- Email: [email protected]
Coherent signal-subspace processing of acoustic vector sensor array for DOA estimation of wideband sources
ARTICLE IN PRESS
Signal Processing 85 (2005) 837–847 www.elsevier.com/locate/sigpro
Coherent signal-subspace processing of acoustic vector sensor array for DOA estimation of wideband sources Huawei Chena,, Junwei Zhaob a
State Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing, 210093, People’s Republic of China b Institute of Acoustic Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, People’s Republic of China Received 3 July 2003; received in revised form 19 July 2004
Abstract This paper deals with the estimation of the direction of arrival (DOA) for wideband coherent sources using acoustic vector sensor array. By extending the idea of wideband focusing into acoustic vector sensor array processing, a wideband DOA estimation approach based on coherent signal-subspace processing is developed for acoustic vector sensor array. Finally, simulation results are presented, to compare the proposed method with its counterpart based on acoustic pressure sensor array, the advantages of the proposed algorithm are identified. r 2004 Elsevier B.V. All rights reserved. Keywords: Acoustic vector sensor; Wideband sources; Array signal processing
1. Introduction In recent years, acoustic vector sensor array processing has been drawn much attention in the literature [3–5,10,13–15]. An acoustic vector sensor measures both pressure and particle velocity of the acoustic field at a point in space; whereas a traditional pressure sensor can only extract the pressure information. Various types of acoustic vector sensors with different design technologies are now commercially available [15]. Vector sensor technologies have been used in the field of underCorresponding author.
E-mail address: [email protected] (H. Chen).
water acoustics for decades and currently attracts reinvigorated attention for underwater source location problems [2–5,13–15]. By taking advantage of the extra information, arrays of vector sensors are able to improve source location performance without increasing array aperture size. Nehorai and Paldi [10] have developed the measurement model of the acoustic vector sensor array for dealing with narrowband sources, and introduced it to the array processing research community. Much work has been done for applying acoustic vector sensor array or a single acoustic vector sensor to direction finding problems [2–5,13–15]. Hawkes and Nehorai [3] investigated the conventional beamforming and
0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.07.030
ARTICLE IN PRESS 838
H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
Nomenclature að2Þ ðYk Þ array manifold of a 2-D acoustic vector sensor að3Þ ðYk Þ array manifold of a 3-D acoustic vector sensor ap ðf i ; Yk Þ direction vector of an acoustic pressure sensor array av ðf i ; Yk Þ direction vector of an acoustic vector sensor array Ap ðf i Þ direction matrix of an acoustic pressure sensor array Av ðf i Þ direction matrix of an acoustic vector sensor array B frequency bandwidth of the signals c sound velocity in the medium d inter-element spacing ei generalized eigenvector corresponding to li fc central frequency of the signals f0 focusing frequency Im m m identity matrix K the number of source signals M the number of array elements
Capon direction estimation for acoustic vector sensor array. ESPRIT and Root-MUSIC algorithms have recently been applied by Wong and Zoltowski to arrays of acoustic vector sensors [13–15]. However, these direction finding algorithms for acoustic vector sensor arrays are all based on the narrowband assumptions, moreover, most of them cannot effectively deal with correlated sources and may fail in fully coherent source scenarios. Wong and Zoltowski [13] applied the method of spatial smoothing to solve the narrowband coherent sources for acoustic vector sensor array, but unfortunately this method reduced the effective array aperture. In practice, the sources are usually wideband in most scenarios, and may be completely correlated with each other. Therefore, it is necessary to investigate the problem of the direction of arrival (DOA) estimation for wideband coherent sources with acoustic vector sensor array. Motivated by this, we studied the
Nðf i Þ noise vector pk ðr; tÞ acoustic pressure corresponding to the kth source at location r and time t Pn ðf i Þ autocovariance matrix of noise vector Ps ðf i Þ autocovariance matrix of signal vector Rx ðf i Þ autocovariance matrix of Xðf i Þ Sðf i Þ signal vector T p ðf i Þ focusing matrix of acoustic pressure sensor array T v ðf i Þ focusing matrix of acoustic vector sensor array uk unit bearing vector of the kth source vk ðr; tÞ acoustic particle velocity corresponding to the kth source at location r and time t Xðf i Þ received data vector of the array li generalized eigenvalue r ambient density s2p noise variance of the acoustic pressure sensor s2v noise variance of the acoustic particle velocity sensor Yk DOA of the kth source signal with elevation angle yk and azimuth angle jk
wideband DOA estimation using acoustic vector sensor array for wideband coherent sources. Wang and Kaveh [12] proposed the coherent signal-subspace (CSS) method for wideband direction finding with acoustic pressure sensor array, it has been shown that the CSS method can treat the fully coherent sources without reducing the effective array aperture, and a lower detection and resolution signal-to-noise ratio (SNR) thresholds can be achieved for wideband sources. From then on, various methods of wideband DOA estimation have been put forward based on CSS processing [1,8]. In this paper, based on the idea of wideband CSS processing, we develop a 2-D wideband DOA estimation algorithm for acoustic vector sensor array. This paper is organized as follows. Section 2 formulates the mathematical data models of the acoustic vector sensor and acoustic vector sensor array, respectively. Section 3 develops an
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
algorithm of wideband DOA estimation based on CSS processing for acoustic vector sensor array. Simulation results as well as performance analysis and comparison with pressure sensor array are presented in Section 4. Finally, Section 5 concludes the paper.
2. Mathematical data model and problem formulation We here consider the scenario of K wideband sources with bandwidth B and central frequency f c traveling from far-field through a homogeneous isotropic fluid medium. Letting Yk ¼ ðyk ; jk Þ denote the DOA of the kth source, with elevation angle yk 2 ½0 ; 180 defined clockwise relative to the z-axis and azimuth angle jk 2 ð 180 ; 180 defined counterclockwise relative to the x-axis, as shown in Fig. 1. The objective of the problem is to determine the DOA parameter vector H ¼ ½Y1 ; Y2 ; ; YK : 2.1. Data model for an acoustic vector sensor The relationship between acoustic pressure and particle velocity for a planewave at point r and
z
DOA(θk,ϕk)
θk y
o ϕk
x Fig. 1. Configuration of DOA of kth source signal.
839
time t can be expressed as [11] vk ðr; tÞ ¼
pk ðr; tÞ uk ; rc
(1)
where vk ðr; tÞ and pk ðr; tÞ denote the acoustic particle velocity and acoustic pressure of the kth source, respectively, r is the ambient density, c is the sound velocity in the medium, uk is the unit bearing vector of the kth source with uk ¼ ½ sin yk cos jk ; sin yk sin jk ; cos yk T ; the superscript T denotes the matrix transpose. An acoustic vector sensor is composed of a pressure sensor and a pair or a triad of spatially co-located but orthogonal velocity sensors which measure the two or three orthogonal components of acoustic particle velocity. The array manifold of an acoustic vector sensor composed of a pressure sensor and a pair of spatially co-located but orthogonal velocity sensors is 2 3 2 3 1 1 6 7 6 7 að2Þ ðYk Þ ¼ 4 ukx 5 ¼ 4 sin yk cos jk 5: (2) uk y sin yk sin jk Correspondingly, the array manifold of an acoustic vector sensor composed of a pressure sensor and a triad of spatially co-located but orthogonal velocity sensors is 3 2 2 3 1 1 7 6 sin yk cos jk 7 6 uk x 7 6 7 7¼6 að3Þ ðYk Þ ¼ 6 ; (3) 6 6 uky 7 4 sin yk sin j 7 5 k 5 4 uk z cos yk where ukx ¼ sin yk cos jk ; uky ¼ sin yk sin jk and ukz ¼ cos yk are the three direction cosine components of uk along the x-axis, y-axis and z-axis, respectively. In the following part, for acoustic vector sensor with array manifold að2Þ ðYk Þ; we call it two-dimensional (2-D) acoustic vector sensor since it deals with the two orthogonal components of ukx and uky : Correspondingly, 3-D acoustic vector sensor refers to the one with array manifold að3Þ ðYk Þ: In this paper, we mainly discuss the array of 2-D acoustic vector sensors, but it is not difficult to generalize the results to the case of arrays of 3D acoustic vector sensors.
