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A constant problem is to localize a number of acoustic sources, to separate their individual signals and to estimate their strengths in a propagation medium. An acoustic receiving array with signal processing algorithms is then used. The most widely used algorithm is the conventional beamforming algorithm but it has a very low resolution and high sidelobes that may cause a signal leakage problem. Several new signal processors for arrays of sensors are derived to evaluate the strengths of acoustic signals arriving at an array of sensors. In particular, we present the covariance vector estimator and the pseudoinverse of the array manifold matrix estimator. The covariance vector estimator uses only the correlations between sensors and the pseudoinverse of the array manifold matrix estimator operates with the minimum eigenvalues of the covariance matrix. Numerical and experimental results are presented.

Arrays of sensors are used in many fields to detect weak signals, to estimate the bearing and the strengths of signals arriving from different directions. For example, in industrial environment an array of microphones is used to localize and to determine the strength of polluting noise sources. Conventional ways of noise source identifications include sound intensity measurement [

The approach taken here is to assume that the signal field at the array is comprised of P independent plane-wave arrivals from P known directions, as shown in

In practice, of course, the directions are rarely known

exactly, however this difficulty can be overcome by using the standard MUSIC algorithm [5,6], which constitutes an angular pseudo-spectrum and an indicator of directions of arrival of different signals. The problem then reduces to estimating the signal powers from each of the P directions.

This study focuses on developing estimators which are used to identify the distribution of signal power generated by acoustic sources. This paper is organized as follows. An array signal model and the spatial covariance matrix of the sensor outputs are formulated in Section 2. The conventional beamformer and adaptive beamformers are presented respectively in Sections 3 and 4. A technique to obtain the strengths of signals arriving at an array of sensors based on the covariance vector of signals is developed in Section 5. A signal power estimation obtained by the pseudoinverse of the array manifold matrix is studied in Section 6. Numerical and experimental results showing the effectiveness and the weakness of different algorithms in signal power estimation are presented in Section 7. This paper is briefly concluded in Section 8.

Consider a uniform linear array with N sensors (

where the meanings of the various parameters are as follows: is the complex envelope of the ith signal source at the first sensor; d is the space between two adjacent sensors; is the signal wavelength corresponding to frequency f and is the additive noise output of the kth sensor.

The complex envelope of the ith source is a zero-mean complex random variable. Its variance, denoted p_{i}, characterizes the signal power of the ith source which we wish to estimate

Here, is the expectation operator and the superscript ^{*} represents the complex conjugate.

Equation (1) can also be expressed in the vector form as the N-dimensional vector [

where

Here, T denotes transpose, and the direction of arrival of the ith signal source is represented by the N-dimensional complex vector. The noise is assumed to be spatially white (uncorrelated from sensor to sensor) and the same power level is present in each receiver. With these assumptions, the covariance matrix for the noise alone is given bywhere is the noise power, the identity matrix and the superscript H denotes the Hermitian transpose operation. Equation (3) may be rewritten in the matrix form

is the array manifold matrix containing the manifold vectors for different sources as its columns,. For any single plane wave arrival, the outputs from the N individual receivers will differ in phase by an amount determined by the geometry of the array and the arrival direction. In other words, the elements A_{kr} of the matrix A are known functions of the signal arrival angles and the array elements locations. It can readily be seen that the output signal from the qth sensor may be written as

Since the P arrivals are by assumption independent, the source covariance matrix is given by

and the diagonal elements are the powers of the sources , from the P directions, which we wish to estimate. The spatial covariance matrix of the receiver outputs can be expressed, for signals uncorrelated of each other and of noise, as

In practice, the spatial covariance matrix is estimated by a finite number of time domain samples (snapshots) and the following estimated form is used

where is the array signal vector sampled at time t_{i} and T is the number of such samples. The caret (^) denotes an estimated value. We can now derive a variety of processors to estimate the strengths of P independent signals arriving at array of N sensors, when the arrival directions are known. Note that the number of sources can be obtained by examining the singular values of the covariance matrix [

The conventional beamformer, also called the “time delay and sum” or “unweighted add-squared” beamformer, consists of a system of delay and sum networks which are designed to make the signals from the beamformer direction in phase at each sensor. The directional data from direction j must be estimated from the sensors output data vector. The usual approach is to find a matrix, such that reconstructs the directional data, and for a conventional beamformer the equation is [

A power estimate for the signals can be found by forming the covariance matrix

and the strength of the ith source estimated by the conventional beamformer is

However, this leads to a biased estimate, as can be seen by substituting (7) into (10)

So unless and, neither of which is generally the case, then the estimate will be biased.

