Acoustic Beamforming Algorithms and Their Applications in Environmental Noise

Abstract

Purpose of Review

Rather than broadly investigating the beamforming field, the present work has the distinctive feature of analyzing the most common algorithms through both a theoretical presentation and a report of their most recent applications to real cases. The intent is to take a step forward towards the harmonization of the sector with a combined approach that could prove to be useful for academics willing to understand the theory and for technicians needing to choose the best algorithms to use in different measurement conditions.

Recent Findings

In recent years, the sector has seen the growth of studies published on the use of beamforming techniques and their applications to real cases. Unfortunately, different authors and research groups developed so many different algorithms that a literature review turns out to be essential to increase awareness and to avoid confusion for both scientists and technicians.

Summary

Nowadays, acoustic cameras have been proven to be powerful instruments that combine a video acquisition with a microphone array to obtain real-time information about the location of noise sources. Different beamforming techniques can be applied to sound signals allowing their visualization or distinguishing the contribution of multiple different emitters. This quality, historically used in different branches of acoustics, is now spreading into environmental noise protection, especially where it is needed to locate the emitters or to better study sources that have not yet been characterized. Acoustic cameras can also be used to identify the responsible for noise limits exceedings at receivers when traditional microphone measurements are not enough, or to identify potential leakages occurred in the installation of noise abatement measurements.

Appendix. Basic Concepts of Beamforming

A.1 Introduction

In acoustics, the computation of acoustic pressure in some points of the space, knowing position and strength of the sources, is called direct problem. Beamforming deals with the inverse problem, namely the estimation of position and strength of the sources, starting from measurements taken in some control points of the space. When starting to deal with this kind of problem, an operator could be brought to use a single microphone, as in other kind of acoustic measurements is done. Anyway, a sound measurement performed with a single microphone does not allow the spatial localization of the emitter, as a single microphone can only record the sound pressure field. The response of a microphone, with respect to the arrival angle of the sound, depends on the directivity of the transducer under analysis. The main typologies of directivity are omnidirectional (if the response is equal for every arrival angle of the sound), bidirectional (if 0° and 180° are favored), unidirectional (if only frontal direction is favored), and superdirectional (if the favored direction is frontal with a narrower opening angle with respect to unidirectional one). A single microphone usually shows a very broad polar response. The only exception is represented by superdirectional microphones, used to identify the exact localization of the source. Even though, the great directivity index shown by this typology of microphones is not sufficient to build functional and reliable methods for source localization. The reason is that it would be necessary to look for the maximum response angle and then move the microphone according to this information. It is straightforward to understand that the resulting technique would be slow and inefficient. This awareness brings to the idea of building an array of microphones in order to exploit the delays between microphones and then finding the arrival angle.

Consider the simplest explanation of the problem before moving on to a more articulated treatment. Assuming sources at great distance from the array, the polar angle can be neglected and only the azimuthal one can be considered. This brings to a simplification of the problem into a linear array. Anyway, the implied assumption is that the source is still enough distant from the array to allow to consider the wave front as a plane wave front. The unidimensional treatment of this challenge is showed in Fig. 3.

Fig. 3

Scheme of the interaction between a plane wave and a linear microphone array

The plane wave front impacts the different microphones in different times because of the inclination of the wave front with respect to the array. In particular, when the wave front impacts the first microphone, the sound perturbation is still distant from the second microphone for a distance equal to l. Calling $\alpha$ the angle between the wave front and the microphone array, which is the unknown quantity of the problem, and calling d the distance between two consequent microphones of the array, which is known, the distance l is equal to $l = d \sin \alpha$. Furthermore, the distance l is traveled by the wave front in a time equal to $\Delta t = l/c$, where c is the sound speed. Therefore, the delay between the first and the second microphone can be computed, using known quantities, as:

$$\begin{aligned} \Delta t = \frac{d \sin \alpha }{c} \end{aligned}$$

(70)

Straightforwardly, the delay between the first microphone and the other ones from the third onwards is a multiple of the time interval computed in Eq. (70). In this way, measuring the delays with which the peak is registered in two consequent microphones of the array, which are equal to $\Delta t$, is possible to compute the azimuthal arrival angle, using the inverse of Eq. (70)

$$\begin{aligned} \alpha = \arcsin \bigg ( \frac{c \Delta t}{d} \bigg ) \end{aligned}$$

