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Application of wave field synthesis to active control of highly non-stationary noise
Applied Acoustics 131 (2018) 220–229
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Application of wave field synthesis to active control of highly non-stationary noise
MARK
⁎
Alessandro Lapinia, , Francesco Borchia, Monica Carfagnia, Fabrizio Argentib a b
Dipartimento di Ingegneria Industriale, Università di Firenze, via Santa Marta, 3 I-50139 Firenze, Italy Dipartimento di Ingegneria dell’Informazione, Università di Firenze, via Santa Marta, 3 I-50139 Firenze, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Active noise control Wave field synthesis Non-stationary noise
Active Noise Control (ANC) methods have been successfully applied to the cancellation of stationary noise. Classical ANC systems use adaptive filtering techniques to produce a waveform that is opposite, or counterphase, to the signal noise we would like to cancel. However, when the noise is of short duration, adaptive filtering cannot be used since convergence is not achieved. In this paper, a novel active control technique for non-stationary noise is presented. The method uses wave field synthesis (WFS) for the construction of the canceling waveform. The system is tailored for an outdoor environment. The noise acoustic field is acquired by microphones and processed by a WFS engine to pilot a linear array of secondary sources. Experimental results, obtained from both simulations and true tests, demonstrate that the proposed method is able to diminish the overall noise perceived in the area covered by the system.
1. Introduction
When the noise is highly non-stationary, for example, when it appears as an impulse having large-amplitude pressure levels with respect to the background and a very short duration, the adaptive filters within an ANC system may not have enough time to reach convergence, making noise reduction ineffective. Examples of such type of noise are represented by man-made disturbances, such as sounds generated by manufacturing plants, vehicle transit, punching machines or construction sites as well as gunshots in shooting ranges. In this paper, we use the ANC paradigm, that is generating a counter-phase signal, to cancel an impulse-like noise. Since in the literature the term impulsive noise is often used to indicate heavy-tailed distributed disturbances that affect the operations of adaptive ANC systems [9–14], in order not to generate ambiguities, we also term the primary source signal as highly non-stationary noise (HNN) and the proposed scheme as active HNN control (AHNNC) system. To put ideas into a real world scenario, consider as an example of such a noise the one produced by a gunshot within a shooting range. This example allows other objectives of this study to be delineated as follows. The HNN we would like to cancel is a short duration wideband signal. Furthermore, the HNN is not assumed as deterministic, even though several features, both in the time and frequency domain, can be derived by the analysis of few realizations of the noise. The scenario in which the AHNNC system should work is the outdoor and the dimensions of the area in which it should reduce the noise are of the order of tens of meters.
Active noise control (ANC) techniques have been widely used to reduce the impact of noise in different environments. Given a source of noise, the aim of an ANC system is that of piloting one or more secondary sources to generate a waveform that is opposite, or in counterphase, to that of the noise. Thus, thanks to the principle of superposition, the primary source of noise is cancelled. An ANC system is particularly effective when the region where it operates is limited, e.g., in headset, ducts, car and airplane cabins, and is more convenient to diminish low frequency noise than passive methods [1,2]. A classical ANC system is based on adaptive filtering. Its basic components are reference microphones, to acquire the primary source of noise; secondary loudspeakers, to generate the counter-phase waveform; and error microphones, to acquire the resulting signal coming from primary and secondary sources in the area where the system is supposed to be effective. The error signal is used to update the coefficients of adaptive filters that drive the secondary sources. Adaptive algorithms, such as least mean square (LMS), filtered-x LMS (FxLMS) [3] and their extensions [4,5], are capable to catch up the variations of the noisy acoustic field and, consequently, to minimize the variance of the residual acoustic signal. Real-time ANC systems can be effectively exploited in scenarios where the noise can be modeled as a stationary or slowly varying process with respect to the reaction time of the ANC systems [6–8].
⁎
Corresponding author. E-mail addresses: alessandro.lapini@unifi.it (A. Lapini), francesco.borchi@unifi.it (F. Borchi), monica.carfagni@unifi.it (M. Carfagni), fabrizio.argenti@unifi.it (F. Argenti).
https://doi.org/10.1016/j.apacoust.2017.10.027 Received 24 July 2017; Received in revised form 25 October 2017; Accepted 27 October 2017 0003-682X/ © 2017 Elsevier Ltd. All rights reserved.
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the acoustic field on a plane B passing through r0 , a simplified solution can be also considered. The 2D1/2 Rayleigh I integral, proposed in [18], states that the wave field synthesis can be approximated by means of a linear array L laying on B, that is
Since, as it is apparent from the above discussion, adaptive filtering techniques cannot be used, the proposed AHNNC system is based on the theory of wave field synthesis (WFS) to generate the counter-phase secondary signal and cancel the primary HNN, where reference microphones are used to acquire the HNN acoustic field and to pilot the secondary loudspeakers. WFS-based applications are usually designed for constructive acoustic field synthesis [15]: by using appropriate driving signals, an array of secondary sources can be set up to reproduce, in a specific area, the sound as it were generated by one or more virtual primary sources. Unfortunately, WFS theory strictly requires an infinite array of elementary infinitesimal sources, so that several approximations are usually introduced in order to obtain a practical implementation. The idea of using the WFS framework to synthesize a virtual anti-source whose acoustic field is used in a destructive manner has been proposed in [16,17]. In this paper, we extend the previous works by formalizing the problem for the case of HNN and by considering several practical issues that arise when the system operates in a free-field environment. Several experimental results, obtained from both simulations and real tests, are presented to validate the proposed AHNNC system and to assess its performance. The paper is organized as follows: Section 2 summarizes the concepts of WFS theory. The formalization of the proposed WFS-based ANC system and its implementation issues are presented in Sections 3 and 4, respectively. The results obtained by means of computer simulations as well as by using an experimental testbed are discussed in Section 5. Some concluding remarks are presented in Section 6.
P (r,ω) ≈ −g0
g0 =
1 2π
∫L G (rL|r,ω) ∇r P (rL,ω)·nr dL. L
L
(5)
In the scenario considered in this paper for the AHNNC system, the primary noise source is represented by an acoustic monopole placed at r0 in a homogeneous medium and excited by a signal s (t ) characterized by a short duration and, thus, by a wideband Fourier transform S (ω) . The source induces a pressure field at the location r , belonging to the area surrounding the source, given by p (r,t ) , that can be expressed by using its Fourier transform P (r,ω) as
p (r,t ) =
1 2π
∞
∫−∞ P (r,ω) e jωtdω,
(6)
The goal of the proposed AHNNC system is reproducing a canceling pressure field p ̂(r,t ) = −p (r,t ) in an N-D half-space not containing the source. According to Eq. (6) and the Rayleigh I integral, in a 2-D scenario the canceling linear array placed on the line L should produce
(1)
p ̂(r,t ) = −
1 2π
∫−∞ − 21π ∫L G (rL|r,ω)·∇r P (rL,ω)·nr dL e jωt dω. ∞
L
L
(7)
For practical purposes, a discrete-sources version of (7) must be considered. Let ΔL be the spacing among the secondary sources on L. Thus, (7) can be rewritten as
p ̂(r,t ) =
1 2π
∫−∞ ∑ G (ri|r,ω)· 2ΔπL ∇r P (ri,ω)·nr e jωt dω, ∞
i
i
i
(8)
where ri represents the position of the monopoles on the sampling grids over L. Equality in the previous relation strictly holds if ΔL ⩽ cπ / ωmax , being ωmax the maximum frequency contained in S (ω) ; otherwise, antialiasing strategies have to be considered [18]. Eq. (8) states that the acoustic field p ̂(r,t ) is synthesized by an infinite number of monopoles excited by Si ̂ (ω) , given by
Si ̂ (ω) =
ΔL ∇r P (ri,ω)·n ri. 2π i
(9)
If the canceling linear array is used in the 3-D space, according to the 2D1/2 Rayleigh I integral, the excitations can be rearranged as
(2)
Si ̂ (ω) = S (ω) G (r0|ri,ω) jω ΔL ·g0
An analogous formulation for a 3-D space is
1 P (r,ω) = − ∮ G (rA|r,ω) ∇rA P (rA,ω)·n rA dA, 4π A
ΔC r/(ΔC r + ΔC r0) ,
3. WFS-based AHNNC
In Eq. (1), P (r,ω) denotes the pressure field at point r ∈ V and having frequency ω;rS ∈ S is a point on the surface S;n rS represents the normal to the surface in rS , pointing inwards V ;∇rS is the gradient evaluated in rS and G (rS |r,ω) is the appropriate Green function for the considered ND space. The above relation allows a virtual source (or more virtual sources) to be synthesized for a listener inside V, by knowing the values of the pressure field P (rS,ω) and the gradient ∇S P (rS,ω) that it induces on the surface S; such values are used to excite monopole, i.e., G (rS |r,ω) , and dipole, i.e., ∇rS G (rS |r,ω) , secondary sources. In order to be useful in practice, Eq. (1) is usually arranged under some convenient geometrical configuration. A typical example is the Rayleigh I integral. Indeed, for a 2-D space, it can be shown that the pressure in the entire half-plane not containing the primary sources can be synthesized by an infinitely extended linear array of elementary monopoles laying on a line L, that is
P (r,ω) = −
(4)
being ΔC r and ΔC r0 the distance between r and C and between r0 and C, respectively. Since g0 depends on r , the approximation given by the 2D1/2 Rayleigh I integral is more accurate for listeners on the plane B and placed at a given distance from L. Moreover, the approximation becomes increasingly poorer also when r moves farther from the plane B.
