Broadband performance of superdirective delay-and-sum beamformers steered to end-fire Open Access

Data-independent beamforming is a well-known approach to the spatial process of propagating acoustic wavefields collected by transducer arrays. When an isotropic noise field and plane wave signal are considered, the array directivity measures the improvement in the signal-to-noise ratio obtained using the array instead of a single omnidirectional transducer.¹ For this reason, directivity maximization is one of the most common objectives in beamforming optimization. This problem is particularly challenging in array systems that are subjected to stringent size constraints because the array aperture is shorter than or comparable to the wavelength. In these cases, superdirective^2–4 solutions should be considered. Superdirectivity theory allows significant directivity values to be achieved for short arrays, particularly linear arrays steered to end-fire.⁵ However, to reduce the sensitivity to array imperfections and fluctuations, the directivity maximization is typically performed by introducing a constraint on the sensitivity factor,^1,2 which is defined as the inverse of the white-noise gain¹ (WNG). In fact, the WNG measures the array gain against spatially uncorrelated noise, like electrical transducer self-noise, transducer imperfections, and mismatches. The performance attained under this constraint is referred to as constrained directivity (CD).

In this letter, we focus on superdirective data-independent beamforming applied to low-cost, small-size systems aimed at processing far-field waves gathered by a linear equispaced array. These systems are assumed to have a fixed steering direction and are pointed to end-fire. Under these conditions (which occur frequently in microphone arrays³ and occasionally in hydrophone arrays⁴), the maximum CD is achieved using complex-valued weight coefficients for the beamforming process.⁵ The weight coefficients adopted here do not include the phase terms corresponding to the time delays required to steer the main lobe toward end-fire. The optimum weight coefficients depend on the frequency. Therefore, a filter-and-sum beamforming structure,^6,7 which is designed to implement the optimum weight coefficients at each frequency, is necessary to reach the maximum CD over a broad band.

However, the deployment of a filter for each transducer can be impractical in array systems that are subject to stringent cost and size constraints. In these systems, delay-and-sum (DAS) techniques, whose implementation is only modestly demanding, are preferred. The weight coefficients used in DAS techniques are optimized at only a single frequency because they are fixed. At different frequencies, the directivity decreases with respect to the maximum CD are unavoidable. Nevertheless, DAS beamforming has been used to process broadband signals with adequate results.⁷ Indeed, DAS techniques have been considered in numerous papers that address microphone arrays (particularly for hearing-aid applications, e.g., Kates and Weiss⁸) and are a common choice in data-independent sonar systems.

To the authors’ knowledge, an analysis of the directivity provided by DAS techniques compared to the maximum achievable CD in a filter-and-sum structure is missing from the literature on end-fire superdirective arrays. This letter is devoted to addressing this gap. In this work, we examine the broadband performance of three DAS techniques. The first, referred to as complex weighting (CW), includes DAS beamforming carried out with fixed complex-valued weight coefficients that maximize the CD at a given frequency. The second, referred to as narrowband approximation (NA), includes DAS beamforming in which the signals are weighted by the modulus of the aforementioned complex weights and delayed depending on the phase of such weights. The third, referred to as optimized oversteering (OO), includes DAS beamforming in which the signals are weighted by real coefficients (which are optimized ad hoc) and are delayed using additional terms introduced to narrow the main lobe by pushing it past end-fire.^2,4,5,8 The definitions, characteristics, and implementation issues for these three techniques are described and discussed below.

The arrays considered in this letter can be used for the low-cost design of directional hydrophones (particularly for locating and capturing underwater sound sources, natural or anthropic) and directional microphones (particularly for wearable personal assistive listening devices).

Let us consider a linear equispaced array composed of N omnidirectional point-like transducers centered at the coordinate origin and placed on the x axis. The nth transducer is placed at the position x_n and generates the real signal s_n(t), whose complex analytic representation is $s̃n(t)$⁠. In CW DAS beamforming, the complex analytic beam signal $b̃(t)$ steered toward the direction θ₀ is computed in the time domain as follows:

where $w̃n$ is the weight coefficient associated with the nth transducer. For far-field plane waves, the delay τ_n is defined as τ_n = x_nsinθ₀/c = x_nu₀/c, where c is the sound speed, θ₀ is an angle measured with respect to the y axis (i.e., θ₀ = 90° for end-fire steering), and u₀ = sin θ₀. Several papers, e.g., Cox et al.,² have described how to compute the weight coefficients $w̃n$ for a given frequency f₀ to maximize the CD. These weights are generally complex. For a plane wave with a frequency f and direction of arrival θ, the complex beam pattern steered to end-fire (i.e., u₀ = 1) is

where u = sin θ.

