International Journal of Engineering Trends and Technology (IJETT) – Volume 28 Number 1 - October 2015
A Speech Quality Enhancement Using Time Varying Least
Mean Square and Recursive Least Square Algorithm
1
2
Pratiksha Gupta , Rajesh Mehra , Lalita Sharma
1
3
PG Scholar, 2Associate Professor, 3PG Scholar, Dept. of ECE, NITTTR, Chandigarh, India1
Abstract- Adaptive filters are used to improve the
quality of the speech. Noise free signals are
necessary to enhance the speech quality. To obtain
the noise free signal adaptive interference
cancellation is used. In this method the interfering
signal is subtracted from the corrupted signal. When
the input of adaptive filter is non stationary, we can
measure the performances of the LMS and RLS by
changing the filter coefficient. It gives better results
of the signal after each interaction. The algorithms
mentioned above are the most common and efficient
algorithm that are widely used in this area. This
paper is about the analysis of LMS and RLS in terms
of interference cancellation from speech signal. The
effect of both the algorithm has been measured in
terms of filter length and step size. These algorithms
have been tested for their adaptive noise cancellation
capabilities.
Keywords: LMS, LMS (NTVLMS), Adaptive filter, RLS,
I. INTRODUCTION
Interference cancellation is a technique plays keys
role in the field of signal processing. It is especially
prominent for speech signal transmission and
processing due to the ever expanding application of
telephone
and
mobile
communication.
Communication interference cancellation can be
accomplish with the help of a filter, which self adjust
its own transfer function according to an optimizing
algorithm [Honing and Messerschmitt] [1].So in this
category of such adaptive algorithm is the Least
Mean Square (LMS) algorithm[widows and huff].A
special case of LMS recursion ,which is known as
Normalized Mean Square algorithm .These algorithm
manages the variation in the signal level at the filter
output[2].
In adaptive signal processing, Least Mean Square
(LMS)
is one of the significant algorithms. It was developed
by
widows
and still, used in adaptive signal processing for
its ease of implementation, simplicity ,less
computational complexity, and having good
property. The LMS algorithm is express by following
equation.
T
(n)W(n)
ISSN: 2231-5381
(1)
(2)
Where w(n) is filter coefficient at time n ,µ is step
size parameter, e(n) is adaptation error d(n) is the
desired signal and X(n) is the filter input respectively.
Equation (2) shows that LMS algorithm uses the error
signal e(n) and a step size parameter µto update the
filter coefficient. The selection of µ is very important
to the convergence and stability of the algorithm. The
value of µ has to satisfy by following equation.
(3)
Where tr(R) is the trace of the autocorrelation matrix
of input X and λmax is the maximum Eigen value of
R.
In general, smaller value of step size leads to a small
steady state mis adjustment (SSM) but convergence
rate is slower. Larger Step size gives faster
convergence but large SSM. It is the drawback of
LMS algorithm. so there are various step size LMS
algorithm were proposed to improve the performance
[3] [4].In this article ,RLS, TVLMS[5][6] and
NTVLMS algorithms are used along with LMS
algorithm to implement adaptive noise cancellation
for Speech quality enhancement .Adaptive filter is a
filter that self adjust its transfer function according to
the optimization algorithm Driven by an error
signal[7]. Using adaptive filtering in real time, it is
possible to accurately determine the distance between
the mobile communication node and fixed
communication node[8].
II. VARIABLE STEP SIZE ALGORITHMS
I.NLMS Algorithm
The NLMS algorithm[1] is improved version of LMS
algorithm given by (4)
µ(n) =
(4)
Where, δ 0 and 0
.the value of the constant µ
is divergent from step size parameter in LMS
alogorithm.
II.RVSSLMS Algorithm
The Robust variable step size (RVSS) LMS[4]
algorithm is defined based on variable step LMS
(VSLMS) [5] algorithm. According to the square of
the time averaged estimate of the auto correlation of
error function e(n) and e(n-1),the step size parameter
adjusted in such type algorithm. The estimate is a
time average of e(n) and e(n-1) given by following
equation.
http://www.ijettjournal.org
Page 40
International Journal of Engineering Trends and Technology (IJETT) – Volume 28 Number 1 - October 2015
P(n)
(5)
βp(n-1)+(1-β)e(n)e(n-1)
=
The step size improved equation is given by
following equation
µ(n+1) = αµ(n)+λp2 (n)
where 0<α<1 and λ>0.
(6)
III.TVLMS algorithm
In such type of algorithm the step sixe parameter is
derived by following equation
µ(n)=α(n)µo
(7)
In TVLMS algorithm The step size parameter can be
calculated as follows
µ(n)=α(n)µ0
(8)
µ0 = step size parameter of LMS algorithm
this µ0 can be used to modify µ(n) where as α(n) is
known as decaying factor.
The decaying law can be represented as
here C, a, b are positive constants that determines the
magnitude and the rate of decrease of α(n). According
to the above equation , C has to be a positive number
larger than 1. When C = 1, α(n) will be equal to 1 and
the TVLMS algorithm will be the same as
conventional LMS algorithm.
