On the design of differential loudspeaker arrays with broadside radiation patterns

Directional sound radiation,^1–3 which is achieved by an array of loudspeakers, has a wide range of applications such as acoustic contrast control in multi-zone reproduction (or personal sound zones),⁴ room reverberation reduction,⁵ and efficiency improvement in spatial active noise control.⁶ This problem can be regarded as a reciprocal problem of microphone array beamforming.^7,8 The resulting arrays can be broadly divided into two categories, i.e., additive arrays, which are based on the sum operator, and differential arrays, which are based on the differential operator.⁸ 

An additive array generates a radiation pattern (or beampattern) using the constructive interference of the sound fields generated by each element of the loudspeaker array. It has advantages of high radiation efficiency and high robustness against system mismatches. However, due to the diffraction limit,⁹ the low-frequency directivity of the additive array is limited by its aperture size with respect to the wavelength and, as a result, a large-sized array is needed to achieve directional sound radiation at low frequencies.

In contrast, a differential array of small size can be used to generate a narrow beampattern. The differential configuration can be applied on each element of the array, i.e., using two or more closely spaced transducers to form a directional source/sensor as the array element,¹⁰ or on the entire array, which has a small aperture size to achieve superdirectivity.^11,12 It has the advantage of high array directivity, but has low radiation efficiency and is sensitive to system mismatches,^11,12 known as the low white-noise-gain (WNG) problem.⁹ Therefore, recent efforts have been focusing on designing robust differential arrays by formulating the design problem as one of optimization with additional constraints to improve WNG.^4,13 Another issue with differential arrays is that it is difficult to generate broadside patterns with a linear array (i.e., the direction of the mainlobe is perpendicular to the line that connects all the loudspeakers as shown in Fig. 1). So far, only the case of using three elements to generate a second-order differential broadside pattern has been investigated,^8,9 where the differential and delay-and-sum patterns are combined to improve its efficiency.

This paper is devoted to the design of broadside patterns with linear differential loudspeaker arrays, which was shown to be possible with linear microphone arrays through numerical results of steering the mainlobe direction into the broadside direction.¹⁴ In this work, we first derive a general formula of the broadside differential radiation pattern for linear loudspeaker arrays, where we assume that the number of loudspeakers is odd (⁠$≥3$⁠) so that the broadside differential pattern is always symmetric and has a gain of one at the broadside direction and an even number of nulls. Then, the design problem is formulated as one of optimization. Three kinds of solutions are subsequently deduced, which include the method with equality constraints of the broadside direction and distinct nulls, the method with maximum WNG to improve radiation efficiency, and a trade-off approach. The effectiveness of the proposed techniques is demonstrated through simulations and compared to a reference method of combining the differential and delay-and-sum (DAS) patterns to improve radiation efficiency.

As depicted in Fig. 1, an M-element uniform linear loudspeaker array lies on the x axis, whose center is located at the origin, with an interelement spacing equal to σ. In this work, small-spacing loudspeaker arrays with differential beampatterns are considered for low-frequency sound radiation and the size of each loudspeaker and the spacing between neighboring loudspeakers are assumed to be much smaller than the wavelength. Therefore, the loudspeaker array is compact and the loudspeakers are modelled as omnidirectional point sources. In this case, we focus on the sound pressure generated by the array at a far-field location of distance r and angle θ written as follows:

where $ı$ is the imaginary unit, $wm(k)$ denotes the weight of the mth loudspeaker at $xm≜mσ$ with $m=−M0,−M0+1,…,M0−1,M0$ and $M0≜(M−1)/2$⁠, the superscript $(·)∗$ stands for the complex-conjugate operator, the angle θ is defined with respect to the positive y axis, $k≜2πf/c$ is the wave number, f denotes the frequency, and c is the speed of sound in air (around 340 m/s).

Putting Eq. (1) into a vector form gives

where the superscript $(·)H$ is the conjugate-transpose operator,

and the superscript $(·)T$ is the transpose operator.

