Remote sensing for wave-based nonreciprocal active control

Joe Tan 1

University of Southampton University Road, Southampton, SO17 1BJ

Jordan Cheer 2

University of Southampton University Road, Southampton, SO17 1BJ

ABSTRACT Reciprocity is an acoustic property that describes the symmetry of sound transmission between two points. However, this property is undesirable in certain applications, and this has led to significant interest in the development of nonreciprocal acoustic devices that achieve one-way sound transmission. These devices typically achieve nonreciprocal sound transmission by introducing nonlinearities or directional biasing. Previously proposed nonreciprocal acoustic devices generally have limitations; for example, they may not be fully tuneable, they can introduce signal distortions such as additional harmonics, or they can only exhibit nonreciprocal behaviour over a narrow bandwidth. To overcome these challenges, previous work has demonstrated how a wave-based active control system can be used to drive an array of acoustic sources to achieve reversible and broadband non-reciprocal behaviour. However these wave-based active control systems use external far-field pressure sensors to achieve broadband nonreciprocal behaviour and, thus, these active control systems are not self-su ffi cient. This paper therefore presents an experimental investigation into how remote sensing techniques can be incorporated into the previously proposed wave-based active control systems to create more self-contained nonreciprocal acoustic devices that still achieve broadband nonreciprocal behaviour in a one-dimensional acoustic system.

1. INTRODUCTION

Reciprocity is an acoustic property, which is considered to be inherent in linear acoustics, that describes the symmetry in sound transmission between two points. For example the sound transmission between an acoustic source and observer is equal to the sound transmission when the two swap positions. In acoustics, reciprocity has been used to simplify measurement processes in a variety of di ff erent applications [1, 2]. However, this property is undesirable in certain acoustic applications, which has led to the development of nonreciprocal acoustic devices. These devices have the ability to block or reflect acoustic waves that are propagating in one direction, whilst

1 j.tan@soton.ac.uk

2 j.cheer@soton.ac.uk

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

acoustic waves propagating in the opposite direction can propagate freely. Previously proposed nonreciprocal acoustic devices generally break the symmetry of sound transmission by introducing nonlinearities [3–7], fluid motion [8] or space-time modulated resonant cavities [9–11]. Although these various approaches do exhibit nonreciprocal sound transmission, they typically have limitations: nonlinear nonreciprocal acoustic device generally require large input power and can introduce signal distortions; fluid motion is practically realised using fans, which introduce additional harmonics; the bandwidth of the nonreciprocal behaviour is often narrow when using resonant cavities; and finally, these systems generally create a preferential direction for nonreciprocal behaviour and, thus, they are not fully tuneable. Some of these limitations have been addressed through the realisation of a non-local active metamaterial consisting of an array of sensor-actuator pairs, which can create the directional biasing required to achieve broadband nonreciprocal sound transmission [12, 13]. However, this nonreciprocal device is still not fully tuneable because the sensor-actuator arrangement still creates a preferential direction for the nonreciprocal behaviour. More recently, it has been shown that adaptive feedforward wave-based active control systems can exhibit nonreciprocal behaviour in one and two dimensional spaces [14–16]. The advantage of these systems is that they are fully tuneable, such that the nonreciprocal behaviour can be reversed by changing the reference and error signals for these controllers. However, these nonreciprocal acoustic devices rely on a wave separation method, which requires external far-field pressure sensors, to calculate the propagating wave components needed for nonreciprocal control. Although these external pressure sensors could be moved closer to the control sources, this would require additional pressure sensors to separate the positive and negative propagating and evanescent wave components. Alternatively, this paper investigates how a remote sensing method can be combined with the previously proposed wave-based active control systems to create a more self-contained nonreciprocal acoustic device that still achieves broadband nonreciprocal sound transmission and absorption. This paper is structured as the follows: Sections 2 and 3 describes the experimental acoustic system and remote sensing method used in this paper. Section 4 describes the active control formulations for the wave-based active controllers that are used to control the wave components when remote sensing is integrated. Section 5 presents the results of the o ffl ine experimental investigation into the performance of the proposed wave-based active control systems and finally, Section 6 presents the conclusions.

