, the main lobe width and the signal to noise ratio (SNR) of the BP play the role of quality parameters. For our study a typical raised cosine window (used in some echographic equipment for medical applications) and a window optimized by a procedure based on simulated annealing (SA), have been considered. The optimized window has been adopted because it allows to achieve a better BP with respect to the one obtainable with the raised cosine configuration. To perform our analysis of stability, adequate probability density functions have been defined to generate the random variables that perturb the weighting coefficients of the array. In our model, each of these random variables acts on the coefficients like a multiplicative
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noise, according to a set of experimental measures carried out on a typical echographic equipment. The final results are expressed in terms of the main lobe width and of the SNR versus the perturbation variance. The paper is organized as follows. Section 2 contains the definition of the wideband BP adopted in our research, and the description of the method used for optimizing the weighting window; in Section 3 the statistical models of the perturbations applied to the weighting coefficients are described; Section 4 deals with the discussion of the results obtained for both weighting fluctuations and envelope change; finally, in Section 5 some conclusions are drawn.
2. Optimizing the weighting window Assuming the acoustic waves received from a given incidence angle by each sensor of the array to be plane (far-field hypothesis), the signals received by each sensor can be modeled as a replica of the transmitted pulse, delayed on the basis of the incidence angle and of the position of the specific linear array element [4]. Let t indicate the time, h the incidence angle of the plane wave, h0 the steering direction, and u the arbitrary variable (defined as u ¼ sin h sin h0 ) that ranges in [)2, 2]. The angle of incidence and the steering angle are both measured according to the perpendicular to the array baseline. Considering the array composed of M pointlike omnidirectional elements, the beam signal, bðt; uÞ, can be computed as follows: M1 X idu idu bðt; uÞ ¼ wi A t þ exp jx ð1Þ c c i¼0 where wi is the weight coefficient applied to the i-th element, AðtÞ is the envelope of the transmitted acoustic pulse (possibly after pulse compression), d is the interelement spacing among the array sensors (pitch), x is the angular frequency (x ¼ 2pfc , where fc is the carrier frequency of the transmitted pulse), and c is the sound velocity in the medium considered. The exponential term in (1) is due to the quadrature reception hypothesized on the signals, thus these are represented in terms of their envelope and phase information [1]. Among all of its possible definitions in wideband, in our research the BP for a given pair of incidence and steering angles, bpðuÞ, has been computed as the maximum over time of the beam signal absolute value: bpðuÞ ¼ maxfjbðt; uÞjg t
The weight coefficients synthesis method proposed in this paper is based on the simulated annealing concept [7]. The SA is a stochastic iterative procedure suitable for facing combinatorial minimization problems. As required by any optimization process, for the SA an energy function (must be defined) to be minimized. In our research, the energy has been defined and computed according to the criteria and the equation proposed in [8]. Essentially, a desired BP should be defined a priori: the energy function penalizes those array configurations that result in a considerable difference between the desired BP and the current one. If the energy function was lowered to zero, the constraints in the desired BP would be perfectly fulfilled. The SA algorithm is particularly suitable as this function is nonlinear, has many variables, and presents a large number of local minima. More details on the concept and implementation of SA can be found in [7]. To test the effectiveness of the proposed method, a phased array composed of 128 elements with a pitch equal to 0.17 mm has been addressed. Two different acoustic pulses have been considered during our study: a truncated Gaussian envelope and an experimentally measured one. Both pulses last 5.25 ls, have a carrier frequency of 2.5 MHz, and a fractional bandwidth B of 25%, measured at )6 dB. The sampling frequency of the signals was fixed at 20 MHz during the optimization step and at 200 MHz for result-testing and BP-visualization purposes. Fig. 1 shows the values of the optimized weighting coefficients, and the (interpolated) profile of the resulting window. The SA-based approach described in [8] has been used for optimizing the weighting window, fixing as the desired BP a flat side lobe profile at )50 dB till u ¼ 1:2. It can be observed that the weighting window is symmetric with respect to the central element of the array. This characteristic has been imposed a priori to
ð2Þ
It can be observed that bpðuÞ is an even function of the variable u, hence it is sufficient to study and visualize the BP over the range 0 6 u 6 2 (u ¼ 2 when h ¼ 90° and h0 ¼ 90°). The conventional visualization of the BP is on a logarithmic scale, normalized to 0 dB.
Fig. 1. Optimized weighting coefficients.
