Abstract
One of the problems concerned with active noise control is the existence of acoustical feedback between the control value (“active” loudspeaker output) and the reference signal. Various experiments show that such feedback can seriously decrease effects of attenuation or even make the whole ANC system unstable. This paper presents a detailed analysis of one of possible approaches allowing to deal with acoustical feedback, namely, virtual unidirectional sound source. With this method, two loudspeakers are used together with control algorithm assuring that the combined behaviour of the pair makes virtual propagation of sound only in one direction. Two different designs are presented for the application of active noise control in an acoustic duct: analytical (leading to fixed controller) and adaptive. The algorithm effectiveness in simulations and real experiments for both solutions is showed, discussed, and compared.
1. Introduction
Active noise control is mainly concerned with low-frequency sounds that cannot be suppressed in passive manner at reasonable cost. Other prerequisite for using ANC is that it does not obstruct media flow in ducts, for example, air conditioning ducts. Unfortunately, almost all duct applications have acoustic feedback; a phenomenon that can heavily decrease or even destroy the results of otherwise properly set-up control system [1, 2].
There are several methods for avoiding acoustic feedback in duct applications [2–6]. The best one that avoids acoustic feedback completely is to use nonacoustic reference sensor. Unfortunately, this usually limits the application to cancel the tonal sounds and does not allow for the reduction of broadband noise. Another approach to acoustic feedback is to internally process signal from the reference microphone in a way allowing to compensate for this effect [2, 4–6]. This technique, usually called “feedback neutralisation,” uses a model of acoustic feedback path to filter the controller output and then subtract it from the reference signal. In the simplest case, the model is obtained offline and does not account for time variation of the feedback path. Moreover, the method requires a model of the acoustic feedback path and therefore is computationally more expensive.
The approach to acoustic feedback cancellation presented in this paper is called virtual unidirectional sound source (VUSS). The method, in authors opinion, has some advantages over well-known two loudspeaker active noise control system in duct applications. First, it clearly separates the subsystem responsible for neutralisation of undesired feedback from active noise control part. Second, its adaptive version can also be applied in case of varying acoustic path responses. Last, virtual unidirectional sound source can also be applied in more general active control of sound applications, not only in active noise control. The paper attempts to explain the theory behind VUSS as well as to give some results showing its excellent performance in case of ANC.
The paper is organised as follows. Section 2 summarises physical basics necessary to understand principles of VUSS operation and its advantage over one-loudspeaker applications. In Section 3, the idea of VUSS is explained together with the original detailed analysis of optimal (fixed) design (Section 3.1) and adaptive approach (Section 3.2). Finally, Section 4 gives the results of simulations and real experiments of feedforward active noise control system using virtual unidirectional sound source.
2. Acoustical Phenomena in Ducts
The simplest solution to acoustic feedback cancellation problem in case where omnidirectional microphone must be used as the reference sensor is called feedback neutralisation [2, 4–6]. With this approach, only one loudspeaker is used, but the signal obtained from reference microphone is processed in the way allowing for compensation of its influence. If no compensation is performed, severe problems can result. In the simplest case of attenuation of one tone, the pressure distribution upstream secondary source [3] takes shape of standing wave, presented with solid line on Figure 1. If, for particular frequency of this tone, the reference microphone is placed in standing wave nodal point, it will give measurement equal to zero regardless of waves' actual magnitude.
Another situation occurs when two loudspeakers are used, as presented in Figure 2. Assume first loudspeaker is placed in the duct at position and the second at position . Denote as the complex pressure [3] at location produced by primary source emitting tonal sound: where is the complex amplitude of the sound wave and where acoustic wave number is equal to , with being wave angular frequency and being the sound speed in air.
Assume the two loudspeakers on Figure 2 produce sound waves propagating upstream the duct, marked as and , and sound waves propagating downstream the duct, marked as and . Assume also that the sound wave produced by the primary source is propagating downstream the duct, as marked on the figure.
Assume that the complex pressure of sound wave generated by first “active” loudspeaker is given by and the complex pressure of sound wave produced by the second “active” loudspeaker is given by where and are the complex amplitudes of the sound waves.