The theory of WFS directly arises from the Kirchhoff-Helmoltz integral [15], which states that the pressure field inside a volume V generated by a distribution of primary acoustic sources can be reproduced by a continuous distribution of elementary secondary sources placed over the boundary of V, i.e., the surface S. Mathematically, it is equivalent to
1 ∮ [G (rS |r,ω) ∇rS P (rS,ω)−P (rS,ω) ∇rS G (rS |r,ω)]·n rS dS. 4π S
jω|rL−r0| cosφinc ·G (r0|rL,ω) G (rL|r,ω) dL, 2πc
where c is the speed of sound in the medium; φinc is the angle between the normal to the array line L laying on B and the line passing through r0 and r . In order to define g0 , consider the plane C normal to B and passing through L; then
2. Overview of wave field synthesis theory
P (r,ω) = −
∫L S (ω)
|ri−r0| cosφinc . 2πc
(10)
Similar formulations can be written for the case of a canceling planar array in the 3-D space; however, they are not considered in this paper since planar arrays are assumed to be of difficult deployment in practical applications. Henceforward, we will refer to (10) as the basic formula to synthesize the counter-phase acoustic field. Several issues related to a practical use of (10) are discussed in the
(3)
which states that the pressure can be reproduced by an infinitely extended planar array of elementary monopoles laying on a plane A. In a homogeneous 3-D scenario, if the primary source is a monopole located in r0 having excitation S (ω) and if we are mainly interested in
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and c = 343 m/s, then we have ΔL ⩽ 57.1 mm, which represents a heavy constraint on the dimensions of the secondary sources. Aliasing avoidance can be theoretically achieved by modifying the monopole sources on the array, in order to exclude the synthesis of higher frequency components of the acoustic field as θ grows. By straightforward extension of the Shannon theorem to the angle–time domain, Start [18] proposes an antialiasing strategy involving the adoption of secondary sources having suitable frequency-dependent directivity patterns, such that the reconstructed wavefront at frequency ω is zero for |θ| > θω , being θω a frequency dependent cut-off angle. Nevertheless, the realization of acoustic sources having frequency-dependent directivity introduces hard manufacturing problems and is not addressed here. Furthermore, the dimensions of loudspeakers are dictated by mechanical and electrical constraints; this prevents from the construction of very small secondary sources and, consequently, imposes a minimum value on ΔL . Therefore, (11) can also represent a trade-off between maximum reconstruction frequency and maximum reconstruction angle.
following subsections. Some of them are recalled in the next sections, where solutions are proposed and/or experimental validation are presented. 3.1. Vertical acoustic field synthesis In the 2D1/2 WFS formulation [18], the synthesis of the acoustic field is perfect only if the primary source and the line of secondary sources are co-planar. The acoustic field synthesized by (10) in a point outside this plane is only an approximation of the true acoustic field generated by the primary source, with an error that depends on the elevation angle of the point from the plane. Thus, HNN cancellation performance degrades when we move from this plane. However, if we are interested in canceling noise in an area that is far from the secondary sources array, we may assume that such an elevation angle is small. 3.2. Finite number of secondary sources From (8) it is apparent that the number of secondary sources deployed on the line L should be, theoretically, infinite and, therefore, the array needs to be truncated in actual implementations. This argumentation straightforwardly extends to the 2D1/2 Rayleigh I integral. This generates a limitation of the coverage area, i.e., the region where the AHNNC system produces a noise cancellation, with errors that become significant at its borders.
3.6. Primary signal acquisition The driving signal of each secondary source is proportional to the sound pressure level generated by the primary source at the secondary source location, i.e. S (ω) G (r0|ri,ω) . In order to reduce the number of components of the AHNNC system, the primary source is acquired by using a single microphone and the acoustic field in proximity of each secondary source is predicted from this.
3.3. Sources directivity In the derivation of 2D1/2 Rayleigh I integral, the sources are assumed to be monopoles. This is not the case of actual primary sources as well as of secondary loudspeakers. However, if they have an approximately constant directivity with respect to the coverage area, then the loss of performance may be assumed to be limited. As to the specific case of the secondary sources, their directivity plays also a role to tackle spatial aliasing (see below).
4. AHNNC system implementation In this section, some practical issues related to the proposed AHNNC system are discussed. The scheme of the system is depicted in Fig. 1. The primary source (PS) produces the impulse-like noise we would like to cancel. The signal acquired by the reference microphone (RM) is sampled and processed by the processing unit (PU), composed of a preamplifier, an analog-to-digital converter (ADC), a programmable device, in which the signal processing takes place, a digital-to-analog converter (DAC), and an amplifier. The signal processing includes the WFS filtering as well as the equalization of the signal, which will be described successively. All digital signals are processed at a sampling frequency fs = 25 kHz. The signals generated by the PU drive an array of loudspeakers that constitute the secondary sources (SS); a different signal is needed for each SS. In order to verify and measure the cancellation performance of the system, one or more error microphones (EM) are placed in the area in front of the SS array. Several issues related to the implementation of the AHNNC scheme are discussed in the following subsections.
3.4. Homogeneous medium One of the fundamental hypothesis of WFS is the medium homogeneity. This may not hold in several applications in which, for example, multipath due to ground reflections is present. In order to avoid performance losses, multipath must be either eliminated or compensated. In the former case, physical tools may be used, such as acoustic barriers and/or acoustic absorbing material for terrain coverage. In the latter case, which has been considered in this paper, signal processing techniques aiming at compensating the effect of terrain reflections by means of linear time-invariant (LTI) filtering are applied. Further details are given in Section 4.5. 3.5. Aliasing The discretization of the linear array in the WFS theory introduces spatial aliasing in the secondary sources acoustic field [18]. Let θ (|θ| ⩽ π /2 ) be the orientation of a planar wavefront with respect to the normal of the linear array. In accordance to the Shannon sampling theorem, the correct synthesis of the wavefront is feasible by means of monopole sources iff
ΔL ⩽
πc . ω sinθ
(11)
Eq. (11) represents a constraint on the spacing ΔL between adjacent secondary sources, given the wave frequency ω and the orientation θ . Consider now a lowpass signal with maximum frequency ωmax ; in this case, perfect reconstruction for all orientations and frequencies is achieved if ΔL ⩽ cπ / ωmax . Incidentally, this corresponds to the relation stated for the validity of Eq. (8). For example, if ωmax = 2π·3000 rad/s
Fig. 1. Scheme of the AHNNC system.
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0.6
frequency of 51.2 kHz. The time duration is less than 5 ms. From the cumulative energy measured in the frequency domain, we observe that about 85% of the signal energy is concentrated below fm = 2.5 kHz. The upper limit of the band of the signal is an important parameter that influences several aspects of the AHNNC implementation.
0.4
Amplitude (V)
0.2 0
4.2. WFS filters
-0.2
According to Eq. (10), WFS theory imposes that the signal driving the ith SS is obtained by processing the PS output by an LTI filter whose frequency response is given by
-0.4 -0.6
GWFSi (f ) = G (f ) Ki
-0.8 0
1
2
3
4
where
5
G (f ) =
Time (ms)
jf
Ki = ΔL g0
30
(13)
|ri−r0| cosφinc c
(14)
where f = ω/2π . The frequency response G (f ) is a nonlinear phase function that has been implemented by using a finite impulse response (FIR) filter with N1 = 201 coefficients and designed by means of the frequency sampling method [19]. In order to obtain a causal filter, a delay of 100 samples has been set; moreover, a lowpass shape has been imposed in the frequency domain (stopband at 2.5 kHz) to avoid the enhancement of high frequency noise. The impulse and frequency responses of the resulting filter are shown in Fig. 3. Interestingly, this filter does not depend on
20 10
Magnitude (dB)
(12)
0 -10 -20 -30 -40 0
5
10
15
20
25
20
25
Frequency (kHz)
Cumulative energy (%)
100
80
60
40
20
0 0
5
10
15
Frequency (kHz)
Fig. 2. Shotgun signal characteristics: (a) time domain; (b) frequency domain; (c) frequency domain cumulative energy.
4.1. Time and frequency domain characteristics of HNN The proposed system is thought to cancel HNN, whose generic characteristics in the time and frequency domain are short duration and wideband, respectively. In order to make the discussion less abstract, we will consider henceforward the sound emitted by a shotgun as a real world example of primary noise. In Fig. 2, the time and frequency domain representations of a competition shotgun, as well as its frequency domain cumulative energy, are shown. The signal has been acquired with a sampling
Fig. 3. Wave field synthesis filter G (f ) : (a) impulse response, (b) frequency response.
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10
(ZF) equalizer [22] is designed by imposing
h (t ) ∗heq (t ) = δ (t )
8
The ZF solution produces unstable results in proximity of the zeros of H (f ) and is quite sensitive to noise; thus, other solutions have been considered here. The weighted and penalized MMSE approach [23] searches the function heq (t ) that minimizes the function
Amplitude (V)
6 4 2
2 2 J (heq) = ∥S (f )−Y (f ) F {heq } ∥W + ∥F {heq } ∥W 1 2
0
-4 -6 0
1
2
3
4
(16)
where S (f ) is the Fourier transform of the excitation signal (an impulse, for instance); Y (f ) is the Fourier transform of the signal measured by the EM; F is the Fourier transform operator; ∥·∥W1 is the weighted L2 -norm defined by the nonnegative weighting function W1 (f ) . The first component in (18) is a fidelity term, where the weighting function W1 (f ) is chosen to put more emphasis to errors in the band of interest; the second term represents the penalizing function, where W2 (f ) can be chosen to shape the frequency response of the equalizer filter. By considering a discrete version of Eq. (16), we have
-2
5
Time (ms)
J (h eq) = ∥s−YFh eq ∥2W1 + ∥Fh eq ∥2W2 10
(17)
where s and h eq are vectors containing the discrete versions of S (f ) and heq (t ) , respectively; Y , W1 and W2 are diagonal matrices containing the corresponding values of Y (f ),W1 (f ) and W2 (f ) , in that order; F is the direct discrete Fourier transform matrix. Since the problem is quadratic, the solution of Eq. (17) can be derived in closed form as
8 6
Amplitude (V)
(15)
4
h eq = (FT Y T W1YF + FT W2F)†FT Y T W1s,
2
(·)T
(18)
(·)†
and the transpose and Moore–Penrose pseudo–inverse being operator, respectively. In the AHNNC system, the loudspeaker equalization is performed by empirically setting in (18)
0 -2
[N (f ) ∫ N (f )−1df ]−1, 200 Hz < f < 3200 Hz W1 (f ) = ⎧ ⎨ otherwise, ⎩ 0, 0, 200 Hz < f < 3200 Hz W2 (f ) = ⎧ 2 ⎨ ⎩10 , otherwise,
-4 -6 0
1
2
3
4
5
Time (ms)
where N (f ) is the estimated power spectrum of the background noise. In Fig. 4b, the effect of the application of a 251 taps equalizer filter is demonstrated. Perfect equalization should exactly reproduce the signal before distortion (an impulse in this case). Actually, since we are operating equalization only on a limited frequency interval, the equalized signal looks like a “sinc” function (a band-limited version of an impulse), as shown in Fig. 4b. Notice that, in order to guarantee the causality of the equalization filter, a processing delay of 63 samples is introduced for each SS channel.
Fig. 4. Loudspeaker equalization: (a) response of the PU-SS-EM chain to an impulse; (b) application of the equalizer filter to the response signal depicted in (a).
the SS index and, thus, it is applied once for all the SS driving signals. On the contrary, Ki is a specific constant for the ith SS because it includes spatial correction terms depending on the reciprocal position of the primary and each secondary source.