IV
ADAPTIVE
NOISE
CANCELLATION
SYSTEM
The schematic block diagram of adaptive noise
cancellation as shown by figure (1)
PS
d(n)=s(n)+xo(n)
Inpute(n)
signal
s(n)
Noise
signal
x(n)
Ʃ
X(n)
Wc(n)
Adaptive
Filter
In this system the signal x(n) is processed by the filter
and this filter will automatically adjust its weights by
the above mentioned algorithms with respect to the
error signal e(n). The output signal of the adaptive
filter y(n) is given by
y(n)=
(10)
here
X(n)= [X(n),
N+1) ]T
W(n)=
1(n) ]T
X(n-1),………………X(n-
[W0(n),W1(n),………………Wn-
where N is the order of the filter, wi is the filter
coefficient. The error signal e(n) at the system output
is given by
e (n) = d(n)–y(n)
(11)
It is very important to choose the parameters such as
filter order, initial values of the filter coefficients, and
value of the step size parameter for better
implementation of LMS based adaptive noise
cancellation system.
Y(n)
V. RLS ALGORITHM
SS
Fig. 1
The above system needs two sensors i.e. microphones
so that it can receive corrupted signal and noise
signals separately. In other words a signal s(n) is
transmitted over a channel and is received by the
receiver i e. primary sensor (PS) with uncontrolled
noise x0 (n). Desired signal can be form by
combining the signal s(n) and noise x0 (n) i.e. d(n)=
ISSN: 2231-5381
s(n)+ x0 (n). second signal is obtained by the
secondary sensor (SS) that is actually the noise
signal x(n) which is uncorrelated with the signal but
correlated in some unknown way. It requires that the
noise signal x0 (n) and x(n) have high coherence. This
will act like a limiting factor because the sensor needs
to be separated to avoid speech being included in the
secondary sensor. This large separation is going to
limit the performance of the system. Here we model
the noise path from the noise source to secondary
sensor as unknown FIR channel WC (n). The noise
x(n) applied as an input to the adaptive filter to
produce an output y(n) is close enough to the replica
of
x0(n).
In RLS algorithm is known for its excellent
performance which recursively finds the filter
coefficients that minimize a weighted linear least
squares cost function relating to the input signals[9].
In this algorithm, the input signal are considered
deterministic, while or the LMS and similar they are
considered stochastic. The RLS algorithm calculate at
each instant an exact minimization of the sum of the
squares of the desired signal estimation errors[10].To
initialize the algorithm P(n) should be made equal to
δ-1 where δ is small positive constant.
http://www.ijettjournal.org
Page 41
International Journal of Engineering Trends and Technology (IJETT) – Volume 28 Number 1 - October 2015
Y(n)=w^H (n).u(n)
e(n) =d(n)- y(n)
(12)
(13)
Correct the filter coefficient by using the following
equation:
W^(n+1)=w^(n) +e(n).k^(n)
(14)
where w^(n) is the filter coefficient vector and k^(n)
is the gain vector
k^(n) =
(15)
I) Effect of filter length on LMS
For filter length = 11, around 100th sample,
MSE converges to zero

For filter length = 15, around 110th sample, MSE
converges
to zero.

For filter length = 21, around 120th sample, MSE
converges to zero. For filter length = 32, around
130th sample, MSE converges to zero
λ=forgetting factor and p(n)=the inverse correlation
matrix of input signal.The RLS algorithm uses
following equation to update the inverse correlation
matrix.
P(n+1)= (P(n)-k^(n)uH(n).
(16)
VI.SIMULTION
AND
RESULTS
Fig 3
DISCUSSION
In this paper the two different type of signals are
mixed with the different type of noise signals. We
have taken the speech signal into the consideration.
The same noise signal is applied into the signals and
then various parameters have been analyzed. The
convergence parameter of the LMS and RLS
algorithms are analyzed. The purpose of this section
is to show the behavior and characteristic of these
two algorithms and subsequent section contains the
comparison between these two. Figure show various
graphs of original signal, noise signal and original
signal state after adding noise. Further graph shows
the recovered signals obtained by LMS and RLS. The
lowermost four graphs show the error convergence
of the LMS and RLS algorithm.
III) Effect of filter length on RLS
For filter length = 11, around 15th
MSE converges to zero
For filter length = 15, around 30th
MSE converges to zero.
For filter length = 21, around 40th
MSE converges to zero.
For filter length = 32, around 50th
MSE converges to zero.
sample,
sample,
sample,
sample,
Fig 4
IV) Effect of filter length on FTF
For filter length = 11, around 14th
MSE converges to zero
For filter length = 15, around 29th
MSE converges to zero.
For filter length = 21, around 39th
MSE converges to zero.
For filter length = 33, around 51th
MSE converges to zero.
sample,
sample,
sample,
Fig 5
Fig 2
ISSN: 2231-5381
sample,
http://www.ijettjournal.org
Page 42
International Journal of Engineering Trends and Technology (IJETT) – Volume 28 Number 1 - October 2015
The summary of the observations of the above
figures are given below in the tabulated form.