2. SYSTEM DESCRIPTION

Figure 1 shows the experimental arrangement of the proposed self-contained wave-based nonreciprocal active system, which combines the previously proposed wave-based active control system with remote sensing techniques, to realise broadband nonreciprocal sound transmission and absorption in a one-dimensional environment. The system shown in Figure 1 contains an impedance tube with a passive termination and primary disturbance source at each end and active control unit located at the centre of the duct. From Figure 1, it can be seen that there are two monitoring pressure sensors, P m 1 and P m 2 , which are in close proximity to the control sources, and four virtual far-field pressure sensors, which are labelled by the coe ffi cients P v 1 to P v 4 . The active control unit consists of the control sources and wave-based active controller that filters the signals measured at each monitoring sensor to drive the control source to achieve the desired nonreciprocal behaviour. The positive and negative propagating wave components are also presented in Figure 1, which are indicated by the coe ffi cients A to D . The coe ffi cients, A to D , represent di ff erent wave components depending on the incident wave direction. When the positive primary source generates the incident sound field, A is the positive propagating incident wave, C is the positive propagating transmitted wave component, whilst B and D are the negative propagating upstream and downstream reflected wave components. However when the negative primary source generates the incident sound field, D is the negative propagating incident wave, B is the negative propagating transmitted wave, whilst C and A are now the positive propagating upstream and downstream reflected wave components

respectively.

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Figure 1: The experimental configuration of the nonreciprocal sound transmission and absorption control systems with remote sensing for a one-dimensional environment.

The geometric parameters that define the experimental duct system shown in Figure 1 are presented in Table 1. The frequency range of interest for the study presented in this paper is between 600Hz and 1500Hz. To ensure higher order duct modes are not present in the duct, the upper frequency limit has been chosen to make sure that the shortest considered acoustic wavelength is larger than the duct diameter presented in Table 1. The low frequency limit has been selected based on the lower frequency performance of the Kingstate KDMG200008 [18] loudspeakers used to create the control source shown in Figure 1.

x v 1 x v 2 x v 3 x v 4 x m 1 x m 2

Control source thickness

Duct diameter

0.1 m 0.02 m -0.286 m -0.243 m 0.243 m 0.286 m 0.05m 0.07m

Table 1: The system parameters used to define the system setup shown in Figure 1.

By combining the wave-based active controllers and remote sensing method, the two monitoring pressure sensors, P m 1 and P m 2 , shown in Figure 1 are used to estimate the disturbance signals at each virtual far-field pressure sensor via an observation filter. Since the incident wave is detected at the virtual far-field sensors before the monitoring pressure sensors in the system presented in Figure 1, the ideal observation filters for the standard remote sensing method will be non-causal. To improve the causality between the monitoring and virtual pressure sensors in the system presented in Figure 1, modelling delays can be introduced to both the pressure measured at the virtual far-field pressure sensors and the internal model used within the wave-based active controllers to generate the filtered reference signals. This improved remote sensing method is also known as the delayed remote sensing method, and this will be described further in Section 3. As in [14, 16], to control the transmitted and reflected wave components independently, a wave separation method has also been used within the wave-based nonreciprocal active systems to separate the positive and negative propagating wave

components from the delayed pressure estimated at the virtual far-field pressure sensors, ˆ e e i ( n − δ ), by processing the vector of monitoring pressure sensor signals, e m , using the delayed optimal observation filters. This paper uses the integration wave separation method, which is summarised in [17], to calculate the propagating wave components from ˆ d e i ( n − δ ). This wave separation method has been selected because it completely separates the positive and negative propagating wave components and, therefore, the incident wave component can be used as the reference signal for broadband feedforward control. As in [14, 16], there are two di ff erent wave-based active controllers will be investigated in this paper: a single monopole control source that is driven to minimise the positive propagating transmitted wave, C + ( n ), to achieve nonreciprocal sound transmission and a pair of monopole control sources that are driven to minimise the positive propagating transmitted, C + ( n ), and reflected, B + ( n ), wave components to achieve nonreciprocal sound absorption.

