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Designing Superdirective Microphone Arrays With a Frequency-Invariant Beam Pattern Stefania Repetto, Member, IEEE, and Andrea Trucco, Senior Member, IEEE
Abstract—Frequency-invariant beam patterns are often required in systems using an array of sensors to process broadband signals. Although several methods have been proposed to design a broadband beamformer [typically realized with a finite-impulse-response (FIR) filter for each sensor] with a frequency-invariant beam pattern (FIBP), until now the case in which the spatial aperture is shorter than the involved wavelengths has very rarely been addressed. In such a case, the use of a superdirective beam pattern is essential for attaining an efficient system. In this paper, a novel method to design a broadband beamformer that produces an FIBP for a data-independent superdirective array is proposed and compared with other potential approaches. The method generates a far-field beam pattern that reproduces the desired profile over a very wide frequency band, also if the array is shorter than the wavelength. Two steps are necessary: 1) the generation of many apodizing windows at different frequency values by a stochastic method and 2) the synthesis of the FIR filters with the Parks–McClellan technique. At the end of the design chain, the very simple implementation and the robustness of the attained broadband beamformer to array imperfections increases the applicability of the system, for instance, in audio signal processing using microphone arrays. Index Terms—Broadband beamforming, data-independent beamforming, frequency-invariant beam pattern (FIBP), microphone arrays, simulated annealing, superdirective arrays.
I. INTRODUCTION YSTEMS using sensor arrays are very often involved in processing broadband signals. In some interesting cases, it is important that the performance of the array processor be adequately constant over the entire frequency band of the signals. If the array processor is a beamformer [1], its performance is mainly measured by the beam pattern, so a frequency-invariant beam pattern (FIBP) is required. An FIBP allows one to receive the broadband signal of interest without any distortion, also if it comes from a different direction from that of the array looking direction. In recent decades, some papers have described the general structure of a broadband beamformer [essentially, a generalized version of the narrowband beamformer in which traditional delay blocks and apodizing coefficients are replaced by finite-impulse-response (FIR) filters] and have proposed methods to optimize the beamformer in such a way that an FIBP may be
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Manuscript received January 18, 2005; revised June 28, 2005. The associate editor coordinating the review of this paper and approving it for publication was Prof. Fabien Josse. The authors are with the Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, 16145 Genova, Italy (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/JSEN.2006.874019
obtained [1]–[9]. However, although many different operating conditions and array geometries have been successfully applied, the array aperture has always been assumed to be larger than . all the involved wavelengths is unfavorable and quite unAlthough the case common, it is nevertheless unavoidable in several important applications. This is one of the reasons why, in the past, the superdirectivity theory has been developed that is aimed at increasing the directivity obtainable by an array, exceeding that produced by uniform apodization [10]. Uniform apodization provides the maximum directivity for a narrowband linear array of half-wavelength-spaced isotropic sensors. (Typical apodization windows used to reduce the sidelobes in narrowband applications also reduce the directivity.) In arrays with closer element spacing, directivities can be obtained that are higher than those provided by uniform apodization. This is true even when the interelement spacing is much less than and, globally, . For these arrays, superdirectivity represents the unique chance to produce a beam pattern with an acceptable profile and a useful directivity value. According to the methods proposed in the literature and reviewed in [10], the generation of a superdirective beam pattern is achieved by synthesizing a specific apodization window. A few papers have addressed the design of superdirective arrays by working with broadband signals [11]–[13], but they have not been devoted to the generation of an FIBP, in particular, when the condition is stressed. The cited papers aimed at maximizing the directivity over the signal band (while maintaining a constant gain in the array looking direction), without including the beam pattern profile in the optimization process. Apart from the rigorous definition of directivity [10], [14], in arrays without high sidelobes, directivity is basically a measure of the mainlobe width. If one increases the frequency, the (maximized) directivity also increases, and, as a result, the attained broadband beam patterns are generally frequency varying. Only under specific circumstances is a beam pattern obtained that is almost frequency invariant [12], but this results from the limited extension of the used band and cannot be considered a feature of the optimization method. To the best of our knowledge, only very recently has an approach been proposed [15] that can be used to directly synthesize the FIR filters necessary to produce an FIBP by using a superdirective array and that is also robust enough to deal with errors in the array characteristics. The procedures introduced in [15] take into account the statistics of the gain and phase errors of the array elements during the synthesis phase, by optimizing, for instance, the mean performance of the broadband beamforming, in a least-squares sense, for all the feasible element characteristics. In this way, broadband
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beam patterns (possibly, FIBPs) are obtained that fit the desired profiles in the best way allowed by the statistics of the errors characterizing the array elements. In this paper, we would like to contribute to the very scarce literature on this topic by proposing a method to design a broadband beamformer that produces an FIBP for a data-independent superdirective array, assuming the noise field to be spherically isotropic. The method allows one to produce an FIBP also if occurs, the number of array sensors is the condition quite limited, a very wide frequency band is applied, and limited array imperfections (e.g., element position uncertainty or element response uncertainty) arise. Due to the specific goal of this paper, it is not possible to compare the proposed method and the related performance with the methods devoted to maximizing the array gain, to maximally reduce jamming signals, or to cancel multiple echoes, as proposed in some previous papers (e.g., [11]–[13]). In fact, here the goal is to yield a beam pattern that reproduces the desired profile (imposed at the beginning of the design process) over the entire frequency band in a far-field condition and a fixed looking direction. In addition, as the considered beamforming structure is data independent, the resulting system allows for a very simple implementation, since the beamformer is composed of just an FIR filter for each sensor and a final summation block. The method can be divided into two steps. 