4.4. Feedback avoidance 4.3. Loudspeaker equalization The secondary loudspeakers used in this study have been designed to propagate towards the half space in front of the array. Nevertheless, a residual rear acoustic emission is present and a feedback path from SS to RM may create echoes that must be avoided. Feedback path modeling and neutralization are well-established issues in the literature [2] and represent crucial aspects in the practical realization of feedforward ANC systems. The feedback path induces a closed loop gain on the RM input and the overall feedforward ANC system must be modeled as an infinite impulse response (IIR) filter; solutions based on offline path estimation [2] and on-line adaptive filtering [24–26] have been proposed. However, severe issues concerning stability arise when adopting such anti-feedback techniques in our scenario and a simpler but effective solution has been devised. Since the noise we would like to cancel has a very short duration, the problem has been solved by silencing the RM after the acquisition of the PS. The algorithm is quite simple: when the system detects the beginning of a noise (PS) signal by an amplitude trigger, a counter is activated; when the counter reaches a first
Ideally, in the absence of WFS or any other type of filtering, the chain composed by RM, PU, and a SS (denoted by RM-PU-SS) should behave as a pure delay multiplied by a constant. Since this is not true in practice, this chain has been modeled as a distorting LTI system with a specific frequency response H (f ) . This fact has been experimentally verified by using the chain PU-SS-EM, that is by generating an impulse with the PU, reproducing it by a SS and acquiring it by the EM in condition of free-space propagation. Since RM and EM are identical microphones, the chains RM-PU-SS and PU-SS-EM exhibit the same H (f ) , apart from a delay and a constant gain. The obtained signal is shown in Fig. 4a. An equalization of each SS is therefore needed. Equalization is a typical problem in telecommunication systems design [20,21], where a communication channel degrades the transmitted signal and an equalizer is needed to invert the channel response before detection. Let h (t ) and heq (t ) be the impulse responses of the RM-PU-SS chain and of the equalizer filter we are looking for, respectively. A zero-forcing 224
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Amplitude (V)
5
0
-5
Fig. 5. Combination of main paths contributing to the acoustic field on a listener point P when considering hard ground terrain reflections (lateral view): primary source’s direct path (solid blue line), primary source’s reflected components (dashed and dotted blue lines), secondary sources’ direct paths (solid red line), secondary sources’ reflected components (dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0
10
20
30
40
50
40
50
Time (ms)
5
dRS > 2cτ
Amplitude (V)
threshold (in a time corresponding to about the duration of the noise), then the RM is silenced in order to avoid to record the feedback of the SS canceling field; when the counter reaches a second threshold, the RM is activated and the counter is reset. The procedure succeeds as long as the duration τ of the incoming noise signal is less than twice the wave propagation time between the RM and the SSs, so that, if we denote with dRS the distance between RM and SS, we have the following constraint:
0
(19)
For example, for τ = 5 ms and c = 343 m/s, we have dRS > 3.43 m. -5 0
4.5. Ground reflections compensation
10
20
30
Time (ms) One of the major difficulties encountered in the design of the AHNNC system is the presence of multiple acoustic paths from the PS (and the SSs) and a receiver in the cancellation area (see Section 5.1 for a simulation example). Therefore, given a point P in the middle range within the cancellation area, the acoustic channels should be modeled as frequency-selective functions, that is, with reference to Fig. 1, HPR, HSE , and HPE depend on the frequency f. Fig. 5 depicts a schematic view of this scenario. The theoretical framework considered so far contemplates only line-of-sight (LOS) components from both the PS and SSs towards the listener point P. When reflections are considered, three further multipath components influence the ANC system. The first one is the reflected PS field that superimposes on the RM, misleading the acquired signal: it depends on the Snell angle θ given by the mutual positions of PS and RM and affects HPR . The other ones are due to both PS and SSs reflections that superimpose on the listener: they depend on the position of the listener with respect to the PS and the SSs and affect HPE and HSE , respectively. Assuming fixed positions, the multipath effects could be theoretically compensated by an a-priori modeling of the air-terrain interface; nevertheless, simple conditions (e.g., hard bound, soft bound) are not encountered in real scenarios, nor a useful model of the actual air-terrain interface can be easily estimated due to high number of involved physical parameters [27]. Introduction of acoustical barriers has been also investigated [28], but this solution can be very expensive in practice. In this study, we propose to tackle the multipath impairments by introducing a new filter, namely a ground equalization filter hGE . Let the error microphone EM be located in the point P in the coverage area. Assume that it acquires the primary noise y1 (t ) and the canceling waveform y2 (t ) . The overall signal measured by the EM is given by
y (t ) = y1 (t ) + y2 (t ).
Fig. 6. Signals recorded by the EM for the estimation of the ground equalization filters: (a) superimposed primary and secondary source signals: this is the normal operating condition of the system, but cancellation may not occur due to the presence of multipath; (b) separate primary and (delayed) secondary source signals: the knowledge of these components allows the ground equalization filter to be designed.
According to the notation given in Fig. 1, we have
y1 (t ) = F−1 {S (f ) HPE (f )}
(21)
and
y2 (t ) = F−1 {S (f ) HPR (f ) HRS (f ) HSE (f )},
(22)
where S (f ) is the excitation signal and HRS (f ) includes the WFS and loudspeaker equalization filtering described in 4.2 and 4.3, respectively. The ground equalization filter hGE is inserted in the processing chain of the SSs and has to be designed so that the signal measured by the EM becomes approximately zero. Therefore, we would like to have
y1 (t ) + hGE (t ) ∗y2 (t ) = 0
(23)
If y1 (t ) and y2 (t ) were known, estimation of hGE could be accomplished by means of the equalization techniques discussed in Section 4.3. However, Eq. (20) cannot be used in practice because y1 (t ) and y2 (t ) are not known individually, but only its superimposition is observable; an illustration of the problem is depicted in Fig. 6a. In order to overcome this limitation, we devised a procedure to measure y1 (t ) and y2 (t ) separately and, consequently, to obtain an estimation of the ground equalization filter. Specifically, an extra delay te > 0 is inserted in the processing chain of y2 (t ) , yielding
(20) 225
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y2̂ (t ) = y2 (t −te ).
(24)
being te chosen such that
y1 (t ) y2̂ (t ) = 0
∀ t.
(25)
Hence, the measured signal becomes
y (t ) = y1 (t ) + y2̂ (t ),
(26)
where y1 (t ) and y2̂ (t ) can now be achieved since they are separated in time, as depicted in Fig. 6b. By substituting (24) into (23), the relation among observable signals and the ground equalization filter is given by
y1 (t ) + hGE (t ) ∗y2̂ (t + te ) = 0.
(27)
The estimation of hGE (t ) is obtained by minimizing the weighted and penalized MMSE function introduced in Section 4.3, that is ∼ 2 2 J (hGE ) = ∥Y1 (f ) + Y2 (f ) F {hGE } ∥W + ∥F {hGE } ∥W . (28) 1 2
Fig. 7. Scheme of the simulation setup of the proposed WFS based AHNNC.
∼ where Y1 (f ) = F {y1} and Y2 (f ) = F {y2̂ }exp(j2πfte ) . Accordingly, the solution, similar to that presented in (18), is given by ∼T ∼ ∼T h GE = (FT Y 2 W1Y2F + FT W2F)†·FT Y 2 W1y1,
related results are discussed. Then, the experimental testbed setup is presented and, eventually, the related results are shown and commented.
(29)
being h GE and y1 the vectors containing discretized hGE (t ) and y1 (t ), ∼ respectively, whereas Y2 is the diagonal matrix containing the discrete ∼ version of Y2 (f ) . The hGE filter has been designed as a 400 order FIR filter and introduces 100 samples of delay in the processing chain. In the design, we empirically set.
5.1. Simulations A set of numerical simulations, obtained by using the k-Wave [29] software tool, has been preliminarily carried out in order to assess the feasibility of the proposed approach. Fig. 7 depicts the simulation setup for the proposed ANC system using a linear array in a 3-D space. The array of SSs is placed dsa = 4 m far from the PS, simulated as an emitting monopole. The primary noise feeding the PS is the shotgun signal shown in Fig. 2a, subsampled by a factor ten, that is at a sampling frequency of f s′ = 5120 Hz. Both primary and secondary sources are placed at h = 1.5 m from the ground. The array aperture is set to a = 10 m and it is composed by discrete elementary monopoles excited according to Eq. (9) and spaced by da = λ/2 ≈ 67 mm, being λ = 2c / f s′ the shorter wavelength associated to the primary acoustic field. Such a value of da is the upper bound on spatial spacing and prevents from aliasing effects in the synthesized field. Virtual reference microphones are co-located with each SS; for the sake of simplicity, perfect synchronization, no ground equalization filter and absence of environmental disturbances were assumed. The sound energy flowing in the half space in front of the SS array is measured by a planar microphone array located at dsm = 12 m from the primary source. As to the performance index, the attenuation obtained when the ANC system is switched on with respect to when it is switched off – i.e., the ratio between the sound intensities measured on the microphone array – is considered. The proposed ANC system has been initially simulated by considering sound waves propagating in a homogeneous medium, that is, the terrain has been assumed constituted by air too. The sound intensity attenuations are reported in Fig. 8a. Attenuation levels greater than 10 dB in the middle region can be observed. A performance degradation emerges in the lateral regions due to the finite aperture of the array. As to the performance along the vertical direction, it is mainly limited by the assumptions of the 2D1/2 Rayleigh I integral and a sharp loss is clearly visible as the elevation increases (the contribution of the ANC system causes up to a 6 dB increment of the PS noise in the uppermost areas). In order to evaluate the impairments due to the presence of terrain reflections, the simulation has been repeated by considering a perfectly hard ground. The measured sound intensity attenuations are depicted in Fig. 8b; in accordance to theoretical considerations, a remarkable performance loss is observed, which justifies the need for solutions to mitigate the terrain reflections effect.
0, 200 Hz < f < 3200 Hz W1 (f ) = 1 W2 (f ) = ⎧ 2 ⎨ 10 , otherwise. ⎩ An advantage of the proposed estimation procedure is that it automatically adjusts the synchronization and the gain of the canceling system, compensating for possible discrepancies between actual and nominal parameters (e.g., RM-SSs distance, loudspeakers and microphones gains). As a limitation, the procedure is dependent on the choice of the point P in the coverage area. Therefore, different equalization filters should be designed and used according to where we would like to locate the silence area. We have, however, experimentally verified that, for the considered frequency band, selecting P in the middle range continues to produce benefits in a large area, from the middle to the long range. 4.6. Processing chain delay The proposed system works in real-time and, therefore, causality is required in the processing chain. Let nWFS , nLE and nGE be the delays introduced by the WFS, the loudspeaker equalization and the ground equalization filters, in that order, expressed in number of samples at the sampling frequency fs . Moreover, let nC be the processing delay due to computing operations. The overall system delay must be lower than the traveling time of the wave from RM to SSs, that is we must have
dRS ⩾
c (nWFS + nLE + nGE + nC ), fs
(30)
where dRS is the distance between RM and SS. Eqs. (19) and (30) represent two constraints on the geometry of the system. Even though no maximum bound for dRS is derived from theoretical analysis, in practice a performance loss occurs if an excessive value is chosen. Indeed, the farther the RM, the poorer the estimation of the PS field on each SS location. 5. Results In this section, simulations and experiments aiming at validating the proposed method are presented. Firstly, the simulation scenario and the 226
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Fig. 8. Sound intensity attenuations (dB) measured on the microphone array in the simulations of different scenarios: (a) homogeneous medium; (b) in the presence of hard ground reflections.