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
840
2.2. Data model for acoustic vector sensor array
where
Consider an acoustic vector sensor array composed of M identical 2-D acoustic vector sensors located at r1 ; r2 ; ; rM ; respectively. The received 3M 1 array output spectral vector for the ith frequency bin f i ðf c B=2pf i pf c þ B=2Þ can be expressed as
Av ðf i Þ ¼ ½av ðf i ; Y1 Þ; av ðf i ; Y2 Þ; ; av ðf i ; YK Þ
Xðf i Þ ¼
K X
ap ðf i ; Yk Þ að2Þ ðYk Þ S k ðf i Þ þ Nðf i Þ;
k¼1
(4) Here denotes the Kronecker product, i ¼ 1; 2; ; J; J denotes the number of frequency components to be analyzed, and Xðf i Þ ¼ ½X 1p ðf i Þ; X 1vx ðf i Þ; X 1vy ðf i Þ; ; T
X Mp ðf i Þ; X Mvx ðf i Þ; X Mvy ðf i Þ :
ð5Þ
where X mp ðf i Þ; X mvx ðf i Þ and X mvy ðf i Þ denote the ith frequency components of the acoustic pressure and two orthogonal particle velocity components for the mth acoustic vector sensor, respectively. S k ðf i Þ is the ith frequency component of the kth source signal. ap ðf i ; Yk Þ is the M 1 direction vector of acoustic pressure sensor array with the same geometry as the acoustic vector sensor array for the kth source at the ith frequency bin, it is given by T
ap ðf i ; Yk Þ ¼ ½ ej2pðr1 uk Þ ;
T
ej2pðr2 uk Þ ;
;
T
ej2pðrM uk Þ T :
(6) The 3M 1 noise vector Nðf i Þ can be expressed as
ð7Þ
where N mp ðf i Þ; N mvx ðf i Þ and N mvy ðf i Þ represent the noise present at the mth acoustic vector sensor. It should be noted that in Eq. (4), without loss of generality, we have normalized the particle velocity measurements by multiplying with a constant rc which is assumed to be known, according to Eq. (1). In vector notation, Eq. (4) can be rewritten as Xðf i Þ ¼ Av ðf i ÞSðf i Þ þ Nðf i Þ;
að2Þ ðY2 Þ; ; ap ðf i ; YK Þ að2Þ ðYK Þ
ð9Þ
is the 3M K direction matrix of acoustic vector sensor array, where av ðf i ; Yk Þ ¼ ap ðf i ; Yk Þ að2Þ ðYk Þ
1pkpK
(10)
is the 3M 1 direction vector of acoustic vector sensor array corresponding to the kth source. And Sðf i Þ contains the ith frequency component of the K source signals Sðf i Þ ¼ ½ S1 ðf i Þ;
S 2 ðf i Þ;
;
SK ðf i Þ T :
(11)
Here, we make the following commonly used assumptions on the array model Eq. (8) for further development: (1) The signals and the noise are independent and identically distributed (i.i.d.), ergodic, zero mean stationary random processes. (2) The noise is statistically independent of the signals and temporally white. (3) The noise is spatially white, say, the noise between different sensor components of an acoustic vector sensor, as well as between all components of acoustic vector sensors located at different places, are uncorrelated with each other. The covariance matrix of Xðf i Þ is given by Rx ðf i Þ ¼ E½Xðf i ÞX H ðf i Þ
Nðf i Þ ¼ ½N 1p ðf i Þ; N 1vx ðf i Þ; N 1vy ðf i Þ; ;
N Mp ðf i Þ; N Mvx ðf i Þ; N Mvy ðf i ÞT
¼ ½ap ðf i ; Y1 Þ að2Þ ðY1 Þ; ap ðf i ; Y2 Þ
(8)
¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ þ Pn ðf i Þ;
ð12Þ
where H denotes the complex conjugate transpose, Ps ðf i Þ ¼ E½Sðf i ÞS H ðf i Þ and Pn ðf i Þ ¼ E½Nðf i ÞN H ðf i Þ are the covariance matrices of source signals and noise at the ith frequency bin, respectively. Based on the above assumptions about the noise, we have " 2 # 0 sp Pn ðf i Þ ¼ I M ; (13) 0 s2v I 2
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
where s2p and s2v denote the noise variance present at pressure sensor and particle velocity sensor, respectively. I m is an m m identity matrix. Under the assumption of homogeneous isotropic medium, for a single 2-D acoustic vector sensor, it holds that s2p ¼ 3s2v
(14)
for a detailed discussion, see [2,5]. Therefore, the covariance matrix Rx ðf i Þ can be rewritten as Rx ðf i Þ ¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ þ P n ðf i Þ ¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ " # 1 0 2 þ sp I M : 0 1=3I 2
ð15Þ
3. Wideband DOA estimation for acoustic vector sensor array 3.1. Wideband focusing for acoustic vector sensor array Wang and Kaveh [12] have proposed the coherent signal-subspace (CSS) method for wideband DOA estimation using the acoustic pressure sensor array. The central idea of CSS method is wideband focusing concept, and they have proved the existence of focusing matrices for acoustic pressure sensor array. In the CSS method, the wideband array data are decomposed into several narrowband components via discrete Fourier transform at first. Focusing matrices are then constructed to transform each of the narrowband covariance matrices into the one corresponding to the reference frequency bin which is referred to focusing frequency. The technique of wideband focusing has been an important basis for several wideband processing methods. Based on the idea of wideband focusing for pressure sensor array, here we extend wideband focusing into acoustic vector sensor array for wideband coherent array processing. Theorem 1. Assume that the focusing matrix for Melement acoustic pressure sensor array is T p ðf i Þ; the
841
focusing frequency is f 0 : For M-element acoustic vector sensor array with the identical array geometry and focusing frequency, it holds that (1) there must exist a focusing matrix T v ðf i Þ; satisfying T v ðf i ÞAv ðf i Þ ¼ Av ðf 0 Þ; (2) T v ðf i Þ ¼ T p ðf i Þ I 3 . Proof. See Appendix.