1) As in [3] and [8], the broadband problem is split into many narrowband problems: the goal is to find apodizing windows that provide narrowband beam patterns similar to the desired one, at different frequency values. Each narrowband problem is solved by a stochastic approach based on the simulated annealing algorithm, similar to that we proposed in [16] and robust enough to deal with array imperfections. 2) Starting from the sequence of narrowband apodizing windows previously synthesized, a set of linear-phase and equiripple FIR filters are designed with a suitable literature method. Without loss of generality, we adopted the Parks–McClellan algorithm for this purpose. The successful combination of the above two steps, given , is likely to result from several original procedures that represent a key aspect of this paper and will be described subsequently in this paper. We would like to emphasize that the approach proposed in [16] is devoted only to the synthesis of a weighting window for a narrowband superdirective array. Therefore, it represents only a step in the synthesis procedure of the FIR filters. Moreover, differently from [16], here the results (both the narrowband and the broadband ones) are assessed by means of a set of parameters well tailored to measure the performance and evaluate the robustness to array imperfections. Differently from the approach proposed in [15], our method does not consider the statistics of the array element errors in the synthesis phase; it simply adopts a synthesis strategy that is inherently quite robust to typical array imperfections. The actual robustness of the obtained broadband beam pattern to element errors should be assessed a posteriori. In this paper, the method we are proposing and the approach proposed in [15] are used to attain a superdirective FIBP and their related robustnesses are compared. To perform the comparison, several parameters
IEEE SENSORS JOURNAL, VOL. 6, NO. 3, JUNE 2006
(i.e., the directivity, the white noise gain, and the expected floor level) are introduced to assess the performance and robustness over the frequency. Therefore, the novelty and the importance of this paper consist of introducing a new method to produce superdirective FIBPs (a problem very poorly investigated to date), a metrics to assess performance and robustness, and a comparison of the results obtained using different methods. It also contributes to enlarging our general knowledge of broadband superdirective arrays, whose potential is not yet well known. One can assume that the method proposed here represents a simple and valuable alternative to the approach proposed in [15], as the obtainable FIBPs are satisfactory in terms of both directivity and robustness, and potentially have important advantages in specific situations. A major topic, on which we have focused our attention, concerns audio signal processing, especially speech. In this context, the directivity of a microphone array can be used to pick up a sound emitted by a distant source, while suppressing noise and reverberation arriving from other directions [3], [17], [18]. Because the signal of interest can arrive from a generic direction, also different from the array looking direction, the frequency-invariant behavior of the beam pattern is extremely important in order to avoid any distortion of such a signal. In this context, multimicrophone techniques are a growing research field, since beamformers and related techniques have great potential for noise reduction by using spatial information in adverse environments. However, the array design is often restricted by the dimensions of the array itself; hence, solutions requiring small apertures are mandatory. Therefore, we have chosen to use a microphone array with a spatial aperture much shorter than, or very close to, the wavelengths of the involved frequencies so as to increase the number of potential applications and, at the same time, to bridge a literature gap. The techniques proposed in this paper can also be very effective in application fields different from audio processing. For instance, underwater broadband communications systems, installed on small autonomous underwater vehicles (AUVs), often involve using transducer arrays to abate the effects of multipath propagation [19], [20]. The reduced dimensions of some vehicles (forcing the array aperture to be rather limited) and the frequency values chosen to successfully communicate over distances of several kilometers result in the . condition This paper is organized as follows. Section II describes the methods for the synthesis of a narrowband superdirective beam pattern that is similar to the desired pattern and defines the problems linked to the synthesis of a broadband superdirective FIBP, proposing a possible solution. Section III presents the results obtained for a very short array over a bandwidth of three octaves and evaluates the outcomes of the proposed method. Finally, in Section IV some conclusions are drawn. II. SUPERDIRECTIVE FREQUENCY-INVARIANT BEAMFORMING Tapped delay-line architectures, where each sensor feeds a transversal filter and the filter outputs are summed together to produce the beam signal, are typically used to design broadband beamformers [1]. Let us consider an array composed of sensors. If each sensor feeds an th-order FIR filter, then the ( being frequency band of the filtered signal, from 0 to
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the sampling frequency), can be considered to be split into equally spaced subbands. For each subband, the apodizing coefficient assigned to a given sensor is provided by the discrete Fourier transform of the impulse response (i.e., the sequence of the tap coefficients) of the filter connected to such a sensor [3]. of each subband is the following: The width
is defined as . The white noise gain broadside array can be defined [14] as follows:
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for a
(3)
(1) The higher the order of the filter, the larger the number of frequency subbands and the smaller the width of each subband. This means that, once the bandwidth of the original signal has been fixed, it is important to select a filter order that enables one to obtain subbands narrow enough to allow the beam pattern to be quasi-constant inside each of them. On the basis of the above considerations, it is possible to move narrowband problems, refrom one broadband problem to frequency bins defined as the edge frequencies of lated to the subbands. To produce an FIBP, it is possible to synthesize vectors of apodizing coefficients and to assign each vector to the elements of the array in order to produce the desired narrowband beam pattern at the related frequency bin. We refer to such vectors of coefficients as apodizing windows. If apodizing windows, we can we succeed in synthesizing the impulse responses of the th-order move from them to the FIR filters. In the next subsection, we describe a useful method for the synthesis of an apodizing window producing the desired narrowband beam pattern for a superdirective array.