5.2. Testbed setup The experimental testbed that was setup to demonstrate the effectiveness of the proposed method is composed by the following components: 1. National Instruments CompactRIO-9039 controller (1.91 GHz, Quad-Core CPU), equipped with 325T FPGA. The controller is the core of the processing unit, where the previously described algorithms have been implemented by using Labview FPGA. The controller mounts a 4-channel ADC NI9215 module (simultaneous sampling, 100 Ksample/s, 16 bit/sample) and a 16-channel DAC NI9264 module (16 bit, 25 Ksample/s); 2. Behringer ECM8000 microphones, used as RM and EM; the microphones signals are supplied and pre-amplified by means of a PreSonus DigiMax FS (8-channels) system; 3. prototype horn loudspeakers, specifically designed for the testbed1, as shown in Fig. 9, and capable to reproduce a shotgun signal with peak SPL@1 m ≈132 dB. Such an elevated acoustic pressure was necessary to obtain reliable measurements in the middle/long range. The diameters of the magnet, coil and throat aperture in the compression driver are 180, 90 and 50 mm, in that order. The nominal power of the amplifier (one for each loudspeaker) is 300 W. The horn has a shape of truncated pyramid, is made of wood and is long 1050 mm. The (internal) section of the horn is square and its throat and final apertures are 50 and 270 mm, respectively. In order to avoid coil overrun due to uncontrolled low frequency noise entering the ANC system – and, consequently, possible damages to the driver – a highpass filter having cutoff frequency at 300 Hz has been inserted in the digital processing chain.
Fig. 9. Horn loudspeakers used in the tests.
Fig. 10. Schematic view from above of the considered scenario for the testbed measurements (distances are not in scale for the sake of clarity).
0.9 m from the ground; in a second set of experiments, denoted as B, the PS and SSs were at 1.6 and 0.9 m from the ground, respectively. The ground reflection compensation was preliminarily performed placing the EM at 16 m from the SSs array center at the height of the PS in both scenarios. Subsequently, the EM was used to evaluate cancellation performance in the same location (dam = 16 m), as well as at 60 m (dam = 60 m) far from the SSs. In the latter case, different lateral offsets were also considered. A view from above of the geometry of the measurements scenario is depicted in Fig. 10. The presented results are related to data acquired in mild weather conditions, that is, sunny days, air temperature between 18 and 22 degrees, in the presence of a variable breeze (rarely gusts).
The experimental tests were carried out in open field, in the absence of natural and artificial obstacles; the terrain was covered by grass. We used one loudspeaker as PS and five loudspeakers as SSs, at a distance from the PS dsa = 15 m. The loudspeakers are spaced by 0.3 m. The RM was placed at a distance drs = 5.5 m from the SSs. In a first set of experiments, namely scenario A, the PS and the SSs were coplanar at
1
The loudspeakers were produced by Look Line s.r.l., Massa Finalese, Modena, Italy.
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50
Table 1 Attenuation results (dB) in the frequency band 400–1500 Hz, measured in the two considered experimental scenarios: PS and SSs at 0.9 m from the ground (A), PS and SSs at 1.6 and 0.9 m from the ground, respectively (B). Measurement positions are referred to Fig. 10.
A (best) B (best) A (median) B (median)
E0
E1
E2
E3
10.8 14.3 8.6 8.5
8.9 7.6 5.8 5.5
8.1 7.3 5.7 5.2
1.4 1.3 0.2 −0.3
30
dB
Scenario
40
20
10
5.3. Attenuation results
Average (ANC off) Average (ANC on) Median (ANC off) Median (ANC on) Background noise
0
In a first set of experiments, the PS was fed with the shotgun signal shown in 2a repeated 20 times with a period of 2 s. Attenuation results in the frequency band 400–1500 Hz are reported in Table 1, in terms of best and median values achieved in each recording session. As to the “best” results for scenario A, the performance loss between the attenuations at 60 m (central position) and 16 m is about 2 dB. A contained decrement of 0.8 dB is observed when moving from the central position to an offset of 4 m; the attenuation sharply decays when the offset becomes 8 m. Also “median” results demonstrate a good performance of the attenuation system, 2–3 dB lower than the best ones (a sharp loss is still present for the largest lateral offset). The results related to the measurements with offset induce some considerations about the coverage area of the proposed system. Consider the footprint of the SS array when illuminated by the PS and let α be the angle in the vertex where PS is located, as show in Fig. 10, such that tanα /2 = a/2dsa . From the geometry of the experimental scenario, we can see that the measurement point E2 is about on the edge of the footprint, whereas E3 is outside. This fact, along with the observation of the attenuation results in Table 1, suggests that the coverage area of the system and the footprint approximately coincide. As to scenario B, the “best” results are worse than those relative to scenario A up to 1.3 dB (the attenuation at 16 m, where an improvement is experienced, is considered as a positive outlier), whereas “median” results are only at most 0.5 dB worse than those of scenario A. A performance loss is expected for scenario B, since the EM is not coplanar with the PS-SS plane; however, the measured loss is quite small at these elevation angles from the plane. Analogous considerations regarding the footprint of the system stated for scenario A hold also for scenario B. In a second set of experiments, we analyzed the performance by studying the frequency representations statistics of the waveforms recorded when the ANC system is active with respect to when it is switched off. The setup of the scenario B is referred to (similar conclusions are valid for the scenario A). Fig. 11a demonstrates the squared amplitude spectra,2 averaged, for each frequency, over 60 recorded realizations of the shot, related to the distance of 16 m (in solid blue3 and red lines when the system is switched off and on, respectively). In the lower part of the spectra, i.e., 300 < f < 450 Hz, the curves are superimposed due to the transition band of the loudspeakers’ protection filter; in the middle range frequency, 500 < f < 1500 Hz, the attenuation level varies from 4 to 12 dB; from 1500 to 1900 Hz a lower performance is observed (from 4 to 7 dB); eventually, the transition band of the WFS mainly characterizes the ANC system’s response starting from 2000 Hz. The previous results can be compared to the average spectra acquired at 60 m, as reported in Fig. 11b. Considering the case when the system is switched off and comparing the blue solid line with the corresponding one in Fig. 11a, we observe similar spectra of the PS signals,
-10 500
1000
1500
2000
2500
2000
2500
Hz
45 40 35
dB
30 25 20
Average (ANC off) Average (ANC on) Median (ANC off) Median (ANC on) Background noise
15 10 500
1000
1500
Hz
Fig. 11. Average and median spectra of the primary source signal (ANC off) and the residual signal (ANC on), measured at 16 m (a) and 60 m (b), respectively.
10 8 6
Amplitude (V)
4 2 0 -2 -4 ANC off
-6
ANC on
-8 0
1
2
3
4
5
6
7
Time (ms) Fig. 12. Time signals related to one shot recorded by the error microphone at 60 m, with the AHNNC system switched off and on (bandpass versions – filtered between 300 and 3100 Hz – are shown). (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)
2
The reported dB values are not absolute SPL. For interpretation of color in Fig. 11, the reader is referred to the web version of this article. 3
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even though some changes appear confirming that the acoustic channel has a different frequency selective transfer function. The attenuation achieved by switching on the ANC system at 60 m is lower with respect to the previous case (about 3–4 dB gap), but follows a similar pattern across frequencies. Two shot signals registered on the EM at 60 m when the proposed ANC system is off and is on, respectively, are depicted in Fig. 12. As can be seen, in the latter case a remarkable amplitude reduction is experienced. From the results shown above, we observe that the proposed system is capable to achieve a noticeable attenuation of the impulsive noise in a wide area in front of the secondary sources array. Incidentally, even though the compensation of ground reflections is calibrated to work in the middle range, the benefits can be appreciated also in the long range.
