International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 10 - May 2014
Implementation of Cascaded Adaptive Filters for
Noise Cancellation
Mr. Prashant Singh
Master of technology student,Amity University,Noida
Sector-125,Noida,Uttar Pradesh-201303,India
Abstract- Adaptive filters are the filters that self adjust their
transfer
functions
according
to
the
error
signal
requirement.There are two categories of adaptive filters : Linear
and non linear.Linear adaptive filters are those adaptive filters
that obey the principle of superposition when the parameters are
kept fixed. Non linear adaptive filters are those that do not follow
the principle of superposition.In this paper I have analyzed the
cascaded combination of adaptive filters which would then be
used for the purpose of noise cancellation.I have used the least
mean square algorithm
Keywords- Least Mean Square, cost function,step size,mean
squared error
I. INTRODUCTION
In this paper I have discussed the cascaded combination of the
Least Mean Square algorithm.The Least Mean Square
algorithm has a very important feature that it is extremely
simple in nature. This Least Mean Square algorithm does not
involve the computation and evaluation of the correlation
functions and also it does not need the calculation of the
inverse of the matrices. The Least Mean Square algorithm is
generally the benchmark which serves as the basis for the
other algorithms that are used in the adaptive filtering
processes.The noise cancellation using cascaded adaptive
LMS filter combination is better performed than the adaptive
filter using LMS algorithm.
II.
LMS ALGORITHM
The Least Mean Square algorithm is based on the stochastic
gradient algorithms. The term stochastic gradient algorithm
has been used in this context so as to distinguish the Least
Mean Square algorithm from the steepest descent method in
which the determininstic gradient is used as a recursive
component so as to compute the Wiener filter parameters and
Wiener solution for inputs that are stochastic in nature.
These two processes work together in a feedback loop.This
involves the basic Least mean square algorithm that is applied
to a linear filter and this is used to perform the filtering
process.Secondly, we need a certain mechanism that can help
to modify or alter the filter parameters according to the
necessity of the filtering process.
Fig 1 Least Mean Square algorithm
The Least Mean Square algorithm consists of two basic
processes :
Filtering: This process involves the computation of the output
of a linear filter which is in response to the input to the linear
filter and then this output is compared with a desired response
to generate the estimation error.[4]
Adaptive Filtering: It involves the adjustment of the linear
adaptive filter parameters according to the estimation error
that has been computed.[4]
ISSN: 2231-5381
Fig 2 Signal Flow representation of the LMS Algorithm
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 10 - May 2014
We conclude from this signal flow graph that there are only
2M+1 complex multiplications and 2M complex additions per
iteration.Here, M is the total number of tap weights applied to
the linear transversal adaptive filter.We note that the least
mean square algorithm is basically recursive in nature and this
algorithm averages each value of the estimate during the
process of adaptative filtering.Least Mean Square algorithm is
basically based on two basic situations: Deterministic
environment and non stationary environment.In a
deterministic environment the tap weights assigned to the
linear transversal adaptive filter and the response that we
should obtain(desired response) are deterministic signals and
non stationary environment implies that the process is non
stationary and involves the tracking of the statistical variations
in the environment.[4]
The adaptive noise canceller acts as an adaptive notch filter
and the null point of the adaptive filter is provided by the
angular frequency of the sinusoidal interference applied to the
filter.
The noise canceller can be tuned and the tuning frequency can
be made proportional to the angular frequency of the
sinusoidal interference. The value of the step size parameter is
taken to be a very small value so that the notch in the
frequency response of the adaptive noise canceller comes out
to be extremely sharp at exactly the frequency of the
sinusoidal interference. We generally have a lot more control
over the frequency response of the adaptive noise canceller
than an ordinary notch filter.[4]
The Least Mean Square algorithm involves the use of
parameter M which is the number of tap weight inputs applied
to the adaptive transversal filter and parameter n which is the
step size.[4]
M and n are related as
0 < n < 2/M S (max) [4]
Here , S(max) is the maximum value of the power spectral
density of the tap inputs.
The value of the tap weights has to be assigned arbitrarily or
initialized. The initial value of the tap weight vector is taken
to be 0 ( w(0) is sometimes taken to be 0 ). Using the data
available to us we can calculate the desired response at time
n.Here, the data available is the M by 1 tap input vector at
time n.The estimate calculation comprises of the calculation
of the tap weight input vector at time n+1.[4]
Fig 4 Single frequency adaptive noise canceller using LMS algorithm
IV.
RESULTS
The error is computed as the difference between the desired
response and the product of the input signal and the tap input
weight vector.[4]
III. NOISE CANCELLATION
Fig 5 Design of cascaded adaptive filters
In a cascade adaptive filter design we have provided the signal
output of the first adaptive filter as input to the second
adaptive filter. Here, we have implemented the cascaded
combination of two adaptive filters with Least Mean Square
algorithm.
Fig 3 Adaptive noise canceller
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 10 - May 2014
V.
Steps
Parameters
0 < < 2 / S (max) M
M is the number of taps and is the step size
S(max) is the maximum power spectral density of the tap
inputs
Initialization
Set w (0) = 0
CONCLUSION
I have implemented a cascaded combination of the adaptive
filters using Least Mean Square algorithm.This combination
of cascaded adaptive filters have been used for the application
of the cancellation of the noise signal in the adaptive filters
and to improve the signal clarity by minimizing the error
signal that is the difference between the desired response
signal and the input signal to the adaptive filter.The noise
cancellation performance of the cascaded adaptive filter
combination was better than the single adaptive filter noise
cancellation performance.
Data
u(n) is the tap input signal and it is given.It has dimensions of
M by 1.
d (n) is the desired response at time instant n.
w(n+1) has to be computed
w(n+1) is the tap weight vector estimate at time instant n+1.
REFERENCES
[1] P.S.R.
Diniz , “ Adaptive filtering algorithms and practical
implementation”,1997.
[2] W.D. Davenport ,” An introduction to the theory of random signals and
noise”,1987
[3] Bourrejeny and Gazor,”Performance of LMS based adaptive filters in
tracking a time varying plant”,vol 44,pp 2866-2870 , 1996.
[4] Simon Haykin,” Adaptive Filter theory”,1991.
[5] Howel and Clarkson,” A class of order statistics LMS Algorithm”,vol 40
pp 46-55,1992.
[6] Kim and Wilde,” Performance analysis of the DCT-LMS Adaptive
filtering algorithm”,vol 80, pp 1631 – 1645, 2000.
Computation
e (n) = d (n) – w (n) u (n)
w (n+1) = w (n) + u (n) e *(n)
e(n) is the error estimate of the system. [4]
3.5
3
2.5
Error 2
1.5
1
0.5
0
0
0.001
0.002
0.003
0.004 0.005 0.006
Time (seconds)
0.007
0.008
0.009
0.01
Fig 6 Error minimization for LMS algorithm
3
2.5
2
Error
1.5
1
0.5
0
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (seconds)
0.007
0.008
0.009
0.01
Fig 7 Error minimization for Cascaded LMS Algorithm
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