A. Characterization and Optimization of Narrowband Beam Patterns The beamformer performance can be derived from the array gain and the white noise gain. The array gain indicates the improvement in the signal-to-noise ratio (SNR) provided by the array for ambient noise. For an isotropic noise field and plane waves, the array gain is called “directivity” [10], [14], [21], [22]. The white noise gain indicates the improvement in the SNR provided by the array for sensor self-noise, assumed to be white [11], [14], [22]. The inverse of the white noise gain is called “sensitivity factor” [11], [12], [22] and indicates the sensitivity of the array beam pattern to array imperfections (e.g., element position errors and element response errors). For a linear array of isotropic elements equispaced at a distance , assuming a broadside looking direction and an isotropic noise field, the directivity can be expressed [10], [14] as follows:
(2)
where is the apodizing coefficient of the th element of the array, is the wavelength of the monochromatic wave, and
The absolute maximum of is and is achieved only if uniform apodization is applied. The maximization of can be carried out by the following matrix formulation: (4) with being the vector of the apodizing coefficients, the vector defined as , where the superscript T indicates the transpose operation, and the matrix defined as
(5) where
, and . Due to the nonsingularity of the matrix , it is possible to obtain the vector of the apodizing coefficients that maximizes the directivity of the array (i.e., the vector that minimizes the denominator in (2) while the sum of the apodizing coefficients is set equal to 1, in the following way: (6)
However, to avoid an excessive decrease in the white noise gain, can be performed imposing the conthe maximization of straint , . The problem is solved by the Lagrange multiplier method [10], [14], where a Lagrange multiplier is incorporated in the matrix equation (4), precisely at the main diagonal of the matrix except for the first row and is adjusted so as to satisfy the constraint . The maximization of at different frequency bins produces narrowband beam patterns with the mainlobe width decreasing as the frequency increases. However, it is possible to tune the Lagrange multiplier in order to obtain a constant directivity over the frequency (allowing the white noise gain to vary), thus producing beam patterns having approximately the same mainlobe width. As a result, a sequence of apodizing windows producing the same beam pattern at different frequency bins can be synthesized. Unfortunately, this procedure is affected by three major problems. 1) Although narrowband beam patterns with the same directivity are generally similar to one another (in particular, the mainlobe width is almost the same), it is not possible to directly control or bound the differences among them.
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2) The relationship between and the value of the directivity is not clear, and its solution requires many attempts. 3) A superdirective apodizing window contains jumps in sign. In general, such jumps are not the same for all the frequency bins; therefore, a given sensor may have a positive coefficient at a certain frequency bin and a negative coefficient at a close frequency bin. This means that the frequency response of the related filter contains jumps in sign and, possibly, a steep slope, making it very difficult to design an FIR filter with good characteristics. Considering the above-mentioned difficulties, it is preferable to synthesize the apodizing windows by an approach we recently proposed [16], in which the desired beam pattern (the same at all the frequency bins) is introduced and the apodizing window minimizing the distance between the actual and desired beam patterns is computed. To obtain this, a least-squares approach be the number of points used in discan be adopted. Let matrix, an cretizing the direction-of-arrival axis, a vector, and a vector. is a matrix for which the is a vector containing the values of the resulting product narrowband beam pattern, as described in [1]; is the vector of the apodizing coefficients to be optimized, and represents the desired beam pattern. The optimization problem can be posed in the following form:
the apodizing coefficients should be fixed a priori according to the signs of the coefficients contained in the window at the lowest frequency bin. This means that, during the synthesis of the first window, the stochastic algorithm, iteration by iteration, affects the value of each coefficient in spite of zero-crossing occurrences. Instead, during the syntheses of the remaining windows, the signs of the coefficients are assigned before starting, and the stochastic algorithm should affect the value of each coefficient and prevent zero-crossing occurrences. In the end, the signal acquired by whatever sensor is processed coefficients) that by a frequency response (composed of is entirely positive or negative. B. Sensitivity Analysis We have stated that the inverse of the white noise gain (i.e., the sensitivity factor) indicates the sensitivity of the beam pattern to array imperfections. Consequently, an excessive decrease in the white noise gain value cannot be accepted. According to [22], the sensitivity of the beam pattern to array imperfections can be investigated by statistically modeling the array errors and analyzing the mean of the beam power pattern. The errors of the element response (in terms of gain and phase) and of element position can be modeled as follows:
