Department of Mechanical Engineering, Faculty of Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Abstract
An active noise control (ANC) system is model dependent/independent if its controller transfer function is dependent/independent on initial estimates of path models in a sound field. Since parameters of path models in a sound field will change when boundary conditions of the sound field change, model-independent ANC systems (MIANC) are able to tolerate variations of boundary conditions in sound fields and more reliable than model-dependent counterparts. A possible way to implement MIANC systems is online path modeling. Many such systems require invasive probing signals (persistent excitations) to obtain accurate estimates of path models. In this study, a noninvasive MIANC system is proposed. It uses online path estimates to cancel feedback, recover reference signal, and optimize a stable controller in the minimum
H2 norm sense, without any forms of persistent excitations. Theoretical analysis and experimental results are presented to demonstrate the stable control performance of the proposed system.
1. Introduction
Most active noise control (ANC)
systems are model dependent. Let 
and 
denote estimates of primary and secondary path
transfer functions 
and 
. Either 
or both 
and 
must be obtained by initial system identification for model-dependent ANC systems. Controller
transfer function 
of a model-dependent ANC system is
either designed by minimizing 
, or adapted with the aid of 
[1, 2].
If estimates 
and 
contain too much error, a model-dependent ANC system may generate constructive instead of
destructive interference. This is mathematically equivalent to 
even if 
is minimized. If phase error in 
exceeds 
in some frequency, an ANC
system adapted by the filtered-
least mean square (FXLMS) algorithm may become
unstable [3–5].
An operator of a model-dependent ANC system must have the knowledge and skill
to obtain accurate estimates of path models by initial system
identification for each individual application.
During the operation of an ANC
system, changes of environmental or boundary conditions may cause significant
changes to path transfer functions 
and 
. Since a model-dependent
ANC system only remembers initial path estimates 
and 
,
variation of 
and 
may cause mismatch
with initial estimates 
and 
to degrade ANC performance. In cases of severe
mismatch between path transfer functions and their initial estimates, a model-dependent
ANC system may generate constructive instead of destructive interference, or
even become unstable.
Model-independent ANC (MIANC) systems
depend on online path modeling or invariant properties of sound fields to update
or design controllers [6–8].
These systems avoid initial path modeling and are adaptive to variations of
environmental or boundary conditions of sound fields. Many adaptive MIANC systems require invasive persistent excitations to obtain accurate path estimates and ensure closed-loop stability
[6, 7, 9, 10].
Noninvasive MIANC systems are able to ensure closed-loop stability without persistent excitations,
which are
possible by a recently developed algorithm, known as orthogonal
adaptation [11, 12],
if the primary noise signal is directly available as the reference signal.
In many real applications, the primary
noise signal is not necessarily available and the reference signal must be recovered
from the
sound field [1, 2]. When an ANC system is active, a
measured signal is a linear combination of primary and secondary signals. Feedback
of ANC
signal in the measurement is mathematically modeled by a feedback transfer
function 
from the controller to the reference sensor. Accurate estimation
of 
is as important as accurate estimation of 
or 
[9, 13]. A complete noninvasive MIANC (CNMIANC) system must be able to
suppress the noise signal without injecting probing signals for online modeling
of 
, 
, and 
. Most
available methods for adaptive feedback cancellation require persistent
excitations [9, 13]. In
this study, a new method is presented for adaptive feedback cancellation without
persistent excitations.
It was proposed to use a pair of sensors
to measure pressure signals in ducts, from which traveling waves are resolved
[14, 15]. The
outbound wave could be used directly as the reference signal without cancelling
feedback signals if an infinite-impulse-response (IIR) controller could be
implemented accurately [14, 15]. In reality, it is very difficult to implement
a stable ideal IIR ANC controller [16]. Most practical ANC systems use
finite-impulse-response (FIR) controllers. The outbound wave in a duct is a
linear combination of primary noise and reflected version of feedback signal. Instead
of using the outbound wave directly as the reference, the least mean square
(LMS) algorithm is applied in this study to cancel feedback signals in the
outbound wave before using it as the reference. Orthogonal adaptation is combined
with the proposed ANC configuration to implement a CNMIANC system. Experimental
result is presented to demonstrate the performance of the CNMIANC system.
2. System Configuration and Model
Figure 1 illustrates the configuration of the proposed ANC system.
