4.3 Fast Computation of Wideband Beam Pattern
4.3.3 Performance Evaluation
The proposed method is compared with some existing methods, including the time-domain method, the frequency-time-domain direct method and the ultrasound simulator Field II [21,22] through computing wideband beam patterns of 2-D arrays. All the results are obtained on the same personal computer (HP Pro3000 MT E5700) and all the methods are implemented by Matlab. In the implementation, there are no any optimized libraries or accelerating operations such as multithreading, parallel processes being used. Figure4.6shows the excitation signal, whose carrier frequency and the bandwidth are 500 and 300 kHz respectively. The expression of the excitation signal is given [20] by serve as point sources. The 2-D array shown in Fig. 4.7is considered as equally spaced for comparisons. The interspacing between two adjacent elements isλ0/2, whereλ0is the wavelength corresponding to the central frequency f0.
For the purpose of comparison, the method that computes (4.16) directly in the frequency domain, and then uses the IDFT with the length ND F T to obtain the time-domain maximum values is applied and referred to as the frequency-time-domain direct method, which is accurate but high in time-consumption. The IDFT is realized by the IFFT, whose length NI F F T is the first power of two which is greater than or equal to ND F T. The steps of the frequency-domain direct method are as follows: complete the computation of (4.16) directly, then apply the IFFT with the length NI F F T to extract the accurate time-domain maximum values.
The wideband beam patterns are also computed in the time domain, which is referred to as the time-domain method. The sampling frequency should be high enough to obtain the accurate results of the domain expressions (4.9). The time-domain method adopted computes all the sampled waveforms that every element radiates in all the directions. Unfortunately, computing waveforms on the fly takes too much time. When the wideband 2-D arrays are relatively large, the time cost by the time-domain method may be excessively long. Here, we set a criterion that if the
Fig. 4.6 The wideband excitation signal:
a time-domain waveform;
b spectrum
66 4 Design of Underwater Large Sparse 2-D Arrays
Fig. 4.7 The uniform 2-D array with N0elements (the inter-element spacing of the uniform arrays isλ0/2)
time exceeds 24 h, namely 8.64× 104 s, the time-domain method is thought to be failed for computing wideband beam pattern.
To evaluate the accuracy of different methods, the results B PD Mof the frequency-domain direct method are taken as reference. The relative error is given by
Er
B P− B PD M
B PD M
× 100%. (4.31)
The sampling frequency of the time-domain method aforementioned is set as 20 MHz, which is larger than the central frequency 500 kHz to guarantee accuracy.
For Field II [21,22], the sampling frequency is 20 MHz and the elements are set to be punctiform to achieve the fast computational speed.
Since the proposed method is an approximated one, a simulation is conducted to evaluate the accuracy of the algorithm. As analyzed in previous sections, the key parameters that determine the computational accuracy and time are the oversampled frequency σ and the interpolated factor J of NUFFT. In order to calculate error with (4.31), the accurate beam patterns needs to be computed with the frequency-domain direct method. A 2-D array with 100× 100 elements is chosen, as shown in Fig.4.7. Table4.1present the computational relative errors and time consumptions at different values ofσ and J for the 2-D array. From Table4.1, it can be found that the relative error and the computational time are mainly determined byσ and slightly dependent on J. With the increase ofσ , the relative error decreases dramatically and the computational time increases moderately. Whenσ 2, the relative errors become small enough and remain nearly constant (0.15%) with the increase of J, while the computational time increases slightly. Thus,σ 2 and J 4 are chosen for the following computation of this book.
Table 4.1 Computational time and relative errors of the proposed method at different values ofσ and J for the 2-D array with 100×100 elements
σ 1, J
σ: the oversampled factor; J: the interpolated factor
Table 4.2 Computing time and relative errors of wideband beam pattern using the frequency-domain direct method, the time-frequency-domain method, Field II and the proposed method for 2-D arrays
N0 8 × 8
Time domain 12.04 0.58 3.21× 104 1.12 Failed Failed
Field II 0.08 0.39 70.35 2.14 11.2×102 2.46
Proposed1 0.13 0.04 8.85 0.12 35.30 0.15
Proposed2 0.04 0.04 0.93 0.12 7.02 0.15
Proposed1: the proposed method with only NUFFT; Proposed2: the proposed method using both NUFFT and short IFFT; N0: the number of elements; P× Q: the number of beams
The next simulation is to evaluate the computational speeds of the proposed method for different arrays. Since the proposed method employs two techniques:
NUFFT and the short IFFT, the respective contributions of NUFFT and the short IFFT are separately evaluated. In Table 4.2, Proposed1 represents the proposed method with only NUFFT, and Proposed2 represents the proposed method using both NUFFT and the short IFFT.
