Underwater Acoustic Noise Cancellation Scheme Combining Kalman
Filter With Adaptive FxNLMS
MD. MOSTAFIZUR RAHMAN1, SUMONTO SARKER2
1Electronics and Communication Engineering (ECE), Hajee Mohammad Danesh Science & Technology University,
BANGLADESH
2 Electronics and Communication Engineering (ECE), Hajee Mohammad Danesh Science & Technology University,
BANGLADESH
*Correspondence: mostafiz.rahman.r@gmail.com
Abstract: Underwater environments are more challenging than that of terrestrial. The performance of a controller or the augmented
system as a whole depends on the real measured data, so noise on data readings can be fatal. To effectively and adaptively control
lower and higher frequency noise, the Active Noise Cancellation (ANC) was developed. Designing a system and the parameter of
modified FxLMS for reducing noise, and disturbance of sensor data is the primary focus of this paper. The required equation will be
analyzed and discussed briefly. Moreover, the system will be simulated in MATLAB and then the filtered result will be analyzed.
Based on the simulation results, the proposed model can filter out signals with noise, particularly when there is a significant variation
in the data and no knowledge of the noise frequency that might affect sensor readings.
Keywords: FxLMS; Kalman Filter; Signal Denoising; Signal Enhancement
Received: June 19, 2022. Revised: May 19, 2023. Accepted: June 18, 2023. Published: July 12, 2023.
1. Introduction
It becomes very challenging to eliminate noise
without losing some signal information if noise and
signal share the same frequency band. As a result, it is
difficult to remove noise from a signal in an underwater
environment without losing some of its original
characteristics. Acoustic noise cancellation (ANC) has
received a lot of attention as a technique for removing
noise from signals. The adaptive filter is a critical
component of ANC because it provides noise reduction
without prior knowledge of the noise and signal [1]. In
ANC, a 180-degree phase signal (anti-noise) is
generated and used to interfere destructively with the
unnecessary noise. Bernard Widrow et al. pioneered the
core concept [2].
Various methods have been proposed in order to
improve the performance of ANC. RLC, LMS, and
their variants (NLMS, VLMS, and so on) are popular
because they have fewer complications [1]. Despite
maintaining an excellent rate of convergence, the RLS
algorithm fails to track the Estimation because the
algorithm is dependent on its model, input data as the
computation progresses, and the correlation matrix [3].
As a result, the LMS and its successor algorithms are
most likely the most widely used algorithms. The
Filtered-X LMS (FxLMS) algorithm is a simple variant
of the LMS algorithm, which was developed
independently in the context of adaptive control
systems, and originally introduced as a modification in
applications where an intervening system exists in the
error path [4] [5].
LMS is based on the steepest descent method but
does not account for secondary path effects, making it
impossible to generate a precise anti-noise signal. The
FxLMS algorithm is computationally simple and
includes secondary path effects [6]. Several ANC
algorithms with improved convergence properties have
been proposed, including ANC systems in the
frequency domain (Kuo & Tahernezhadi, 1997);
Recursive Least Squares (RLS) based algorithms called
filtered-x RLS (FxRLS) (Kuo & Morgan, 1996) and
Filtered-x fast transversal filter (FxFTF) (Bouchard &
Quednau, 2000); Lattice ANC systems (Park &
Sommerfeldt, 1996) and Infinite impulse response (IIR)
filter based LMS algorithms called filtered-u recursive
LMS (FuRLMS) (L. J. Eriksson and Allie, 1987), and
filtered-v algorithms (Crawford & Stewart, 1997). The
basic problems in the above approaches are inherent
stability problems in IIR-based structures, increment in
the computational requirement and numerical instability
problems in RLS based ANC systems [7]. For these
reasons, FxLMS remains a viable option for ANC
applications.
The step size is the most inherent feature of the
Least Mean Squares (LMS) algorithm that FxLMS
inherited, and it requires careful adjustment. The Small
step size, required for small excess mean square error,
results in slow convergence. Large step size, needed for
fast adaptation, may result in loss of stability. For
controlling the step size and making it variable rather
than fixed, we used the Kalman filter. Kalman filter was
proposed by R. E. Kalman in 1960 [8] is popular for
having easy computation, memory requirements and
good capability on overcoming noises. There are
various types of Kalman Filter, such as standard
Kalman Filter, Extended Kalman Filter, Unscented
Kalman Filter etc [9]. The paper used standard Kalman
filter since it contains enough part of equation for noise
reducing.