In the situation described above it is possible to set another condition on the system regardless of primary sound source attenuation requirement. In the case of unidirectional sound source, it may be the requirement of self-cancellation of the secondary sound waves propagating upstream the duct:
In this case, the secondary sources have to be driven in the way providing [3]In the same time to achieve the main ANC goal, which is to cancel the primary sound downstream the second loudspeaker completely, it is necessary to set the complex amplitude: The acoustic pressure level distribution for two loudspeakers case has been shown with dashed line on Figure 1. The figure shows that the sound pressure has been perfectly cancelled at coordinates while the absolute value of the pressure level has not been affected by the secondary sources at negative coordinates . The zone between the secondary loudspeakers () is a transient zone.
Although two-loudspeaker system offers significant improvement over one loudspeaker system, it has its drawbacks too. Specifically, the above control law says that for some frequencies for which is close to zero, both the secondary sources will have to produce sound waves with amplitude very large, compared with the amplitude of the primary waveform. Consider, for example, the case where the loudspeakers are separated by the distance m. In this case the sound waves of frequency 570 Hz and all its multiplies will result in equal to, or close to, zero, depending on the actual speed of sound, and thus on temperature, humidity, and so on. This fact will certainly influence the performance of ANC system trying to attenuate such sound waves (see, e.g., the drop in attenuation just below 600 Hz on Figure 10).
3. Virtual Unidirectional Sound Source
It is well known that introducing the second “active” loudspeaker allows to deal with many one-loudspeaker systems difficulties [4]. Moreover, the analysis outlined in Section 2 proves that it allows also for (hypothetically) perfect cancellation of the acoustic feedback. One of possible systems using two secondary loudspeakers is called virtual unidirectional sound source.
The idea of virtual unidirectional sound source (VUSS) is to use digital signal processing algorithm to drive two loudspeakers in such way that the sound produced by them propagates only downstream the duct [7]. In fact, although the sound generated by each loudspeaker propagates in both directions, the processing algorithm tries to assure that the sound waves propagating upstream the duct are actively cancelled by themselves while those propagating downstream the duct are amplified. The additional advantage of this approach is that it is sometimes possible to equalise the amplitude spectrum of the secondary path transfer function (transfer function between signals and on Figure 3) that plays very important role in active control of sound [4].
The block diagram of VUSS has been shown in Figure 3. The system contains two “active” loudspeakers, L2a and L2b, as well as two microphones: M1, generally called reference microphone, and M2 called error microphone (the names of the microphones correspond to the role they play in active noise control). The signal to be processed by VUSS is the set point value (or antinoise signal) produced by ANC algorithm. It is filtered by two filters with transfer functions, and , and sent to the loudspeakers, L2a and L2b, respectively. The error signals are obtained by comparing values acquired from the microphones M1 and M2 with their required values (set points). As the goal of VUSS system is to cancel the influence of the set point value at the microphone M1 point, the signal is compared with zero (). Similarly, as the desired signal at the microphone M2 point is the delayed set point value, the second error signal is calculated as . The need for delaying the set point value comes from condition on causality of the system: no filtration can compensate for the delay time introduced by the acoustic path between the loudspeakers and the M2 microphone. The value of the delay ensuring causality of the algorithm should be chosen before running the algorithm. The error signals and are used by adaptive algorithm to update parameters of the transfer functions and (in case of optimal design they are used only for observing the performance of the system).
3.1. Optimal and Suboptimal Filter Designs
In this subsection, an attempt to derive analytical optimal and suboptimal filter formulae will be made. The need for suboptimal formula will be explained and feasibility of both the solutions will be discussed.
3.1.1. Formulation Using Transfer Functions
Suppose that the whole system presented in Figure 3 is linear and the signal to be cancelled is deterministic, or random wide-sense stationary [8]. Suppose that and are infinite impulse response (IIR) filters. Figure 3 shows that the control values and can be expressed as a filtration of set point value by the and filters. Using the notation of difference equation, this filtration can be expressed aswhereIn the above equation, and are th elements of the and filter impulse response functions, and are also called filter coefficients. Using this notation, the is treated as discrete shift operator so that .
From Figure 3 it also follows that the reference signal due to the secondary sources measured by the M1 microphone can be expressed with the following difference equation:where and are impulse response functions of electroacoustical paths between the reference signal and the control values, including amplifiers, filters, loudspeakers, and others.