Unfortunately, the complex weight coefficients require processing of the complex analytic representations of the real signals that are generated by the transducers. Because this task complicates the system considerably, some approximations have been proposed.^2,4,5 In the NA, the signals are assumed to have a narrow-band spectrum centered at f₀. The phases φ_n of the weight coefficients $w̃n$⁠, $w̃n=|w̃n|exp(jφn)$⁠, are used as additional delays, and the real signals s_n(t) are weighted by $|w̃n|$ to produce a real beam signal b(t),

The resulting complex beam pattern steered to end-fire is

and is equal to B_CW(u,f) if f = f₀. The two beam patterns differ if f ≠ f₀.

The term τ_n is linear with respect to the transducer position and, for equispaced arrays, is linear with n. In contrast, the term φ_n is not linear with n. The oversteering^2,4,6,8 approximation has been proposed to simplify the practical implementation of the total delay. In the OO technique, real weight coefficients w_n are used to scale the real signals, and additional delays $τno$ (linear with n) are inserted to narrow the main lobe by pushing it past end-fire. The beam signal is computed as follows:

where $τno=xnε/c$⁠, and ε is the amount of oversteering expressed in a scale comparable with u₀ = sin θ₀. The resulting complex beam pattern steered to end-fire is

As described by Trucco et al.,⁵ it is possible to compute the weight coefficients w_n and the oversteering amount ε that maximize the CD. Those authors also demonstrated that the CD attained by the OO technique is only slightly lower than the maximum CD achieved by the CW and NA techniques.

From a system implementation perspective, the CW technique is the most demanding (requiring the computation of complex analytic signals and operations with fixed complex weights), whereas the OO technique is the simplest because the real signals are processed and the total delays (i.e., $τn+τno$⁠) can be implemented by shifting each signal by an integer number of samples⁵ (assuming that the proper sampling frequency is set).

Finally, the directivity of a linear array steered toward the direction θ₀ is defined as follows:

where B(u, f) is a generic complex beam pattern function. By setting u₀ = 1, substituting Eqs. (2), (4), and (6) into Eq. (7), and performing mathematics, the following equations are obtained for the end-fire directivities of the three DAS techniques:

where * denotes the complex conjugate and sinc(η) = sin(πη)/(πη).

A linear array composed of N = 8 omnidirectional point-like transducers is initially considered, and the WNG is constrained to values greater than or equal to 0 dB. The transducers are equispaced at an inter-element distance d. The end-fire directivity of this array is evaluated as a function of the normalized spacing, i.e., the ratio d/λ = f·d/c, where λ is the wavelength. Additionally, inter-element distances smaller than λ/2 are considered.

Figure 1(a) compares the maximum CD that can be obtained using complex weights optimized at each frequency value (i.e., the performance achieved by an ideal filter-and-sum beamformer) and the directivity obtainable with uniform weights. The maximum CD curve has an absolute maximum at a frequency f* of f*·d/c = 0.36. The optimum complex weights provide a gain with respect to the uniform weights (referred to as superdirectivity) of approximately 5 dB over a wide frequency interval. In addition, Fig. 1(a) presents the performance achieved by the CW technique when the optimum, fixed, complex weights, computed at frequency f₀, are used over a broad band. Two cases are analyzed: f₀·d/c = d/λ₀ = 0.2 and f₀·d/c = d/λ₀ = 0.35. In both cases, at f = f₀, the directivity D_CW is equal to the maximum CD. For f < f₀, the directivity D_CW slowly decreases and remains close to the maximum CD curve. For f > f₀, the directivity D_CW rapidly decreases and falls below the directivity obtained with uniform weights. We verified that this behavior also occurs when applying the NA and OO techniques and for different values of N and f₀. Therefore, to achieve a directivity close to the maximum CD curve using the DAS techniques, it is convenient to set f₀ equal to the upper bound of the frequency band and to use the optimized weights for f ≤ f₀.

Figure 1(b) compares the directivity D_CW, computed for f ≤ f₀, over three different f₀ values: f₀·d/c = d/λ₀ = 0.16; f₀·d/c = d/λ₀ = 0.35; and f₀·d/c = d/λ₀ = 0.42. The CW technique provides a curve that is close to the maximum CD curve over a very large band if f₀ is less than the frequency f*. In contrast, if f₀ is greater than f*, the directivity D_CW does not increase considerably when the frequency is reduced, and the performance deviates considerably from the maximum CD curve. Therefore, it is convenient that the upper bound of the frequency band is less than the frequency f*.