S. Curletto et al. / Ultrasonics 42 (2004) 997–1003
Fig. 2. BP profile obtained by the weights shown in Fig. 1 (solid line) and BP obtained using a typical raised cosine weighting window (dashed line).
reduce the computational load of the optimization process. Fig. 2 shows the BP obtained using the weights shown in Fig. 1 (solid line), and the BP due to a typical raised cosine weighting window (dashed line). The optimized weights result in a thinner main lobe width (about 20%) than that measured on the BP obtained with the raised cosine window, although the side lobe level in the latter case is lower. The performances achievable from these two acoustic systems may present important differences.
3. Characterizing sensitivity fluctuations To evaluate the stability of the BP with respect to the fluctuations of the channels used to receive the acoustic echoes, a noise model based on experimental measurements has been determined. Particularly, on each channel we have considered two noise types having different natures: one from gain variations in the echographic system receiving channels, and one from gain variations in the probe elements. The channels’ sensitivity has been measured during some tests carried out on an echographic equipment having 192 channels. The gain of each channel has been defined as the ratio between the amplitude of the acquired echo and the amplitude of the excitation pulse, provided that the echo is just an attenuated replica of the electrical excitation pulse. Together these assumptions allow to avoid the generation of the acoustic pulse, and therefore to perform the measurements independently from the fluctuations of the transducers’ sensitivity. For each channel, the mean of the values obtained after many repetitions has been computed. Two different
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versions of a phased array composed of 128 elements (the same described in Section 2) have been considered to show the variability of the array transducers’ gains. The gain of each transducer has been experimentally measured by tank experiments. Once the values of the gains of the channels and of the transducers were collected, they were normalized, and a statistical model was selected. For this purpose, a histogram has been generated for each type of the above mentioned random variables. Fig. 3 shows the histogram of the receiving channel gains, computed by defining 38 columns in the range [0.65, 1.0]. The two limits of the interval are the minimum and the maximum of the normalized receiving channels’ values. In order to obtain a valid statistical model, the probability density function (PDF) has been hypothesized for each of the random variables considered. Given that the number of the available realizations for each random variable was low, a rigorous estimation was not possible. Therefore, an heuristic approach has been adopted that was based on several comparisons between the histograms of the experimental data (e.g., Fig. 3) and those obtained for synthetic random variables, having a uniform or a Gaussian PDF. This approach has suggested to assume a uniform PDF in [0.71, 0.95] for the spread of the receiving channel gains. The two limits of the interval have been chosen to match the mean value and the variance of the experimental data, and in accordance to the profile of the histogram in Fig. 3. An analogous procedure has been adopted to determine the PDF of the array transducers’ gains. For the first version of phased array considered, each transducer gain has been modeled as a Gaussian random variable, with a mean value equal to 0.94, and a standard deviation equal to 0.03. In the second version of phased array considered, each
Fig. 3. Histogram of the receiving channel gains, computed by using 38 columns in the range [0.65, 1.0].
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transducer gain is a random variable that presents a histogram which is more similar to that of a uniform random variable ranging in [0.87, 0.98]. As two different PDFs for the transducers’ gains have been adopted, two different global models have been determined. The first one (denoted as ‘‘UG’’) has been obtained by multiplying the channels’ gains (uniform PDF) by the transducers’ gains of the first version (Gaussian PDF), while the second (denoted as ‘‘UU’’) has been obtained by multiplying the channels’ gains (uniform PDF) by the transducers’ gains of the second version (uniform PDF). Weighting windows taking into account the fluctuations caused by the receiving channels and the array transducers, were obtained by multiplying each element of the original window for one realization of the UG or UU noise model. To produce the results presented in the next Section, we adopted the following approach: (1) to generate many realizations of a UG variable and of a UU random variable; (2) to apply the realizations obtained at the previous step to the optimized and to the raised cosine weighting windows; (3) to compare the original BP and the BP after perturbation, for each of the two noise models, and for each of the two weighting windows.
