Kernel Adaptive Filtering Method for Noise ReductionAkshay Nagashetti
Abstract An active noise cancellation system have been built
and implemented. Speech and ultrasound data were both used to
verify the system. MATLAB/Simulink is used to design and test
a least mean square (LMS) and a recursive least square (RLS)
adaptive lter for the project. It requires more careful veri cation
for numerous simulations for the accuracy of design. Thus results
obtained with the nite precision from the MATLAB model
to ne-tune the lter.
There are four types of FIR structures
were investigated, we investigate the enhancement of speech
by applying kernel adaptive lter. Removal of noise is very
necessary for many applications such as conversation of telephone
, recognition of speech , etc. Further Kernel methods have been
showing good results for other applications like recognition of
handwriting , weightings of inverse distance , etc.
In this paper mainly focus for the enhancement of audio signal
for speech from noise and then compare Least Mean-Square
(LMS) and Recursive Least Square (RLS) algorithms.
Index Terms Adaptive noise cancellation (ANC), LMS algo-
A DAPTIVE lters are one of the best useful in cases where
conditions of signal or parameters of system are slower
changes and for lters for the adjustment to compromise for
the change. The simple but very powerful lter can be called
as linear adaptive combiner, for which will be nothing more
compared to adjustable FIR lter. The LMS criterion can be a
search algorithm which will be using for providing the strategy
for adjustment for the coef cients of lters.
In the FIR and IIR conventional digital lters, which will be
assumed for the parameters process for determination of the
lter characteristics is known. They will be changed according
to time, but the nature of the variation is assumed to be
known. In various practical problems, the coef cients of an
adaptive lter can be adjusted to compensate for changes in
input signal, output signal or system parameters. In spite of
being rigid, an adaptive system could be learned by the signal
characteristics and track slow changes. An adaptive lter will
be very helpful for the uncertainty about the characteristics of
a signal or when these characteristics change.
II. N O I S ECA N C E L L AT I O N U S I N G KE R N E L ADA P T I V E
The Single channel for feedback adaptive Active Noise
Cancellation (ANC) system works by the process of the noise
acoustical (the target noise) that could be liked for reducing, by
the production of an anti-noise which will cancels out for the
noise component by the method of this adaptive ltering. Thus,
the main aim of an ANC system is for reduction component of the noise for the signal of interest. Figure 1 shows schematic
diagram of a single channel feedback active noise cancellation
system Fig. 1. Single channel feedback active noise cancellation
The anti-noise waveform will be similar for that of the target
noise, except that its phase can be reversed by 180 degrees.
If such of the waveforms will be combined together. It will
also result in a much weaker residual waveform (this waveform
would be of zero amplitude if the anti-noise matches perfectly
The residual waveform can be what is picked up by the
microphone is shown in gure2. The feedback ANC system
produced an anti-noise by predicting the incoming target noise Fig. 2. Description of control active noise
There can be a lot of approach reported in the literature
relating for enhancement of speech. Since for the last ten years,
adaptive lters shall be effective and popular approaches for
the enhancement of speech. The main advantage of Adaptive
lters is that the detection of time varying potentials and also
tracking the dynamic variations of the signals. Before going
for the LMS an introduction to adaptive lter is given as
follows. As the name is adaptive lters it can be important
to understand the meaning of the terms adaptive and lter in a
very general sense. The adjective adaptive can be understood
by considering a system which is tried to adjusting itself
so as to respond for some phenomenon that is taking place
in its surrounding. In some words the system tries to make
adjustment in the parameters with a aim of meeting some well
de ned goals or target on which depends upon the state for
the system as well as its surrounding. This is what adaptation
means. Such that there will be a need to have a set of steps or
certain procedure by which this process of adaptation is carried
out. Thus the system in which that will carry out or undergoes
the process of adaptation is called by the more technical name
III. N O I S ECA N C E L L AT I O N U S I N G KE R N E L ADA P T I V E
A. Least Mean Square (LMS) Algorithm The Least Mean Square (LMS) algorithm, brings into use
for the rst time by Widrow and Hoff, is an adaptive algorithm
using kernel techniques. LMS algorithm can be used the
calculation of the gradient vector in the data available. The
LMS includes an computational procedures in making the
corrections of the weighted vector for the direction of the
negative of the gradient vector which can be eventually may
leads to the minimum mean square error. When Comparing
to other algorithms, the LMS algorithm can be considered
for simpler because of it which will not require correlation
functions for calculations nor does it require matrix inversions.