&
It shows that the wideband focusing technique can also be applied to vector sensor arrays. From this point of view, a wideband array processing method for vector sensor array similar as pressure sensor array can be established. There are many classes of focusing matrices available have being proposed in the literature for wideband processing of pressure sensor arrays [6,7,9]. From the above theorem, it shows that the wideband focusing matrices for acoustic vector sensor array can be constructed by those for pressure sensor array with the same array geometry. Thus, this provides us a method for construction of wideband focusing matrices for acoustic vector sensor arrays. From the above derivation, for the array composed of 3-D vector sensors, it holds that T v ðf i Þ ¼ T p ðf i Þ I 4 : 3.2. The DOA estimation based on coherent signalsubspace processing The array output after focusing transform is (16)
Yðf i Þ ¼ T v ðf i ÞXðf i Þ:
The covariance matrix corresponding to the ith frequency bin is Ry ðf i Þ ¼ E½Yðf i ÞY H ðf i Þ ¼ T v ðf i ÞE½Xðf i ÞX H ðf i ÞT H v ðf i Þ ¼ Av ðf 0 ÞPs ðf i ÞAH v ðf 0 Þ ( " þ s2p T v ðf i Þ I M
1
0
0
1=3I 2
#) TH v ðf i Þ: ð17Þ
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
842
The coherently averaged covariance matrix is given by Ry ¼
J X
Ry ðf i Þ
i¼1
"
¼ Av ðf 0 Þ
J X
þ s2p
PMUSIC ðYÞ ¼ P3M
# (
"
T v ðf i Þ I M
i¼1
#)
1
0
0
1=3I 2
TH v ðf i Þ: ð18Þ
For the ease of representation, Eq. (18) can be rewritten as 2 Ry ¼ Av ðf 0 ÞRs AH v ðf 0 Þ þ sp Rn
(19)
with Rs ¼
J X
(20)
Ps ðf i Þ
i¼1
and Rn ¼
J X i¼1
( T v ðf i Þ I M
"
1 0
1
2 H i¼Kþ1 jei av ðf 0 ; YÞj
Ps ðf i Þ AH v ðf 0 Þ
i¼1 J X
With Theorem 2 at hand, we can construct the MUSIC spatial spectra for acoustic vector sensor array as
0 1=3I 2
#) TH v ðf i Þ: (21)
Based on the idea of wideband focusing, we have the following coherent signal-subspace theorem for acoustic vector sensor array. Theorem 2. Let li ði ¼ 1; ; 3MÞ denote the generalized eigenvalue of matrix pencil ðRy ; Rn Þ; l1 Xl2 X XlK XlKþ1 X Xl3M ; ei is the generalized eigenvector associated with li ; the focusing frequency is f 0 : It holds that (1) there must exist 3M2K smallest eigenvalues lKþ1 ¼ lKþ2 ¼ ¼ l3M ¼ s2p ; (2) the column space of the matrix E n ¼ ½eKþ1 ; eKþ2 ; ; e3M is orthogonal to the column space of the matrix Av ðf 0 Þ ¼ ½av ðf 0 ; Y1 Þ; av ðf 0 ; Y2 Þ; ; av ðf 0 ; YK Þ: Proof. It can be obtained by modifying the proof in [12], and thus omitted here. &
:
(22)
The DOA estimation for wideband sources can be given by the peaks of PMUSIC ðYÞ: The steps of the proposed DOA estimation algorithm are summarized as follows: (1) Perform FFT to obtain the array output spectral vector Xðf i Þ: (2) Set the focusing frequency and corresponding focusing matrix, compute Yðf i Þ from Eq. (16) and calculate the sample covariance matrix Ry ðf i Þ: (3) Compute the averaged covariance matrix Ry ¼ PJ R ðf i¼1 y i Þ; compute Rn from Eq. (21). (4) Calculate the noise subspace E n from generalized eigendecomposition of the matrix pencil (Ry ;Rn ). (5) Compute the MUSIC spatial spectra PMUSIC ðYÞ from Eq. (22), the peaks of the spectra give the DOAs of the sources.
4. Simulation results To verify the performance of the proposed method, simulations are carried out in the following. The proposed method of wideband DOA estimation based on CSS processing for acoustic vector sensor array is compared with its counterpart for pressure array proposed in [12]. We consider a uniform linear array (ULA) composed of eight 2-D acoustic vector sensors and an eightelement acoustic pressure sensor ULA, both arrays have the same geometry with sensor elements located at x-axis. The signals of interest consisted of 33 narrowband components lying in the frequency band 802120 Hz: The focusing frequency is chosen as f 0 ¼ 80 Hz: The spatially resampled focusing method [9] is used, for it requires no preliminary DOA estimation of the source signals. For acoustic vector sensor ULA, with the focusing matrix for acoustic pressure
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
843
sensor ULA at hand, the focusing matrix for acoustic vector sensor ULA can be obtained by Theorem 1 in Section 3. The additive array noise which is independent of source signals, is stationary zero-mean bandpass white Gaussian process with the same pass band with the source signals. Here the SNR is defined as the ratio of the power of each source signal to that of the noise at a single sensor. For the two kinds of arrays, the total 4096 successive data samples at each sensor are collected for each simulation trial, which are divided into 64 segments, each having 64 samples, to estimate the covariance matrices. Example 1. Consider 8-element acoustic vector sensor ULA and 8-element acoustic pressure sensor ULA, both have the same inter-element spacing d ¼ lmin =2; where lmin is the wavelength corresponding to the maximum frequency of interest band. Two coherent wideband source signals s1 ðtÞ arriving from Y1 ¼ ðy1 ; j1 Þ ¼ ð20 ; 45 Þ and s2 ðtÞ arriving from Y2 ¼ ðy2 ; j2 Þ ¼ ð20 ; 70 Þ impinge the two arrays simultaneously. The source signal s2 ðtÞ is the time delayed version of s1 ðtÞ by 0.0625 s. The SNR of the received signal for each element of pressure sensor ULA is 10 dB, and the SNR for the acoustic pressure sensors of acoustic vector sensor ULA are also all set to be 10 dB. Thirty independent trials are carried out for each figure in the following. The MUSIC spatial spectra for acoustic pressure sensor ULA and vector sensor ULA are shown in Figs. 2 and 3, respectively. There are two points to be noted from the results of the simulations. One is that the CSS processing-based DOA estimation for pressure sensor ULA can only estimate the azimuth angle, the other is that it has azimuth angle estimation ambiguities. Comparatively, as Fig. 3 shows, there appear spectrum peaks only at the true source bearings for acoustic vector sensor ULA. Therefore, it shows that we can locate the 2-D DOA of wideband coherent sources with a vector sensor ULA. For the ease of comparisons in the following, we shall concentrate on the azimuth angle estimation of the source signals, and the elevation angle is set to be 901. But it should be noted that the vector sensor ULA can also determine the elevation angle as well.
Fig. 2. 2-D spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA. Signals at Y1 ¼ ð20 ; 45 Þ and Y2 ¼ ð20 ; 70 Þ; the interelement spacing d ¼ lmin =2:
Fig. 3. 2-D spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA. Signals at Y1 ¼ ð20 ; 45 Þ and Y2 ¼ ð20 ; 70 Þ; the interelement spacing d ¼ lmin =2:
Example 2. In this part, we analyze the performance of the wideband DOA estimation methods when the source signal is close to endfire direction (near j ¼ 0 or 180 ). The arriving angles of the two wideband signals s1 ðtÞ and s2 ðtÞ are j1 ¼ 10 and j2 ¼ 70 ; respectively. The other conditions for the source signals and the arrays are the same
ARTICLE IN PRESS 844
H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
as those in Example 1. Figs. 4 and 5 show the array output MUSIC spatial spectra for pressure sensor ULA and vector sensor ULA, respectively. It is seen that when the source signals are close to endfire direction, the wideband DOA estimation for acoustic pressure sensor ULA has not only DOA estimation ambiguities at the central sym-
metric points of true DOAs of source signals, but also ambiguities near 180 and 180 ; simultaneously. Therefore, it cannot be determined that which end the signal source comes from. As it is shown in Fig. 5, the wideband DOA estimation for acoustic vector sensor ULA exhibits a better performance when dealing with the signal sources with arriving angle close to endfire. When the DOA of source signal is close to endfire, the width of the spatial spectrum becomes slightly larger, but there exists no ambiguities, and can still give an accurate DOA estimation for the wideband sources.