The paper is organized in the following way:
Section II presented all the necessary equations of
FxLMS and linear Kalman filter, and also devoted to
the developing of a modified FxLMS with Kalman filter
to reduce active noises. Section III is devoted to the
simulation and discussions of the obtained results. The
conclusion section closes the paper.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
32
Volume 3, 2023
2. Materials and Methods
The aim of this paper is to demonstrate a
simulation, where we proposed a noise reduction model
by modifying existing FxLMS algorithm, employing
Kalman filter. The step size of FxLMS needs to be
carefully adjusted. We replaced step size
with kalman
gain in each iteration. There are two step sizes in
FxLMS, one in LMS part and another in the secondary
noise path. So, in that case, the whole simulation based
on three assumptions: using
during LMS and
Kalman-Gain at secondary noise path, using Kalman-
Gain during LMS and
at secondary noise path, and
using Kalman-Gain during both in LMS and in
secondary noise path. Finally, we compared data with
novel FxLMS algorithm based on Signal to noise ratio
(SNR). Our experimental data shows that after using
kalman, it filtered noisy signal more efficiently than
before. All the simulations are carried away by
MATLAB R2015b on a Windows 10 PC (x64) with an
Intel i3-7100U CPU and an Nvidia GeForce 920MX
GPU card.
2.1. FxLMS Algorithm
Figure 1. Block diagram of FxLMS algorithm
Widrow, Shur & Shaffer proposed the integration
of a secondary path model in the reference signal path
(from speaker to error microphone) [11]. Figure 1
shows the block diagram of FXLMS algorithm on how
the noise reduction algorithm works and the definition
of each symbol is shown in Table 1.
ANC has commonly used in two different
configurations of the FxLMS algorithm, one is a
feedback ANC approach proposed by Olson and May in
1953 [12], in this model, a microphone is used as an
error sensor, and also as a reference sensor. The second
is a feed-forward ANC approach, which uses two
sensors, an error sensor, and a reference sensor. This
setup is used for narrow-band noise control using a non-
acoustic reference sensor [13].
Table 1. Symbols and Definitions for Figure 1
Symbols
Definitions
x(k)
Noise signal
xs(k)
Noise signal combined with assumed Sh(z) based on S(z)
P(z)
Primary path transfer function
yp(k)
Primary noise signal at the error microphone
e(k)
Modified error signal
S(z)
Secondary path transfer function
C(z), Sh(z)
Controller for the FxLMS algorithm
yw(k)
Generated noise based on C(z) controller
ys(k)
Output of adaptive filter
Figure 1 shows the general how the feedforward
approach uses a different microphone to measure the
signal at the output. It is significant to know the
software elements that are part of the ANC controller,
these are the LMS adaptive algorithm that updates the
coefficients of the W(z) adaptive filter, which in this
case is represented as an FIR filter. The C(z) filter
represented the secondary path estimation or the
transfer function between the secondary source (control
source) and the error microphone. To derive the FxLMS
algorithm, a similar method of the LMS algorithm is
utilized but with the steepest descent, the following
update equation can lead to this minimization:
(
1)
Where W is the controller weight error,
µ is an adaption step size (scalar),
J(n) is the power of error signal.
The derivation of ,
Where E{.} denotes statistical expectation
operator and E{.} is a theoretical function. To avoid this
operator, J(n) is approximated by
Then, estimate as follows,
(
2)
Now to estimate e(n), the derivation is as follows
based on the block diagram,
Where s(n) is the secondary path impulse
response.
(
3)
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
33
Volume 3, 2023
Now to estimate , the derivation is as
follows based on the block diagram,
Where W is the controller weight vector and x is
the reference signal tap vector (of the same length as the
controller length)
Now can be expressed by,
(
4)
Substitute (4) into (3)
(
5)
Substitute (5) into (2)
(
6)
Substitute (6) into (1)
(
7)
The reference signal is filtered by ŝ(n) before
passing through the standard LMS algorithm.
Therefore, resulting the compensation for secondary
path. ŝ(n) should be estimated through off-line or online
secondary path techniques. If ŝ(n) denotes an estimate
of s(n), then
OR
The stability of the FxLMS algorithm is highly
dependent on the x_f(n) power where it directly
proportional to the step-size . So, Step-size is
indirectly proportional to the steady state performance.
FxLMS is simple, fast, and surprisingly robust.
Despite its straightforwardness, FxLMS acquired the
most central feature of the Least Mean Squares (LMS)
algorithm is the step size, and it undoubtedly requires
precise adjustment. To properly control step size, we
utilized the Kalman filter.
2.2. Kalman Filter
The paper will use a standard Kalman filter since
it contains enough parts of the equation for noise
cutting. Kalman Filter has two parts, the predicted part,
and the update part. The standard Kalman Filter
equation is shown in (11) – (15).