3. DELAYED-REMOTE SENSING METHOD

This section describes the remote sensing method used in this paper to calculate the optimal observation filters required to estimate the delayed pressure measured at each virtual far-field pressure sensor from the monitoring pressure sensor signals. As mentioned in Section 2, the delayed remote sensing method has been used in this paper to ensure that causal delayed optimal observation filters are calculated for the system shown in Figure 1. There are two main stages in the delayed remote sensing method: these are the identification and control stages. This section will be describe the identification stage of the delayed remote sensing method and the control stage will be described in Section 4. In the identification stage, the delayed optimal observation filters are designed prior to real-time implementation, since this process requires the pressure measured at the virtual pressure sensors, which are not used in the control stage. The least square estimate of the i -th delayed optimal observation filter can be obtained by minimising the mean squared di ff erence between the estimated disturbance signal, o opt i d m , and the delayed disturbance signal, d e i ( n − δ ), measured at the i -th virtual far-field pressure sensor, which can be given as

J o i = E  ( d e i ( n − δ ) − o opt i d m ( n )) T ( d e i ( n − δ ) − o opt i d m ( n ))  . (1)

Expanding Eq. 1, the cost function can also be expressed as

J o i = o T opt i E  d T m ( n ) d m ( n )  o opt i − o T opt i d T m ( n ) d e i ( n − δ ) − d T e i ( n − δ ) o opt i d m ( n ) + d T e i ( n − δ ) d e i ( n − δ ) . (2)

Taking the derivative of the cost function given by Eq. 2 with respect to the corresponding optimal observation filter, the derivative in this case can be calculated as

∂ J o i ∂ o opt i = 2 E  d m ( n ) T d m ( n )  − 2 E  d m ( n ) T d e i ( n − δ )  . (3)

Since the two monitoring pressure sensors are used to estimate signal measured at each virtual far- field pressure sensor individually, the derivative can be equated to zero and the i -th delayed optimal observation filter can be calculated as

o opt i =  E  d m ( n ) T d m ( n )  − 1 E  d m ( n ) T d e i ( n − δ )  . (4)

The same procedure detailed in this section has been repeated to obtain an observation filter for each virtual far-field pressure sensor and each primary source shown in Figure 1.

4. WAVE-BASED ACTIVE CONTROL FORMULATIONS WITH REMOTE SENSING

Using the delayed optimal observation filters calculated using the procedure described in Section 3, the delayed pressure can be estimated at the virtual pressure sensor locations using the signals detected at both monitoring pressure sensors. This section will describe the wave-based active control formulations when remote sensing is also integrated. As mentioned in Section 2, there two wave- based active control systems that have been investigated: a single monopole control source that is driven to minimise the estimated positive propagating transmitted wave, C + ( n − δ ), and a pair of monopole control sources that are driven to minimise the estimated positive propagating transmitted, C + ( n − δ ), and upstream reflected, B + ( n − δ ), wave components. The aim of these wave-based active control systems is to use remote sensors to control the respective wave components when subject to a positive propagating incident wave and allow a negative propagating incident wave to propagate unimpeded and, thus, achieve broadband nonreciprocal behaviour. Both of these wave-based active controllers use a feedforward filtered-reference least mean squares (FxLMS) algorithm to adaptively calculate the optimal control signals that drive the control sources to minimise the corresponding wave components at the virtual sensor locations, where the reference signal for both controllers is the estimated positive propagating incident wave, A + ( n − δ ). Figures 2(a) and 2(b) show the block diagrams of the proposed transmitted wave and absorption controllers with remote sensors to achieve nonreciprocal sound transmission and absorption respectively. It is worth noting that the wave-based active control systems shown in Figure 2 are completely tuneable, such that the the direction of the nonreciprocal behaviour can be reversed by changing the reference and error signals, however, this is not demonstrated in this paper for conciseness. In a real-time implementation, the disturbance signals, ˆ d m , that are measured at the monitoring pressure sensors can be calculated by subtracting the estimated control source contribution from the total pressure measured at the monitoring pressure sensors, which can be calculated as