(7) (9) for which, provided that is larger than , the least-squares solution is well known. However, as in the case of the unconstrained directivity maximization, the obtained apodizing window (after normalizing the coefficients to have their sum equal to 1) typically contains very large absolute values, producing an excessive decrease in the white noise gain. To avoid this fact, in [16], we have proposed the minimization of the following energy function: (8) where is the main diagonal of the related matrix and, in the above equation, is used to compute the squared beam pattern, i.e., the beam power pattern. Whereas the expression in (7) admits just one minimum, the function in (8) has many local minima. In [16], it has been shown that some of the minima have a good balance between a small energy value (i.e., a satisfactory similarity between the actual and desired beam power pattern) and a limited decrease in the white noise gain. The stochastic minimization algorithm proposed and extensively described in [16] applies the simulated annealing strategy and is very effective in finding one of the minima with a good balance. This solves the above-mentioned problem (1) and, partially (as the simulated annealing algorithm needs an easy tuning of a few parameters), problem (2). To overcome problem (3), we suggest starting to synthesize the apodizing window at the lowest frequency bin. Of all the frequency bins, this is the most difficult, as the ratio between and is the lowest. Subsequently, in performing the other syntheses, the signs of
is the actual value of the th apodizing coefficient, where is its nominal value, and is a zero-mean Gaussian is the actual random variable representing the response error; position of the th array element, is its nominal position, is a zero-mean Gaussian random variable representing and is the actual value of the phase of the the position error; th element, is its nominal value; and is a zero-mean Gaussian random variable representing the phase error. All the random variables of this model are assumed to be statistically independent. Analyzing the behavior of the beam pattern [22], one can deduce that the mean of the actual beam power pattern is like the nominal one plus a term equal to
(10) where , , and denote the variances of the corresponding random variables. All the values are assumed to be small. Concerning the errors of position, it is worth noting the multiplica, where is the arrival angle belonging to the tive term interval 90 90 . Such a term is due to the modeling of errors only along the array direction, and it means that, if the value is fixed, the effect of the position errors increases as the of arrival angle moves away from the broadside direction. The term establishes a relationship between the sensitivity factor (i.e., the inverse of the white noise gain) and the magnitude of the errors. Moreover, the conversion of into the logarithmic scale provides an expected value below which the side-
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lobes of the beam power pattern cannot fall. In other words, represents the level of the floor [22] for the expected value of the beam power pattern. C. From Narrowband Solutions to a Broadband Beamformer At a given frequency bin, the amplitudes of the frequency responses of the FIR filters should be equal to the apodizing coefficients computed for that bin. In other words, given a single apodizing coefficients related to it represent sensor, the amplitude samples (one for each frequency bin) of the frequency response that should characterize the FIR filter applied to such a sensor. Therefore, the synthesis of the FIR-filter taps apodizing coefficients associated with a spestarts from the cific sensor. More precisely, the design of the FIR filters is based on the Parks–McClellan algorithm [23]. This algorithm uses the Remez exchange technique and the Chebyshev approximation theory to design filters with an optimal fit between the desired and actual frequency responses. The filters are optimal in the sense that the maximum error between the desired frequency response and the actual one is minimized. Filters designed in this way show a linear phase and an equiripple behavior of their tap coefficients of an th-order frequency responses. The FIR filter will satisfy the following symmetry relation:
(11) is the value of the th tap coefficient. This allows one where to design FIR filters that have the same (linear) phase profile. Despite their equiripple nature, if sharp variations in the desired frequency response between some pairs of consecutive frequency bins occur, some difficulties may arise in the filter design process. To handle this point, two strategies have been adopted: 1) completely avoiding changes in sign between adjacent frequency bins, which frequently produce steep slopes of the frequency response, and 2) adequately weighting the importance to achieve a good fit in each frequency subband, as allowed by the Remez exchange technique [23], [24]. To do this, a vector of real, positive weights, one per subband, has been used during the filter design process, according to the rules described in [24]. Essentially, the weight coefficient should be proportional to the desired adherence between the obtained and the desired frequency responses inside a given subband. In order to achieve a globally acceptable result, such an adherence may be relaxed for the subbands that, after some trials, show particular difficulties. Therefore, the weights assigned to the subbands may be inversely proportional to the difficulty experienced for each of them. As described earlier, the sensitivity to array imperfections is proportional to the squared values of the apodizing coefficients: after normalizing the coefficients to make their sum equal to 1, the higher the denominator in (3), the lower the white noise apodizing coefficients gain. At a given frequency bin, the computed for that bin turn out to be equal to the amplitudes of FIR filters. Thus, at a given the frequency responses of the frequency, the sensitivity to array imperfections is proportional to the squared amplitudes of the frequency responses. For this reason, it is commonly affirmed that the higher the energy of the FIR filters, the higher the sensitivity to array imperfections [12].
Fig. 1. Beam pattern obtained by the unconstrained directivity maximization at a frequency of 513 Hz for a linear array of eight sensors over a spatial aperture 0.12 m, working in the air.