Fig. 4. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 10 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
Example 3. Consider the same two wideband coherent source signals as in Example 1 except that the arriving angles for s1 ðtÞ and s2 ðtÞ are j1 ¼ 65 and j2 ¼ 70 ; respectively. The same ULAs as used in Example 1 are employed except that the inter-element spacing increases to d ¼ lmin for both arrays. Fig. 6 shows the MUSIC spatial spectrum for pressure sensor ULA with inter-element spacing d ¼ 0:5 lmin ; Figs. 7 and 8 show the MUSIC spatial spectra for pressure sensor ULA and vector sensor ULA both with inter-element
Fig. 5. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 10 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
Fig. 6. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
Fig. 7. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin :
845
pressure sensor array, when inter-element spacing d ¼ 0:5lmin ; the two sources can not be resolved, with inter-element spacing d increased to d ¼ lmin ; the two sources can be clearly resolved. Comparatively, the wideband DOA estimation for acoustic vector sensor ULA can also clearly resolve the two wideband coherent sources, while it does not suffer from DOA estimation ambiguity when the interelement spacing exceeds the Nyquist half-wavelength upper limit, as shown in Fig. 8. For the acoustic pressure sensor array, when the aperture size is enlarged, there are more sensor elements for avoiding the appearance of grating lobes. It shows that the aperture enlarging for vector sensor array can be achieved by spacing the vector sensor greater than half-wavelength while no additional sensors are needed, therefore a better array resolution and DOA estimation precision can be obtained with no extra cost and computational load to be needed.
5. Conclusions
Fig. 8. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin :
spacing d ¼ lmin ; respectively. As it is shown in Fig. 7, when the inter-element spacing exceeds the Nyquist half-wavelength upper limit, there appear the undesired grating lobes which have the same amplitudes with those of the true spectrum peaks. As we know, an enlarged aperture size generally offers enhanced array resolution and DOA estimation precision. As Figs. 6 and 7 show, for
In this paper, by extending the concept of wideband focusing into wideband array processing for acoustic vector sensor array, a wideband DOA estimation method for acoustic vector sensor array is proposed based on CSS processing, which can handle the problems of wideband coherent sources without reducing the effective array aperture size. Simulations were carried out to compare the proposed method with its counterpart for pressure sensor array when the source signals are wideband coherent. It is demonstrated that the proposed method for acoustic vector sensor array not only can solve the wideband DOA estimation for coherent sources, but also preserves the good properties of vector sensor array embodied in the narrowband scenario. The simulation results show that the proposed approach for acoustic vector sensor linear array are able to: realize unambiguous direction finding in the whole space, thus suffers no left/right ambiguity problem; allow the uniform inter-element spacing to exceed the Nyquist half-wavelength upper limit. Therefore, this allows the use of spatially undersampled arrays for extending the effective aperture, enhances array
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
846
resolution and the DOA estimation precision without increasing the computational load. Thus the shortcoming of poor performance as encountered by pressure sensor linear array when dealing with direction estimation of sources impinging from the endfire directions can be overcome.
A, B, C and D, it holds that ðA BÞðC DÞ ¼ ðACÞ ðBDÞ:
(28)
Considering the above property, Eq. (27) can be rewritten as fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ fT p ðf i Þap ðf i ; Yk Þg fI 3 að2Þ ðYk Þg
Acknowledgements The research in this paper was supported by the National Science Foundation of China under Grant 10304008 and the China Postdoctoral Foundation under Grant 2004036415.
¼ fT p ðf i Þ I 3 gfap ðf i ; Yk Þ að2Þ ðYk Þg:
ð29Þ
Substitute Eq. (29) into Eq. (26), we have fT p ðf i Þ I 3 gfap ðf i ; Yk Þ að2Þ ðYk Þg ¼ ap ðf 0 ; Yk Þ að2Þ ðYk Þ:
ð30Þ
Appendix
Finally, comparing Eq. (30) with Eq. (25), it shows that
The proof of Theorem 1. The proof of (1) is simple as [12]. Here we call T v ðf i Þ wideband focusing matrix for acoustic vector sensor array. According to the concept of wideband focusing, we have, respectively,
T v ðf i Þ ¼ T p ðf i Þ I 3 :
T p ðf i Þap ðf i ; Yk Þ ¼ ap ðf 0 ; Yk Þ
(23)
and T v ðf i Þav ðf i ; Yk Þ ¼ av ðf 0 ; Yk Þ:
(24)
Substitute Eq. (10) into Eq. (24), one can obtain T v ðf i Þfap ðf i ; Yk Þ að2Þ ðYk Þg ¼ ap ðf 0 ; Yk Þ að2Þ ðYk Þ:
ð25Þ
Appling Kronecker product operation to the two sides of Eq. (23) by að2Þ ðYk Þ simultaneously, we have fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ ap ðf 0 ; Yk Þ að2Þ Yk Þ:
ð26Þ
The left side of Eq. (26) is equivalent to fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ fT p ðf i Þap ðf i ; Yk Þg fI 3 að2Þ ðYk Þg:
ð27Þ
In the following, we shall use a property for Kronecker product operation. That is, for matrices
(31)
References [1] K.M. Buckley, L.J. Griffiths, Broad-band signal-subspace spatially spectrum (BASS-ALE) estimation, IEEE Trans. ASSP. 36 (7) (July 1988) 953–964. [2] H. Feng, G. Liang, J. Hui, The direction estimation using combined sensor with pressure and particle velocity, Acta Acustica (in Chinese) 25 (6) (2000) 516–520. [3] M. Hawkes, A. Nehorai, Acoustic vector-sensor beamforming and Capon direction estimation, IEEE Trans. Signal Processing 46 (9) (September 1998) 2291–2304. [4] M. Hawkes, A. Nehorai, Acoustic vector-sensor processing in the presence of a reflecting boundary, IEEE Trans. Signal Processing 48 (11) (November 2000) 2981–2993. [5] M. Hawkes, A. Nehorai, Acoustic vector-sensor correlations in ambient noise, IEEE J. Oceanic Eng. 26 (3) (July 2001) 2291–2304. [6] H.S. Hung, M. Kaveh, Focussing Matrices for coherent signal-subspace processing, IEEE Trans. ASSP 36 (8) (August 1988) 1272–1281. [7] H.S. Hung, C.Y. Mao, Robust coherent signal-subspace processing for directions of arrival estimation of wideband sources, IEE Proc. Radar Sonar Navig. 141 (5) (1994) 256–262. [8] J. Krolik, D.N. Swingler, Multiple broad-band source location using steered covariance matrices, IEEE Trans. ASSP 37 (10) (October 1989) 1481–1494. [9] J. Krolik, D. Swingler, The detection performance of coherent wideband focusing for a spatially resampled array, Proc. IEEE ICASSP (1990) 2827–2830.
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847 [10] A. Nehorai, E. Paldi, Acoustic vector-sensor array processing, IEEE Trans. Signal Processing 42 (9) (September 1994) 2481–2491. [11] P.M. Se, K.U. Ingard, Theoretical acoustics, McGrawHill, New York, 1968. [12] H. Wang, M. Kaveh, Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources, IEEE Trans. ASSP. 33 (4) (April 1985) 823–831. [13] K.T. Wong, M.D. Zoltowski, Extended-aperture underwater acoustic multisource azimuth/elevation direction-
847
finding using uniformly but sparsely spaced vector hydrophones, IEEE J. Oceanic Eng. 22 (4) (April 1997) 659–672. [14] K.T. Wong, M.D. Zoltowski, Root-MUSIC-based azimuth-elevation angle-of-arrival estimation with uniformly spaced but arbitrarily oriented velocity hydrophones, IEEE Trans. Signal Processing 47 (12) (December 1999) 3250–3260. [15] K.T. Wong, M.D. Zoltowski, Self-initiating MUSIC-based direction finding in underwater acoustic particle velocityfield beamspace, IEEE J. Oceanic Eng. 25 (2) (February 2000) 262–273.
