Fast Computation Method

In what follows, how the fast method computes wideband beam patterns for 2-D arrays by using the adjoint operator of the 2-D nonuniform fast Fourier transform (NUFFT) and the short inverse fast Fourier transform (IFFT) is shown in the follow-ing.

The discrete-time expression of (4.7) is

p(u, v, l)  index. The DFT of (4.9) is given as

P(u, v, fk)

where S( fk) are the DFT coefficients of s(l/fs), k is the frequency index, and the discrete frequency is given by

fk k fs/K (4.11)

where K is the length of the DFT. Assume that u and v have P and Q values respectively and are uniformly distributed between−1 and 1 [2,13,17]. Then, the respective minimum spacing of u and v is given by

u  2

P− 1, (4.12)

and

respectively. Substituting (4.14) and (4.15) into (4.10), it has

P(up, vq, fk)

It is found that (4.16) has a form which is similar to the definition of the adjoint 2-D NU2-DFT (3.46) when am,kand bm,kof (4.16) are taken as the arbitrary frequencies ω1mandω2mof (3.46) respectively. Therefore, (4.16) can be computed by using the adjoint operator of the 2-D NUFFT, shown in Chap.3. Since the distribution of the 2-D array elements is not specifically detailed in the definitions of am,kin (4.18) and bm,kin (4.19), the computation of (4.16) is suitable for arbitrary 2-D arrays.

After the computation of (4.16) at each frequency points, the inverse discrete Fourier transform (IDFT) is needed to acquire the maximum value of the amplitude of the radiated acoustic pressure field in the time domain to obtain the wideband beam patterns. The IDFT coefficients are written as

p(up, vq, l) 

62 4 Design of Underwater Large Sparse 2-D Arrays

Fig. 4.4 The waveforms of the radiation pattern versus time at a (u, v) (0, 0) and b (u, v)  (1, 1)

where NI D F T is the length of the IDFT. Then the wideband beam pattern can be given by

B P(up, vq) max

l

p(up, vq, l). (4.21) Obviously, the length NI D F T influences the computation speed of extracting the maximum values.

Since the number of non-zero discrete frequency points of the excitation signal is crucial to ensure the accuracy and efficiency of the wideband computation in the frequency domain, the length of the DFT and the required points of the IDFT in the fast computation method are discussed in what follows.

Assume that the side length of a square 2-D array aperture is D₁, the central frequency and the nominal bandwidth of the excitation signal s(t) are f₀ and B respectively. The time-domain waveforms of the radiation pattern at (u, v) (0, 0) and (u, v) (1, 1) are shown in Fig.4.4. The waveform at (u, v) (1, 1) is the longest one among the waveforms of all the values of u and v, the length of which is 2τmax, withτmax being the delay between the two signals radiated by the senor at the apex of the array and the central element. The expression ofτmax should be

τmax 

√2D1

2c . (4.22)

To make sure that all maximum values of the amplitude of the radiated acoustic pressure over u and v can be extracted, the length of the DFT ND F T should be larger than or equal to that of the longest waveform at (u, v) (1, 1). Thus, it has

ND F T

fs ≥ 2τmax. (4.23)

The number of the non-zero frequency points in passband is given by

Nf  B

From (4.25), it can be found that the number of the non-zero frequencies Nf is determined by the bandwidth of the excitation signal B, the 2-D array aperture D₁ and the sound velocity c, while it is irrelevant to the sampling frequency fs. While for a circular 2-D array with the diameter D1, similarly, the number of the non-zero frequency points should be

B/f0 is the relative bandwidth of the excitation signal. Hence, when the array aperture can be described by the number of elements and the wavelength of the central frequency, it can be concluded that the number of the non-zero frequencies Nf is determined by the relative bandwidth and the number of elements.

Usually, to ensure that the waveform obtained by IDFT has the comparable accu-racy with that computed by (4.9) in the time domain, NI D F T should be larger than or equal to ND F T. However, since ND F T is usually overlong, it will slow the computa-tion of wideband beam pattern. From (4.21), it can be found that only the time-domain maximum values corresponding to all directions are needed for computing the wide-band beam pattern, which means if the maximum values are extracted correctly, the distortion of the time-domain samples caused by the IDFT will be irrelevant to the beam pattern results. Since the waveform envelope is enough to extract the correct maximum values, the passband spectrum centered at f0 can be shifted to around f  0 to obtain the waveform envelope with the IFFT of NI D F T points, as oper-ated in [18,19]. NI D F T can be much smaller than ND F T due to the lowering of sampling rate. It is seen from (4.20) that the discrete passband frequency points are around K0  f0/( fs/NI D F T), by which all the discrete passband frequency points

64 4 Design of Underwater Large Sparse 2-D Arrays

Fig. 4.5 Steps of the proposed fast method of computing wideband beam pattern (BP)

are shifted, the size of the IDFT can be reduced from larger than or equal to ND F T

to Nf.

The short IDFT can be expressed as

p^(up, vq, l) 

where k1represents that the k1th frequency point is the first non-zero one in the pass band. With the short IDFT, the time-domain envelop of the radiated waveform can be obtained and the maximum value of wideband beam pattern computation can be extracted accurately and efficiently. The wideband beam pattern can be given by

B P(up, vq) max

l

p^(up, vq, l). (4.29) The short IDFT can be realized by the IFFT with the length NI F F T, where NI F F T

is the first power of two, greater than or equal to Nf. In this chapter, the IFFT operation is referred to as the short IFFT, which is applied in order that the time required by extracting the maximum values can be reduced dramatically.

In summary, the proposed fast method of computing wideband beam pattern consists of three steps: (i) transform the acoustic pressure signal in the time domain to the frequency domain according to the number of the non-zero frequency points of the excitation signal; (ii) apply the adjoint operator of the 2-D NUFFT to complete the frequency-domain spatial computation of (4.10); (iii) use the short IFFT to transform the frequency-domain results to the time-domain envelopes and extract the maximum values in the time domain. The proposed fast method is shown in Fig.4.5.