Similarly, the signal measured by the microphone M2 due to the secondary sources can be given byConsidering the above and the definitions of error signals given in the previous subsection, the error signals can be expressed aswhere the dependence of transfer functions on has been omitted to simplify the notation.
The goal of the tuning algorithm is to drive error signals to zero:
Substituting (7) and (11) into (12), recognising that gives for all where samples delay time was introduced to assure causality of the system.
Solving the above equation set gives the transfer functions of optimal filters: Unfortunately, (14) cannot be used in practise due to non-minimumphase nature of transfer functions leading to instability of VUSS filters. Moreover, even when all the transfer functions are minimumphase, the subtraction in denominator of (14) can still result in instability of the whole filter. This is referred to in the literature as “unconstrained controller” [3].
The suboptimal design of virtual unidirectional sound source can be performed after omitting the second condition from (12) set in mathematical derivations. In consequence, it means we do not expect the transfer function from set point value to microphone M2 signal to be pure delay, but we allow it to take more complex form. Furthermore, it means that the amplitude spectrum of this transfer function will not be flat over the frequency range under consideration. However, we still request cancellation of the secondary sources influence on the reference signal .
With control goal stated above, there are two equivalent solutions that will be called suboptimal, namely,orMoreover, assuming the transfer functions are of FIR type, the suboptimal solutions are also FIR type.
It is, however, important to notice that in case of suboptimal solution the secondary path transfer function (transfer function between signals and on Figure 3) is given by(Again, the dependence of transfer functions on has been omitted to clarify the notation.) This shows that in case of no demand for equalisation of the secondary path transfer function amplitude spectrum it can assume arbitrary shape, potentially “worse” than in case of single secondary source. This effect must be taken into account when estimating secondary path models required by active noise control algorithm (e.g., by using longer filters).
3.1.2. Formulation Using Reference Signal Matrix
The following subsection presents result obtained using the approach presented in [3, 9], which is based on impulse response of appropriate electroacoustic paths and on the use of filtered-reference signals. Using this approach, the th error signal ( in this case) can be expressed as a sum of desired signals and contributions from secondary sources (with in this case) as:where is the th coefficient of impulse response of electroacoustic path between the th sensor and the th actuator and is desired value of the th sensor signal. The length of impulse response can be arbitrary number guaranteeing desired accuracy.
In the same manner, the signals driving actuators can be expressed aswhere is the th parameter of the th order FIR filter impulse response.
Substituting (19) into (18) yields To derive a matrix formulation for the above equation it is necessary to reorganise the order of filtration of the set point value . To do this, it is necessary to assume that the filters and are time invariant. If the filtered reference signal is defined asthe error signals can be written asAfter definingEquation (22) can be expressed asFinally, the vector of two error signals can be defined asand the vector of two desired signals can be defined asleading to the conclusive formulawhereis called matrix of filtered reference signals, andis the vector containing all the coefficients of both and filters (the filters are time independent).
It is interesting to notice that in case of virtual unidirectional sound source the vector of desired signals can be instantiated asFollowing the methodology presented in [9] the optimal filter solution can be expressed aswhere denotes expectation operator.
The matrix to be inverted is of dimension , where is the filter length. Fortunately, it can be proved that the matrix is block Toeplitz matrix [3], so effective iterative methods can be applied for inversion. The expectation operator tells that statistical properties of filtered set point value will be taken into account. The expression under the right-hand side expectation operator, in the above equation, takes particularly the simple form of
3.1.3. Formulation in Frequency Domain
Applying the Fourier transform to (7) and (11) and substituting the transformed (7) into (11) yieldswith , where is a sampling period.
The above equations can be expressed in more convenient matrix form aswhere , and are transformed vectors defined by (25), (30), and (29), respectively, and the matrix of transfer functions is defined as The unconstrained controller can be found by minimising the cost function equal to expectation of the sum of squared errors, independently at each frequency [3]:Thus, the optimal controller is equal towhere matrices of power and cross spectral densities are defined as Assume now that the set point value used during VUSS learning phase is a sequence of white noise with expected value . The power spectral density matrix is then equal toindependent on , and the cross spectral density matrix is equal toAfter evaluating inverse of transfer function block matrix , the optimal controller obtained with white noise excitation becomeswhere the dependence of transfer function on has been omitted to clarify the notation.