D=
This statement is very general as the relationship of the sensitivity to the frequency is lost, and the energy of the FIR filters represents a sort of averaged figure. In this paper, we preferred to check the value of the white noise gain as it is a function of the frequency and has a very clear physical meaning. III. RESULTS AND DISCUSSION The proposed array is linear, composed of eight equispaced pointlike omnidirectional microphones with a spatial aperture equal to 12 cm and an interelement spacing equal to 1.714 cm. It has been designed to work in the air with acoustic waves trav340 m/s. eling at a speed A. Narrowband Solutions Fig. 1 shows the narrowband beam pattern obtained by the unconstrained maximization of the directivity for the array previously described, at a frequency of 513 Hz. It is worth noting is 0.181 and the ratio that at such a frequency, the ratio is 0.026; the directivity produced by the uniform apodization is 0.20 dB, and the white noise gain is 9.03 dB. After the directivity maximization, the directivity value increases up to 6.80 dB, but the white noise gain decreases to 135.52 dB, i.e., an unacceptably small value. Moving from unconstrained to constrained directivity maximization, Fig. 2 shows an example of the narrowband beam pattern obtained (solid line). In this case, the directivity is 3.25 dB, whereas the white noise gain is 17.85 dB. The balance is now more acceptable, and the beam pattern shows a nice profile. Increasing the frequency up to 1083 Hz, by tuning the Lagrange multiplier, it is possible to again produce a superdirective beam pattern with a directivity of 3.25 dB (that produced by the uniform apodization is 0.85 dB). The related white noise gain is 3.96 dB, higher than before as the ratio is also higher (0.382 against 0.181). Fig. 2 confirms that, as a consequence of the similarity of the directivity values, the obtained beam pattern (dashed line) is also very similar to the previous one (solid
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Fig. 2. Beam patterns obtained by the constrained directivity maximization at three frequency values for a linear array of eight sensors over a spatial aperture 0.12 m, working in the air: 513 Hz (solid line); 1083 Hz (dashed line); and 2220 Hz (dotted line).
D=
Fig. 3. Desired beam pattern profile for an eight-sensor linear array with a spatial aperture 0.12 m, working in the air.
D=
line), especially in the 50 50 interval. However, this is no longer true if a third, much higher, frequency value is used. For instance, if a frequency of 2220 Hz is chosen and the Lagrange multiplier is tuned to produce a superdirective window with the same directivity as achieved at lower frequencies (i.e., 3.25 dB against a directivity of 2.96 dB due to the uniform apodization), a very different beam pattern profile can be observed, as shown by the dotted line in Fig. 2. Therefore, there is no guarantee that producing the same directivity results in similar narrowband beam patterns. It may only happen over restricted frequency bands. This fact confirms the importance of problem (1), faced in Section II-A. Moreover, comparing the signs of the apodizing coefficients at a couple of frequency bins, one can verify that a change in sign occurred for a few sensors, as anticipated in problem (3), described in Section II-A.
Fig. 4. Apodizing windows provided by the simulated annealing-based method at different frequencies. (a) 513 Hz (solid line) and 627 Hz (dashed line). (b) 1083 Hz (solid line) and 2907 Hz (dashed line).
To apply the stochastic approach, the desired narrowband beam pattern should be defined. We used that shown in Fig. 3. 50 ) was obtained The central part of it (included in 50 by the uniform apodization of the array at a frequency of 2390 ). As we shall describe in greater deHz (i.e., at a ratio tail later on, the desired beam pattern was used to synthesize the apodizing window at many frequency bins, ranging from 456 to 3648 Hz. Of these, we present the results obtained at 513, 627, 1083, and 2907 Hz. Fig. 4 shows the apodizing windows obtained at such frequency bins. It is worth noting that the signs of the apodizing coefficients contained in the windows are the same as those contained in the window at the lowest frequency bin, i.e., 513 Hz. At 513, 627, 1083, and 2907 Hz, the directivity values are 3.25, 3.24, 3.25, and 3.28 dB, respectively, and the white noise gains are 19.60, 15.40, 6.84, and 0.01 dB, respectively. The increase in the white noise gain over frequency
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More generally, the lower the frequency, the smaller the white noise gain and the higher the sensitivity to array imperfections. To quantify the latter effect, the term , representing the floor level for the expected beam power pattern, can be computed at the above frequency bins. For this, the values of the variances of the element response (gain and phase) and the element position , rad errors have been chosen to be and mm , respectively. In order to maximally stress the effect of the position errors, the value of in (10) has been set equal to 90 . At 513, 627, 1083, and 2907 Hz, the term turns out to be equal to 7.5, 11.5, 20, and 26 dB, respectively. The expected floor level does not significantly reduce the performance of the nominal beam patterns shown in Fig. 5. Only at the lowest frequency bins is the oscillation between the mainlobe and the sidelobe statistically reduced and limited to a value that is slightly higher than the maximum of the sidelobe itself. Therefore, the obtained values of the white noise gain and of the expected floor level are not so negative as to prevent a real application of the array. However, special attention should be given to not exceeding the magnitude of the modeled imperfections and to keep the self-noise limited, especially over the lowest frequency band. B. Broadband Beamforming Design
Fig. 5. Beam patterns obtained by the apodizing coefficients provided by the SA-based method at different frequencies. (a) 513 Hz (solid line) and 627 Hz (dashed line). (b) 1083 Hz (solid line) and 2907 Hz (dashed line).