4. Results and discussion 4.1. Weighting fluctuations Fig. 4 shows an example of the effects of the fluctuations described above on the BP profile, for both the raised cosine window and the optimized window. These results refer to a given vector of 128 realizations of the UU noise model: the UG model has not been considered
in Fig. 4, since the worsening in terms of BP quality is similar in both cases. The BPs displayed in Fig. 4 have been obtained with the experimentally measured envelope mentioned in Section 2. Examining Fig. 4, it can be observed that the BP quality decreases when the perturbations are applied to the weights of the array sensors: while the main lobe profile remains roughly the same, the side lobes level becomes higher in both cases. The BP obtained with the optimized window shows a worsening in the side lobes level that is smaller than that observed in the BP generated by the raised cosine window. The sharp worsening of the BP generated by the raised cosine reveals that the nice profile of the unperturbed BP is never exploited in real systems where fluctuations of this extent surely occur. The perturbed BPs shown in Fig. 4 present several similarities that are due to the application of the same vector of 128 realizations to both the raised cosine window and the SA-optimized window. However, it is important to notice that, despite the similarity, the better angular resolution obtained by the SA-optimized window is maintained also after perturbation. In order to carry out a more rigorous analysis, the effects of the weighting fluctuations have been quantified in terms of average SNR and average angular resolution, both measured at different reference levels. Once a reference dB level has been chosen, the SNR for that level is defined as the ratio of the BP area taken from u ¼ 0 to the value of u corresponding to the dB level chosen, and the BP area from this point to u ¼ 2. The angular resolution is equivalent to the main lobe width, measured at different dB level. The averages of the SNR and the angular resolution have been obtained by applying many different realizations of the random fluctuations (following the UG model or the UU model) to the
Fig. 4. (a) Comparison between the BPs obtained by using the raised cosine window perturbed (dashed line) and unperturbed (solid line). (b) The same BP comparison of the panel (a) when the optimized weighting window is considered.
S. Curletto et al. / Ultrasonics 42 (2004) 997–1003
weights of the considered windows (i.e., raised cosine and optimized by the SA-based method) and analyzing the obtained BP. Fig. 5(a) shows the comparison between the angular resolution obtained by the unperturbed raised cosine window (dashed line) and by the unperturbed optimized window (solid line). The improvement due to the opti-
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mized window is retained when the weights are perturbed by the random variables that follow the UU model or the UG model, as shown, respectively, in Fig. 5(c) and (e). Comparing the panels (a), (c), and (e) one can also notice that the weights fluctuations do not alter the main lobe width till )30 dB, whereas they produce a growth of the width measured at )40 dB.
Fig. 5. (a) Angular resolution of the BPs obtained with the optimized window (solid line) and the raised cosine window (dashed line). (b) SNR of the BPs obtained with the optimized window (solid line) and the raised cosine window (dashed line). The worsening of the average angular resolution when fluctuations following the UU model and the UG model are presented in panels (c) and (e), respectively. The worsening of the average SNR when fluctuations following the UU model and the UG model are presented in panels (d) and (f), respectively.
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Fig. 6. (a) Comparison of the BPs obtained using the raised cosine window in conjunction with the Gaussian envelope (solid line) or the experimentally measured envelope (dashed line). (b) The same comparison of panel (a) when the optimized window is considered.
Concerning the SNR, when the weights are unperturbed, the optimized window offers a poorer performance than the raised cosine window, as shown in Fig. 5(b). However, this disadvantage is sharply reduced when the weights are perturbed by random variables that follow the UU model or the UG model, as shown in Fig. 5(d) and (f), respectively. It can be observed that, once the fluctuations are applied to the weights, the worsening of this parameter is lower for the optimized window ( 4 dB for a reference level of )40 dB) than for the raised cosine window ( 15 dB for a reference level of )40 dB). This worsening is related to the increase in the side lobe level due to the perturbation of the weights’ values, as it can be observed in Fig. 4(a) and (b). In particular, the nice side lobes profile produced by the raised cosine window is not maintained when fluctuations of realistic entity are applied, as shown in Fig. 4(a). The reason for which the solution provided by the SA-based method is more robust than the raised cosine relies on the conceptual nature of SA. This algorithm tries to find the global minimum of the energy function by successive refinements of the current status; in other words, it approaches the final solution by a sequence of little approximations. Due to this fact, the final solution is rarely unstable. For instance, if the global minimum is unstable, SA will probably not succeed in finding the global minimum; it will stop inside the basin of a stable local minimum with an energy value close to that of the global minimum. This means that the BP produced by the weighting window synthesized by the SA-based method is most frequently very robust to random fluctuations. Instead, the raised cosine window follows an exact mathematical equation: the effects produced on the related BP by random fluctuations are difficult to foresee.