The above equation is consistent with the result obtained in Section 3.1.1. It is, however, still the unconstrained form of the controller which is not feasible for practical implementation.
3.2. Adaptive Filter Solution
This subsection presents an alternative approach to optimal and suboptimal filter designs discussed above. Now we will try to develop an adaptive algorithm with the goal defined by set of (12). We will assume that and are in form of finite impulse response filters so the filter coefficients can be stored in a vector similar to this defined by (29), but now with coefficients varying slowly (compared to timescales of plant dynamics) in time.
Although there are many algorithms that can be used to tune filter coefficients [4], two-channel filtered-x LMS algorithm was chosen for the following derivation and experiments. Its advantages are simplicity and robustness, even if speed of convergence is not the best. Using two-channel FXLMS algorithm, the update equation is given bywhere is step size and and are defined by (28) and (25), respectively.
The matrix of reference signals is generated by filtration of the set point value by electroacoustic path transfer functions to . In practise, the latter can be only estimated yielding a matrix of estimated plant responses . The estimation must take place before running FXLMS algorithm and is usually done by separate identification procedure. The same procedure can also be turned on during active noise control phase after detecting substantial changes in plant dynamics (e.g., changes in air temperature).
To study convergence properties of the above algorithm, it is necessary to substitute (27) into (42) with estimated plant responses matrix, producingIf the adaptive algorithm is stable, it will converge to the solution setting the expectation value of the term in bracket to zero [3]. The steady-state vector of filter coefficients will therefore be equal to The above result for the adaptive algorithm steady state is in coincidence with the optimal solution presented in Section 3.1.2 (31) if and only if the estimated matrix of reference signals is equal to the true matrix of reference signal. This validates, however, the methodology.
The precise consequences of nonperfect matching of true plant responses contained in the matrix of reference signals on adaptation process are still unknown. The strongest result has been shown by Wang and Ren [10], the sufficient condition for stability of adaptation. Nevertheless it is not a necessary condition and it has been obtained with the small step size assumption.
During this research it was useful to introduce an additional weight parameter to specify which of the goals defined in (12) should have more bearing on adaptation process. For such case the definition of error vector was modified as follows:When is close to zero, the algorithm is better excited along the modes responsible for neutralisation of the acoustic feedback effect, while when is close to one the algorithm puts more stress on equalising the secondary path transfer function.
Introducing the weight parameter modifies the steady-state vector of filter coefficients givingThe maximum step size parameter, , assuring convergence of the whole algorithm is given bywhere denotes maximum eigenvalue of matrix rather than as was in the original solution [3]. The weight parameter can therefore influence, to some degree, the eigenvalues of the matrix under consideration.
4. Active Noise Control with VUSS
4.1. Active Noise Control Algorithms
Feedforward active noise control system using virtual unidirectional sound source has been shown in Figure 4. This system uses a finite impulse response filter as control filter with different adaptation algorithms. As in case of all feedforward algorithms, only the reference value is used by the control filter to produce the set point value . The set point value is then processed by VUSS filters and to give two control values and that are amplified and sent to the loudspeakers (via amplifiers and reconstruction filters not shown on the figure). The adaptation algorithm on the other hand uses the reference signal as well as the error signal to tune the control filter coefficients.
The duct used in experiments described below was made out of wood. It was 4 m long, with m rectangular section. One of the duct ends was terminated with noise generating loudspeaker, while the other was opened. The attenuating loudspeakers were of 0.16 m diameter and were situated approximately in the middle of the duct. The distance between the middles of the loudspeakers was equal to 0.3 m. The reference microphone was located 1.23 m from the middle of L2a loudspeaker, while the error microphone was separated from L2b loudspeaker by the distance of 0.23 m.