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I-109–I-112. http://dx.doi.org/10.1109/ICASSP.2007.366628. [8] Borchi F, Carfagni M, Martelli L, Turchi A, Argenti F. Design and experimental tests of active control barriers for low-frequency stationary noise reduction in urban outdoor environment. Appl Acoust 2016;114:125–35. http://dx.doi.org/10.1016/j. apacoust.2016.07.020. [9] Leahy R, Zhou Z, Hsu Y-C. Adaptive filtering of stable processes for active attenuation of impulsive noise. In: 1995 International conference on acoustics, speech, and signal processing, 1995. ICASSP-95, vol. 5; 1995. p. 2983–6. http://dx. doi.org/10.1109/ICASSP.1995.479472. [10] Sun X, Kuo SM, Meng G. Adaptive algorithm for active control of impulsive noise. J Sound Vib 2006;291(1–2):516–22. http://dx.doi.org/10.1016/j.jsv.2005.06.011. [11] Akhtar MT, Mitsuhashi W. Improving robustness of filtered-x least mean p-power algorithm for active attenuation of standard symmetric–stable impulsive noise. 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Development of the filteredu algorithm for active noise control. J Acoust Soc Am 1991;89(1):257–65. http://dx.doi.org/10.1121/1.400508. [25] Akhtar MT, Abe M, Kawamata M. On active noise control systems with online acoustic feedback path modeling. IEEE Trans Audio Speech Lang Process 2007;15(2):593–600. http://dx.doi.org/10.1109/TASL.2006.876749. [26] Akhtar MT, Abe M, Kawamata M, Mitsuhashi W. A simplified method for online acoustic feedback path modeling and neutralization in multichannel active noise control systems. Sign Process 2009;89(6):1090–9. http://dx.doi.org/10.1016/j. sigpro.2008.12.013. [27] Attenborough K, Li KM, Horoshenkov K. Predicting outdoor sound. Boca Ratonr (FL, USA): CRC Press; 2006. [28] Lapini A, Borchi F, Carfagni M, Argenti F. Simulation and design of active control systems for acoustic pulse noise. Int J Interactive Des Manuf (IJIDeM), http://dx. doi.org/10.1007/s12008-017-0409-9. [29] Treeby BE, Rendell AP, Cox BT. 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6. Conclusions In this paper, the problem of canceling highly non-stationary (or pulse-like) noise by means of active systems has been addressed. The noise considered in this study is characterized by a very short duration, so that classical ANC techniques, based on adaptive filtering, are ineffective since convergence cannot be reached. Therefore, a system based on the wave field synthesis (WFS) theory to construct the canceling acoustic field has been presented. The key elements of the proposed system are an array of secondary loudspeakers that generate the canceling waveform, a processing unit that generates the driving signals for the array and a reference microphone to acquire the signal to be canceled. The system has been thought to work in the outdoor. The major issues related to the actual implementation of the system – such as WFS filters design, loudspeakers equalization, finite dimension of the array, feedback avoidance – have been analyzed. The presence of ground reflections induces a mismatch between the theoretical model at the basis of the system and the actual working environment. In order to reduce the consequent performance loss, a ground reflections compensation method has been devised: even though such a procedure solves exactly the problem only in one measurement point, its beneficial effects can be experienced also in the middle/long range. Experimental results were presented to validate the effectiveness of the proposed system. First, the correctness of the basic idea of using WFS in active noise control systems was established by means of computer simulations. An experimental testbed was also setup and the results of onthe-field measurements were presented. The experiments demonstrate that the proposed system is actually able to diminish the overall noise perceived in the area in front of the secondary loudspeakers array till a distance of tens of meters. Acknowledgements The authors would like to thank Fabbrica d’Armi Pietro Beretta S.p.A. for supporting the research undertaken in this study. The authors would like also to thank Mr. Claudio Bergamini and Look Line s.r.l. for the realization of the horn loudspeakers and the related amplifiers as well as for many discussions and support during the development of the present study. References [1] Elliott SJ, Nelson PA. Active noise control. IEEE Sign Process Mag 1993;10(4):12–35. http://dx.doi.org/10.1109/79.248551. [2] Kuo S, Morgan D. Active noise control: a tutorial review. Proc IEEE
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Application of wave field synthesis to active control of highly non-stationary noise
MARK
⁎
Alessandro Lapinia, , Francesco Borchia, Monica Carfagnia, Fabrizio Argentib a b
Dipartimento di Ingegneria Industriale, Università di Firenze, via Santa Marta, 3 I-50139 Firenze, Italy Dipartimento di Ingegneria dell’Informazione, Università di Firenze, via Santa Marta, 3 I-50139 Firenze, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Active noise control Wave field synthesis Non-stationary noise
Active Noise Control (ANC) methods have been successfully applied to the cancellation of stationary noise. Classical ANC systems use adaptive filtering techniques to produce a waveform that is opposite, or counterphase, to the signal noise we would like to cancel. However, when the noise is of short duration, adaptive filtering cannot be used since convergence is not achieved. In this paper, a novel active control technique for non-stationary noise is presented. The method uses wave field synthesis (WFS) for the construction of the canceling waveform. The system is tailored for an outdoor environment. The noise acoustic field is acquired by microphones and processed by a WFS engine to pilot a linear array of secondary sources. Experimental results, obtained from both simulations and true tests, demonstrate that the proposed method is able to diminish the overall noise perceived in the area covered by the system.
1. Introduction
When the noise is highly non-stationary, for example, when it appears as an impulse having large-amplitude pressure levels with respect to the background and a very short duration, the adaptive filters within an ANC system may not have enough time to reach convergence, making noise reduction ineffective. Examples of such type of noise are represented by man-made disturbances, such as sounds generated by manufacturing plants, vehicle transit, punching machines or construction sites as well as gunshots in shooting ranges. In this paper, we use the ANC paradigm, that is generating a counter-phase signal, to cancel an impulse-like noise. Since in the literature the term impulsive noise is often used to indicate heavy-tailed distributed disturbances that affect the operations of adaptive ANC systems [9–14], in order not to generate ambiguities, we also term the primary source signal as highly non-stationary noise (HNN) and the proposed scheme as active HNN control (AHNNC) system. To put ideas into a real world scenario, consider as an example of such a noise the one produced by a gunshot within a shooting range. This example allows other objectives of this study to be delineated as follows. The HNN we would like to cancel is a short duration wideband signal. Furthermore, the HNN is not assumed as deterministic, even though several features, both in the time and frequency domain, can be derived by the analysis of few realizations of the noise. The scenario in which the AHNNC system should work is the outdoor and the dimensions of the area in which it should reduce the noise are of the order of tens of meters.
Active noise control (ANC) techniques have been widely used to reduce the impact of noise in different environments. Given a source of noise, the aim of an ANC system is that of piloting one or more secondary sources to generate a waveform that is opposite, or in counterphase, to that of the noise. Thus, thanks to the principle of superposition, the primary source of noise is cancelled. An ANC system is particularly effective when the region where it operates is limited, e.g., in headset, ducts, car and airplane cabins, and is more convenient to diminish low frequency noise than passive methods [1,2]. A classical ANC system is based on adaptive filtering. Its basic components are reference microphones, to acquire the primary source of noise; secondary loudspeakers, to generate the counter-phase waveform; and error microphones, to acquire the resulting signal coming from primary and secondary sources in the area where the system is supposed to be effective. The error signal is used to update the coefficients of adaptive filters that drive the secondary sources. Adaptive algorithms, such as least mean square (LMS), filtered-x LMS (FxLMS) [3] and their extensions [4,5], are capable to catch up the variations of the noisy acoustic field and, consequently, to minimize the variance of the residual acoustic signal. Real-time ANC systems can be effectively exploited in scenarios where the noise can be modeled as a stationary or slowly varying process with respect to the reaction time of the ANC systems [6–8].
⁎
Corresponding author. E-mail addresses: alessandro.lapini@unifi.it (A. Lapini), francesco.borchi@unifi.it (F. Borchi), monica.carfagni@unifi.it (M. Carfagni), fabrizio.argenti@unifi.it (F. Argenti).
https://doi.org/10.1016/j.apacoust.2017.10.027 Received 24 July 2017; Received in revised form 25 October 2017; Accepted 27 October 2017 0003-682X/ © 2017 Elsevier Ltd. All rights reserved.
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the acoustic field on a plane B passing through r0 , a simplified solution can be also considered. The 2D1/2 Rayleigh I integral, proposed in [18], states that the wave field synthesis can be approximated by means of a linear array L laying on B, that is
Since, as it is apparent from the above discussion, adaptive filtering techniques cannot be used, the proposed AHNNC system is based on the theory of wave field synthesis (WFS) to generate the counter-phase secondary signal and cancel the primary HNN, where reference microphones are used to acquire the HNN acoustic field and to pilot the secondary loudspeakers. WFS-based applications are usually designed for constructive acoustic field synthesis [15]: by using appropriate driving signals, an array of secondary sources can be set up to reproduce, in a specific area, the sound as it were generated by one or more virtual primary sources. Unfortunately, WFS theory strictly requires an infinite array of elementary infinitesimal sources, so that several approximations are usually introduced in order to obtain a practical implementation. The idea of using the WFS framework to synthesize a virtual anti-source whose acoustic field is used in a destructive manner has been proposed in [16,17]. In this paper, we extend the previous works by formalizing the problem for the case of HNN and by considering several practical issues that arise when the system operates in a free-field environment. Several experimental results, obtained from both simulations and real tests, are presented to validate the proposed AHNNC system and to assess its performance. The paper is organized as follows: Section 2 summarizes the concepts of WFS theory. The formalization of the proposed WFS-based ANC system and its implementation issues are presented in Sections 3 and 4, respectively. The results obtained by means of computer simulations as well as by using an experimental testbed are discussed in Section 5. Some concluding remarks are presented in Section 6.
P (r,ω) ≈ −g0
g0 =
1 2π
∫L G (rL|r,ω) ∇r P (rL,ω)·nr dL. L
L
(5)
In the scenario considered in this paper for the AHNNC system, the primary noise source is represented by an acoustic monopole placed at r0 in a homogeneous medium and excited by a signal s (t ) characterized by a short duration and, thus, by a wideband Fourier transform S (ω) . The source induces a pressure field at the location r , belonging to the area surrounding the source, given by p (r,t ) , that can be expressed by using its Fourier transform P (r,ω) as
p (r,t ) =
1 2π
∞
∫−∞ P (r,ω) e jωtdω,
(6)
The goal of the proposed AHNNC system is reproducing a canceling pressure field p ̂(r,t ) = −p (r,t ) in an N-D half-space not containing the source. According to Eq. (6) and the Rayleigh I integral, in a 2-D scenario the canceling linear array placed on the line L should produce
(1)
p ̂(r,t ) = −
1 2π
∫−∞ − 21π ∫L G (rL|r,ω)·∇r P (rL,ω)·nr dL e jωt dω. ∞
L
L
(7)
For practical purposes, a discrete-sources version of (7) must be considered. Let ΔL be the spacing among the secondary sources on L. Thus, (7) can be rewritten as
p ̂(r,t ) =
1 2π
∫−∞ ∑ G (ri|r,ω)· 2ΔπL ∇r P (ri,ω)·nr e jωt dω, ∞
i
i
i
(8)
where ri represents the position of the monopoles on the sampling grids over L. Equality in the previous relation strictly holds if ΔL ⩽ cπ / ωmax , being ωmax the maximum frequency contained in S (ω) ; otherwise, antialiasing strategies have to be considered [18]. Eq. (8) states that the acoustic field p ̂(r,t ) is synthesized by an infinite number of monopoles excited by Si ̂ (ω) , given by
Si ̂ (ω) =
ΔL ∇r P (ri,ω)·n ri. 2π i
(9)
If the canceling linear array is used in the 3-D space, according to the 2D1/2 Rayleigh I integral, the excitations can be rearranged as
(2)
Si ̂ (ω) = S (ω) G (r0|ri,ω) jω ΔL ·g0
An analogous formulation for a 3-D space is
1 P (r,ω) = − ∮ G (rA|r,ω) ∇rA P (rA,ω)·n rA dA, 4π A
ΔC r/(ΔC r + ΔC r0) ,
3. WFS-based AHNNC
In Eq. (1), P (r,ω) denotes the pressure field at point r ∈ V and having frequency ω;rS ∈ S is a point on the surface S;n rS represents the normal to the surface in rS , pointing inwards V ;∇rS is the gradient evaluated in rS and G (rS |r,ω) is the appropriate Green function for the considered ND space. The above relation allows a virtual source (or more virtual sources) to be synthesized for a listener inside V, by knowing the values of the pressure field P (rS,ω) and the gradient ∇S P (rS,ω) that it induces on the surface S; such values are used to excite monopole, i.e., G (rS |r,ω) , and dipole, i.e., ∇rS G (rS |r,ω) , secondary sources. In order to be useful in practice, Eq. (1) is usually arranged under some convenient geometrical configuration. A typical example is the Rayleigh I integral. Indeed, for a 2-D space, it can be shown that the pressure in the entire half-plane not containing the primary sources can be synthesized by an infinitely extended linear array of elementary monopoles laying on a line L, that is
P (r,ω) = −
(4)
being ΔC r and ΔC r0 the distance between r and C and between r0 and C, respectively. Since g0 depends on r , the approximation given by the 2D1/2 Rayleigh I integral is more accurate for listeners on the plane B and placed at a given distance from L. Moreover, the approximation becomes increasingly poorer also when r moves farther from the plane B.