4.2. Envelope shape In our research we also considered the effects on the BP when changing the envelope shape. In particular, we evaluated the BP using a Gaussian envelope instead of the experimentally measured one, with the same global characteristics (i.e., B ¼ 25%, sampling frequency equal to 200 MHz, time duration equal to 5.25 ls, and carrier frequency of 2.5 MHz) maintained for the two pulses. Fig. 6(a) shows the comparison between the BP obtained with the raised cosine window and with the Gaussian envelope (solid line) or with the experimental envelope (dashed line). Fig. 6(b) show the same comparison when the optimized window is applied. Examining Fig. 6(b), it clearly appears that the optimized window is robust to the fluctuations of the envelope shape, providing very similar BPs in the two cases. Instead, the Gaussian envelope in conjunction with the raised cosine window produces a BP improvement (see Fig. 6(a)) that is just virtual, as it disappears if realistic fluctuations of the weights are introduced. In any case, this (virtual) improvement stresses the fact that the BP produced with the raised cosine window is less stable with respect to the envelope changes than that produced by the optimized window.
5. Conclusions Theoretical performances of wideband transducer arrays often disagree with their real behaviour; in particular, fluctuations of the channel sensitivities (due to a few different causes) could be harmful for the entire imaging process.
S. Curletto et al. / Ultrasonics 42 (2004) 997–1003
In this paper, a study on the worsening of the BP caused by random fluctuations, having statistical characteristics similar to those experimentally observed, has been presented, considering both a typical raised cosine weighting window and a window optimised by a SAbased method. Also the consequences of the alteration of the shape of the acoustic pulse have been briefly studied. The analysis of the obtained results provides useful hints about the amount and the kind of the beam pattern worsening, when realistic fluctuations occur. Moreover, the results show that the weighting windows provided by the SA-based method are particularly robust to such fluctuations and to pulse shape alterations. Therefore, the SA-based method is well suited to synthesize a weighting window that produces a theoretical BP close to the BP measured in real systems. In other words, the obtained system performances will be similar to the expected ones. On the contrary, the nice BP profile theoretically yielded by the raised cosine window is actually not achieved in real systems, where gain fluctuations and pulse alteration surely occur; consequently, measured performances can be much poorer than the expected ones.
Acknowledgements This work has been funded by Esaote Biomedica S.p.A, Genoa, Italy. The authors would like to thank
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Prof. F. Bertora, Dr. P. Pellegretti, and Dr. A. Questa for their valuable support in collecting and analysing experimental data. Moreover, the authors would like to thank the anonymous reviewers, whose constructive comments have surely contributed to improve the quality of this paper.
References [1] R.O. Nielsen, Sonar Signal Processing, Artech House, Boston, 1991. [2] V. Murino, A. Trucco, Three-dimensional image generation and processing in underwater acoustic vision, The Proceedings of the IEEE 88 (2000) 1903–1946. [3] A. Macovski, Ultrasonic imaging using arrays, IEEE Proceedings 67 (1979) 484–495. [4] V. Murino, A. Trucco, A. Tesei, Beam pattern formulation and analysis for wide-band beamforming systems using sparse arrays, Signal Processing 56 (2) (1997) 177–183. [5] J.L. Schwartz, B.D. Steinberg, Ultrasparse, ultrawideband arrays, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 45 (2) (1998) 376–393. [6] G. Cardone, G. Cincotti, M. Pappalardo, Design of wide-band arrays for low side-lobe level beam patterns by simulated annealing, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 49 (8) (2002) 1050–1059. [7] S. Kirkpatrick, C.D. Gellatt Jr., M.P. Vecchi, Optimization by simulated annealing, Science 220 (4598) (1983) 671–680. [8] A. Trucco, A stochastic approach to optimise wide-band beam patterns, in: M. Halliwell, P.N.T. Wells (Eds.), Acoustical Imaging, vol. 25, Kluwer Academic/Plenum Press, New York, 2000, pp. 123– 130.