It should be emphasised that the goal of active noise control algorithm is in contradiction to the goal of the virtual unidirectional sound source adaptation algorithm [11]. The former tries to assure the compensation of sound waves from both the primary and secondary sources at the microphone M2 point without taking any notice of what happens at the microphone M1 point. The goal of the latter is to compensate sound waves from the secondary sources only at the microphone M1 point and try to assure the set point value appearing without any alternation at the microphone M2 point. This leads to a conclusion that both the algorithms should never operate at the same time. Indeed, the experiments show that when both the adaptation algorithms are in operation the whole system goes unstable. Therefore, the VUSS adaptation algorithm is usually run on the beginning of experiments and switched off after adaptation is completed. Next, only ANC algorithm is in use. Only when significant changes in the environment are detected (e.g., temperature change above some level), the ANC adaptation is temporarily frozen and the VUSS is retuned.
FXLMS Algorithm
The first of
ANC filter adaptation algorithms tested was Filtered-x LMS algorithm [4, 9]. Assume that the is an FIR filter with coefficients (see Figure 4)
in the th sample of time stored in a form of a vector .
In that case, the FXLMS algorithm update equation is given bywhere is the step size and is the vector of reference signal samples filtered by the secondary path
transfer function estimate, that is, an estimate of the transfer function
between set point value signal and error signal .
If VUSS algorithm was working perfectly, the secondary
path transfer function would be equal to simple time delay. Unfortunately, the
study from Section 3.1 leads to a conclusion that such perfect situation is
impossible: the VUSS algorithm has to “pay more attention” to neutralising
acoustic feedback and secondary path equalisation is only its additional task.
RLS Algorithm
The
second algorithm of
ANC filter adaptation algorithms tested was recursive least squares (RLS)
algorithm (see, e.g., [9]). RLS algorithm uses very similar update equation in
the formwhere is the gain vector showing how much the value
of will modify different filter coefficients. The
gain vector is calculated aswhere is a matrix updated in each step according to
the following equation:
The RLS algorithm usually converges faster than the
LMS algorithm [12].
Because RLS algorithm needs to update matrix of size equal to the number of filter
coefficients in each adaptation step, it requires computational effort of ,
whereas LMS algorithm requires computational effort of only [13]. There is, however, a family of RLS algorithm
implementations called fast RLS algorithms that allows to omit computation of matrix and use a selection of row vector
instead [13, 14]. This work used one of such
fast algorithms, called fast
transversal filter [15]. The algorithm was parametrised as follows. The
initial value of the minimum sum of backward a posteriori prediction-error
squares was equal to 1 and the exponential weighting factor was equal to
0.9999. The algorithm showed no stability issues, as it was expected on
floating-point arithmetic platform.
Both the FXLMS and RLS algorithms were implemented in
C programming language. Testing and debugging were performed using developed
simulation platform for Linux operating system. The platform allowed to emulate
DSP processor board behaviour and therefore the next step, moving the code
onto Texas Instruments TMS320C31 processor board, was purely automatic.
Another PC-computer program was used to tune various algorithm parameters and
to acquire the results. The sampling frequency of 2 kHz was chosen as frequency
band up to 1 kHz was of the author interest. It appeared that, due to hardware
limitations, appliable filter lengths were up to 70 with this sampling
frequency.
4.2. Testing Sounds
The set of signals chosen for experiments is presented with dashed line on Figures 12–17. Testing signals N1 and N2 were generated offline as white noise filtered with high-order bandpass filter. Bandpass was 200–300 Hz in case of N1 signal and 700–800 Hz in case of signal N2. The former was chosen to show low-frequency attenuation capabilities and the latter to show high frequency attenuation. The N3 is a signal acquired in close vicinity of a food processor. It has the dominant frequency of about 270 Hz, but with substantial amount of broadband noise. The N4 is a signal collected near a power transformer. It has all the harmonics of 50 Hz visible with 250 Hz being the dominant. The N5 signal was recorded in water power plant turbine proximity. It has the harmonics of 100 Hz distinguishable among broadband noise. The last signal N6 was recorded in small bureau with an electric propeller fan turned on. It is similar to N5, but the dominant frequency is higher.
All the signals were filed using 1000 samples. This ensemble was repeated many times and (after amplification) sent to loudspeaker L2 (see Figure 4). The results of attenuation were measured with microphone M2 and calculated as
4.3. Simulation Results
The simulations were performed using 150th order FIR filters models of the duct paths, identified offline using methodology presented in [16]. The algorithm for suboptimal VUSS filter solution described in Section 3.1 was tested first. As expected, it allowed for very effective acoustical feedback cancellation but at the cost of the secondary path being substantially degenerated. The amplitude spectrum of transfer function between set point value and both the reference signal and error signal is presented on Figure 5.