The primary source is represented by the upstream speaker and the secondary
source is the midstream speaker. Cross-sectional area of the duct is small
enough such that sound field in the duct can be modeled by a 1D sound field in
the frequency range of interest. Three microphone sensors are installed in the
duct, measuring signals 
, 
, and 
, respectively. Since the
primary noise signal is not available to the ANC system, the reference signal is recovered
from 
and 
, while 
is the error signal to be
minimized by the ANC system.
Figure 1: Configuration of the proposed MIANC system.
Let d denote the axial distance between 
and 
. The acoustical two-port theory [16, 17] has been applied by many
ANC researchers for the design and analysis of ANC systems. It is adopted here
as an analytical tool. An equivalent acoustical circuit is shown in
Figure 2 to
model the two-microphone system. The upstream part, from the primary source to
location of 
, is
equivalent to an acoustical source with strength 
and impedance 
.
The downstream part, from location of 
to the outlet, is represented by another acoustical source with strength us and impedance 
. Characteristic
impedance of the duct is represented by 
.
Figure 2: (a) Acoustical two-port circuit in the duct
system, (b) contribution by controller, and (c) contribution by primary source.
The linear system theory allows one to solve 
and 
in Figure 2(a) by focusing on acoustical circuits of
Figures 2(b) and 2(c) before adding two solutions together as the final solution
of Figure 2(a). For the case of 
,
which is represented by Figure 2(b), one obtains 
(1) where k is the wave number. One can solve, from (1),
(2)
(3)Similarly, for the case of 
, which is represented by
Figure 2(c), one obtains 
(4) from which one can solve
(5)
(6)Adding (2) and (6), one may write
(7)The same method is applicable to
(3) and (5) for
(8)The next step
is to use complex factor 
to simplify (7) and (8). The results read
(9) Since 
and 
,
(9) can be written as
(10)Let
(11)
(12) represent the in- and outbound
waves in the duct. By comparing (10)
with (11) and (12), one can see
that (10) are equivalent to
(13) The in- and outbound waves can be
resolved from 
and 
via
(14)
In a digital
implementation of ANC system, it is recommended to select sampling interval 
such that
its product with sound speed c satisfies 
. As a result, the delay operator 
becomes an exact one-sample delay for discrete-time
ANC systems.
3. Feedback Cancellation
It is indicated
by (12) that the outbound wave contains feedback from 
that must be cancelled to recover the reference signal. Let 
denote the upstream reflection coefficient. By
multiplying 
to (11), one obtains
(15)A subtraction of (15) from (12) enables
one to
write
(16)where
(17) is only contributed by the
primary source 
.
Using 
and 
,
one can see that the common denominator of 
, 
, and all transfer
functions in the duct is
(18)Substituting (18) into the
definition of α (immediately after (8)), one obtains
(19)A further substitution of (19)
into (17) leads to
(20)This is the reference signal to
be recovered by the proposed ANC system.
A question to
be answered here is why not recovering the reference signal from a pressure
signal such as 
. The hint is
(8) that may be expressed as 
.
In view of (8), the acoustical feedback transfer function is
(21)Since 
is a transfer function with resonant
poles, it has an infinite impulse response (IIR). In many ANC systems, a finite-impulse-response
(FIR) filter 
is used to approximate 
. This means inevitable approximation
errors in the first place.
Besides, all
transfer functions in a duct are sensitive to values of 
, 
,
and 
. In particular, 
is the impedance of the
entire downstream segment from the location of 
to the duct outlet. Objects moving near the duct
outlet could cause changes of 
.
A fracture in any downstream part may also cause a significant change to 
as well. If initial
estimate 
is remembered by an ANC system, it is a
stability issue how significant will 
turn
out as a result of a small variation of 
. An indicative answer might be
(22)The common denominator of 
, 
, and all transfer functions in the duct has an
alternative form in (18), which is equivalent to
(23) where 
represents the downstream reflection coefficient.
Since resonant
frequencies of the duct are roots of the common denominator, it is suggested by
(22) and (23) that all transfer functions in the duct, including the feedback
transfer function 
, are
sensitive to variance of 
at the resonant peaks. The stronger the resonance, the more sensitive of
transfer functions with respect to 
.
If an ANC system recovers the reference signal from a pressure signal like 
, a small online variation
of 
may cause a
significant mismatch between 
and
initial estimate 
.
As a result, closed-loop stability is sensitive to possible variation of 
.
If the
reference signal is recovered from traveling waves with (16), the situation
will be different. In a discrete-time implementation, one may rewrite (16) to 
, where the acoustical feedback transfer function is a delayed version of
upstream reflection coefficient 
. Here, 
is only sensitive to 
and 
.