is in accordance with the decrease in the absolute values of the apodizing coefficients over frequency. Fig. 5 shows the narrowband beam patterns obtained at the above frequency bins. Such patterns exhibit very small differ50 , as compared with the deences in the interval 50 sired beam pattern, whereas limited differences are evident elsewhere. It is worth noting that, if beam patterns with very similar mainlobes are produced at different frequency bins, the same directivity value is obtained. As observed before, the contrary is not always true. With regard to the white noise gain, we verified that the values provided by the stochastic approach were about 1.5–3.5 dB smaller than those yielded, at the same frequency bins, by the constrained directivity maximization. The higher the frequency, the larger the discrepancies. This is probably due to the imposition of the coefficient signs observed at the lowest frequency bin over a very wide band.
In our design, each sensor feeds a seventieth-order FIR filter with a sampling frequency equal to 8 kHz; this means that the frequency band of the processing signals, from 0 to 4 kHz, is divided into 35 equally spaced subbands. The width of each subband is 114 Hz; this is narrow enough to consider the beam pattern quite constant at all the frequencies used. Indeed, we restricted the frequency interval for the design of the FIBP to about 3.1 kHz, ranging from 513 to 3591 Hz (i.e., about three octaves). Only 27 out of the aforesaid 35 subbands belong to such an interval and are taken into account. The remaining ones are disregarded. Thus, 28 frequency bins have been defined as the edge frequencies of the related subbands, and 28 narrowband optimization problems have been addressed and solved by the stochastic approach. It is very important to note that the ratio is below unity until 2835 Hz and above unity at higher frequencies. Moreover, due to the characteristics of the desired beam pattern shown in Fig. 3, the FIBP we would design can be considered superdirective until 2390 Hz (i.e., the frequency at which the directivity due to the uniform apodization is 3.27 dB). Therefore, the design we address is characterized by a very and a wide band, most of which is characterized by superdirective beam pattern. Starting from the 28 vectors of the narrowband apodizing windows, eight FIR filters of the seventieth order have been synthesized by the Parks–McClellan algorithm. The weighting of the subbands in the Remez exchange algorithm includes values ranging from 1 to 20, according to the fit accuracy desired for each subband and the difficulty actually met in obtaining it. A tuning process, based on an iterative reasoning, has been adopted to set such values. Fig. 6 shows the broadband beam pattern, computed by using the tap coefficients, as described in [1]. It is worth noting that it is quite similar to a replication of the desired narrowband beam pattern shown in Fig. 3, and that it
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Fig. 6. Broadband beam pattern for an eight-sensor linear array with an aper0.12 m, working in the air, obtained by the FIR filters designed on ture the basis of the apodizing windows provided by the proposed method.
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Fig. 8. Directivity versus frequency for the broadband beam pattern shown in Fig. 6.
Fig. 9. White noise gain versus frequency for the broadband beam pattern shown in Fig. 6. Fig. 7. Broadband beam pattern obtained by the same FIR filters used in Fig. 6 and applying a postbeamforming filter to equalize the frequency response in the broadside direction.
is satisfactorily frequency invariant, mainly in the central part of the mainlobe. However, even though from a macroscopic point of view an FIBP has been obtained over three octaves (most ), some residual oscillaof which is characterized by tions are present. To flatten the frequency response applied to a signal coming from broadside (i.e., the beam pattern profile for a direction of arrival equal to zero), it is possible to use a postbeamforming filter to flatten the overall response. Fig. 7 shows the broadband beam pattern obtained after introducing an ideal postbeamforming filter by performing a perfect equalization of the frequency response for the broadside signals. The performance of the broadband beam pattern obtained by the proposed method has been assessed by evaluating the directivity, the white noise gain, and the expected floor level over the entire frequency band considered, as shown in Figs. 8, 9, and 10, respectively. While the directivity is quite constant around the desired value of 3.2 dB, as expected, the lower the frequency, the smaller the white noise gain and the higher the sensitivity
Fig. 10. Expected floor level K , versus frequency for the broadband beam pattern shown in Fig. 6. The variance of the element response (gain and phase) and the element position errors are 0:001, = 0:001 rad and = 0:25 mm , respectively.
=
to array imperfections. The values of these three parameters obtained for the narrowband solutions (some of them have been
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Fig. 11. Results obtained by a method proposed in [15]. (a) Broadband FIBP; (b) directivity versus frequency; (c) white noise gain versus frequency; and (d) expected floor level versus frequency.