(71)

Up to this point, the treatment has been restricted to a single angle and a single frequency problem, just to explain in a simple way the task. For nearest sources, the azimuthal angle is not sufficient and it is necessary to introduce the polar angle, switching to a bidimensional planar array. These few lines aim to introduce the problem and explain why a bidimensional array is required for an optimal resolution of the beamforming task. Another remarkable detail regards the disposition in space of the microphones. Indeed, it is common to see bidimensional microphone arrays with irregular shape. The reason lies in the will to avoid spatial aliasing, that is, a sampling error due to inadequate distance of the microphones for some wavelengths. It is the same effect occurring in the time domain when the sampling frequency is not adequate. It has been found out that severe sampling problems occur when periodic arrays are employed, and that these problems can be solved using arrays with irregular shape [30•].

A.2 Mathematical Formulation of Beamforming

After this brief and simple introduction to the problem, it is now possible to switch to a more advanced treatment of the topic, which introduces the basic concepts of beamforming. An exhaustive treatment of the subject can be found in Allen et al. [32]. Suppose having an array composed by N microphones, so that a generic microphone is identified with the letter n, and a grid of points in the space to be investigated because they can contain the sources. A generic grid point is identified with $x_s$. As input, beamforming algorithms take sound pressure signals acquired by the array of microphones. Each signal acquired by a microphone is thought as the real signal emitted by a source, modified by the path traveled. To reflect this idea, for each point in the space, the signal acquired by a microphone is modeled as the product between a propagation factor $C_n(x_s)$ and a source function $S(x_s,t)$. The propagation factor is equal to

$$\begin{aligned} C_n(x_s) = \frac{e^{-i \omega \sigma (x_n,x_s)}}{D(x_n,x_s)} \end{aligned}$$

(72)

where the numerator accounts for the delay while the denominator for the damping of the signal during the path. The source function $S(x_s,t)$, instead, represents the distribution of sources in space. The basic assumption is that sources associated with different points in space are uncorrelated, namely

$$\begin{aligned} \langle S^*(x'_s,t)S(x_s,t) \rangle = q(x_s)\delta (x'_s-x_s) \end{aligned}$$

(73)

where $q(x_s)$ is the mean square source strength at $x_s$ and $\langle \rangle$ is the time average defined by

$$\begin{aligned} \langle \,f(t) \rangle = \frac{1}{T} \int _0^T f(t) dt \;. \end{aligned}$$

(74)

For an extended source, the total signal $u_n(t)$ at each microphone is modeled as

$$\begin{aligned} u_n(t) = \int C_n(x_s) S(x_s,t) d^3x_s + E_n(t) \end{aligned}$$

(75)

where $E_n(t)$ is the background noise. In the inverse problem framework, the signal $u_n(t)$ acquired by each microphone is the known variable, while source function $S(x_s,t)$ is the unknown variable to be determined. The signal acquired by each microphone is both represented by concrete data and by a theoretical model (75), and the beamforming goal is to estimate the function $S(x_s,t)$ extrapolating it from (75). Collecting all the $u_n(t)$, $C_n(x_s)$, and $E_n(t)$ in vectors of dimension N, $\vec {u}(t)$, $\vec {C}(x_s)$, and $\vec {E}(t)$, it is possible to write Eq. (75) in vectorial notation as

$$\begin{aligned} \vec {u}(t) = \int \vec {C}(x_s) S(x_s,t) d^3x_s + \vec {E}(t) \; . \end{aligned}$$

(76)

At this point, a very important tool is introduced, that is, the set of microphone weight vectors $\vec {w}(x_b)$, defined for every potential source of interest. The task assigned to these vectors is to delay the signal of each microphone of the array, in order to “steer” the array towards the direction which is supposed to be the direction of arrival of the signal. For this reason, they are also called “steering vectors.” By definition, they are normalized, i.e.,

$$\begin{aligned} \vec {w}^{\dag }(x_b)\vec {w}(x_b)=1 \;. \end{aligned}$$

(77)