Signal Processing 85 (2005) 837–847 www.elsevier.com/locate/sigpro
Coherent signal-subspace processing of acoustic vector sensor array for DOA estimation of wideband sources Huawei Chena,, Junwei Zhaob a
State Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing, 210093, People’s Republic of China b Institute of Acoustic Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, People’s Republic of China Received 3 July 2003; received in revised form 19 July 2004
Abstract This paper deals with the estimation of the direction of arrival (DOA) for wideband coherent sources using acoustic vector sensor array. By extending the idea of wideband focusing into acoustic vector sensor array processing, a wideband DOA estimation approach based on coherent signal-subspace processing is developed for acoustic vector sensor array. Finally, simulation results are presented, to compare the proposed method with its counterpart based on acoustic pressure sensor array, the advantages of the proposed algorithm are identified. r 2004 Elsevier B.V. All rights reserved. Keywords: Acoustic vector sensor; Wideband sources; Array signal processing
1. Introduction In recent years, acoustic vector sensor array processing has been drawn much attention in the literature [3–5,10,13–15]. An acoustic vector sensor measures both pressure and particle velocity of the acoustic field at a point in space; whereas a traditional pressure sensor can only extract the pressure information. Various types of acoustic vector sensors with different design technologies are now commercially available [15]. Vector sensor technologies have been used in the field of underCorresponding author.
E-mail address: [email protected] (H. Chen).
water acoustics for decades and currently attracts reinvigorated attention for underwater source location problems [2–5,13–15]. By taking advantage of the extra information, arrays of vector sensors are able to improve source location performance without increasing array aperture size. Nehorai and Paldi [10] have developed the measurement model of the acoustic vector sensor array for dealing with narrowband sources, and introduced it to the array processing research community. Much work has been done for applying acoustic vector sensor array or a single acoustic vector sensor to direction finding problems [2–5,13–15]. Hawkes and Nehorai [3] investigated the conventional beamforming and
0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.07.030
ARTICLE IN PRESS 838
H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
Nomenclature að2Þ ðYk Þ array manifold of a 2-D acoustic vector sensor að3Þ ðYk Þ array manifold of a 3-D acoustic vector sensor ap ðf i ; Yk Þ direction vector of an acoustic pressure sensor array av ðf i ; Yk Þ direction vector of an acoustic vector sensor array Ap ðf i Þ direction matrix of an acoustic pressure sensor array Av ðf i Þ direction matrix of an acoustic vector sensor array B frequency bandwidth of the signals c sound velocity in the medium d inter-element spacing ei generalized eigenvector corresponding to li fc central frequency of the signals f0 focusing frequency Im m m identity matrix K the number of source signals M the number of array elements
Capon direction estimation for acoustic vector sensor array. ESPRIT and Root-MUSIC algorithms have recently been applied by Wong and Zoltowski to arrays of acoustic vector sensors [13–15]. However, these direction finding algorithms for acoustic vector sensor arrays are all based on the narrowband assumptions, moreover, most of them cannot effectively deal with correlated sources and may fail in fully coherent source scenarios. Wong and Zoltowski [13] applied the method of spatial smoothing to solve the narrowband coherent sources for acoustic vector sensor array, but unfortunately this method reduced the effective array aperture. In practice, the sources are usually wideband in most scenarios, and may be completely correlated with each other. Therefore, it is necessary to investigate the problem of the direction of arrival (DOA) estimation for wideband coherent sources with acoustic vector sensor array. Motivated by this, we studied the
Nðf i Þ noise vector pk ðr; tÞ acoustic pressure corresponding to the kth source at location r and time t Pn ðf i Þ autocovariance matrix of noise vector Ps ðf i Þ autocovariance matrix of signal vector Rx ðf i Þ autocovariance matrix of Xðf i Þ Sðf i Þ signal vector T p ðf i Þ focusing matrix of acoustic pressure sensor array T v ðf i Þ focusing matrix of acoustic vector sensor array uk unit bearing vector of the kth source vk ðr; tÞ acoustic particle velocity corresponding to the kth source at location r and time t Xðf i Þ received data vector of the array li generalized eigenvalue r ambient density s2p noise variance of the acoustic pressure sensor s2v noise variance of the acoustic particle velocity sensor Yk DOA of the kth source signal with elevation angle yk and azimuth angle jk
wideband DOA estimation using acoustic vector sensor array for wideband coherent sources. Wang and Kaveh [12] proposed the coherent signal-subspace (CSS) method for wideband direction finding with acoustic pressure sensor array, it has been shown that the CSS method can treat the fully coherent sources without reducing the effective array aperture, and a lower detection and resolution signal-to-noise ratio (SNR) thresholds can be achieved for wideband sources. From then on, various methods of wideband DOA estimation have been put forward based on CSS processing [1,8]. In this paper, based on the idea of wideband CSS processing, we develop a 2-D wideband DOA estimation algorithm for acoustic vector sensor array. This paper is organized as follows. Section 2 formulates the mathematical data models of the acoustic vector sensor and acoustic vector sensor array, respectively. Section 3 develops an
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
algorithm of wideband DOA estimation based on CSS processing for acoustic vector sensor array. Simulation results as well as performance analysis and comparison with pressure sensor array are presented in Section 4. Finally, Section 5 concludes the paper.
2. Mathematical data model and problem formulation We here consider the scenario of K wideband sources with bandwidth B and central frequency f c traveling from far-field through a homogeneous isotropic fluid medium. Letting Yk ¼ ðyk ; jk Þ denote the DOA of the kth source, with elevation angle yk 2 ½0 ; 180 defined clockwise relative to the z-axis and azimuth angle jk 2 ð 180 ; 180 defined counterclockwise relative to the x-axis, as shown in Fig. 1. The objective of the problem is to determine the DOA parameter vector H ¼ ½Y1 ; Y2 ; ; YK : 2.1. Data model for an acoustic vector sensor The relationship between acoustic pressure and particle velocity for a planewave at point r and
z
DOA(θk,ϕk)
θk y
o ϕk
x Fig. 1. Configuration of DOA of kth source signal.
839
time t can be expressed as [11] vk ðr; tÞ ¼
pk ðr; tÞ uk ; rc
(1)
where vk ðr; tÞ and pk ðr; tÞ denote the acoustic particle velocity and acoustic pressure of the kth source, respectively, r is the ambient density, c is the sound velocity in the medium, uk is the unit bearing vector of the kth source with uk ¼ ½ sin yk cos jk ; sin yk sin jk ; cos yk T ; the superscript T denotes the matrix transpose. An acoustic vector sensor is composed of a pressure sensor and a pair or a triad of spatially co-located but orthogonal velocity sensors which measure the two or three orthogonal components of acoustic particle velocity. The array manifold of an acoustic vector sensor composed of a pressure sensor and a pair of spatially co-located but orthogonal velocity sensors is 2 3 2 3 1 1 6 7 6 7 að2Þ ðYk Þ ¼ 4 ukx 5 ¼ 4 sin yk cos jk 5: (2) uk y sin yk sin jk Correspondingly, the array manifold of an acoustic vector sensor composed of a pressure sensor and a triad of spatially co-located but orthogonal velocity sensors is 3 2 2 3 1 1 7 6 sin yk cos jk 7 6 uk x 7 6 7 7¼6 að3Þ ðYk Þ ¼ 6 ; (3) 6 6 uky 7 4 sin yk sin j 7 5 k 5 4 uk z cos yk where ukx ¼ sin yk cos jk ; uky ¼ sin yk sin jk and ukz ¼ cos yk are the three direction cosine components of uk along the x-axis, y-axis and z-axis, respectively. In the following part, for acoustic vector sensor with array manifold að2Þ ðYk Þ; we call it two-dimensional (2-D) acoustic vector sensor since it deals with the two orthogonal components of ukx and uky : Correspondingly, 3-D acoustic vector sensor refers to the one with array manifold að3Þ ðYk Þ: In this paper, we mainly discuss the array of 2-D acoustic vector sensors, but it is not difficult to generalize the results to the case of arrays of 3D acoustic vector sensors.