K
reported in the last subsection) are almost confirmed in the wideband design. Although the lowest portion of frequencies is the most difficult to handle and a higher sensitivity to array imperfections and self-noise arises, the obtained values of the white noise gain and of the expected floor level are not so negative as to prevent a real application of the array. Finally, although the frequency interval over which and the beam pattern is superdirective is the most difficult to handle, the proposed method properly performed also in the remaining part of the chosen frequency band, thus making it possible to design a broadband beamformer partly superdirective and partly not. C. Comparisons For the sake of comparison with the results attainable by the approach proposed in [15], the synthesis of the FIR filters, using the weighted least-squares cost function jointly with the probability density functions of the array imperfections, has been carried out. The same array and the same desired beam pattern as those described above have been assumed. The adherence of
the obtained FIBP to the desired one strictly depends on the variance of errors, hypothesized to be Gaussian. Assuming adequately small errors, a very nice fit is achieved, producing the broadband beam pattern shown in Fig. 11(a). This FIBP has a better profile than that obtained with our method, the necessity of a postbeamforming equalizing filter is less stringent, and the directivity is more constant over frequency, as shown in Fig. 11(b). However, Fig. 11(c) and (d) shows that the white noise gain and the expected floor level (computed by using the same error variances set in Section II-A) are very bad, preventing a real application of the synthesized beamformer. By assuming heavier error variances during the optimization stage, one can find broadband solutions having white noise gain and expected floor level profiles similar to those obtained by our method. In this case, however, the obtained beam pattern does not follow the desired one, it is not frequency-invariant any more, and a decrease in the directivity at the lowest frequencies is observed. This kind of behavior is also reported for the results obtained with our method: if errors occur, the directivity of the beam pat-
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IEEE SENSORS JOURNAL, VOL. 6, NO. 3, JUNE 2006
tern will decrease at the lowest frequencies, as can be predicted by observing the profile of the expected floor level. However, if errors do not occur or are very small, the beam pattern is undistorted, keeping the desired frequency-invariant profile over the entire band, and an acceptable white noise gain. This fact represents an important advantage of the method proposed in this paper.
IV. CONCLUSION A method for designing a broadband data-independent beamformer that produces an FIBP for a superdirective array has been proposed and tested. The obtained far-field beam pattern has reproduced the desired profile imposed at the beginning of the design process over a very wide frequency band. The method is occurs over the entire efficient also if i) the condition band or just a portion of it and ii) the desired beam pattern is superdirective with respect to all the frequencies of the addressed band or just a portion of them. The sensitivity to array imperfections and self-noise has been evaluated by means of the white noise gain and the expected floor level and can be considered acceptable. A very simple implementation of the beamformer is possible if one uses FIR filters, making the designed beamformer more attractive for real applications such as audio signal processing using microphone arrays. Finally, the results obtained have been discussed and compared with those produced by an alternative approach. Among many considerations, the comparison has shown that the key feature of the method proposed here is to yield a broadband beam pattern that is very similar to the desired one when the actual array errors are small and moderately loses its directivity (especially at the lowest frequencies) when the actual array errors increase.
ACKNOWLEDGMENT The authors would like to thank LINEAR S.r.l., Genoa, Italy, for supporting the work presented in this paper and also M. Crocco for his valuable assistance in implementing the algorithms and in discussing the related results. The authors would like also to thank the anonymous reviewers for their valuable comments and suggestions.
[6] D. B. Ward, R. A. Kennedy, and R. C. Williamson, “FIR Filter design for frequency invariant beamformers,” IEEE Signal Process. Lett., vol. 3, no. 3, pp. 69–71, Mar. 1996. [7] R. A. Kennedy, T. D. Abhayapala, and D. B. Ward, “Broadband nearfield beamforming using a radial beampattern transformation,” IEEE Trans. Signal Process., vol. 46, no. 8, pp. 2147–2156, Aug. 1998. [8] S. Van Gerven, D. Van Compernolle, P. Wauters, and W. Verstraeten et al., “Multiple beam broadband beamforming: Filter design and realtime implementation,” presented at the IEEE Workshop Applications Signal Processing to Audio Acoustics, New Paltz, NY, Oct. 1995. [9] D. B. Ward, R. A. Kennedy, and R. C. Williamson, “Constant directivity beamforming,” in Microphone Arrays, M. Brandstein and D. B. Ward, Eds. New York: Springer-Verlag, 2001, pp. 3–18. [10] R. C. Hansen, Phased Array Antennas. New York: Wiley, 1998. [11] J. M. Kates, “Superdirective arrays for hearing aids,” J. Acoust. Soc. Amer., vol. 94, pp. 1930–1933, Oct. 1993. [12] D. P. Scholnik and J. O. Coleman, “Superdirectivity