Experimental study of superdirective frequencyinvariant beamforming for a circular hydrophone array Yong Wang*
Yixin Yang, Member, IEEE, Shaohao Zhu, Yang Shi, Long Yang, Zhixiong Lei
School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an, China, 710049
School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China, 710072
Abstract—This paper first presents a superdirective frequency-invariant beamforming method for circular sensor arrays. The weighting vector of this method is in close-form, which is derived using the criterion of minimizing the mean square error between the desired and synthesized beampatterns. A detailed experimental study of this superdirective frequencyinvariant beamforming method is then followed, which is used to further demonstrate its performance. In the experiment, the superdirective frequency-invariant broadband beamformers with a specific response are designed so that they can be applied to effectively suppress the ambient noise. The experimental data are processed in the time domain in which finite impulse response filters are designed according the obtained superdirective frequency-invariant broadband beamformers. The results show that this superdirective frequency-invariant beamforming method provides good signal-to-noise ratio enhancement. Keywords—circular invariant beamforming
arrays;
I.
superdirectivity;
frequency-
INTRODUCTION
Superdirective frequency-invariant beamforming is required in many applications ranging from sonar, radar, audio engineering, to communications, in which broadband signals are often needed to be processed without any distortion using sensor arrays whose apertures are much smaller than wavelengths [1,2]. Circular arrays are widely used in many applications, since they have simple configuration, no left-right ambiguities, and advantage to form uniform beams along with 360° azimuthal directions. The performance of this sort of arrays has been investigated intensively in the literature [3-8], and many approaches, such as the statistics approaches [1,2], the optimization-based methods [9,10], and the modal method [11,12] were proposed to apply for designing superdirective frequency-invariant (SFI) beamformers for circular arrays. However, these methods cannot fully exploit the advantages of this array type and thus cannot provide precise and simple closed-form solutions. By contrast, the optimal solutions were derived for circular sensor arrays based on the criterion of minimizing the mean square error between the desired and synthesized beampatterns [13]. Specifically, the optimal This work was supported by the China Postdoctoral Science Foundation (Grant No. 2016M592782) and the National Natural Science Foundation of China (Grant Nos. 11274253 and 11527809) . *Corresponding author:
[email protected] 978-1-5090-1537-5/16/$31.00 ©2016 IEEE
weighting vector, the output beam, and the minimum mean square error are all expressed in closed-form exactly when the desired beampattern is properly formulated. These results provide a more effective method for designing practical SFI beamformers, and it was also extended to baffled circular arrays [14]. Although simulations and experimental results have demonstrated the good performance of this SFI method in [13], they only focused on synthesizing FI broadband beampatterns. Some other performance measures, such as practical array gain, are still needed to be evaluated. This paper also presents an experimental study of the previously proposed SFI method, with the main purpose of further analyzing its broadband beamforming performance and practical array gain. Since the ambient noise field shows a stable directivity in the experiment, a desired beampattern with a specific spatial response is achieved which can effectively suppress the ambient noise. The SFI broadband beampatterns are then straightforwardly derived and the corresponding finite impulse response (FIR) filters are also designed to process the experimental data with the least distortions. The results are analyzed in detail which are also compared with other methods. II.
SUPERDIRECTIVE FREQUENCY-INVARIANT BEAMFORMING
As shown in Fig. 1, an M-element uniform circular array with radius a is considered in this paper. A unit-magnitude plane-wave impinges from direction (θ ,φ ) , and the pressure received by the mth-sensor located at (rm ,φm , π / 2) is
pm (θ ,φ ) = e−ika sinθ cos(φ −φm ) ,
(1)
where i = −1 , the wavenumber is k = 2π / λ , λ denoting the wavelength. The manifold vector is P (θ , φ ) = [ p0 , p1 , , pM −1 ]T .
(2)
In this paper, only the horizontal beampatterns are desired and there is θ = π / 2 . The output beampattern will be
respectively, where φm = mβ and β = 2π / M , J0 is the 0th z
order cylindrical Bessel function, Δrs is the distance between
Sound source
the mth and m'th sensors, and s = m − m′ . III.
θ
m
a
rm
y
φ M-1
0
1
x Fig. 1. Coordinates of circular hydrophone array.
B (φ ) = w H (φ0 ) P (φ ),
(3)
where w is the weighting vector, φ0 is the preset steering direction, and (⋅) H denotes the Hermitian transpose. The synthesis method is to find the least square error approximation to the desired beampattern Bd (φ ) which can be obtained using an M-element uniform circular array with radius ar . The desired beampattern can be formulated as [15,16]
Bd (kr ar , φ ) =
M −1
ω
∗ m
(kr ar ) Em (kr ar , φ ),
SIMULATIONS AND EXPERIMENTAL RESULTS
A. Experimental setup A lake experiment was conducted to demonstrate the performance of the SFI method and the results were analyzed in detail. The experimental uniform circular array consists of 6 omnidirectional hydrophones and its diameter is 5 m. In the experiment, both the circular array and the sound source were placed at a depth of 20 m below the lake surface (see Fig. 2), and the water depth and sound speed at this location were about 70 m and 1430 m/s, respectively. The sound source was far-field. The transmitted signals were linear frequency modulated (LFM) signals with frequency ranging from 70 Hz to 200 Hz, and the incident direction is (θ0 , φ0 ) = (90 ,150 ) . The sampling frequency was 39 kHz. The directivity of ambient noise can be estimated using the minimum variance distortionless response (MVDR) method. The spatial spectrum is calculated using the following equation [17]:
(4)
QMVDR (φ ) =
m=0
where kr denotes the reference wavenumber, superscript ∗ denotes complex conjugation and
Em (φ ) =
1
M −1
M
s =0
e
− ismβ
⋅ ps .