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
840
2.2. Data model for acoustic vector sensor array
where
Consider an acoustic vector sensor array composed of M identical 2-D acoustic vector sensors located at r1 ; r2 ; ; rM ; respectively. The received 3M 1 array output spectral vector for the ith frequency bin f i ðf c B=2pf i pf c þ B=2Þ can be expressed as
Av ðf i Þ ¼ ½av ðf i ; Y1 Þ; av ðf i ; Y2 Þ; ; av ðf i ; YK Þ
Xðf i Þ ¼
K X
ap ðf i ; Yk Þ að2Þ ðYk Þ S k ðf i Þ þ Nðf i Þ;
k¼1
(4) Here denotes the Kronecker product, i ¼ 1; 2; ; J; J denotes the number of frequency components to be analyzed, and Xðf i Þ ¼ ½X 1p ðf i Þ; X 1vx ðf i Þ; X 1vy ðf i Þ; ; T
X Mp ðf i Þ; X Mvx ðf i Þ; X Mvy ðf i Þ :
ð5Þ
where X mp ðf i Þ; X mvx ðf i Þ and X mvy ðf i Þ denote the ith frequency components of the acoustic pressure and two orthogonal particle velocity components for the mth acoustic vector sensor, respectively. S k ðf i Þ is the ith frequency component of the kth source signal. ap ðf i ; Yk Þ is the M 1 direction vector of acoustic pressure sensor array with the same geometry as the acoustic vector sensor array for the kth source at the ith frequency bin, it is given by T
ap ðf i ; Yk Þ ¼ ½ ej2pðr1 uk Þ ;
T
ej2pðr2 uk Þ ;
;
T
ej2pðrM uk Þ T :
(6) The 3M 1 noise vector Nðf i Þ can be expressed as
ð7Þ
where N mp ðf i Þ; N mvx ðf i Þ and N mvy ðf i Þ represent the noise present at the mth acoustic vector sensor. It should be noted that in Eq. (4), without loss of generality, we have normalized the particle velocity measurements by multiplying with a constant rc which is assumed to be known, according to Eq. (1). In vector notation, Eq. (4) can be rewritten as Xðf i Þ ¼ Av ðf i ÞSðf i Þ þ Nðf i Þ;
að2Þ ðY2 Þ; ; ap ðf i ; YK Þ að2Þ ðYK Þ
ð9Þ
is the 3M K direction matrix of acoustic vector sensor array, where av ðf i ; Yk Þ ¼ ap ðf i ; Yk Þ að2Þ ðYk Þ
1pkpK
(10)
is the 3M 1 direction vector of acoustic vector sensor array corresponding to the kth source. And Sðf i Þ contains the ith frequency component of the K source signals Sðf i Þ ¼ ½ S1 ðf i Þ;
S 2 ðf i Þ;
;
SK ðf i Þ T :
(11)
Here, we make the following commonly used assumptions on the array model Eq. (8) for further development: (1) The signals and the noise are independent and identically distributed (i.i.d.), ergodic, zero mean stationary random processes. (2) The noise is statistically independent of the signals and temporally white. (3) The noise is spatially white, say, the noise between different sensor components of an acoustic vector sensor, as well as between all components of acoustic vector sensors located at different places, are uncorrelated with each other. The covariance matrix of Xðf i Þ is given by Rx ðf i Þ ¼ E½Xðf i ÞX H ðf i Þ
Nðf i Þ ¼ ½N 1p ðf i Þ; N 1vx ðf i Þ; N 1vy ðf i Þ; ;
N Mp ðf i Þ; N Mvx ðf i Þ; N Mvy ðf i ÞT
¼ ½ap ðf i ; Y1 Þ að2Þ ðY1 Þ; ap ðf i ; Y2 Þ
(8)
¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ þ Pn ðf i Þ;
ð12Þ
where H denotes the complex conjugate transpose, Ps ðf i Þ ¼ E½Sðf i ÞS H ðf i Þ and Pn ðf i Þ ¼ E½Nðf i ÞN H ðf i Þ are the covariance matrices of source signals and noise at the ith frequency bin, respectively. Based on the above assumptions about the noise, we have " 2 # 0 sp Pn ðf i Þ ¼ I M ; (13) 0 s2v I 2
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
where s2p and s2v denote the noise variance present at pressure sensor and particle velocity sensor, respectively. I m is an m m identity matrix. Under the assumption of homogeneous isotropic medium, for a single 2-D acoustic vector sensor, it holds that s2p ¼ 3s2v
(14)
for a detailed discussion, see [2,5]. Therefore, the covariance matrix Rx ðf i Þ can be rewritten as Rx ðf i Þ ¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ þ P n ðf i Þ ¼ Av ðf i ÞPs ðf i ÞAH v ðf i Þ " # 1 0 2 þ sp I M : 0 1=3I 2
ð15Þ
3. Wideband DOA estimation for acoustic vector sensor array 3.1. Wideband focusing for acoustic vector sensor array Wang and Kaveh [12] have proposed the coherent signal-subspace (CSS) method for wideband DOA estimation using the acoustic pressure sensor array. The central idea of CSS method is wideband focusing concept, and they have proved the existence of focusing matrices for acoustic pressure sensor array. In the CSS method, the wideband array data are decomposed into several narrowband components via discrete Fourier transform at first. Focusing matrices are then constructed to transform each of the narrowband covariance matrices into the one corresponding to the reference frequency bin which is referred to focusing frequency. The technique of wideband focusing has been an important basis for several wideband processing methods. Based on the idea of wideband focusing for pressure sensor array, here we extend wideband focusing into acoustic vector sensor array for wideband coherent array processing. Theorem 1. Assume that the focusing matrix for Melement acoustic pressure sensor array is T p ðf i Þ; the
841
focusing frequency is f 0 : For M-element acoustic vector sensor array with the identical array geometry and focusing frequency, it holds that (1) there must exist a focusing matrix T v ðf i Þ; satisfying T v ðf i ÞAv ðf i Þ ¼ Av ðf 0 Þ; (2) T v ðf i Þ ¼ T p ðf i Þ I 3 . Proof. See Appendix.