Predict:
(
8)
(
9)
Update:
10)
11)
12)
where is estimated state, is state transition
matrix, is control variables, is control matrix, is
state variance matrix, is process variance matrix, is
measurement variables, is measurement matrix, is
Kalman gain, is measurement matrix, | is current
time period, - 1| - 1 is previous time period, and | -
1 is intermediate steps.
To implement Kalman Filter algorithm, so that it
can be used to reduce noise of sensor-readings, some
adjustments for the conditions are needed. Those
adjustments are as follows [14].
2.2.1. Predicting the state
On this stage, adjustments are done in (11) by
giving the score Ft = 1 because there is no state
transition. Thus, reducing the system’s input component
Bt because the used system does not have any input ut.
The adjusted equation is shown in (6).
13)
2.2.2. Predicting the error
Since Ft = 1, then (9) becomes (14)
14)
2.2.3. Updating the state value
From (10), t = 1 since the sensor data that will be
filtered is only consisted of one sensor reading. Hence,
the equation can be written as (15).
15)
2.2.4. Calculating the gain of Kalman
Since Ht=1, then (11) can be written as (16)
16)
2.2.5. Updating the error value
Since Ht=1, then (12) can be written as (17)
17)
The Kalman Filter equation can be modified to
reduce sensor reading noise once the necessary
adjustments have been performed. The weights given to
the data and the current-state estimate are represented
by the Kalman-gain (at eq. 19), which can be "adjusted"
to get a specific performance. We replaced the step-size
in the FxLMS with this Kalman gain so that the step
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
34
Volume 3, 2023
size is flexible according to the signal elements rather
than being fixed.
As we replaced step size, with Kalman-gain in
the original FxLMS algorithm, we needed to declare
some necessary variables to calculate Kalman-gain out
of the noisy signal. During calculating Kalman-gain, we
had to specify the values for Q (process noise
covariance) and R (measurement noise covariance).
The value of Q and R are chosen according to the
system operations. Covariance Q and R states may not
be in general observable but the measurements should
be related to the states [16].
Q, the process noise covariance, contributes to the
overall uncertainty. When Q is large, the Kalman Filter
more closely tracks large changes in the data than when
Q is small. The measurement noise covariance R
determines how much information is used from the
measurement. When R is large, the Kalman Filter
considers the measurements to be inaccurate. The three
images below visualize the positional data. The red
lines represent the measurement data, the green lines
are the estimated states. [17]
Q small, R large
Q and R both equal
Q large, R small
Figure 2. Relations between Q and R.
We need to balance between Q and R according to
our needs. The vast majority of noise estimation
methods were designed with the assumption of
uncorrelated state and measurement noise in mind [18].
For example, if kalman used in tracking cars on a road,
then the constant velocity model should be reasonably
good, and the entries of Q should be small. Else if it is
used tracking people's faces, they are not likely to move
with a constant velocity, so the Q need to cranked up
[19].
In [14], author used kalman filter to denoising
signals. During their operations, they discovered that
the greater the difference between R and Q, the greater
the mean error values. Furthermore, the same and
values result in similar value of mean error, whatever
the values of and . According to their analysis, the
best parameters that provide results with their original
data characteristics have mean error values ranging
from 40 to 55 in the table below.
Table 2. Ratio between R and Q and their yielding mean error
No. of
Analysis
Kalman Filter
Parameter Value
R and Q
Ratio
Mean
Error
R
Q
1
1
1
1
26.0677
2
1
0.1
10
44.7392
3
1
0.01
100
53.4466
4
10
0.1
100
53.4541
5
100
0.1
1000
56.9959
They experimentally showed that in case of signal
denoising, the kalman filter yields best results if the
ratio between R and Q are in 100:1. Therefore, in our
operation, we kept R, Q ratio 100:1 too.
The flowchart of our modified FxLMS is given
below. When we replace step size
with kalman gain at
secondary path operation, we get the best output by far.
Reference signal x(n) is propagating from the
source to the sensor, through the fluid medium P(z).
The sensor measures the arriving noise as p(n). To
reduce noise, we generate another 'noise' y(n) using the
controller W(z). We hope that it destructively interferes
x(n). It means that the controller has to be a model of
the propagation medium P(z). Least mean square
algorithm is applied to adjust the controller
coefficient/weight. However, there is also fluid medium
S(z) that stay between the actuator and sensor. We
called it the secondary propagation path. So, to make
the solution right, we need to compensate the
adjustment process, estimate of Ŝ(z).
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
35
Volume 3, 2023
Figure 3. Modified FxLMS algorithm.