ˆ d m ( n ) = P m − ˆ G m u ( n ) (5)

where ˆ G m is the matrix of estimated plant responses, which are modelled by finite impulse response (FIR) filters, and u ( n ) is the control source voltage. Using the remote sensor method, the delayed estimated disturbance signal at the i -th virtual far-field sensor location is obtained by filtering ˆ d m ( n ) by the i -th delayed optimal observation filter given by Eq. 4, which can be expressed as

ˆ d e i ( n − δ ) = o opt i ˆ d m ( n ) . (6)

By adding the delayed estimated contribution from the control sources, ˆ G e , to the delayed estimated disturbance signals given by Eq. 6, the delayed total pressure can be estimated at the i -th virtual far-field sensor location, which can be written as

ˆ P e i ( n − δ ) = ˆ d e ( n − δ ) + ˆ G e u ( n − δ ) . (7)

Using the estimated total pressure given by Eq. 7, the positive and negative wave components in the upstream and downstream sections of the duct can be estimated using the wave separation method summarised in [17]. These estimated wave components can then be individually controlled by the feedforward wave-based active controllers to achieve the desired nonreciprocal behaviour and the following sections will describe the active control formulations for the transmitted wave and absorption controllers.

4.1. Nonreciprocal transmission active controller In the first instance, a transmitted wave controller can be used to achieve nonreciprocal sound transmission by driving a single monopole control source to minimise the estimated positive propagating transmitted wave, C + ( n − δ ), whilst allowing the negative propagating incident wave,

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Figure 2: The block diagram of the transmission (a) and absorption (b) controllers with remote sensing used to achieve broadband nonreciprocal sound transmission and absorption. The thin and thick lines relate to single channel and multichannel signals respectively and the dashed red lines show the remote sensing component.

D − ( n ), to be perfectly transmitted. The error signal in this case is the estimated positive propagating transmitted wave, C + ( n − δ ) calculated by the wave separation method and remote sensing method described in Section 4, which can be written in terms of the control source responses as

ˆ e T ( n − δ ) = ˆ d T ( n − δ ) + ˆ R T ( n ) w T ( n ) (8)

where ˆ d T ( n − δ ) = ˆ d e 3 ( n − δ ) + ˆ d e 4 ( n − δ )

Z T s

4 + c 0

0 ˆ d e 3 ( n − δ ) − ˆ d e 4 ( n − δ ) dt , (9)

2 ∆ x

is the transmitted wave due to the positive primary source, calculated using the wave separation method [17], ∆ x is the microphone spacing and ˆ d e 3 ( n − δ ) and ˆ d e 4 ( n − δ ) are the delayed estimated disturbance signals at the third and fourth virtual far-field pressure sensor locations;

w T ( n ) = [ w T 0 , ..., w T n − I − 1 ] T , (10)

is the vector of I FIR control filter coe ffi cients;

R T ( n ) = [ r T ( n ) , r T ( n − 1) , ..., r T ( n − I + 1)] T (11)

is the vector of transmitted wave components calculated using the filtered reference signals and

Z T s

r T ( n − δ ) = r 3 2 ( n ) + r 4 2 ( n )

4 + c 0

0 r 3 2 ( n ) − r 4 2 ( n ) dt , (12)

2 ∆ x

is the transmitted wave component calculated from the filtered reference signals, which are given by

J − 1 X

r l 2 =

j = 0 g l j A + ( n − δ − j ) , (13)

where A + is the reference signal and g l j is the j -th FIR filter coe ffi cient of the J coe ffi cient filter representing the plant response between the control source and the l -th pressure sensor. The cost function in this case is the mean squared value of the error signal corresponding to the estimated transmitted wave, which can be defined as

J T ( n ) = | e T ( n − δ ) | 2 . (14)