On the stability of wideband beam patterns with respect to weighting and envelope fluctuations Simone Curletto a, Caterina Parodi b, Andrea Trucco a
a,*
Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, Via Opera Pia 11A, Genova 16145, Italy b Esaote Biomedica S.p.A., Genoa, Italy
Abstract This paper presents a study on the worsening caused by a random perturbation of the weighting window (modeling, in this way, the uncertainty on the channel sensitivity), or by an alteration of the shape of the transmitted pulse, for a wideband array. In the stability evaluation, main-lobe width and signal to noise ratio evaluated at different dB levels of the beam pattern play the role of quality parameters. Two different profiles of the weighting window have been used for this analysis: a typical raised cosine, and a window optimized by a Simulated Annealing procedure. Also for the pulse envelope, two distinct shapes have been chosen: perfectly Gaussian and experimentally measured. The analysis of the obtained results provides useful hints about the amount and the kind of the beam pattern worsening, when realistic fluctuations occur. Moreover, the results show that the weighting windows provided by the simulated annealing procedure are particularly robust to such fluctuations. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Wideband beam pattern; Weighting window; Pulse envelope; Sensitivity fluctuations
1. Introduction This paper refers to wideband linear beamforming [1,2]. Among all of the applications which the beamforming method is applicable to, here we consider the one referring to active acoustical imaging systems [2,3]. In wideband conditions the beam pattern (BP) depends on both the weighting window and the shape of the pulse envelope adopted [3–5], and pulses of short time duration (i.e., wideband signals) are required. Images obtained by using wideband pulses are generally of a higher quality (e.g., the range resolution improves) and potentially contain more information than those obtained when narrow-band signals are employed [2,3]. However, in wideband conditions there are more than one definition of BP [4–6]. Among all of its possible definitions, the one chosen here considers the BP as the maximum over time of the beam signals’ absolute value. The shape of the BP is strongly related to the performances of the imaging system. In particular, the main *
Corresponding author. Fax: +39-010-353-2134. E-mail address: [email protected] (A. Trucco).
0041-624X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2003.12.006
lobe width and the side lobes level are related respectively to the angular resolution and to the presence of artifacts in the acoustic images generated. This paper presents a study on the worsening caused by a random perturbation of the weighting window (modeling, in this way, the uncertainty on the channel sensitivity), or by an alteration of the shape of the transmitted pulse, for a wideband array. In the stability evaluation, the main lobe width and the signal to noise ratio (SNR) of the BP play the role of quality parameters. For our study a typical raised cosine window (used in some echographic equipment for medical applications) and a window optimized by a procedure based on simulated annealing (SA), have been considered. The optimized window has been adopted because it allows to achieve a better BP with respect to the one obtainable with the raised cosine configuration. To perform our analysis of stability, adequate probability density functions have been defined to generate the random variables that perturb the weighting coefficients of the array. In our model, each of these random variables acts on the coefficients like a multiplicative
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S. Curletto et al. / Ultrasonics 42 (2004) 997–1003
noise, according to a set of experimental measures carried out on a typical echographic equipment. The final results are expressed in terms of the main lobe width and of the SNR versus the perturbation variance. The paper is organized as follows. Section 2 contains the definition of the wideband BP adopted in our research, and the description of the method used for optimizing the weighting window; in Section 3 the statistical models of the perturbations applied to the weighting coefficients are described; Section 4 deals with the discussion of the results obtained for both weighting fluctuations and envelope change; finally, in Section 5 some conclusions are drawn.
2. Optimizing the weighting window Assuming the acoustic waves received from a given incidence angle by each sensor of the array to be plane (far-field hypothesis), the signals received by each sensor can be modeled as a replica of the transmitted pulse, delayed on the basis of the incidence angle and of the position of the specific linear array element [4]. Let t indicate the time, h the incidence angle of the plane wave, h0 the steering direction, and u the arbitrary variable (defined as u ¼ sin h sin h0 ) that ranges in [)2, 2]. The angle of incidence and the steering angle are both measured according to the perpendicular to the array baseline. Considering the array composed of M pointlike omnidirectional elements, the beam signal, bðt; uÞ, can be computed as follows: M1 X idu idu bðt; uÞ ¼ wi A t þ exp jx ð1Þ c c i¼0 where wi is the weight coefficient applied to the i-th element, AðtÞ is the envelope of the transmitted acoustic pulse (possibly after pulse compression), d is the interelement spacing among the array sensors (pitch), x is the angular frequency (x ¼ 2pfc , where fc is the carrier frequency of the transmitted pulse), and c is the sound velocity in the medium considered. The exponential term in (1) is due to the quadrature reception hypothesized on the signals, thus these are represented in terms of their envelope and phase information [1]. Among all of its possible definitions in wideband, in our research the BP for a given pair of incidence and steering angles, bpðuÞ, has been computed as the maximum over time of the beam signal absolute value: bpðuÞ ¼ maxfjbðt; uÞjg t