The theory of WFS directly arises from the Kirchhoff-Helmoltz integral [15], which states that the pressure field inside a volume V generated by a distribution of primary acoustic sources can be reproduced by a continuous distribution of elementary secondary sources placed over the boundary of V, i.e., the surface S. Mathematically, it is equivalent to
1 ∮ [G (rS |r,ω) ∇rS P (rS,ω)−P (rS,ω) ∇rS G (rS |r,ω)]·n rS dS. 4π S
jω|rL−r0| cosφinc ·G (r0|rL,ω) G (rL|r,ω) dL, 2πc
where c is the speed of sound in the medium; φinc is the angle between the normal to the array line L laying on B and the line passing through r0 and r . In order to define g0 , consider the plane C normal to B and passing through L; then
2. Overview of wave field synthesis theory
P (r,ω) = −
∫L S (ω)
|ri−r0| cosφinc . 2πc
(10)
Similar formulations can be written for the case of a canceling planar array in the 3-D space; however, they are not considered in this paper since planar arrays are assumed to be of difficult deployment in practical applications. Henceforward, we will refer to (10) as the basic formula to synthesize the counter-phase acoustic field. Several issues related to a practical use of (10) are discussed in the
(3)
which states that the pressure can be reproduced by an infinitely extended planar array of elementary monopoles laying on a plane A. In a homogeneous 3-D scenario, if the primary source is a monopole located in r0 having excitation S (ω) and if we are mainly interested in
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and c = 343 m/s, then we have ΔL ⩽ 57.1 mm, which represents a heavy constraint on the dimensions of the secondary sources. Aliasing avoidance can be theoretically achieved by modifying the monopole sources on the array, in order to exclude the synthesis of higher frequency components of the acoustic field as θ grows. By straightforward extension of the Shannon theorem to the angle–time domain, Start [18] proposes an antialiasing strategy involving the adoption of secondary sources having suitable frequency-dependent directivity patterns, such that the reconstructed wavefront at frequency ω is zero for |θ| > θω , being θω a frequency dependent cut-off angle. Nevertheless, the realization of acoustic sources having frequency-dependent directivity introduces hard manufacturing problems and is not addressed here. Furthermore, the dimensions of loudspeakers are dictated by mechanical and electrical constraints; this prevents from the construction of very small secondary sources and, consequently, imposes a minimum value on ΔL . Therefore, (11) can also represent a trade-off between maximum reconstruction frequency and maximum reconstruction angle.
following subsections. Some of them are recalled in the next sections, where solutions are proposed and/or experimental validation are presented. 3.1. Vertical acoustic field synthesis In the 2D1/2 WFS formulation [18], the synthesis of the acoustic field is perfect only if the primary source and the line of secondary sources are co-planar. The acoustic field synthesized by (10) in a point outside this plane is only an approximation of the true acoustic field generated by the primary source, with an error that depends on the elevation angle of the point from the plane. Thus, HNN cancellation performance degrades when we move from this plane. However, if we are interested in canceling noise in an area that is far from the secondary sources array, we may assume that such an elevation angle is small. 3.2. Finite number of secondary sources From (8) it is apparent that the number of secondary sources deployed on the line L should be, theoretically, infinite and, therefore, the array needs to be truncated in actual implementations. This argumentation straightforwardly extends to the 2D1/2 Rayleigh I integral. This generates a limitation of the coverage area, i.e., the region where the AHNNC system produces a noise cancellation, with errors that become significant at its borders.
3.6. Primary signal acquisition The driving signal of each secondary source is proportional to the sound pressure level generated by the primary source at the secondary source location, i.e. S (ω) G (r0|ri,ω) . In order to reduce the number of components of the AHNNC system, the primary source is acquired by using a single microphone and the acoustic field in proximity of each secondary source is predicted from this.
3.3. Sources directivity In the derivation of 2D1/2 Rayleigh I integral, the sources are assumed to be monopoles. This is not the case of actual primary sources as well as of secondary loudspeakers. However, if they have an approximately constant directivity with respect to the coverage area, then the loss of performance may be assumed to be limited. As to the specific case of the secondary sources, their directivity plays also a role to tackle spatial aliasing (see below).
4. AHNNC system implementation In this section, some practical issues related to the proposed AHNNC system are discussed. The scheme of the system is depicted in Fig. 1. The primary source (PS) produces the impulse-like noise we would like to cancel. The signal acquired by the reference microphone (RM) is sampled and processed by the processing unit (PU), composed of a preamplifier, an analog-to-digital converter (ADC), a programmable device, in which the signal processing takes place, a digital-to-analog converter (DAC), and an amplifier. The signal processing includes the WFS filtering as well as the equalization of the signal, which will be described successively. All digital signals are processed at a sampling frequency fs = 25 kHz. The signals generated by the PU drive an array of loudspeakers that constitute the secondary sources (SS); a different signal is needed for each SS. In order to verify and measure the cancellation performance of the system, one or more error microphones (EM) are placed in the area in front of the SS array. Several issues related to the implementation of the AHNNC scheme are discussed in the following subsections.
3.4. Homogeneous medium One of the fundamental hypothesis of WFS is the medium homogeneity. This may not hold in several applications in which, for example, multipath due to ground reflections is present. In order to avoid performance losses, multipath must be either eliminated or compensated. In the former case, physical tools may be used, such as acoustic barriers and/or acoustic absorbing material for terrain coverage. In the latter case, which has been considered in this paper, signal processing techniques aiming at compensating the effect of terrain reflections by means of linear time-invariant (LTI) filtering are applied. Further details are given in Section 4.5. 3.5. Aliasing The discretization of the linear array in the WFS theory introduces spatial aliasing in the secondary sources acoustic field [18]. Let θ (|θ| ⩽ π /2 ) be the orientation of a planar wavefront with respect to the normal of the linear array. In accordance to the Shannon sampling theorem, the correct synthesis of the wavefront is feasible by means of monopole sources iff
ΔL ⩽
πc . ω sinθ
(11)
Eq. (11) represents a constraint on the spacing ΔL between adjacent secondary sources, given the wave frequency ω and the orientation θ . Consider now a lowpass signal with maximum frequency ωmax ; in this case, perfect reconstruction for all orientations and frequencies is achieved if ΔL ⩽ cπ / ωmax . Incidentally, this corresponds to the relation stated for the validity of Eq. (8). For example, if ωmax = 2π·3000 rad/s
Fig. 1. Scheme of the AHNNC system.
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0.6
frequency of 51.2 kHz. The time duration is less than 5 ms. From the cumulative energy measured in the frequency domain, we observe that about 85% of the signal energy is concentrated below fm = 2.5 kHz. The upper limit of the band of the signal is an important parameter that influences several aspects of the AHNNC implementation.
0.4
Amplitude (V)
0.2 0
4.2. WFS filters
-0.2
According to Eq. (10), WFS theory imposes that the signal driving the ith SS is obtained by processing the PS output by an LTI filter whose frequency response is given by
-0.4 -0.6
GWFSi (f ) = G (f ) Ki
-0.8 0
1
2
3
4
where
5
G (f ) =
Time (ms)
jf
Ki = ΔL g0
30
(13)
|ri−r0| cosφinc c
(14)
where f = ω/2π . The frequency response G (f ) is a nonlinear phase function that has been implemented by using a finite impulse response (FIR) filter with N1 = 201 coefficients and designed by means of the frequency sampling method [19]. In order to obtain a causal filter, a delay of 100 samples has been set; moreover, a lowpass shape has been imposed in the frequency domain (stopband at 2.5 kHz) to avoid the enhancement of high frequency noise. The impulse and frequency responses of the resulting filter are shown in Fig. 3. Interestingly, this filter does not depend on
20 10
Magnitude (dB)
(12)
0 -10 -20 -30 -40 0
5
10
15
20
25
20
25
Frequency (kHz)
Cumulative energy (%)
100
80
60
40
20
0 0
5
10
15
Frequency (kHz)
Fig. 2. Shotgun signal characteristics: (a) time domain; (b) frequency domain; (c) frequency domain cumulative energy.
4.1. Time and frequency domain characteristics of HNN The proposed system is thought to cancel HNN, whose generic characteristics in the time and frequency domain are short duration and wideband, respectively. In order to make the discussion less abstract, we will consider henceforward the sound emitted by a shotgun as a real world example of primary noise. In Fig. 2, the time and frequency domain representations of a competition shotgun, as well as its frequency domain cumulative energy, are shown. The signal has been acquired with a sampling
Fig. 3. Wave field synthesis filter G (f ) : (a) impulse response, (b) frequency response.
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10
(ZF) equalizer [22] is designed by imposing
h (t ) ∗heq (t ) = δ (t )
8
The ZF solution produces unstable results in proximity of the zeros of H (f ) and is quite sensitive to noise; thus, other solutions have been considered here. The weighted and penalized MMSE approach [23] searches the function heq (t ) that minimizes the function
Amplitude (V)
6 4 2
2 2 J (heq) = ∥S (f )−Y (f ) F {heq } ∥W + ∥F {heq } ∥W 1 2
0
-4 -6 0
1
2
3
4
(16)
where S (f ) is the Fourier transform of the excitation signal (an impulse, for instance); Y (f ) is the Fourier transform of the signal measured by the EM; F is the Fourier transform operator; ∥·∥W1 is the weighted L2 -norm defined by the nonnegative weighting function W1 (f ) . The first component in (18) is a fidelity term, where the weighting function W1 (f ) is chosen to put more emphasis to errors in the band of interest; the second term represents the penalizing function, where W2 (f ) can be chosen to shape the frequency response of the equalizer filter. By considering a discrete version of Eq. (16), we have
-2
5
Time (ms)
J (h eq) = ∥s−YFh eq ∥2W1 + ∥Fh eq ∥2W2 10
(17)
where s and h eq are vectors containing the discrete versions of S (f ) and heq (t ) , respectively; Y , W1 and W2 are diagonal matrices containing the corresponding values of Y (f ),W1 (f ) and W2 (f ) , in that order; F is the direct discrete Fourier transform matrix. Since the problem is quadratic, the solution of Eq. (17) can be derived in closed form as
8 6
Amplitude (V)
(15)
4
h eq = (FT Y T W1YF + FT W2F)†FT Y T W1s,
2
(·)T
(18)
(·)†
and the transpose and Moore–Penrose pseudo–inverse being operator, respectively. In the AHNNC system, the loudspeaker equalization is performed by empirically setting in (18)
0 -2
[N (f ) ∫ N (f )−1df ]−1, 200 Hz < f < 3200 Hz W1 (f ) = ⎧ ⎨ otherwise, ⎩ 0, 0, 200 Hz < f < 3200 Hz W2 (f ) = ⎧ 2 ⎨ ⎩10 , otherwise,
-4 -6 0
1
2
3
4
5
Time (ms)
where N (f ) is the estimated power spectrum of the background noise. In Fig. 4b, the effect of the application of a 251 taps equalizer filter is demonstrated. Perfect equalization should exactly reproduce the signal before distortion (an impulse in this case). Actually, since we are operating equalization only on a limited frequency interval, the equalized signal looks like a “sinc” function (a band-limited version of an impulse), as shown in Fig. 4b. Notice that, in order to guarantee the causality of the equalization filter, a processing delay of 63 samples is introduced for each SS channel.