Characteristic impedance 
is a real constant depending on sound speed and cross-sectional area between 
and 
. It seldom changes significantly in online ANC
operations. As for 
, it is
the impedance of the upstream portion from the primary source to the location
of 
. In most
applications, 
and 
are measured as close as
possible to the primary source. Impedance 
is deeply hidden in a very short segment of the duct. Its variation, if any,
would be certainly not as significant as that of 
.
No matter how
significant are the possible variations of 
or 
,
the passive upstream reflection always has a limited magnitude 
. For each pair of fixed 
and 
, 
does not have sharp peaks or dips as a
function of 
. In many cases, 
is constant for a pair of fixed 
and 
. Let 
denote
the Fourier transform of 
, then 
and 
share
many similar properties. For example, if 
is a low-frequency function of 
, then the bandwidth of 
is narrow in terms of 
. Similarly,
if 
is a
“low-frequency” function of 
, then the time duration of 
is short (a narrow bandwidth in terms of 
).
The fact that 
is a
“low-frequency" function of ω for
each pair of fixed 
and 
implies short impulse
responses of 
. It is, therefore, reasonable to
assume that 
can be approximated by a FIR transfer function
with negligible errors (Assumption A1). If both 
and 
are constant, 
is a
single constant. Resonant effects in the duct are hidden in wave signals 
and 
without affecting 
.
This is a major difference between recovering the reference signal from
traveling waves and recovering the reference signal from a pressure signal.
Even if an estimate of 
is obtained by initial
identification, it is less likely that online variations of environmental or
boundary conditions could cause significant mismatch between 
and its initial estimate. The resultant ANC system is semimodel
independent if its reference signal is recovered with (16) in combination with
a MIANC adaptation algorithm such as orthogonal adaptation.
4. Complete Noninvasive MIANC
Noninvasive model-independent feedback cancellation is possible by applying
LMS to (16). With assumption A1, online estimate of the feedback transfer
function is represented by polynomial
(24)where 
is the 
th coefficient for the 
th sample. An estimated version of (16)
would be 
(25)which has a discrete-time domain
expression,
(26)Coefficients of 
are updated with the LMS algorithm as follows:
(27)where 
is a
small constant representing the LMS step size. Since 
by assumption A1, the discrete-time domain
version of (16) is
(28)Subtracting (28) from (27), one
obtains
(29)where 
is the estimation error of 
. Let 
and let 
. It is possible to express
(29) in an inner product
(30)Estimation residues of LMS
algorithms are usually expressed as inner products like (30). It has been
proven that the LMS algorithm is able to drive the convergence of these inner
products towards zero.
If the primary noise signal 
was available, mathematical model of the error signal
may be expressed in the discrete-time 
-transform domain as 
, where the actuation signal would be synthesized as 
. Since 
is actually not available, the ANC system has to
recover 
from the outbound wave and then synthesize 
instead. After the convergence of 
→n (z), one may express the mathematical
model of the error signal to
(31)where (20)
has been substituted. Let 
,
then (31) becomes
(32) It is mathematically
equivalent to another ANC system whose primary source is available to the
controller as 
, with primary path transfer function 
and secondary path
transfer function 
. Orthogonal adaptation is readily
applicable to (32) to implement a noninvasive MIANC system.
It is assumed that 
and 
can be approximated
by FIR filters with negligible errors (Assumption A2). Let 
and 
denote coefficients of 
and 
, respectively, the discrete-time domain version of 
is a discrete-time convolution:
(33)where 
, 
, and 
denote
samples of 
, 
, and 
,
respectively. Introducing coefficient vector 
and regression vector 
, one may rewrite (33) to 
(34) Let 
and 
denote online estimates of 
and 
. Path estimates 
and 
are obtained by minimizing estimation error as
follows:
(35)where 
and 
are online modeling errors. Let 
denote online estimate of 
, then 
and 
represent the coefficients of 
and 
,
respectively. Similar to the equivalence between (34) and 
, (35) has
a discrete-time domain equivalence
(36)where 
is the online coefficient error vector. The entire
CNMIANC system performs three online tasks that are mathematically represented
by the minimization of three inner products. The first is inner product given
in (30); the second one is given in (36); and the third one is 
.
Equations (30)
and (36) contain estimation errors Δr and Δθ. Most
available estimation algorithms, such as LMS and the recursive least squares
(RLS), are very capable of driving inner products like (30) and (36) towards
zero, or at least minimizing their magnitudes [18].