The weight coefficients synthesis method proposed in this paper is based on the simulated annealing concept [7]. The SA is a stochastic iterative procedure suitable for facing combinatorial minimization problems. As required by any optimization process, for the SA an energy function (must be defined) to be minimized. In our research, the energy has been defined and computed according to the criteria and the equation proposed in [8]. Essentially, a desired BP should be defined a priori: the energy function penalizes those array configurations that result in a considerable difference between the desired BP and the current one. If the energy function was lowered to zero, the constraints in the desired BP would be perfectly fulfilled. The SA algorithm is particularly suitable as this function is nonlinear, has many variables, and presents a large number of local minima. More details on the concept and implementation of SA can be found in [7]. To test the effectiveness of the proposed method, a phased array composed of 128 elements with a pitch equal to 0.17 mm has been addressed. Two different acoustic pulses have been considered during our study: a truncated Gaussian envelope and an experimentally measured one. Both pulses last 5.25 ls, have a carrier frequency of 2.5 MHz, and a fractional bandwidth B of 25%, measured at )6 dB. The sampling frequency of the signals was fixed at 20 MHz during the optimization step and at 200 MHz for result-testing and BP-visualization purposes. Fig. 1 shows the values of the optimized weighting coefficients, and the (interpolated) profile of the resulting window. The SA-based approach described in [8] has been used for optimizing the weighting window, fixing as the desired BP a flat side lobe profile at )50 dB till u ¼ 1:2. It can be observed that the weighting window is symmetric with respect to the central element of the array. This characteristic has been imposed a priori to
ð2Þ
It can be observed that bpðuÞ is an even function of the variable u, hence it is sufficient to study and visualize the BP over the range 0 6 u 6 2 (u ¼ 2 when h ¼ 90° and h0 ¼ 90°). The conventional visualization of the BP is on a logarithmic scale, normalized to 0 dB.
Fig. 1. Optimized weighting coefficients.
S. Curletto et al. / Ultrasonics 42 (2004) 997–1003
Fig. 2. BP profile obtained by the weights shown in Fig. 1 (solid line) and BP obtained using a typical raised cosine weighting window (dashed line).
reduce the computational load of the optimization process. Fig. 2 shows the BP obtained using the weights shown in Fig. 1 (solid line), and the BP due to a typical raised cosine weighting window (dashed line). The optimized weights result in a thinner main lobe width (about 20%) than that measured on the BP obtained with the raised cosine window, although the side lobe level in the latter case is lower. The performances achievable from these two acoustic systems may present important differences.
3. Characterizing sensitivity fluctuations To evaluate the stability of the BP with respect to the fluctuations of the channels used to receive the acoustic echoes, a noise model based on experimental measurements has been determined. Particularly, on each channel we have considered two noise types having different natures: one from gain variations in the echographic system receiving channels, and one from gain variations in the probe elements. The channels’ sensitivity has been measured during some tests carried out on an echographic equipment having 192 channels. The gain of each channel has been defined as the ratio between the amplitude of the acquired echo and the amplitude of the excitation pulse, provided that the echo is just an attenuated replica of the electrical excitation pulse. Together these assumptions allow to avoid the generation of the acoustic pulse, and therefore to perform the measurements independently from the fluctuations of the transducers’ sensitivity. For each channel, the mean of the values obtained after many repetitions has been computed. Two different
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versions of a phased array composed of 128 elements (the same described in Section 2) have been considered to show the variability of the array transducers’ gains. The gain of each transducer has been experimentally measured by tank experiments. Once the values of the gains of the channels and of the transducers were collected, they were normalized, and a statistical model was selected. For this purpose, a histogram has been generated for each type of the above mentioned random variables. Fig. 3 shows the histogram of the receiving channel gains, computed by defining 38 columns in the range [0.65, 1.0]. The two limits of the interval are the minimum and the maximum of the normalized receiving channels’ values. In order to obtain a valid statistical model, the probability density function (PDF) has been hypothesized for each of the random variables considered. Given that the number of the available realizations for each random variable was low, a rigorous estimation was not possible. Therefore, an heuristic approach has been adopted that was based on several comparisons between the histograms of the experimental data (e.g., Fig. 3) and those obtained for synthetic random variables, having a uniform or a Gaussian PDF. This approach has suggested to assume a uniform PDF in [0.71, 0.95] for the spread of the receiving channel gains. The two limits of the interval have been chosen to match the mean value and the variance of the experimental data, and in accordance to the profile of the histogram in Fig. 3. An analogous procedure has been adopted to determine the PDF of the array transducers’ gains. For the first version of phased array considered, each transducer gain has been modeled as a Gaussian random variable, with a mean value equal to 0.94, and a standard deviation equal to 0.03. In the second version of phased array considered, each
Fig. 3. Histogram of the receiving channel gains, computed by using 38 columns in the range [0.65, 1.0].