Substituting Eq. (8) into (14), the cost function can also expressed in Hermitian quadratic form as

J T ( n ) = w T T ( n ) R T T ( n ) R T ( n ) w T ( n ) + 2 w T T ( n ) R T T ( n ) d T ( n ) + | d T ( n ) | 2 . (15)

Taking the derivative of the cost function with respect to the vector of FIR control filter coe ffi cients, w T ( n ), the gradient of the cost function can be calculated as

∂ J T ( n )

∂ w T = 2 R T T ( n ) e T ( n − δ ) . (16)

Using the negative gradient given by Eq. 16, the vector of FIR control filter coe ffi cients can then be adapted to control the transmitted wave using an FxLMS algorithm, which can be expressed as

w T ( n + 1) = w T ( n ) − µ R T T ( n ) e T ( n − δ ) (17)

where µ is a convergence gain, which controls the stability and speed of convergence.

4.2. Nonreciprocal absorption active controller Building upon the transmitted wave controller described in Section 4.1, the absorption active controller and remote sensing method shown in Figure 2(b) can be used to drive a pair of monopole control sources to minimise the estimated transmitted and reflected waves when the incident wave is generated by the positive primary source, whilst allowing the incident wave propagating in the opposite direction to be perfectly transmitted and, thus, creating nonreciprocal sound absorption. The vector of error signals in this case is the transmitted, C + ( n − δ ), and reflected, B + ( n − δ ), wave components at the virtual far-field sensor locations, which have been calculated using the wave separation method proposed in [17] and the remote sensing method described in Section 4. Similarly to the transmitted wave controller, the vector of error signals for the absorption active controller can be written in terms of the disturbance and control responses as

ˆ e ( n − δ ) = ˆ d ( n − δ ) + ˆ R ( n ) w ( n ) (18)

where

d ( n − δ ) =  d T ( n − δ ) , d R ( n − δ )  T , (19)

is the vector of disturbance signals consisting of the transmitted and reflected wave components due to the positive primary source, where d T ( n ) is given by Eq. Equation 9 and

Z T s

d R ( n − δ ) = ˆ d e 1 ( n − δ ) + ˆ d e 2 ( n − δ )

4 + c 0

0 ˆ d e 1 ( n − δ ) − ˆ d e 2 ( n − δ ) dt , (20)

2 ∆ x

is the reflected wave component due to the primary source and ˆ d e 1 ( n − δ ) and ˆ d e 2 ( n − δ ) are the delayed estimated disturbance signals at the first and second virtual far-field sensor locations respectively;

w ( n ) =  w 2 0 , w 1 0 , ..., w 2 n − I − 1 , w 1 n − I − 1  T , (21)

is the 2 I × 1 vector of FIR control filter coe ffi cients;

T

 r T 2 ( n ) , r T 1 ( n ) , r T 2 ( n − 1) , r T 1 ( n − 1) , ..., r T 2 ( n − I + 1) , r T 1 ( n − I − 1)



R ( n ) =

, (22)

r R 2 ( n ) , r R 1 ( n ) , r R 2 ( n − 1) , r R 1 ( n − 1) , ..., r R 2 ( n − I − 1) , r R 1 ( n − I − 1)

is the matrix of filtered reference signals, where

Z t

r T m = r 3 m ( n ) + r 4 m ( n )

4 + c 0

0 r 3 m ( n ) − r 4 m ( n ) dt , (23)

2 ∆ x

and

Z t

r R m = r 1 m ( n ) + r 2 m ( n )

4 + c 0

0 r 1 m ( n ) − r 2 m ( n ) dt , (24)

2 ∆ x

are the transmitted and reflected wave components due to the filtered reference signals, which are given by

J − 1 X

r l m =

j = 0 g lm j A + ( n − δ − j ) , (25)

where g lm j is the j -th FIR filter coe ffi cient of the J coe ffi cient filter representing the plant responses from the m -th control source to the l -th pressure sensor. The cost function in this case is defined as the mean squared value of the sum of the transmitted and reflected wave components, which can be expressed as J ( n ) = e T ( n − δ ) e ( n − δ ) . (26)