Fig. 4. Loudspeaker equalization: (a) response of the PU-SS-EM chain to an impulse; (b) application of the equalizer filter to the response signal depicted in (a).
the SS index and, thus, it is applied once for all the SS driving signals. On the contrary, Ki is a specific constant for the ith SS because it includes spatial correction terms depending on the reciprocal position of the primary and each secondary source.
4.4. Feedback avoidance 4.3. Loudspeaker equalization The secondary loudspeakers used in this study have been designed to propagate towards the half space in front of the array. Nevertheless, a residual rear acoustic emission is present and a feedback path from SS to RM may create echoes that must be avoided. Feedback path modeling and neutralization are well-established issues in the literature [2] and represent crucial aspects in the practical realization of feedforward ANC systems. The feedback path induces a closed loop gain on the RM input and the overall feedforward ANC system must be modeled as an infinite impulse response (IIR) filter; solutions based on offline path estimation [2] and on-line adaptive filtering [24–26] have been proposed. However, severe issues concerning stability arise when adopting such anti-feedback techniques in our scenario and a simpler but effective solution has been devised. Since the noise we would like to cancel has a very short duration, the problem has been solved by silencing the RM after the acquisition of the PS. The algorithm is quite simple: when the system detects the beginning of a noise (PS) signal by an amplitude trigger, a counter is activated; when the counter reaches a first
Ideally, in the absence of WFS or any other type of filtering, the chain composed by RM, PU, and a SS (denoted by RM-PU-SS) should behave as a pure delay multiplied by a constant. Since this is not true in practice, this chain has been modeled as a distorting LTI system with a specific frequency response H (f ) . This fact has been experimentally verified by using the chain PU-SS-EM, that is by generating an impulse with the PU, reproducing it by a SS and acquiring it by the EM in condition of free-space propagation. Since RM and EM are identical microphones, the chains RM-PU-SS and PU-SS-EM exhibit the same H (f ) , apart from a delay and a constant gain. The obtained signal is shown in Fig. 4a. An equalization of each SS is therefore needed. Equalization is a typical problem in telecommunication systems design [20,21], where a communication channel degrades the transmitted signal and an equalizer is needed to invert the channel response before detection. Let h (t ) and heq (t ) be the impulse responses of the RM-PU-SS chain and of the equalizer filter we are looking for, respectively. A zero-forcing 224
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Amplitude (V)
5
0
-5
Fig. 5. Combination of main paths contributing to the acoustic field on a listener point P when considering hard ground terrain reflections (lateral view): primary source’s direct path (solid blue line), primary source’s reflected components (dashed and dotted blue lines), secondary sources’ direct paths (solid red line), secondary sources’ reflected components (dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0
10
20
30
40
50
40
50
Time (ms)
5
dRS > 2cτ
Amplitude (V)
threshold (in a time corresponding to about the duration of the noise), then the RM is silenced in order to avoid to record the feedback of the SS canceling field; when the counter reaches a second threshold, the RM is activated and the counter is reset. The procedure succeeds as long as the duration τ of the incoming noise signal is less than twice the wave propagation time between the RM and the SSs, so that, if we denote with dRS the distance between RM and SS, we have the following constraint:
0
(19)
For example, for τ = 5 ms and c = 343 m/s, we have dRS > 3.43 m. -5 0
4.5. Ground reflections compensation
10
20
30
Time (ms) One of the major difficulties encountered in the design of the AHNNC system is the presence of multiple acoustic paths from the PS (and the SSs) and a receiver in the cancellation area (see Section 5.1 for a simulation example). Therefore, given a point P in the middle range within the cancellation area, the acoustic channels should be modeled as frequency-selective functions, that is, with reference to Fig. 1, HPR, HSE , and HPE depend on the frequency f. Fig. 5 depicts a schematic view of this scenario. The theoretical framework considered so far contemplates only line-of-sight (LOS) components from both the PS and SSs towards the listener point P. When reflections are considered, three further multipath components influence the ANC system. The first one is the reflected PS field that superimposes on the RM, misleading the acquired signal: it depends on the Snell angle θ given by the mutual positions of PS and RM and affects HPR . The other ones are due to both PS and SSs reflections that superimpose on the listener: they depend on the position of the listener with respect to the PS and the SSs and affect HPE and HSE , respectively. Assuming fixed positions, the multipath effects could be theoretically compensated by an a-priori modeling of the air-terrain interface; nevertheless, simple conditions (e.g., hard bound, soft bound) are not encountered in real scenarios, nor a useful model of the actual air-terrain interface can be easily estimated due to high number of involved physical parameters [27]. Introduction of acoustical barriers has been also investigated [28], but this solution can be very expensive in practice. In this study, we propose to tackle the multipath impairments by introducing a new filter, namely a ground equalization filter hGE . Let the error microphone EM be located in the point P in the coverage area. Assume that it acquires the primary noise y1 (t ) and the canceling waveform y2 (t ) . The overall signal measured by the EM is given by
y (t ) = y1 (t ) + y2 (t ).
Fig. 6. Signals recorded by the EM for the estimation of the ground equalization filters: (a) superimposed primary and secondary source signals: this is the normal operating condition of the system, but cancellation may not occur due to the presence of multipath; (b) separate primary and (delayed) secondary source signals: the knowledge of these components allows the ground equalization filter to be designed.
According to the notation given in Fig. 1, we have
y1 (t ) = F−1 {S (f ) HPE (f )}
(21)
and
y2 (t ) = F−1 {S (f ) HPR (f ) HRS (f ) HSE (f )},
(22)
where S (f ) is the excitation signal and HRS (f ) includes the WFS and loudspeaker equalization filtering described in 4.2 and 4.3, respectively. The ground equalization filter hGE is inserted in the processing chain of the SSs and has to be designed so that the signal measured by the EM becomes approximately zero. Therefore, we would like to have
y1 (t ) + hGE (t ) ∗y2 (t ) = 0
(23)
If y1 (t ) and y2 (t ) were known, estimation of hGE could be accomplished by means of the equalization techniques discussed in Section 4.3. However, Eq. (20) cannot be used in practice because y1 (t ) and y2 (t ) are not known individually, but only its superimposition is observable; an illustration of the problem is depicted in Fig. 6a. In order to overcome this limitation, we devised a procedure to measure y1 (t ) and y2 (t ) separately and, consequently, to obtain an estimation of the ground equalization filter. Specifically, an extra delay te > 0 is inserted in the processing chain of y2 (t ) , yielding
(20) 225
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y2̂ (t ) = y2 (t −te ).
(24)
being te chosen such that
y1 (t ) y2̂ (t ) = 0
∀ t.
(25)
Hence, the measured signal becomes
y (t ) = y1 (t ) + y2̂ (t ),
(26)
where y1 (t ) and y2̂ (t ) can now be achieved since they are separated in time, as depicted in Fig. 6b. By substituting (24) into (23), the relation among observable signals and the ground equalization filter is given by
y1 (t ) + hGE (t ) ∗y2̂ (t + te ) = 0.
(27)
The estimation of hGE (t ) is obtained by minimizing the weighted and penalized MMSE function introduced in Section 4.3, that is ∼ 2 2 J (hGE ) = ∥Y1 (f ) + Y2 (f ) F {hGE } ∥W + ∥F {hGE } ∥W . (28) 1 2
Fig. 7. Scheme of the simulation setup of the proposed WFS based AHNNC.
∼ where Y1 (f ) = F {y1} and Y2 (f ) = F {y2̂ }exp(j2πfte ) . Accordingly, the solution, similar to that presented in (18), is given by ∼T ∼ ∼T h GE = (FT Y 2 W1Y2F + FT W2F)†·FT Y 2 W1y1,
related results are discussed. Then, the experimental testbed setup is presented and, eventually, the related results are shown and commented.
(29)
being h GE and y1 the vectors containing discretized hGE (t ) and y1 (t ), ∼ respectively, whereas Y2 is the diagonal matrix containing the discrete ∼ version of Y2 (f ) . The hGE filter has been designed as a 400 order FIR filter and introduces 100 samples of delay in the processing chain. In the design, we empirically set.
5.1. Simulations A set of numerical simulations, obtained by using the k-Wave [29] software tool, has been preliminarily carried out in order to assess the feasibility of the proposed approach. Fig. 7 depicts the simulation setup for the proposed ANC system using a linear array in a 3-D space. The array of SSs is placed dsa = 4 m far from the PS, simulated as an emitting monopole. The primary noise feeding the PS is the shotgun signal shown in Fig. 2a, subsampled by a factor ten, that is at a sampling frequency of f s′ = 5120 Hz. Both primary and secondary sources are placed at h = 1.5 m from the ground. The array aperture is set to a = 10 m and it is composed by discrete elementary monopoles excited according to Eq. (9) and spaced by da = λ/2 ≈ 67 mm, being λ = 2c / f s′ the shorter wavelength associated to the primary acoustic field. Such a value of da is the upper bound on spatial spacing and prevents from aliasing effects in the synthesized field. Virtual reference microphones are co-located with each SS; for the sake of simplicity, perfect synchronization, no ground equalization filter and absence of environmental disturbances were assumed. The sound energy flowing in the half space in front of the SS array is measured by a planar microphone array located at dsm = 12 m from the primary source. As to the performance index, the attenuation obtained when the ANC system is switched on with respect to when it is switched off – i.e., the ratio between the sound intensities measured on the microphone array – is considered. The proposed ANC system has been initially simulated by considering sound waves propagating in a homogeneous medium, that is, the terrain has been assumed constituted by air too. The sound intensity attenuations are reported in Fig. 8a. Attenuation levels greater than 10 dB in the middle region can be observed. A performance degradation emerges in the lateral regions due to the finite aperture of the array. As to the performance along the vertical direction, it is mainly limited by the assumptions of the 2D1/2 Rayleigh I integral and a sharp loss is clearly visible as the elevation increases (the contribution of the ANC system causes up to a 6 dB increment of the PS noise in the uppermost areas). In order to evaluate the impairments due to the presence of terrain reflections, the simulation has been repeated by considering a perfectly hard ground. The measured sound intensity attenuations are depicted in Fig. 8b; in accordance to theoretical considerations, a remarkable performance loss is observed, which justifies the need for solutions to mitigate the terrain reflections effect.