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transducer gain is a random variable that presents a histogram which is more similar to that of a uniform random variable ranging in [0.87, 0.98]. As two different PDFs for the transducers’ gains have been adopted, two different global models have been determined. The first one (denoted as ‘‘UG’’) has been obtained by multiplying the channels’ gains (uniform PDF) by the transducers’ gains of the first version (Gaussian PDF), while the second (denoted as ‘‘UU’’) has been obtained by multiplying the channels’ gains (uniform PDF) by the transducers’ gains of the second version (uniform PDF). Weighting windows taking into account the fluctuations caused by the receiving channels and the array transducers, were obtained by multiplying each element of the original window for one realization of the UG or UU noise model. To produce the results presented in the next Section, we adopted the following approach: (1) to generate many realizations of a UG variable and of a UU random variable; (2) to apply the realizations obtained at the previous step to the optimized and to the raised cosine weighting windows; (3) to compare the original BP and the BP after perturbation, for each of the two noise models, and for each of the two weighting windows.
4. Results and discussion 4.1. Weighting fluctuations Fig. 4 shows an example of the effects of the fluctuations described above on the BP profile, for both the raised cosine window and the optimized window. These results refer to a given vector of 128 realizations of the UU noise model: the UG model has not been considered
in Fig. 4, since the worsening in terms of BP quality is similar in both cases. The BPs displayed in Fig. 4 have been obtained with the experimentally measured envelope mentioned in Section 2. Examining Fig. 4, it can be observed that the BP quality decreases when the perturbations are applied to the weights of the array sensors: while the main lobe profile remains roughly the same, the side lobes level becomes higher in both cases. The BP obtained with the optimized window shows a worsening in the side lobes level that is smaller than that observed in the BP generated by the raised cosine window. The sharp worsening of the BP generated by the raised cosine reveals that the nice profile of the unperturbed BP is never exploited in real systems where fluctuations of this extent surely occur. The perturbed BPs shown in Fig. 4 present several similarities that are due to the application of the same vector of 128 realizations to both the raised cosine window and the SA-optimized window. However, it is important to notice that, despite the similarity, the better angular resolution obtained by the SA-optimized window is maintained also after perturbation. In order to carry out a more rigorous analysis, the effects of the weighting fluctuations have been quantified in terms of average SNR and average angular resolution, both measured at different reference levels. Once a reference dB level has been chosen, the SNR for that level is defined as the ratio of the BP area taken from u ¼ 0 to the value of u corresponding to the dB level chosen, and the BP area from this point to u ¼ 2. The angular resolution is equivalent to the main lobe width, measured at different dB level. The averages of the SNR and the angular resolution have been obtained by applying many different realizations of the random fluctuations (following the UG model or the UU model) to the
Fig. 4. (a) Comparison between the BPs obtained by using the raised cosine window perturbed (dashed line) and unperturbed (solid line). (b) The same BP comparison of the panel (a) when the optimized weighting window is considered.
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weights of the considered windows (i.e., raised cosine and optimized by the SA-based method) and analyzing the obtained BP. Fig. 5(a) shows the comparison between the angular resolution obtained by the unperturbed raised cosine window (dashed line) and by the unperturbed optimized window (solid line). The improvement due to the opti-
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mized window is retained when the weights are perturbed by the random variables that follow the UU model or the UG model, as shown, respectively, in Fig. 5(c) and (e). Comparing the panels (a), (c), and (e) one can also notice that the weights fluctuations do not alter the main lobe width till )30 dB, whereas they produce a growth of the width measured at )40 dB.
Fig. 5. (a) Angular resolution of the BPs obtained with the optimized window (solid line) and the raised cosine window (dashed line). (b) SNR of the BPs obtained with the optimized window (solid line) and the raised cosine window (dashed line). The worsening of the average angular resolution when fluctuations following the UU model and the UG model are presented in panels (c) and (e), respectively. The worsening of the average SNR when fluctuations following the UU model and the UG model are presented in panels (d) and (f), respectively.