Following the same procedure described in Section 4.1, the multichannel FxLMS algorithm in this case can be expressed as w ( n + 1) = w ( n ) − µ R T ( n ) e ( n − δ ) . (27)

5. RESULTS

This section presents the results of an o ffl ine experimental investigation into the performance of the two proposed wave-based active control systems described in Section 4 when incident sound field is generated by either the positive or negative primary sources. The pressure measured at the far-field pressure sensors shown in Figure 1 are used to calculate the magnitude of the resulting transmission (blue lines), reflection (red lines) and absorption (black lines) coe ffi cients, which are used to assess the performance of the proposed wave-based active control systems, with respect to the corresponding incident wave direction and these results are presented in Figure 3. Figure 3(a) shows that the transmitted wave controller achieves near-zero transmission with respect to a positive propagating incident wave, whilst the negative propagating incident wave is unaltered by the controller with the magnitude of the normalised transmission coe ffi cient being close to unity as shown in Figure 3(b). The results presented in Figures 3(a) and 3(b) show that the transmitted wave controller with remote sensors can still achieve broadband nonreciprocal sound transmission. It is also worth investigating how this active controller a ff ects the reflection and absorption coe ffi cients. In the positive incident wave case, a proportion of the incident wave is reflected or absorbed by this active controller depending on frequency because the upstream sound radiation generated by the control sources is not constrained to zero. Figures 3(c) and 3(d) show the performance of the proposed absorption controller with remote sensing when the incident sound field is propagating in the positive and negative directions respectively. From Figure 3(c), it can be seen that the absorption controller achieves near-zero transmission and reflection coe ffi cients, which leads to near-perfect absorption of the positive propagating incident wave. However when the absorption controller is subject to a negative propagating incident wave, this incident wave is near-perfectly transmitted with near-zero reflection and absorption as shown in Figure 3(d). These results show that broadband nonreciprocal sound absorption is still achieved by the proposed absorption controller when using remote sensors.

6. CONCLUSIONS

Previously proposed wave-based active control systems have shown the ability to achieve broadband nonreciprocal sound transmission or absorption by independently controlling the transmitted and reflected wave components in a one-dimensional environment. The wave separation method used within these wave-based active control systems require external pressure sensors that are a su ffi cient distance away from the control sources, such that the evanescent waves are negligible, since only

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Figure 3: The o ffl ine experimental performance of the transmitted wave controller (a-b) and absorption controller (c-d) in terms of the magnitude of the normalised transmitted (blue lines), reflected (red lines) and absorbed (black lines) energy for a positive propagating incident wave (a,c) and for a negative propagating incident wave (b,d).

two pressure sensors are used to separate the positive and negative propagating wave components. To create a more self-contained nonreciprocal acoustic device, this paper has investigated how these wave-based active controllers can be combined with remote sensing methods to achieve broadband nonreciprocal behaviour. Through an o ffl ine experimental implementation using measured responses obtain from a practical duct system, this paper has demonstrated that the proposed transmitted wave and absorption controllers can use remote sensors, which are in close proximity to the control sources, to control the respective wave components at the virtual sensor locations to achieve broadband nonreciprocal sound transmission and absorption respectively. It has shown that the proposed wave-based active controllers have the ability to drive a single active subwavelength unit cell to achieve broadband nonreciprocal behaviour in a one-dimensional acoustic system. For future work, this concept could be extended to create an active acoustic metasurface consisting of an array of these active unit cells in order to achieve broadband nonreciprocal sound transmission or absorption in a two or three dimensional spaces. The proposed wave-based active control systems can be used to create future self-contained nonreciprocal acoustic devices that achieve broadband nonreciprocal behaviour.

ACKNOWLEDGEMENTS

This research was supported by the Intelligent Structures for Low Noise Environments EPSRC Prosperity Partnership (EP / S03661X / 1).

REFERENCES

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