0, 200 Hz < f < 3200 Hz W1 (f ) = 1 W2 (f ) = ⎧ 2 ⎨ 10 , otherwise. ⎩ An advantage of the proposed estimation procedure is that it automatically adjusts the synchronization and the gain of the canceling system, compensating for possible discrepancies between actual and nominal parameters (e.g., RM-SSs distance, loudspeakers and microphones gains). As a limitation, the procedure is dependent on the choice of the point P in the coverage area. Therefore, different equalization filters should be designed and used according to where we would like to locate the silence area. We have, however, experimentally verified that, for the considered frequency band, selecting P in the middle range continues to produce benefits in a large area, from the middle to the long range. 4.6. Processing chain delay The proposed system works in real-time and, therefore, causality is required in the processing chain. Let nWFS , nLE and nGE be the delays introduced by the WFS, the loudspeaker equalization and the ground equalization filters, in that order, expressed in number of samples at the sampling frequency fs . Moreover, let nC be the processing delay due to computing operations. The overall system delay must be lower than the traveling time of the wave from RM to SSs, that is we must have
dRS ⩾
c (nWFS + nLE + nGE + nC ), fs
(30)
where dRS is the distance between RM and SS. Eqs. (19) and (30) represent two constraints on the geometry of the system. Even though no maximum bound for dRS is derived from theoretical analysis, in practice a performance loss occurs if an excessive value is chosen. Indeed, the farther the RM, the poorer the estimation of the PS field on each SS location. 5. Results In this section, simulations and experiments aiming at validating the proposed method are presented. Firstly, the simulation scenario and the 226
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Fig. 8. Sound intensity attenuations (dB) measured on the microphone array in the simulations of different scenarios: (a) homogeneous medium; (b) in the presence of hard ground reflections.
5.2. Testbed setup The experimental testbed that was setup to demonstrate the effectiveness of the proposed method is composed by the following components: 1. National Instruments CompactRIO-9039 controller (1.91 GHz, Quad-Core CPU), equipped with 325T FPGA. The controller is the core of the processing unit, where the previously described algorithms have been implemented by using Labview FPGA. The controller mounts a 4-channel ADC NI9215 module (simultaneous sampling, 100 Ksample/s, 16 bit/sample) and a 16-channel DAC NI9264 module (16 bit, 25 Ksample/s); 2. Behringer ECM8000 microphones, used as RM and EM; the microphones signals are supplied and pre-amplified by means of a PreSonus DigiMax FS (8-channels) system; 3. prototype horn loudspeakers, specifically designed for the testbed1, as shown in Fig. 9, and capable to reproduce a shotgun signal with peak SPL@1 m ≈132 dB. Such an elevated acoustic pressure was necessary to obtain reliable measurements in the middle/long range. The diameters of the magnet, coil and throat aperture in the compression driver are 180, 90 and 50 mm, in that order. The nominal power of the amplifier (one for each loudspeaker) is 300 W. The horn has a shape of truncated pyramid, is made of wood and is long 1050 mm. The (internal) section of the horn is square and its throat and final apertures are 50 and 270 mm, respectively. In order to avoid coil overrun due to uncontrolled low frequency noise entering the ANC system – and, consequently, possible damages to the driver – a highpass filter having cutoff frequency at 300 Hz has been inserted in the digital processing chain.
Fig. 9. Horn loudspeakers used in the tests.
Fig. 10. Schematic view from above of the considered scenario for the testbed measurements (distances are not in scale for the sake of clarity).
0.9 m from the ground; in a second set of experiments, denoted as B, the PS and SSs were at 1.6 and 0.9 m from the ground, respectively. The ground reflection compensation was preliminarily performed placing the EM at 16 m from the SSs array center at the height of the PS in both scenarios. Subsequently, the EM was used to evaluate cancellation performance in the same location (dam = 16 m), as well as at 60 m (dam = 60 m) far from the SSs. In the latter case, different lateral offsets were also considered. A view from above of the geometry of the measurements scenario is depicted in Fig. 10. The presented results are related to data acquired in mild weather conditions, that is, sunny days, air temperature between 18 and 22 degrees, in the presence of a variable breeze (rarely gusts).
The experimental tests were carried out in open field, in the absence of natural and artificial obstacles; the terrain was covered by grass. We used one loudspeaker as PS and five loudspeakers as SSs, at a distance from the PS dsa = 15 m. The loudspeakers are spaced by 0.3 m. The RM was placed at a distance drs = 5.5 m from the SSs. In a first set of experiments, namely scenario A, the PS and the SSs were coplanar at
1
The loudspeakers were produced by Look Line s.r.l., Massa Finalese, Modena, Italy.
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Table 1 Attenuation results (dB) in the frequency band 400–1500 Hz, measured in the two considered experimental scenarios: PS and SSs at 0.9 m from the ground (A), PS and SSs at 1.6 and 0.9 m from the ground, respectively (B). Measurement positions are referred to Fig. 10.
A (best) B (best) A (median) B (median)
E0
E1
E2
E3
10.8 14.3 8.6 8.5
8.9 7.6 5.8 5.5
8.1 7.3 5.7 5.2
1.4 1.3 0.2 −0.3
30
dB
Scenario
40
20
10
5.3. Attenuation results
Average (ANC off) Average (ANC on) Median (ANC off) Median (ANC on) Background noise
0
In a first set of experiments, the PS was fed with the shotgun signal shown in 2a repeated 20 times with a period of 2 s. Attenuation results in the frequency band 400–1500 Hz are reported in Table 1, in terms of best and median values achieved in each recording session. As to the “best” results for scenario A, the performance loss between the attenuations at 60 m (central position) and 16 m is about 2 dB. A contained decrement of 0.8 dB is observed when moving from the central position to an offset of 4 m; the attenuation sharply decays when the offset becomes 8 m. Also “median” results demonstrate a good performance of the attenuation system, 2–3 dB lower than the best ones (a sharp loss is still present for the largest lateral offset). The results related to the measurements with offset induce some considerations about the coverage area of the proposed system. Consider the footprint of the SS array when illuminated by the PS and let α be the angle in the vertex where PS is located, as show in Fig. 10, such that tanα /2 = a/2dsa . From the geometry of the experimental scenario, we can see that the measurement point E2 is about on the edge of the footprint, whereas E3 is outside. This fact, along with the observation of the attenuation results in Table 1, suggests that the coverage area of the system and the footprint approximately coincide. As to scenario B, the “best” results are worse than those relative to scenario A up to 1.3 dB (the attenuation at 16 m, where an improvement is experienced, is considered as a positive outlier), whereas “median” results are only at most 0.5 dB worse than those of scenario A. A performance loss is expected for scenario B, since the EM is not coplanar with the PS-SS plane; however, the measured loss is quite small at these elevation angles from the plane. Analogous considerations regarding the footprint of the system stated for scenario A hold also for scenario B. In a second set of experiments, we analyzed the performance by studying the frequency representations statistics of the waveforms recorded when the ANC system is active with respect to when it is switched off. The setup of the scenario B is referred to (similar conclusions are valid for the scenario A). Fig. 11a demonstrates the squared amplitude spectra,2 averaged, for each frequency, over 60 recorded realizations of the shot, related to the distance of 16 m (in solid blue3 and red lines when the system is switched off and on, respectively). In the lower part of the spectra, i.e., 300 < f < 450 Hz, the curves are superimposed due to the transition band of the loudspeakers’ protection filter; in the middle range frequency, 500 < f < 1500 Hz, the attenuation level varies from 4 to 12 dB; from 1500 to 1900 Hz a lower performance is observed (from 4 to 7 dB); eventually, the transition band of the WFS mainly characterizes the ANC system’s response starting from 2000 Hz. The previous results can be compared to the average spectra acquired at 60 m, as reported in Fig. 11b. Considering the case when the system is switched off and comparing the blue solid line with the corresponding one in Fig. 11a, we observe similar spectra of the PS signals,
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even though some changes appear confirming that the acoustic channel has a different frequency selective transfer function. The attenuation achieved by switching on the ANC system at 60 m is lower with respect to the previous case (about 3–4 dB gap), but follows a similar pattern across frequencies. Two shot signals registered on the EM at 60 m when the proposed ANC system is off and is on, respectively, are depicted in Fig. 12. As can be seen, in the latter case a remarkable amplitude reduction is experienced. From the results shown above, we observe that the proposed system is capable to achieve a noticeable attenuation of the impulsive noise in a wide area in front of the secondary sources array. Incidentally, even though the compensation of ground reflections is calibrated to work in the middle range, the benefits can be appreciated also in the long range.
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6. Conclusions In this paper, the problem of canceling highly non-stationary (or pulse-like) noise by means of active systems has been addressed. The noise considered in this study is characterized by a very short duration, so that classical ANC techniques, based on adaptive filtering, are ineffective since convergence cannot be reached. Therefore, a system based on the wave field synthesis (WFS) theory to construct the canceling acoustic field has been presented. The key elements of the proposed system are an array of secondary loudspeakers that generate the canceling waveform, a processing unit that generates the driving signals for the array and a reference microphone to acquire the signal to be canceled. The system has been thought to work in the outdoor. The major issues related to the actual implementation of the system – such as WFS filters design, loudspeakers equalization, finite dimension of the array, feedback avoidance – have been analyzed. The presence of ground reflections induces a mismatch between the theoretical model at the basis of the system and the actual working environment. In order to reduce the consequent performance loss, a ground reflections compensation method has been devised: even though such a procedure solves exactly the problem only in one measurement point, its beneficial effects can be experienced also in the middle/long range. Experimental results were presented to validate the effectiveness of the proposed system. First, the correctness of the basic idea of using WFS in active noise control systems was established by means of computer simulations. An experimental testbed was also setup and the results of onthe-field measurements were presented. The experiments demonstrate that the proposed system is actually able to diminish the overall noise perceived in the area in front of the secondary loudspeakers array till a distance of tens of meters. Acknowledgements The authors would like to thank Fabbrica d’Armi Pietro Beretta S.p.A. for supporting the research undertaken in this study. The authors would like also to thank Mr. Claudio Bergamini and Look Line s.r.l. for the realization of the horn loudspeakers and the related amplifiers as well as for many discussions and support during the development of the present study. References [1] Elliott SJ, Nelson PA. Active noise control. IEEE Sign Process Mag 1993;10(4):12–35. http://dx.doi.org/10.1109/79.248551. [2] Kuo S, Morgan D. Active noise control: a tutorial review. Proc IEEE
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