With the signal model given in Eq. (2), the objective of this work is to design the loudspeaker weights (or driving signals) in $w(k)$ in order to achieve a highly directive or superdirective radiation in the broadside direction, i.e., $θ=0°$⁠. We focus on the design of the low-frequency region using a small-spacing array with the differential excitation function adopted to overcome the diffraction limit.⁹ To achieve this objective, we make the following assumptions: (1) the spacing between neighboring loudspeakers in the array is much smaller than the smallest wavelength in the frequency band of interest, (2) an odd number of loudspeakers (i.e., $M=2N+1$ with N being a positive integer number) is used to design a broadside pattern, which is symmetric with respect to the y axis and has a gain of one at $θ=0°$ and 2 N nulls, and (3) the weights are symmetric, i.e., $wm(k)=w−m(k)$⁠, in order to generate the symmetric broadside

Before ending this subsection, let us define two functions, which will be used in the following for beamforming. The first one is the array excitation function, which is defined as

It represents the driving signals of the loudspeaker array in a continuous form as a function of the wavenumber and spatial position. The second function is the far-field broadside beampattern, which is defined as

In this subsection, we review the formula of the second-order differential pattern, where three point sources are used to form the pattern. The array excitation function can be expressed as

which is a function of the speaker positions and weights, where the superscript $(·)(2)$ denotes the second order and $ξ1≜(kσβ1)2$⁠, with β₁ being a tuning parameter for the broadside beampattern. The excitation function in Eq. (7) corresponds to the case where the three loudspeakers are at $−σ, 0, σ,$ with weight $w(k)=(1/ξ1)×[1−2 cos (kσβ1) 1]T$⁠. Then, the radiation pattern in the far-field¹⁵ can be approximated as⁸ 

The nulls of this pattern are decided by the parameter β₁, i.e., $θnull(1)=±arcsin(β1)$⁠, with $0<β1≤1$⁠.

Now, we derive a general formula of the 2Nth-order broadside differential pattern based on a recurrence relation of the array excitation function. That is, a 2Nth-order excitation function can be written in term of combining three $2(N−1)$ th-order excitation functions, or a spatial convolution of the $2(N−1)$ th-order and the second-order excitation function, i.e.,

where ⊗ denotes the spatial convolution operator. Equation (9) shows that a 2Nth-order excitation function can also be represented as the N-fold spatial convolution of the second-order excitation function with tuning parameters ${βn,n=1,2,…,N}$⁠. It follows then that the radiation pattern can be written as

Note that the deduction of Eq. (10) is based on the fact that the far-field radiation pattern can be regarded as the wavenumber spectrum of the array excitation function (or its spatial Fourier transform with the variable $k′≜k sin θ$⁠).^9,16 Then, the radiation pattern of the N-fold convolution of the second-order excitation function is equivalent to the product of N second-order radiation patterns.

Equation (10) shows that a 2Nth-order differential pattern is uniquely determined by its 2 N nulls located at $θnull(2N)=±arcsin(β1),±arcsin(β2), …,±arcsin(βN)$⁠, where $0<β1≤β2≤⋯≤βN≤1$⁠. Note that at least $2N+1$ loudspeakers are required to generate such a pattern.

The general formula in Eq. (10) can also be represented in a sum form, i.e.,

where $αN,n, n=1,2,…,N$ are also functions of ${βn,n=1,2,…,N}$⁠. The directivity of the higher-order differential pattern is represented by $sin2Nθ$ as in Eq. (11). The higher the order, the narrower is the mainlobe.

The objective of designing loudspeaker array weights is to derive $w(k)$ in Eq. (2) so that its radiation pattern would be as close as possible to an ideal differential pattern in Eq. (11). It is shown from Sec. 2 that an ideal 2Nth-order differential pattern has a one at the broadside direction $θ=0°$ and 2 N nulls. In this section, we study the case where 2 N nulls are distinct. We formulate the design problem as one of optimization as in Ref. 13 to derive the corresponding solutions.

We first study the design of the array weights with $M=2N+1$ speakers using equality constraints (EC), i.e., the mainlobe direction of $0°$ and 2 N distinct nulls in the angle range of $[−90°, 0°)∪(0°, 90°]$⁠. Note that the broadside radiation pattern is symmetric. Therefore, the nulls are symmetric with respect to the mainlobe direction. So, only N distinct nulls $θN,n, n=1,…,N$⁠, located in the range of $(0°,90°]$ need to be identified.

We express these fundamental constraints in a matrix form as

where

$I$ is the identity matrix of size $M0×M0$⁠, and $I˘$ is the flip of the matrix $I$⁠. The matrix $D(k,θ)$⁠, of size $(N+1)×M$⁠, is the mainlobe and null direction constraints. The matrix $C$⁠, of size $M0×M$⁠, is used to maintain the symmetry of the array weights.