After tuning phase, the system was engaged to active noise control algorithms: FXLMS and Fast RLS. The ANC filter having 70 coefficients proved to be long enough to cancel out the noise in case of tonal sounds, so this was the chosen value. However, in case of N1–N6 signals attenuation was only (as for simulations of feedforward controller) 6–16 dB (with FXLMS algorithm, which proved to be better), see Table 1. The examples of spectra of the signals observed at the error microphone before and after attenuation have been presented on Figures 6 and 7.
In the following experiments, adaptive algorithm described in Section 3.2 was responsible for VUSS filters tuning. The value of the parameter (see (45)) has been chosen as 0.5 to put the same effort into satisfying both the goals defined by (12). The attenuation of tonal sounds was also infinite in case of adaptive VUSS filters tuning. But the attenuation of testing signals was slightly better: 9–30 dB in case of FXLMS algorithm and 8–33 dB in case of RLS algorithm, see Table 1. The examples of spectra of the signals observed at the error microphone before and after attenuation have been shown on Figures 8 and 9. The difference in amplitude spectrum of the N2 and N5 signals without attenuation visible on these figures and on Figure 6 and 7 comes from the fact that this is the spectrum measured with the M2 microphone, not the spectrum of the signal driving the L1 loudspeaker.
Table 1 summarises the results obtained during simulations for broadband signals N1–N6. The simulation experiments proved adaptive VUSS filters tuning superiority over analytical suboptimal filter design.
4.4. Real Experiments
To perform the real experiments using analytical filter design it was necessary to identify the models of electroacoustic paths (see Figure 3) first. The identification of FIR models of these paths was performed using LMS algorithm prior to each experiment. Moreover, in case of both the analytical and adaptive filter designs it was necessary to identify the model of the secondary path before the ANC. The identification of FIR model of this path was performed after VUSS was tuned by online identification procedure. The procedure, however, was disabled during the ANC.
The first ANC laboratory experiments used tonal sounds. The frequency range between 100 and 950 Hz has been checked with resolution of 20 Hz. The VUSS filter design was performed using adaptive method. The attenuation obtained during these experiments has been shown on Figure 10. The figure shows that in frequency range from 150 to 950 Hz the attenuation was between 14 and 33 dB (average 25 dB) for FXLMS ANC algorithm and between 15 and 44 dB (average 30 dB) for RLS ANC algorithm.
The drop in attenuation above a frequency of about 420 Hz, especially distinct in case of RLS algorithm, can be explained as the ducts first cut-on frequency appears at 425 Hz. The figure shows that the performances of both FXLMS and RLS algorithms above first cut-on frequency are similar. However, below a frequency of 400 Hz the performance of RLS algorithm is up to 14 dB better than in case of FXLMS algorithm.
The next step was to compare the performance of ANC system in case of analytical and adaptive filter design. During these experiments frequency range from 100 to 900 Hz was tested with resolution of 100 Hz. The results are shown on Figure 11. In all cases the efficiency of ANC algorithm with analytical VUSS tuning was worse than efficiency of the same algorithm with adaptive VUSS tuning. Therefore, only adaptive VUSS design was considered for the following tests.
Finally, the noise signals described in Section 4.2 were used in real experiments. Although both FXLMS and RLS ANC algorithms were checked for performance, the only spectra presented on Figure 12 through Figure 17 are those obtained with RLS algorithm, as this algorithm performance was superior to FXLMS in all experiments. The attenuation factors for both FXLMS and RLS algorithms are presented in Table 2.
5. Conclusions
The idea of virtual unidirectional sound source presented in this paper is based on theoretical study of wave propagation in duct. VUSS itself is a special case of two-reference, two-output system with a detailed analysis presented in Section 3. Its application to active noise control in an acoustic duct proved to be effective, resulting in 20–40 dB attenuation of tonal sounds and 5–16 dB attenuation of complex signals with broadband noise. The best results were obtained with adaptive VUSS design together with RLS active noise control algorithm. These results are comparable with similar results reported by other authors. The results, however, do not show one of the strengths of VUSS, the ability to adapt to time varying feedback paths.