(5)
The elements of the weighting vector ω have the property ω M − m = ( −1) m ωm∗ and they can be calculated according to the desired requirements [15]. Based on the properties of circular arrays, the synthesized beampattern can be derived as [13]
B(ka, φ ) =
λm∗ ∗ ∗ ωm (kr ar ) Em (ka, φ ) m = 0 λm
M −1
(6)
1 , P H (φ )(R n + ζ DL I )P (φ )
where R n is the noise covariance matrix, I is an identity matrix, ζ DL is the diagonal loading value, and the superscript H indicates the Hermitian transpose. The noise covariance matrix R n is estimated using the received noise data, which is expressed as 1 L n(l )n H (l ) , L l =1
Rn =
(10)
where n(l ) is the noise-only vector and L is the number of training snapshots. pre-processing
A/D
signal generator
beamforming
in which λm = s = 0 ρ s eismβ and λm = s = 0 ρ s eismβ are the M −1
(9)
power amplifier
oscillograph
M −1
eigenvalues of different circulant matrixes. The elements of these matrixes are
Experiment Ship lake surface
ρ s = J 0 (k ⋅ Δrs )
(7) circular array
9.2m
20m
7.5m
and
ρs = J 0
(
(kr ar )2 + (ka)2 − 2kr ar ⋅ ka ⋅ cos( sβ )
)
o
(8)
x
standard hydrophone
projector
y
Fig. 2. Configuration of Experiment.
Fig. 3. Noise spatial directivity.
Fig. 6. DIs of the DAS and SFI methods.
ELEMENTS OF THE WEIGHTING VECTOR FOR DESIRED BEAMPATTERN
Desired Beampattern (dB)
TABLE I.
ω0
ω1
0.131
ω2
0.10130.1696*i
-0.28210.4291*i
ω3 0.1599*i
In this paper, a desired beampattern (see Fig. 4) is obtained using a 6-element circular array with ar = 2.5m . The frequency is f r = 100 Hz , the preset steering direction is Fig. 4. Desired beampattern.
(θ0 , φ0 ) = (90 ,150 ) and the sidelobe levels are all below -15 dB. The obtained desired values of ω0 ~ ω3 are listed in Table I. Note that other elements ω4 and ω5 can be calculated by employing the relation ωM − m = (−1) m ωm∗ .The synthesized beampatterns at different frequencies are shown in Fig. 5, in which all of the beampatterns in the given frequency band exhibit good agreement with the desired beampattern and they are frequency-invariant, especially in the mainlobe region. As shown in Fig. 6, the directivity index (DI) of the SFI method is larger than that of the delay-and-sum (DAS) method, and its values approximately equal to 7 dB in the entire frequency band. -5
Fig. 5. Synthesized frequency-invariant broadband beampatterns.
The obtained noise spatial directivity is shown in Fig. 3, in which only the broadband noise with frequency ranging from 70 Hz to 200 Hz is used. The diagonal loading value ζ DL is set to 0. It is found that the ambient noise field shows a stable directivity in the experiment. The noise in the region of ΩSL2 = {(θ , φ ) θ = 90 , φ ∈ [280 ,390 ]} is higher than that in other regions, which means the ambient noise field was not isotropic. One method than can reduce the effect of this anisotropic ambient noise field is to design a suitable beamformer that can minimize the responses in the high noise region while it is subject to some other constraints.
Desired Designed
-10 -15 -20 -25 -30 -35 -40 -45
0
100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
(a)
Phase ( o)
(b) Fig. 7. Frequency responses of designed and desired filters for the DAS method. -5
interest and its phase is linear. By contrast, the magnitude of frequency response of the desired FIR filter for the SFI method is not constant in the frequency range of interest, whereas the phase is still linear. The designed responses for both methods have some errors with the desired ones, but their effects to final results are limited. The output signals of the DAS and SFI methods are shown in Fig. 9. The broadband output signal-to-noise ratio (SNR) of the SFI method is approximately 13.1 dB, whereas the broadband output SNR of the DAS method is 6.0 dB. The SNR of the reference hydrophone is -6.3 dB. It is clear that the SFI is more efficient to suppress ambient noise than the DAS method in this experiment. Since the ambient noise is not isotropic, both array gains of the DAS and SFI methods are larger than DIs of these two methods. The corresponding output signal spectrum is shown in Fig. 10, in which the SFI method shows the best noise suppression ability and the corresponding output signal spectrum shows clearer details than the DAS method.
Desired Designed
-10 -15 -20 -25 -30 -35 -40 -45
0
100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
(a) 200 Desired Designed
150
Fig. 9. Output signals of different methods.
100 50 0 -50 -100 -150 -200
0
100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
(b) Fig. 8. Frequency responses of designed and desired filters for the SFI method.
It has been pointed out that the SFI method can be easily implemented in the time domain and the coefficients of FIR filters can be obtained using the method presented in Ref. [18]. In the following, the corresponding FIR filters of length 361 are designed, and the frequency responses of desired and designed FIR filters are shown in Figs. 7 and 8, respectively. The magnitude of frequency response of the desired FIR filter for the DAS method is constant in the given frequency range of
Fig. 10. Output signal spectrum of different methods.
IV.
CONCLUSIONS
A previously proposed SFI method for circular arrays is experimentally studied in this paper. Because the ambient noise shows a stable directivity, the desired beampattern is deliberately synthesized to have a specific responese so that it can effectively suppress the anisotropic ambient noise and
improve SNR. The SFI broadband beampatterns are then straightforwardly obtained and the corresponding FIR filters are also designed to process the experimental data with the least distortions. The array gains of the DAS and SFI methods are calculated using experimental data and the results show that the SFI method provides good SNR enhancement and is more advantageous than the DAS method. REFERENCES [1] [2] [3] [4] [5]
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