A “beamforming value” related to the measured signal $\vec {u}(t)$ and the point $x_b$ is computed using the steering vectors. Namely, the scalar product between the signal $\vec {u}(t)$ and the steering vector $\vec {w}(x_b)$ is computed, in order to obtain the “steered” version of $\vec {u}(t)$ towards the direction $x_b$. From this product, the “beamforming value” of interest is obtained and plotted for every potential point. In mathematical language, this sentence is equal to

$$\begin{aligned} b(x_b) = \langle |\vec {w}^{\dag }(x_b) \vec {u}(t)|^2 \rangle \end{aligned}$$

(78)

Exploiting Eqs. (73) and (77), it is possible to show that, from a theoretical point of view, it is possible to show that for a direction $x_b$ containing an actual source, $b(x_b)$ is equal to the time-average square source strength q, multiplied by the sum of the squares of the propagation factors. Namely

$$\begin{aligned} b(x_b) = q \; \vec {C}^{\dag }(x_b)\vec {C}(x_b) = q \sum _{n=1}^N |C_n(x_b)|^2 \end{aligned}$$

(79)

The expression $q \; |C_n(x_b)|^2$ can be thought as the time-average square pressure experienced by a generic microphone n due to the signal produced by the source in $x_b$, because it is equal to the time-average square source strength q modified by the path traveled until reaching microphone n. Therefore, Eq. (79) shows that, for an actual source in $x_b$, $b(x_b)$ is equal to the sum over the array of these time-average square pressures due to the presence of the source in $x_b$. Equation (78) can be expanded, obtaining

$$\begin{aligned} b(x_b) = \langle |\vec {w}^{\dag }(x_b) \vec {u}(t)|^2 \rangle = \langle \vec {w}^{\dag }(x_b) \vec {u}(t) \vec {u}^{\dag }(t) \vec {w}(x_b) \rangle \end{aligned}$$

(80)

The product $\langle \vec {u}(t) \vec {u}^{\dag }(t) \rangle$ is equal to the cross-spectral matrix (CSM). CSM is, in the frequency domain, the analogue of the cross-correlation in the time domain. It can be finally obtained

$$\begin{aligned} b(x_b) = \vec {w}^{\dag }(x_b) \; CSM \; \vec {w}(x_b) \end{aligned}$$

(81)

which is the operative equation of beamforming. The goal of the simplest version of beamforming is to find the set of weight vectors $\vec {w}(x_b)$ which allows to get the maximum value of $b(x_b)$ for a specific direction $x_b$, because this would mean that in $x_b$, an actual source is present. In other words, it is necessary to find the set of $\vec {w}(x_b)$ which is able to “steer” the array towards the correct direction. As stated before, the CSM is obtained time-averaging the scalar product between real data collected by microphones and their conjugate transpose. Therefore, the last unknown variables left to compute Eq. (81) are the steering vectors $\vec {w}(x_b)$. Up to this point, an explicit form of $\vec {w}(x_b)$ has not been given. This is the last missing information to complete the theoretical formulation of the problem. Actually, from Eqs. (76) and (77), the choice of the explicit form of the steering vectors is straightforward. Indeed, if, in order to maximize $b(x_b)$, the goal is to maximize the expression

$$\begin{aligned} \vec {w}^{\dag }(x_b) \vec {u}(t) = \int \vec {w}^{\dag }(x_b) \vec {C}(x_s) S(x_s,t) d^3x_s + \vec {E}(t) \end{aligned}$$

(82)

it follows that the scalar product $\vec {w}^{\dag }(x_b) \vec {C}(x_s)$ must be maximized, and this happens when $\vec {w}^{\dag }(x_b)$ and $\vec {C}(x_s)$ are parallel. Given this constraint and also the constraint shown in Eq. (77), it is straightforward to choose the following form of the steering vectors

$$\begin{aligned} \vec {w}(x_b) = \frac{\vec {C}(x_b)}{||\vec {C}(x_b)||} \end{aligned}$$

(83)

Given this expression for the steering vectors, the task of conventional beamforming procedure is to compute the CSM using the data acquired by the microphones and then to find the correct direction $x_b$ wherewith obtaining the maximum value of $b(x_b)$ using Eq. (81). Of course, the most advanced algorithm functioning extends far beyond this simple mathematical formulation and how it is fully shown in the “Mathematical Formulations of Beamforming Algorithms” section. However, it is useful to see this basic treatment of the problem, both because it is the basis of more advanced algorithms and also because it explains the basic idea behind beamforming, from which the current research field originated.