When the number of constraints is equal to the number of weights, $A(k,θ)$ is a square matrix. Given that $θN,n$ are distinct nulls, and $θN,n≠0°,∀n, A(k,θ)$ is of full rank. It follows then that the array weights can be determined uniquely as

The differential loudspeaker array with a highly directive radiation pattern generally suffers from the intrinsic low-radiation efficiency problem as a large number of evanescent sound waves cannot propagate to far-field.⁹ The radiation efficiency is represented by WNG,⁸ which is expressed as

Note that WNG is often expressed in the decibel scale. The higher the value of WNG, the more efficient is the radiation.

A common approach to improving WNG is by using $M>2N+1$ speakers to design a 2Nth-order broadside differential pattern.¹⁷ According to Eq. (19), we formulate the following optimization problem to maximize WNG, i.e.,

Note that in this case, we have N + 1 beampattern constraints and M₀ constraints to maintain the symmetry of the array weights.

The solution to Eq. (20) is known as the minimum norm (MN) solution,^13,17

which gives the maximum WNG. However, as will be shown in Sec. 4, this design may lead to frequency-variant radiation patterns.

To further improve the stability of the design, more constraints can be included, in addition to the N + 1 fundamental beampattern constraints. Assume that we have L more constraints at (⁠$θL,l$⁠, b_l), where $0°<θL,1<⋯<θL,L≤90°$ and $θL,l$ is different from $θN,n$⁠, with b_l being the corresponding value of the desired gain at $θL,l$⁠. Then, with $N+L+1$ beampattern constraints and M₀ constraints to maintain the symmetry of the array weights, a linear system is constructed as follows:¹³ 

where

and

The array uses $M≥2(N+L)+1$ loudspeakers and the problem is also formulated as one of optimization to maximize WNG, i.e.,

whose solution is

which is called the minimum-norm solution with additional constraints (MNA). This gives another WNG improved solution but using additional constraints, which are different from the null constraints. The resulting radiation pattern will better match the desired differential pattern than the MN method.

In this section, we perform simulations to validate the proposed methods and compare the results with the existing approach that combines the differential and DAS patterns to improve the WNG.⁸ The loudspeakers are assumed to be omnidirectional point sources and the interelement spacing is σ = 5 cm.

We first study the design of SLA using the method presented in Sec. 3.1 with $2N+1$ equality constraints, i.e., a gain of 1 at the broadside direction and 2 N distinct nulls. The desired radiation patterns includes the second-order, fourth-order, and sixth-order broadside differential patterns with the maximum directivity index (DI).⁸ With these patterns, the maximum achievable acoustic power is concentrated in the preferred broadside direction given a fixed total radiation power.³ The designed patterns are plotted in Fig. 2. As seen, the radiation patterns designed by the proposed method are almost the same as the desired patterns.

Next, we compare the three proposed methods, i.e., the EC, MN, and MNA solutions, to design a fourth-order broadside differential pattern with the maximum DI, using 21 omnidirectional loudspeakers. For EC, the central 5 loudspeakers are used. For MNA, an additional constraint is set at the angle of $16°$⁠, which is within the 3 dB (or half-power) bandwidth of the mainlobe. The performance measures are WNG and DI. We also plot the results by the method that combines the differential and DAS patterns (abbreviated as DD in Fig. 3)⁸ to improve the WNG for comparison. As shown in Fig. 3(a), the reference method (DD) and the proposed MN and MNA solutions can significantly improve the WNG, i.e., the radiation efficiency, compared with the standard EC solution. The MN method gives the maximum WNG. Figure 3(b) plots the DI of different methods as a function of frequency. While the EC method has an almost constant DI over frequency, the DI of the MN and DD methods increases with frequency, i.e., the generated radiation pattern is variant with frequency. The MNA solution can approximately maintain the same DI as EC and also achieve a higher WNG.

Figure 4 plots the radiation patterns with the MN and MNA methods in a wider frequency range. It can be seen that the mainlobe of the MN solution becomes narrower as frequency increases, while the radiation pattern of the MNA solution keeps almost frequency invariant. In summary, the MN solution achieves the maximum WNG, i.e., the maximum radiation efficiency, while the MNA solution can maintain a frequency-invariant radiation pattern with slightly reduced WNG.

In this paper, we derived the general formula of higher-order broadside differential patterns for linear loudspeaker array. We then proposed three methods for the design of the loudspeaker array weights. The EC method can successfully design the desired radiation pattern, but it has low radiation efficiency. The MN solution gives the maximum WNG but the resulting radiation pattern changes with frequency. In comparison, the MNA solution (at a price of reduced directivity) maintains a frequency-invariant radiation pattern with high radiation efficiency.

This work was partially supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 61671380 and 61831019.