3. Results and Discussions
The research scenario is by generating a signal
with noise then it will be filtered by using both FxLMS
and our modified FxLMS algorithm. The step size,
is
used in two major operations in the FxLMS algorithm;
one in the LMS calculating process which is run in the
secondary path, and another one in the whole secondary
noise path of FxLMS. So, in that case, we run whole
simulation based on three assumptions: a) Use
during
LMS and Kalman-Gain at secondary noise path, b) Use
Kalman-Gain during LMS and
at secondary noise
path, c) Use Kalman-Gain during both LMS and
secondary noise path.
Table 3. SNR analysis between three assumption,: a) Use during LMS and Kalman-Gain at secondary noise path, b) Use Kalman-
Gain during LMS and at secondary noise path, c) Use Kalman-Gain during both LMS and secondary noise path.
Use during LMS and
Kalman-Gain at secondary noise path
Use Kalman-Gain during LMS and
at secondary noise path
Use Kalman-Gain during both LMS and
secondary noise path
Sample
no.
SNR of
FxLMS
SNR of
modified
FxLMS
Sample
no.
SNR of
FxLMS
SNR of
modified
FxLMS
Sample
no.
SNR of
FxLMS
SNR of
modified
FxLMS
1
12.41158
13.40830
1
11.48765
11.48765
1
12.39523
13.34103
2
11.50421
12.46322
2
11.06984
11.06986
2
11.62135
12.43485
3
12.10779
12.72214
3
10.97759
10.97759
3
12.02066
12.42624
4
11.52281
12.35170
4
11.56018
11.56018
4
11.15641
11.55346
5
12.16835
12.79602
5
11.73240
11.73242
5
10.33399
9.478464
6
12.23658
13.26459
6
11.81109
11.81109
6
10.11580
11.03627
7
11.09591
11.38486
7
11.73250
11.73251
7
11.80953
12.60883
8
11.64333
11.96308
8
11.64378
11.64377
8
12.36314
13.29479
9
11.64407
12.45536
9
11.31957
11.31932
9
11.59755
12.55345
10
11.10289
11.52138
10
11.52379
11.52379
10
11.369437
12.10035
In section (a), where we used
during LMS and
Kalman-Gain at secondary noise path, yields better
SNR value. In section (b), where we used Kalman-Gain
during LMS and
at secondary noise path, we almost
got same SNR in novel FxLMS and in our modified
FxLMS algorithm. In section (c), we used Kalman-Gain
during both LMS and secondary noise path. Here we
got slightly better output than that of in section (b), but
in some samples, we got less SNR than novel FxLMS.
So, the operation here is considered unstable.
So, we concluded, if we don’t change
during
LMS operation and use Kalman-gain as a replacement
of
during secondary noise path of FxLMS, we will get
more stable and better output. In figure 4, existing
FxLMS and modified FxLMS plots are shown. In this
particular sample we have shown, having SNR of
FxLMS is 11.706468 and modified FxLMS is
12.226385.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
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Volume 3, 2023
Figure 4. Plot analysis between received noisy signal and filtered signal of (a) FxLMS (SNR = 11.706468), (b) modified FxLMS
(SNR = 12.226385).
As having higher SNR means more information, at
figure 4(b) we got best output. In 4(a), filtered signal
lost more information than that of 4(b). If we analysis
noise residue plots at figure 5, modified FxLMS got
relatively less noise after each iteration, and at the end
of discrete time T, noise figures are way smaller than
before that indicates, our modified FxLMS capable of
reducing noises much efficiently.
Figure 5. Plot analysis of noise residue after each filtering iteration of (a) FxLMS (SNR = 11.706468), (b) modified FxLMS
(SNR = 12.226385).
4. Conclusion
To conclude, this paper demonstrated an
underwater acoustic active noise cancellation (ANC)
system using Filtered-x LMS and kalman filter. Based
on the simulation and test results, proposed model of
modified Filtered-x LMS is able to reduce noise in
received signals. Performance of modified FxLMS is
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
Md. Mostafizur Rahman, Sumonto Sarker
E-ISSN: 2769-2477
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Volume 3, 2023
best when we keep using step size
during LMS and
replace it with kalman gain at secondary noise path
calculation. The future recommendations that can be
taken into consideration is by using various of kalman
filter, multiple frequency tone testing, and also
implementation towards the industrial system.
Acknowledgment
The authors gratefully acknowledge the Institute
of Research and Training (IRT), Hajee Mohammad
Danesh Science & Technology University for their
support.
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The authors equally contributed in the present
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problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.5
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E-ISSN: 2769-2477
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