&
It shows that the wideband focusing technique can also be applied to vector sensor arrays. From this point of view, a wideband array processing method for vector sensor array similar as pressure sensor array can be established. There are many classes of focusing matrices available have being proposed in the literature for wideband processing of pressure sensor arrays [6,7,9]. From the above theorem, it shows that the wideband focusing matrices for acoustic vector sensor array can be constructed by those for pressure sensor array with the same array geometry. Thus, this provides us a method for construction of wideband focusing matrices for acoustic vector sensor arrays. From the above derivation, for the array composed of 3-D vector sensors, it holds that T v ðf i Þ ¼ T p ðf i Þ I 4 : 3.2. The DOA estimation based on coherent signalsubspace processing The array output after focusing transform is (16)
Yðf i Þ ¼ T v ðf i ÞXðf i Þ:
The covariance matrix corresponding to the ith frequency bin is Ry ðf i Þ ¼ E½Yðf i ÞY H ðf i Þ ¼ T v ðf i ÞE½Xðf i ÞX H ðf i ÞT H v ðf i Þ ¼ Av ðf 0 ÞPs ðf i ÞAH v ðf 0 Þ ( " þ s2p T v ðf i Þ I M
1
0
0
1=3I 2
#) TH v ðf i Þ: ð17Þ
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
842
The coherently averaged covariance matrix is given by Ry ¼
J X
Ry ðf i Þ
i¼1
"
¼ Av ðf 0 Þ
J X
þ s2p
PMUSIC ðYÞ ¼ P3M
# (
"
T v ðf i Þ I M
i¼1
#)
1
0
0
1=3I 2
TH v ðf i Þ: ð18Þ
For the ease of representation, Eq. (18) can be rewritten as 2 Ry ¼ Av ðf 0 ÞRs AH v ðf 0 Þ þ sp Rn
(19)
with Rs ¼
J X
(20)
Ps ðf i Þ
i¼1
and Rn ¼
J X i¼1
( T v ðf i Þ I M
"
1 0
1
2 H i¼Kþ1 jei av ðf 0 ; YÞj
Ps ðf i Þ AH v ðf 0 Þ
i¼1 J X
With Theorem 2 at hand, we can construct the MUSIC spatial spectra for acoustic vector sensor array as
0 1=3I 2
#) TH v ðf i Þ: (21)
Based on the idea of wideband focusing, we have the following coherent signal-subspace theorem for acoustic vector sensor array. Theorem 2. Let li ði ¼ 1; ; 3MÞ denote the generalized eigenvalue of matrix pencil ðRy ; Rn Þ; l1 Xl2 X XlK XlKþ1 X Xl3M ; ei is the generalized eigenvector associated with li ; the focusing frequency is f 0 : It holds that (1) there must exist 3M2K smallest eigenvalues lKþ1 ¼ lKþ2 ¼ ¼ l3M ¼ s2p ; (2) the column space of the matrix E n ¼ ½eKþ1 ; eKþ2 ; ; e3M is orthogonal to the column space of the matrix Av ðf 0 Þ ¼ ½av ðf 0 ; Y1 Þ; av ðf 0 ; Y2 Þ; ; av ðf 0 ; YK Þ: Proof. It can be obtained by modifying the proof in [12], and thus omitted here. &
:
(22)
The DOA estimation for wideband sources can be given by the peaks of PMUSIC ðYÞ: The steps of the proposed DOA estimation algorithm are summarized as follows: (1) Perform FFT to obtain the array output spectral vector Xðf i Þ: (2) Set the focusing frequency and corresponding focusing matrix, compute Yðf i Þ from Eq. (16) and calculate the sample covariance matrix Ry ðf i Þ: (3) Compute the averaged covariance matrix Ry ¼ PJ R ðf i¼1 y i Þ; compute Rn from Eq. (21). (4) Calculate the noise subspace E n from generalized eigendecomposition of the matrix pencil (Ry ;Rn ). (5) Compute the MUSIC spatial spectra PMUSIC ðYÞ from Eq. (22), the peaks of the spectra give the DOAs of the sources.
4. Simulation results To verify the performance of the proposed method, simulations are carried out in the following. The proposed method of wideband DOA estimation based on CSS processing for acoustic vector sensor array is compared with its counterpart for pressure array proposed in [12]. We consider a uniform linear array (ULA) composed of eight 2-D acoustic vector sensors and an eightelement acoustic pressure sensor ULA, both arrays have the same geometry with sensor elements located at x-axis. The signals of interest consisted of 33 narrowband components lying in the frequency band 802120 Hz: The focusing frequency is chosen as f 0 ¼ 80 Hz: The spatially resampled focusing method [9] is used, for it requires no preliminary DOA estimation of the source signals. For acoustic vector sensor ULA, with the focusing matrix for acoustic pressure
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
843
sensor ULA at hand, the focusing matrix for acoustic vector sensor ULA can be obtained by Theorem 1 in Section 3. The additive array noise which is independent of source signals, is stationary zero-mean bandpass white Gaussian process with the same pass band with the source signals. Here the SNR is defined as the ratio of the power of each source signal to that of the noise at a single sensor. For the two kinds of arrays, the total 4096 successive data samples at each sensor are collected for each simulation trial, which are divided into 64 segments, each having 64 samples, to estimate the covariance matrices. Example 1. Consider 8-element acoustic vector sensor ULA and 8-element acoustic pressure sensor ULA, both have the same inter-element spacing d ¼ lmin =2; where lmin is the wavelength corresponding to the maximum frequency of interest band. Two coherent wideband source signals s1 ðtÞ arriving from Y1 ¼ ðy1 ; j1 Þ ¼ ð20 ; 45 Þ and s2 ðtÞ arriving from Y2 ¼ ðy2 ; j2 Þ ¼ ð20 ; 70 Þ impinge the two arrays simultaneously. The source signal s2 ðtÞ is the time delayed version of s1 ðtÞ by 0.0625 s. The SNR of the received signal for each element of pressure sensor ULA is 10 dB, and the SNR for the acoustic pressure sensors of acoustic vector sensor ULA are also all set to be 10 dB. Thirty independent trials are carried out for each figure in the following. The MUSIC spatial spectra for acoustic pressure sensor ULA and vector sensor ULA are shown in Figs. 2 and 3, respectively. There are two points to be noted from the results of the simulations. One is that the CSS processing-based DOA estimation for pressure sensor ULA can only estimate the azimuth angle, the other is that it has azimuth angle estimation ambiguities. Comparatively, as Fig. 3 shows, there appear spectrum peaks only at the true source bearings for acoustic vector sensor ULA. Therefore, it shows that we can locate the 2-D DOA of wideband coherent sources with a vector sensor ULA. For the ease of comparisons in the following, we shall concentrate on the azimuth angle estimation of the source signals, and the elevation angle is set to be 901. But it should be noted that the vector sensor ULA can also determine the elevation angle as well.
Fig. 2. 2-D spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA. Signals at Y1 ¼ ð20 ; 45 Þ and Y2 ¼ ð20 ; 70 Þ; the interelement spacing d ¼ lmin =2:
Fig. 3. 2-D spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA. Signals at Y1 ¼ ð20 ; 45 Þ and Y2 ¼ ð20 ; 70 Þ; the interelement spacing d ¼ lmin =2:
Example 2. In this part, we analyze the performance of the wideband DOA estimation methods when the source signal is close to endfire direction (near j ¼ 0 or 180 ). The arriving angles of the two wideband signals s1 ðtÞ and s2 ðtÞ are j1 ¼ 10 and j2 ¼ 70 ; respectively. The other conditions for the source signals and the arrays are the same
ARTICLE IN PRESS 844
H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
as those in Example 1. Figs. 4 and 5 show the array output MUSIC spatial spectra for pressure sensor ULA and vector sensor ULA, respectively. It is seen that when the source signals are close to endfire direction, the wideband DOA estimation for acoustic pressure sensor ULA has not only DOA estimation ambiguities at the central sym-
metric points of true DOAs of source signals, but also ambiguities near 180 and 180 ; simultaneously. Therefore, it cannot be determined that which end the signal source comes from. As it is shown in Fig. 5, the wideband DOA estimation for acoustic vector sensor ULA exhibits a better performance when dealing with the signal sources with arriving angle close to endfire. When the DOA of source signal is close to endfire, the width of the spatial spectrum becomes slightly larger, but there exists no ambiguities, and can still give an accurate DOA estimation for the wideband sources.