The conventional beamformer can be considered as a kind of linear spatial filter with dataindependent coefficients. In contrast, the minimum variance beamformer, called also the standard Capon beamformer [

subject to the constraint

By the method of Lagrange’s multiplier, it can be shown that the optimum weight vector is

and the power of the ith source estimated by the standard Capon beamformer is

The standard Capon beamformer has better resolution than the conventional beamformer provided that the array steering vector corresponding to the signal of interest is accurately known. However, the performance of this traditional adaptive beamformer can degrade seriously in practice when errors exist in the signal of interest steering vector, which may be due to look direction error, array sensor position error and small mismatches in the sensor responses. In such cases the signal of interest might be mistaken as an interference signal and might be suppressed. A robust Capon beamforming algorithm [

subject to the constraint

The optimization problem can be rewritten as the following form

subject to the constraint (18). We consider the solution on the boundary of the constraint set and we reformulate the optimization problem as the following quadratic form with a quadratic equality constraint

subject to (20)

This problem can be solved by the method of Lagrange’s multiplier which is based on the cost function

Differentiating (21) with respect to and equating to zero gives the optimal solution

where and are matrices containing the eigenvectors and eigenvalues of the covariance matrix and is the Lagrange multiplier. Using (22) in the equality constraint of (20) the Lagrange multiplier is obtained as the solution to the constraint equation

The signal power estimation of the ith source using the robust Capon beamformer is then

In summary of this section, the standard Capon beamformer is an optimal spatial filter that maximizes the signal to noise ratio, provided that the true covariance matrix and the array steering vector are accurately known. However, the covariance matrix can be inaccurately estimated due to imperfect array calibrations, gain and phase errors in the sensors. The robust Capon beamformer presented in the paper can then be used in such situations for both signal power estimation and source location as shown in examples given in Section 7. Another power estimator using the covariance vector of signals is presented in the next section.

Since we are interested in the signal powers, the covariance matrix of the data contains all the information about these signal strengths. The correlation between sensor k and l is and from Equation (5) we obtrain

But since signals from different directions must be uncorrelated, we have

The sensor noise power on each sensor is constant and equals p_{n} and for and zero otherwise. Equation (26) may be split into real and imaginary components as

Of the 2N^{2} equations represented by (27) and (28) only equations are independent. Indeed, one gets

Equations (27) and (28) may then be written in the form

where r, B and p are reals. r is the vector which contains the real and imaginary components of and is called the covariance vector; p is the vector containing the signal powers and sensor noise power p_{n}. Note that if the sensor noise is small, it may be desirable to omit the model of sensor noise. B is the matrix which contains all the array geometry terms and if required

The least squares solution to (32) is given by

p_{CV} is the vector containing the strengths of signals by the covariance vector. Performances of this estimator are given in Section 7. Another power estimator based on the pseudoinverse of the array manifold matrix A is presented in the next section.

From Equation (7) we obtain

A possible approach to estimate the signal strengths is to select the P diagonal elements of the matrix R_{s}. We get

where A^{+} is the pseudoinverse of the array manifold matrix. To obtain the noise covariance matrix, or the noise power p_{n}, we consider the eigenvalue decomposition of the covariance matrix R

The rank of is equal to the number of incident signals P and can be determined from the P largest eigenvalues of R. The minimum eigenvalue of R corresponds to the noise power p_{n}_{. }

Numerical simulations and experimental tests are now presented to evaluate the performances of the estimators presented in the paper.

The conventional beamformer (CB), the standard Capon beamformer (SCB), the robust Capon beamformer (RCB), the covariance vector estimator (CV) and the pseudoinverse estimator (PI) are employed to estimate the strengths of signals arriving at an array of receivers. In our simulations, we assume a uniform linear array with N = 6 omnidirectional sensors and half-wavelength sensor spacing. Four point sources are located at bearings of— 30˚, 0˚, 22˚ and 45˚. The source powers are respectively 60 dB, 55 dB, 80 dB and 70 dB and the number of snapshots is T = 4096. The signal to noise ratio is SNR = 20 dB.

tions. The CB estimator has much poorer resolution than both SCB and RCB and the sidelobes of the former give false peaks and false directions of arrival of sources.

The remaining part of the section is focused on the application of the developed algorithms to the experimental identification of noise sources generated by two loudspeakers. The experimental setup is schematically shown in the block diagram of

The receiving acoustic array is linear and formed with six omnidirectional microphones equally spaced, with inter-element spacing of d = 4.5 cm. The two acoustic sources (the loudspeakers) and the acoustic array are in the same horizontal plane. The transmitting loudspeakers generate two typical audio signals at a frequency of 3800 Hz corresponding to a microphone separation distance of one-half wavelength. The number of snapshots is T = 4096. We are able to find the direction of the two sources by using the MUSIC algorithm, however, unlike the methods mentioned earlier, MUSIC does not physically correspond to the signal power. The MUSIC algorithm is only an indicator of directions of arrival of different signals.

Once the arrival angles have been determined we can estimate the power of the two acoustic sources by our proposed algorithms.

The experimental results confirm that the CV and the PI estimators give very similar results and the RCB estimator overestimates very slightly the source powers.