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Fig. 6. (a) Comparison of the BPs obtained using the raised cosine window in conjunction with the Gaussian envelope (solid line) or the experimentally measured envelope (dashed line). (b) The same comparison of panel (a) when the optimized window is considered.
Concerning the SNR, when the weights are unperturbed, the optimized window offers a poorer performance than the raised cosine window, as shown in Fig. 5(b). However, this disadvantage is sharply reduced when the weights are perturbed by random variables that follow the UU model or the UG model, as shown in Fig. 5(d) and (f), respectively. It can be observed that, once the fluctuations are applied to the weights, the worsening of this parameter is lower for the optimized window ( 4 dB for a reference level of )40 dB) than for the raised cosine window ( 15 dB for a reference level of )40 dB). This worsening is related to the increase in the side lobe level due to the perturbation of the weights’ values, as it can be observed in Fig. 4(a) and (b). In particular, the nice side lobes profile produced by the raised cosine window is not maintained when fluctuations of realistic entity are applied, as shown in Fig. 4(a). The reason for which the solution provided by the SA-based method is more robust than the raised cosine relies on the conceptual nature of SA. This algorithm tries to find the global minimum of the energy function by successive refinements of the current status; in other words, it approaches the final solution by a sequence of little approximations. Due to this fact, the final solution is rarely unstable. For instance, if the global minimum is unstable, SA will probably not succeed in finding the global minimum; it will stop inside the basin of a stable local minimum with an energy value close to that of the global minimum. This means that the BP produced by the weighting window synthesized by the SA-based method is most frequently very robust to random fluctuations. Instead, the raised cosine window follows an exact mathematical equation: the effects produced on the related BP by random fluctuations are difficult to foresee.
4.2. Envelope shape In our research we also considered the effects on the BP when changing the envelope shape. In particular, we evaluated the BP using a Gaussian envelope instead of the experimentally measured one, with the same global characteristics (i.e., B ¼ 25%, sampling frequency equal to 200 MHz, time duration equal to 5.25 ls, and carrier frequency of 2.5 MHz) maintained for the two pulses. Fig. 6(a) shows the comparison between the BP obtained with the raised cosine window and with the Gaussian envelope (solid line) or with the experimental envelope (dashed line). Fig. 6(b) show the same comparison when the optimized window is applied. Examining Fig. 6(b), it clearly appears that the optimized window is robust to the fluctuations of the envelope shape, providing very similar BPs in the two cases. Instead, the Gaussian envelope in conjunction with the raised cosine window produces a BP improvement (see Fig. 6(a)) that is just virtual, as it disappears if realistic fluctuations of the weights are introduced. In any case, this (virtual) improvement stresses the fact that the BP produced with the raised cosine window is less stable with respect to the envelope changes than that produced by the optimized window.
5. Conclusions Theoretical performances of wideband transducer arrays often disagree with their real behaviour; in particular, fluctuations of the channel sensitivities (due to a few different causes) could be harmful for the entire imaging process.
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In this paper, a study on the worsening of the BP caused by random fluctuations, having statistical characteristics similar to those experimentally observed, has been presented, considering both a typical raised cosine weighting window and a window optimised by a SAbased method. Also the consequences of the alteration of the shape of the acoustic pulse have been briefly studied. The analysis of the obtained results provides useful hints about the amount and the kind of the beam pattern worsening, when realistic fluctuations occur. Moreover, the results show that the weighting windows provided by the SA-based method are particularly robust to such fluctuations and to pulse shape alterations. Therefore, the SA-based method is well suited to synthesize a weighting window that produces a theoretical BP close to the BP measured in real systems. In other words, the obtained system performances will be similar to the expected ones. On the contrary, the nice BP profile theoretically yielded by the raised cosine window is actually not achieved in real systems, where gain fluctuations and pulse alteration surely occur; consequently, measured performances can be much poorer than the expected ones.
Acknowledgements This work has been funded by Esaote Biomedica S.p.A, Genoa, Italy. The authors would like to thank
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Prof. F. Bertora, Dr. P. Pellegretti, and Dr. A. Questa for their valuable support in collecting and analysing experimental data. Moreover, the authors would like to thank the anonymous reviewers, whose constructive comments have surely contributed to improve the quality of this paper.
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