Fig. 4. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 10 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
Example 3. Consider the same two wideband coherent source signals as in Example 1 except that the arriving angles for s1 ðtÞ and s2 ðtÞ are j1 ¼ 65 and j2 ¼ 70 ; respectively. The same ULAs as used in Example 1 are employed except that the inter-element spacing increases to d ¼ lmin for both arrays. Fig. 6 shows the MUSIC spatial spectrum for pressure sensor ULA with inter-element spacing d ¼ 0:5 lmin ; Figs. 7 and 8 show the MUSIC spatial spectra for pressure sensor ULA and vector sensor ULA both with inter-element
Fig. 5. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 10 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
Fig. 6. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin =2:
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
Fig. 7. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element pressure sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin :
845
pressure sensor array, when inter-element spacing d ¼ 0:5lmin ; the two sources can not be resolved, with inter-element spacing d increased to d ¼ lmin ; the two sources can be clearly resolved. Comparatively, the wideband DOA estimation for acoustic vector sensor ULA can also clearly resolve the two wideband coherent sources, while it does not suffer from DOA estimation ambiguity when the interelement spacing exceeds the Nyquist half-wavelength upper limit, as shown in Fig. 8. For the acoustic pressure sensor array, when the aperture size is enlarged, there are more sensor elements for avoiding the appearance of grating lobes. It shows that the aperture enlarging for vector sensor array can be achieved by spacing the vector sensor greater than half-wavelength while no additional sensors are needed, therefore a better array resolution and DOA estimation precision can be obtained with no extra cost and computational load to be needed.
5. Conclusions
Fig. 8. Spatial spectrum of wideband DOA estimation method based on CSS processing for 8-element vector sensor ULA against scanning angle j 2 ð 180 ; 180 : Signals at j1 ¼ 65 and j2 ¼ 70 ; the inter-element spacing d ¼ lmin :
spacing d ¼ lmin ; respectively. As it is shown in Fig. 7, when the inter-element spacing exceeds the Nyquist half-wavelength upper limit, there appear the undesired grating lobes which have the same amplitudes with those of the true spectrum peaks. As we know, an enlarged aperture size generally offers enhanced array resolution and DOA estimation precision. As Figs. 6 and 7 show, for
In this paper, by extending the concept of wideband focusing into wideband array processing for acoustic vector sensor array, a wideband DOA estimation method for acoustic vector sensor array is proposed based on CSS processing, which can handle the problems of wideband coherent sources without reducing the effective array aperture size. Simulations were carried out to compare the proposed method with its counterpart for pressure sensor array when the source signals are wideband coherent. It is demonstrated that the proposed method for acoustic vector sensor array not only can solve the wideband DOA estimation for coherent sources, but also preserves the good properties of vector sensor array embodied in the narrowband scenario. The simulation results show that the proposed approach for acoustic vector sensor linear array are able to: realize unambiguous direction finding in the whole space, thus suffers no left/right ambiguity problem; allow the uniform inter-element spacing to exceed the Nyquist half-wavelength upper limit. Therefore, this allows the use of spatially undersampled arrays for extending the effective aperture, enhances array
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847
846
resolution and the DOA estimation precision without increasing the computational load. Thus the shortcoming of poor performance as encountered by pressure sensor linear array when dealing with direction estimation of sources impinging from the endfire directions can be overcome.
A, B, C and D, it holds that ðA BÞðC DÞ ¼ ðACÞ ðBDÞ:
(28)
Considering the above property, Eq. (27) can be rewritten as fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ fT p ðf i Þap ðf i ; Yk Þg fI 3 að2Þ ðYk Þg
Acknowledgements The research in this paper was supported by the National Science Foundation of China under Grant 10304008 and the China Postdoctoral Foundation under Grant 2004036415.
¼ fT p ðf i Þ I 3 gfap ðf i ; Yk Þ að2Þ ðYk Þg:
ð29Þ
Substitute Eq. (29) into Eq. (26), we have fT p ðf i Þ I 3 gfap ðf i ; Yk Þ að2Þ ðYk Þg ¼ ap ðf 0 ; Yk Þ að2Þ ðYk Þ:
ð30Þ
Appendix
Finally, comparing Eq. (30) with Eq. (25), it shows that
The proof of Theorem 1. The proof of (1) is simple as [12]. Here we call T v ðf i Þ wideband focusing matrix for acoustic vector sensor array. According to the concept of wideband focusing, we have, respectively,
T v ðf i Þ ¼ T p ðf i Þ I 3 :
T p ðf i Þap ðf i ; Yk Þ ¼ ap ðf 0 ; Yk Þ
(23)
and T v ðf i Þav ðf i ; Yk Þ ¼ av ðf 0 ; Yk Þ:
(24)
Substitute Eq. (10) into Eq. (24), one can obtain T v ðf i Þfap ðf i ; Yk Þ að2Þ ðYk Þg ¼ ap ðf 0 ; Yk Þ að2Þ ðYk Þ:
ð25Þ
Appling Kronecker product operation to the two sides of Eq. (23) by að2Þ ðYk Þ simultaneously, we have fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ ap ðf 0 ; Yk Þ að2Þ Yk Þ:
ð26Þ
The left side of Eq. (26) is equivalent to fT p ðf i Þap ðf i ; Yk Þg að2Þ ðYk Þ ¼ fT p ðf i Þap ðf i ; Yk Þg fI 3 að2Þ ðYk Þg:
ð27Þ
In the following, we shall use a property for Kronecker product operation. That is, for matrices
(31)
References [1] K.M. Buckley, L.J. Griffiths, Broad-band signal-subspace spatially spectrum (BASS-ALE) estimation, IEEE Trans. ASSP. 36 (7) (July 1988) 953–964. [2] H. Feng, G. Liang, J. Hui, The direction estimation using combined sensor with pressure and particle velocity, Acta Acustica (in Chinese) 25 (6) (2000) 516–520. [3] M. Hawkes, A. Nehorai, Acoustic vector-sensor beamforming and Capon direction estimation, IEEE Trans. Signal Processing 46 (9) (September 1998) 2291–2304. [4] M. Hawkes, A. Nehorai, Acoustic vector-sensor processing in the presence of a reflecting boundary, IEEE Trans. Signal Processing 48 (11) (November 2000) 2981–2993. [5] M. Hawkes, A. Nehorai, Acoustic vector-sensor correlations in ambient noise, IEEE J. Oceanic Eng. 26 (3) (July 2001) 2291–2304. [6] H.S. Hung, M. Kaveh, Focussing Matrices for coherent signal-subspace processing, IEEE Trans. ASSP 36 (8) (August 1988) 1272–1281. [7] H.S. Hung, C.Y. Mao, Robust coherent signal-subspace processing for directions of arrival estimation of wideband sources, IEE Proc. Radar Sonar Navig. 141 (5) (1994) 256–262. [8] J. Krolik, D.N. Swingler, Multiple broad-band source location using steered covariance matrices, IEEE Trans. ASSP 37 (10) (October 1989) 1481–1494. [9] J. Krolik, D. Swingler, The detection performance of coherent wideband focusing for a spatially resampled array, Proc. IEEE ICASSP (1990) 2827–2830.
ARTICLE IN PRESS H. Chen, J. Zhao / Signal Processing 85 (2005) 837–847 [10] A. Nehorai, E. Paldi, Acoustic vector-sensor array processing, IEEE Trans. Signal Processing 42 (9) (September 1994) 2481–2491. [11] P.M. Se, K.U. Ingard, Theoretical acoustics, McGrawHill, New York, 1968. [12] H. Wang, M. Kaveh, Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources, IEEE Trans. ASSP. 33 (4) (April 1985) 823–831. [13] K.T. Wong, M.D. Zoltowski, Extended-aperture underwater acoustic multisource azimuth/elevation direction-
847
finding using uniformly but sparsely spaced vector hydrophones, IEEE J. Oceanic Eng. 22 (4) (April 1997) 659–672. [14] K.T. Wong, M.D. Zoltowski, Root-MUSIC-based azimuth-elevation angle-of-arrival estimation with uniformly spaced but arbitrarily oriented velocity hydrophones, IEEE Trans. Signal Processing 47 (12) (December 1999) 3250–3260. [15] K.T. Wong, M.D. Zoltowski, Self-initiating MUSIC-based direction finding in underwater acoustic particle velocityfield beamspace, IEEE J. Oceanic Eng. 25 (2) (February 2000) 262–273.