Table4.2indicates the computing time and the relative errors of wideband beam patterns of 2-D arrays with different number N0of elements and different number P× Q of beams. It is found that for large 2-D arrays, Poposed2 provides about two orders of magnitude faster computing speed compared with Field II, and about three orders of magnitude that compared with the frequency-domain direct method. For the smallest 2-D array with N0 8×8, the speeds of the time-domain method, Field II, Proposed1 and Proposed2 are nearly in the same order of magnitude. Compared with that of Proposed1, the computational time of Proposed2 is reduced by about three and five times at N0 8 × 8 and 100 × 100. It can be seen that the short IFFT helps achieve five-times gain in computational speed for large 2-D arrays. Figure4.8 shows the wideband beam patterns of using different methods for N0 100 × 100 respectively, which demonstrates that the results of different methods match very well. In addition, when N0 100 × 100, the time-domain method fails because the computational time is excessively long.
68 4 Design of Underwater Large Sparse 2-D Arrays
Fig. 4.8 Comparison among the wideband beam patterns obtained by the frequency-domain direct method, the time-domain method, Field II and the proposed fast method for the 2-D array with N0 100 × 100. a The direct method. b Field II. c The proposed fast method. d The diagonal line view of the wideband beam patterns (the time-domain method failed because the computational time is excessive)
For the proposed method of computing wideband beam pattern, the time savings are mainly determined by the bandwidth and the array size when the errors’ limitation is determined. To reflect the influence of the bandwidth, the simulations with different bandwidths are carried out. For the 2-D array with N0 100×100 shown in Fig.4.7, when the bandwidth is 200 kHz, the computational time of the proposed method is 4.63 s withσ 2 and J 4, which is about 2/3 of 7.02 s achieved in the bandwidth of 300 kHz. The comparison verifies that the bandwidth is the key that affects the speed of the proposed method when the errors’ limitation is determined.
From Table4.2, the proposed method shows no obvious advantages for small arrays. For a 2-D array with 8× 8 elements and the number of the beams being 16 × 16, the time consumption of the proposed method is approximate to that of the time-domain method and Field II. The proposed method is superior in saving time when the number of array elements and the number of beams are more than aforementioned values.
At last, a simulation is conducted to verify the feasibility of the fast computation of wideband beam pattern for arbitrary 2-D arrays with the proposed method. A uniform-distribution 2-D random array with 2000 elements [23–25] and a spiral array with 1000 elements [26] are chosen and shown in Fig. 4.9. Owing to the excessively long computing time, the time-domain method cannot be available in this case. Besides, according to the users’ guide [22] of Field II, the standard program of Field II cannot compute the wideband beam pattern of the 2-D random and spiral arrays. Thus, the comparisons are carried out between the proposed method and the
Fig. 4.9 Two non-uniform 2-D arrays: a a random 2-D array with uniform distribution and 2000 elements, b a spiral 2-D array with 1000 elements
Fig. 4.10 Wideband beam patterns of the random 2-D array for a the frequency-domain direct method and b the proposed method
frequency-domain direct method. Figure4.10shows the wideband beam patterns of the random array computed by the two methods, where the relative error is 0.48%.
Figure4.11shows the wideband beam patterns of the spiral array computed by the two methods, where the relative error is 0.41%. Both Figs.4.10and4.11show that the wideband beam patterns obtained by the two methods match perfectly. Moreover, for the random array, the computational time of the proposed method and the frequency-domain direct method is 6.78 s and 1.65× 103s respectively. While for the spiral array, the computational time of the proposed method and the frequency-domain direct method is 6.32 s and 1.03× 103s respectively. The results for the above two 2-D arrays indicate that the proposed method can be employed to fast compute wideband beam patterns of arbitrary 2-D arrays.
70 4 Design of Underwater Large Sparse 2-D Arrays
Fig. 4.11 Wideband beam patterns of the spiral array for a the frequency-domain direct method and b the proposed method