and SNR constraints in wideband array-pattern design,” presented at the 2001 IEEE Int. Radar Conf., Atlanta, GA, May 2001. [13] J. Bitzer and K. Uwe Simmer, “Superdirective microphone arrays,” in Microphone Arrays, M. Brandstein and D. B. Ward, Eds. New York: Springer-Verlag, 2001, pp. 19–38. [14] H. Cox, R. M. Zeskind, and M. M. Owen, “Robust adaptive beamforming,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, no. 10, pp. 1365–1376, Oct. 1987. [15] S. Doclo and M. Moonen, “Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics,” IEEE Trans. Signal Process., vol. 51, no. 10, pp. 2511–2526, Oct. 2003. [16] S. Repetto and A. Trucco, “A stochastic approach for the apodization of very short arrays,” Ultrasonics, vol. 42, pp. 425–429, Apr. 2004. [17] D. B. Ward, “Technique for broadband correlated interference rejection in microphone arrays,” IEEE Trans. Speech Audio Process., vol. 6, no. 4, pp. 414–417, Jul. 1998. [18] K. F. C. Yiu, N. Grbic, K. Teo, and S. Nordholm, “A new design method for broadband microphone arrays for speech input in automobiles,” IEEE Signal Process. Lett., vol. 9, no. 7, pp. 222–224, Jul. 2002. [19] S. G. Chappell, J. C. Jalbert, P. Pietryka, and J. Duchesny, “Acoustic communication between two autonomous underwater vehicles,” in Proc. 1994 Symp. AUV Technology, Cambridge, MA, Jul. 1994, pp. 462–469. [20] M. Stojanovic, “High-speed underwater acoustic communication,” in Underwater Acoustic Digital Signal Processing and Communication Systems, R. S. H. Istepanian and M. Stojanovic, Eds. Boston, MA: Kluwer Academic, 2002, pp. 1–35. [21] R. O. Nielsen, Sonar Signal Processing. Boston, MA: Artech House, 1991. [22] H. L. Van Trees, Optimum Array Processing. Part IV of Detection, Estimation, and Modulation Theory. New York: Wiley, 2002. [23] L. R. Rabiner, J. H. McClellan, and T. W. Parks, “FIR digital filter design techniques using weighted Chebyshev approximations,” Proc. IEEE , vol. 63, no. 4, pp. 595–610, Apr. 1975. [24] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.
REFERENCES [1] B. D. Van Veen and K. V. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Mag. Acoust., Speech, Signal Process., vol. 5, no. 2, pp. 4–24, Apr. 1988. [2] D. B. Ward, R. A. Kennedy, and R. C. Williamson, “Theory and design of broadband sensor arrays with frequency invariant far-field beam pattern,” J. Acoust. Soc. Amer., vol. 97, pp. 1023–1034, Feb. 1995. [3] C. Sydow, “Broadband beamforming for a microphone array,” J. Acoust. Soc. Amer., vol. 96, pp. 845–849, Aug. 1994. [4] J. S. Marciano, Jr. and T. B. Vu, “Reduced complexity beam space broadband frequency invariant beamforming,” Electron. Lett., vol. 36, no. 7, pp. 682–683, Mar. 2000. [5] Y. Yang, C. Sun, and C. Wan, “Theoretical and experimental studies on broadband constant beamwidth beamforming for circular arrays,” in Int. Conf. Oceans 2003 MTS/IEEE, San Diego, CA, Sep. 2003, pp. 1647–1653.
Stefania Repetto (S’03–M’06) was born in Genoa, Italy, in 1975. She received the Laurea degree in electronic engineering from the University of Genoa, Genoa, Italy, in 2001 and the Ph.D. degree in electronic engineering and computer science from the Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, in April 2005. She is involved in the research activities of the Acoustic, Antennas Arrays, and Underwater Signals (A3US) Laboratory of the Signal Processing and Telecommunications Group at DIBE. Her main research interests include advanced hearing aids, array synthesis, array signal processing, and high-resolution radar imaging. In 2005, Dr. Repetto won the second prize at the Student Paper Competition organized by the IEEE Oceans’05 Europe International Conference.
REPETTO AND TRUCCO: DESIGNING SUPERDIRECTIVE MICROPHONE ARRAYS WITH A FREQUENCY-INVARIANT BEAM PATTERN
Andrea Trucco (S’93–M’98–A’99–SM’03) was born in Genoa, Italy, in 1970. He received the Laurea (M.Sc.) degree in electronic engineering from the University of Genoa, Genoa, Italy, in June 1994 and the Ph.D. degree in electronic engineering and computer science from the Department of Biophysical and Electronic Engineering (DIBE) of the same university in June 1998. In 1999, he was appointed Assistant Professor at DIBE, where he teaches several courses in the telecommunications area. He is responsible for the research activities in the Acoustics, Antenna Arrays, and Underwater Signals Laboratory of the Signal Processing and Telecommunications Group at DIBE. His main research interests are array synthesis, array signal processing,
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acoustic imaging methods, acoustic image processing, three-dimensional acoustic vision, and simulation methodologies. Dr. Trucco is a member of the International Association for Pattern Recognition (IAPR). He is involved in the scientific activities related to several research projects funded by the European Commission, the Italian Space Agency, and some industrial companies. From 1987 to 1990, he was three times a finalist for the Philips Awards for European Young Scientists and ranked third twice. In 1995, he won the Student Paper Competition organized by the 9th International Symposium on Unmanned Untethered Submersible Technology; in 1997, he won the Student Paper Competition organized by the MTS/IEEE Oceans’97 International Conference. He is an Associate Editor for the IEEE JOURNAL OF OCEANIC ENGINEERING and for the IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL. He has been a Guest Editor for the Computer Vision and Pattern Recognition journal.