The broadband performances of the CW, NA, and OO techniques are presented in Figs. 2 and 3, where the same analysis is repeated for N = 4 [Fig. 2(a)], N = 8 [Fig. 2(b)], N = 16 [Fig. 3(a)], and N = 24 [Fig. 3(b)]. For each array under consideration, the directivity obtained using uniform weights and the maximum CD curve is added to the figure for comparison. The directivity of the three DAS techniques is computed over a bandwidth of four octaves, from f₀/16 to f₀. The upper bound of the band, f₀, is chosen to be slightly less than f*: f₀·d/c = d/λ₀ = 0.25 for N = 4; f₀·d/c = d/λ₀ = 0.35 for N = 8; f₀·d/c = d/λ₀ = 0.38 for N = 16; and f₀·d/c = d/λ₀ = 0.43 for N = 24.

The constraint WNG ≥ 0 dB is imposed when the weights and oversteering amount (if applicable) are optimized at f = f₀. Whereas the WNG for the CW technique is not frequency dependent, the WNG for the NA and OO techniques gradually increases with decreasing frequency. Therefore, the WNG constraint is satisfied over the entire band under consideration.

The directivity of the CW technique, D_CW, decreases with decreasing frequency, remaining close to the maximum CD curve. For N = 4 and N = 8, the distance between D_CW and the maximum CD curve does not exceed 2.2 dB (Fig. 2), whereas for N = 16 and N = 24, the distance does not exceed 1.5 dB (Fig. 3).

The directivity of the NA technique, D_NA, is equal to D_CW at f = f₀ and lower than D_CW when f < f₀. As the frequency decreases, the corresponding decrease in D_NA is larger in magnitude than the decrease of the maximum CD curve, causing the distance between D_NA and D_CW to increase. Figure 3 illustrates that for N = 16 and N = 24, D_NA falls below the directivity obtained with uniform weights when approaching the lower bound of the band. The significant superdirectivity gain observed at f = f₀ decreases rapidly with decreasing frequency. However, the performance of the CW technique retains significant superdirectivity over the four octaves, and the band over which the NA technique provides substantial superdirectivity is at most two octaves wide.

Using the OO technique, D_OO is almost equal to D_NA for N = 4 and N = 8. However, for N = 16 and N = 24, D_OO is marginally better than D_NA over a wide portion of the considered band. At f = f₀, D_NA is equal to D_CW and D_OO is slightly lower than D_CW, as described in Trucco et al.⁵ This is not visible in Figs. 2 and 3 because the differences are small. Aside from these specific aspects, the remarks made for the NA technique hold for the OO technique as well.

In the context of superdirective beamforming steered to end-fire, we considered three DAS techniques (CW, NA, and OO) that were optimized to yield the maximum CD at a given frequency. We investigated the performance of these techniques over a broad band and concluded that it is convenient to optimize the weights and oversteering amount (if applicable) at the upper bound of the band under consideration. Furthermore, the upper bound should be less than the frequency at which the absolute maximum of the maximum CD curve occurs.

Although the DAS techniques do not allow the beam pattern shape to be controlled over the frequency axis, we demonstrated that for linear arrays of 4, 8, 16, and 24 equispaced transducers, the CW technique provides a directivity that is very close to the maximum CD curve over a bandwidth exceeding four octaves. Therefore, an optimized DAS structure can provide quasi-optimum performance over a broad band, avoiding the deployment of more demanding filter-and-sum structures.

In contrast to the CW technique, the NA and OO techniques work directly on the real broadband signals generated by the transducers, employing real weights, which simplifies the processing system and reduces its overall cost. Although the performance of the NA and OO techniques is optimized in the upper bound of the band, their directivities decrease considerably and the distance from the maximum CD curve increases with decreasing frequency. Nevertheless, we demonstrated that a significant level of superdirectivity is achieved over a bandwidth of approximately two octaves for the arrays mentioned here. Given that the directivity curves for the NA and OO techniques are very similar, the OO technique is preferable because the exact delays can be applied by simply shifting the signals over an integer number of samples⁵ (provided that the array is equispaced and the proper sampling frequency is set).

Low-cost directional microphones and hydrophones (used in hearing aids and underwater acoustic capturing devices, respectively) represent good examples of potential applications for the arrays considered in this letter. To set the bandwidth necessary for a specific application and, consequently, the complexity of the processing system, it can be noted that a band of two octaves (e.g., from 0.85 to 3.4 kHz) is sufficient to achieve 90% of intelligibility for speech sentences.⁹