A difficult problem is how to force Δr → 0 and Δθ→ 0. Available
solutions inject significant levels of “persistent excitations” (invasive probing
signals) to the estimation system [6, 7, 9, 10, 13]. A unique feature of the
proposed CNMIANC is no persistent excitations. The system works well without
requiring Δr → 0 and Δθ→ 0.
For (30),
minimizing the inner product in the right-hand side implies convergence of 
in the left-hand side. It would be great if Δr → 0 as well. Otherwise, Δr may just converge
to a FIR filter that filters out wi from wo. On the other
hand, minimizing the inner product in (36) only implies εt→ 0. The question is what does it further
implies? One may consider the equivalence between (34) and 
, which holds
if one replaces 
, 
,
and 
with respective estimates 
, 
,
and 
.
The equivalence is now between forcing 
and forcing
(37)The CNMIANC system uses online
estimates of 
and 
to solve 
that minimizes 
.
This is equivalent to forcing 
.
One can obtain
(38) by adding 
to both sides of (36). As the CNMIANC system
drives 
→ 0 and forces 
ultimately, it implies ultimate convergence of 
→ 0 even though Δθ does
not necessarily converge to zero [11, 12].
5. Experimental Verification
A CNMIANC system was implemented and
tested in an experiment, with a configuration shown in
Figure 1. Cross-sectional
area of the duct was 
cm2. Two microphones were placed
30 cm downstream
from the primary speaker with a space of d = 10 cm
between 
and 
.
The distance between 
and the secondary speaker is represented by L in
Figure 1. To guarantee a causal ANC system, the value of L must satisfy 
such that the
outbound wave is at least two samples ahead of sound propagation in duct. The
sampling interval of the controller was 0.29 millisecond with a sampling frequency
of 3.448 Hz, which satisfies d = cδt with
m/s and exp(jkd) = z. The cutoff
frequency of antialias filters was chosen to be 1200 Hz. The in- and
outbound waves were recovered from pressure signals with (14). The reference
signal was recovered with (25). Coefficients of 
were adapted with (27). Another online
modeling process used (34) to obtain coefficients of 
and 
. The ANC
transfer function was solved by online minimization of 
. The CNMIANC
system was implemented in a dSPACE 1103 board.
Error signal 
and primary noise 
were collected
as vectors 
and 
for three cases. In case 1, there was no control
action. In case 2, 
was available as the reference signal
for an ANC system to suppress noise in the duct. In case 3, 
was not available and the CNMIANC system had to recover 
from 
and 
for controller
synthesis. For each respective case, power spectral densities (PSD’s) of 
and 
were computed with a MATLAB command called
“pmtm()”. Computational results are denoted as vectors 
and
, where argument vectors 
and 
represent measurement samples of 
and 
. The normalized PSD of 
was calculated as
for all
three cases.
Shown in Figure 3 are normalized PSD of 
for the three cases. For case 1, normalized
PSD of 
is
represented by the dashed-black curve. For case 2, normalized PSD of 
is plotted with
the solid-gray curve. For case 3, normalized PSD of 
is
represented by the solid-black curve. Both ANC systems were able to suppress
noise with good control performance as seen in Figure 3. The proposed CNMIANC
has slightly worse performance since its reference was the recovered signal 
instead of the true primary source 
. This is a small price to pay in case 
is not
available to the ANC system. The proposed CNMIANC system was stable and able to
recover the reference and suppress noise without any persistent excitations.
Figure 3: Normalized PSDs of
for (a) uncontrolled
case (dashed-black), (b) controlled case with
available
(solid-gray), and (c) controlled case with recovered
(solid-black).
The CNMIANC system was robust with respect to sudden parameter change in
the duct. In the experiment, the duct outlet was changed from completely open
to completely closed. Such a sudden change shifted all resonant frequencies in
the duct. Path transfer functions also changed suddenly. The CNMIANC system
remained stable and converged very quickly.
6. Conclusions
The primary source is not necessarily available as the reference signal
for ANC systems in all practical applications. When the primary source is not
available, the ANC system must recover the reference signal from a sound field to
which ANC is applied. Feedback cancellation is an important issue in ANC
systems that recover reference signals from sound fields. In most
MIANC systems, persistent excitations are required for online modeling of
feedback path model and adaptive feedback cancellation [9, 13]. In this study,
a CNMIANC system is proposed that recovers reference signal from traveling
waves without persistent excitations. The corresponding feedback path model is
the upstream reflection coefficient and hence closer to an FIR filter than
pressure feedback transfer functions (IIR path models in resonant ducts).
Theoretical analysis and experimental results are presented to demonstrate the
stable operation of the proposed CNMIANC system.
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