In the LMS algorithm, the coef cients are adjusted from
sample to sample in such a way as to minimize the Mean
Square Error (MSE).The LMS is based on the steepest descent
algorithm where the weight vector is updated from sample to
k are the weight and the true gradient vectors,
respectively at the kth sampling instant. controls the stability
Initially, set each weight to an arbitrary xed value as 0.
For each subsequent sampling instant, k, carryout steps
Compute lter output
Compute the error estimate
Update the next lter weights
Because of its simplicity, the LMS algorithm is one of the
popular adaptive algorithm. However, the LMS algorithm is
very slow and data dependent convergence behaviour. One of
the primary disadvantages of the LMS algorithm is having
a xed step size parameter for every iteration. This requires
an understanding of the statistics of the input signal prior to
commencing the adaptive ltering operation. In practice this
is rarely achievable. Fig. 3. Flowchart of LMS algorithm
Implementation of Least Mean Square Algorithm for noise
Initially, the weight parameter (w) and also the loop variable
are set to zero. Then obtained input signals from the micro-
phone. In the next step the lter output is calculated and is
further used to compute the error estimate signal.
Then the lter weights are updated by xing the step size
value () to be a constant. This procedure is repeated until
the loop parameter becomes equal to the buffer size. This
implementation is depicted in gure 3.
Condition for stability is: 0? =mu? 2 (input signal power)
Larger values for step size
Increases adaptation rate (faster adaptation)
Increases residual mean-squared error
B. Recursive Least Square (RLS) Algorithm Recursive least square (RLS) is another algorithm for adap-
tive lters. This algorithm attempts to directly update the auto
and cross-correlation matrices in order to approach the Wiener-
RLS is relatively complex algorithm as compared to LMS
and NLMS algorithm. Also performance of RLS in terms of
convergence and Mean Square Error (MSE) is better when
The Recursive least square (RLS) adaptive lter is an al-
gorithm which recursively determines the lter coef cients
that reduces a weighted linear least squares cost function
relating to the input signals. The RLS algorithms are known
for their excellent performance when working in time varying
environments but at the cost of an increased computational
complexity along with stability problems.
RLS adaptation algorithm with input signals y(n) and x(n)
is given below. Initial values for RLS algorithm is given
For m = 1,2, lter gain update vector is given by
Error signal equation is given by
(n 1)y(n ) (6)
Filter coef cient adaptation is given by
Inverse correlation matrix update is calculated using
[P (n 1) (
1)k(n )y T
(P (n 1))] (8)
IV. E X P E R I M E N TA L RE S U LT S Fig. 4. Mean Square Error for LMS Fig. 5. Variation Of MSE With Respect To
V. CO M PA R I S I O N O F AL G O R I T H M S
We compare the LMS and RLS algorithm and conclude
which adaptive algorithm is suitable for adaptive lter. Both
computational resource and convergence speed requirements
when choosing an adaptive lter algorithm are important. The
key difference is that LMS algorithm is a Markov process. It
has its present state, but other than that it doesnt remember
data from the past. For time varying signals this is a character-
istic because past data will give you invalid information about
The RLS algorithm uses all of the information, present
and past, but that can be a problem if the past data is
ambiguous for the current parameters. If researcher looking for
a quantitative rule for when to use one or the other, we don’t
have one. RLS algorithm is more computationally intensive
than LMS algorithm, so if LMS is good enough, then that is
the secure one to go with. RLS algorithm converges faster, but
it is more computationally intensive and has the time varying
disadvantage. Fig. 9. Performance Comparison of Kernel Adaptive Algorithm
Active noise cancellation (ANC) scheme employs the adap-
tive digital lter to generate control signals. The adaptive lter
updates its coef cients iteratively to track the best possible
solutions using adaptive algorithms LMS was the simplest
and easiest to implement but it converges at the slowest
rate RLS has rapid rate convergence, compared to LMS.
RLS is computationally more expensive than LMS. The RLS
algorithms are known for their excellent performance when
working in time varying environments but at the cost of
an increased computational complexity and some stability
problems.We have studied the trade-offs between theMSE
performance and the complexity for several state-of-the-art
kernel adaptive ltering algorithms, on three benchmark data
sets. The proposed gures of merit are meaningful indicators
of the relative performances of these algorithms: Since they
allow us to highlight advantages and disadvantages of each
algorithm in different scenarios, they constitute an interesting
tool for the practitioner.As expected, we observed that there is
not a single best performing algorithm for all scenarios. Rather,
the optimal choice of algorithm depends on the target MSE
range, the available computational resources and the particular
data set. In future work we plan to include additional measures,
such as as the convergence speed, which may be of interest
in scenarios with less restrictions on complexity.
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