Sr.
No.
01.
Filter
Length
11
LMS
RLS
110(iterations)
16(iterations)
02.
15
120(iterations)
32(iterations)
03.
23
130(iterations)
38(iterations)
04.
30
120(iterations)
45(iterations)
From the simulation results, it is clear that, when the
filter length is increased ,it shows that number of
iterations of the adaptive algorithms increase to
converge the MSE towards zero .Hence, the
simulation results shows that when the number of
iterations is calculated, then it is conclude that,
adaptive filter with small filter length performs noise
cancellation faster than higher filter length. So filter
length 11 is the best choice for results. Forgetting
factor or exponential weighting factor is an important
parameter of RLS and it controls the stability and the
rate of convergence.
VII. CONCLUSION
The various components of the adaptive noise
eliminator were generated through the simulation.
The software used to simulate the filter is MATLAB
simulink. The efficiency and performance were
measured in terms of various parameters. Different
type of inputs and noise signals has been used to
analyze the behavior. The result of the aforesaid
analysis is that in the LMS and RLS algorithms when
we increase the filter length then it going to increase
the MSE and convergence time also. In this paper we
have compared the results of the two algorithms LMS
and RLS in terms of noise cancellation performance,
convergence time and making the signal to noise
high. It is to be observed that RLS algorithm has
shown better performance than the LMS. If we taken
convergence time in to the consideration the LMS
algorithm will perform best. When there is an abrupt
changes in the amplitude and frequency components
of the signals that are analyzed then RLS algorithm
show poor performance. In the above circumstances
the RLS graph show sudden increase in error while
LMS maintains stability to zero.
REFERENCES
Simon
Haykin
“Adaptive
filter
Theory”,ISBN10:0130901261,Prentice-Hall, pp.231-38320-24,2001.
[2] Simon Haykin “Least Mean Square Adaptive Filters”,ISBN0471-21570-8 Wiley, pp.315,2003.
[3] R.W Harris, D. M Chabries, F.A Bishop “Avariable step size
(VS) adaptive filter algorithm”,IEEE Trans, vol.
ASSP.34,PP.309-319, APRIL1986.
[4] R.H. Kwong And E.W.Johnson “Avariable step size LMS
Algorithm”, IEEE Trans,vol.40,no.7, pp no.1633-1642,
JULY1992.
[1]
ISSN: 2231-5381
[5] W.S Gan,” Fuzzy step size adjustment for the LMS Algorithm”
Elsevier science, signal processing -49, pp no.145-149, JAN
1996
[6] T.Aboulnasr and K. Mayyas “Aroboust variable step size LMS
type analysis and simulation”,IEEE Trans..vol 45
,no.3pp.631-639, MARCH1997.
[7] Srishtee Chaudhary ,Rajesh Choudhary “adaptive filter design
and analysis using least square &least PTH norm”,IJAET vol
6,pp no.836-841
[8] Rajesh Mehra ,Abhishek Singh”Real time RSSI error reduction
in distance estimation using RLS algorithem”IEEE ,pp no.
661-665 ,February 2013.
[9] Shashi Kant Sharma, Rajesh Mehra “LMS and RLS based
adaptive filter design for different signals” IJETE ,pp no.9296, AUGUST 2014.]
[10]Deepika,Isha chauhan “A Simulation & performance analysis
of ANC using Adaptive Filters”IJEIT,pp no.283-287
,AUGUST 2013.
Miss Pratiksha Gupta is currently
associated with Institute of Technology of
CSVTU Bhilai Chhattisgarh , India since
2010.She is currently pursuing M.E
National Institute of Technical Teachers
Training and Research,Chandigarh India.
She has completed her B.E from Pt.
Ravishankar university Raipur,India.She
is having 5.5 years of teaching experience.
Dr.Rajesh Mehra: Dr..Mehra is currently
associated
with
Electronics
and
Communication Engineering Department
of National Institute of Technical
Teachers’
Training
&
Research,
Chandigarh, India since 1996. He has
received his Doctor of Philosophy in Engineering and
Technology from Panjab University, Chandigarh, India in
2015. Dr. Mehra received his Master of Engineering from
Panjab Univeristy, Chandigarh, India in 2008 and Bachelor
of Technology from NIT, Jalandhar, India in 1994. Dr.
Mehra has 20 years of academic and industry experience.
He has more than250papers in his credit which are
published in refereed International Journals and
Conferences. Dr.Mehra has 55 ME thesis in his credit. He
has also authored one book on PLC & SCADA. His
research areas are Advanced Digital Signal Processing,
VLSI Design, FPGA System Design, Embedded System
Design, and Wireless & Mobile Communication. Dr .Mehra
is member of IEEE and ISTE.
Er. Lalita Sharma: Er. LalitaSharma is
currently associated with School of
Engineering & Technology of Shoolini
University, Solan, Himachal Pradesh,
India since 2011. She is currently
pursuing M.Efrom National Institute of
Technical Teachers Training and Research, Chandigarh
India. She has completed her B. Tech from H.P. University,
Shimla, India. She is having six years of teaching and
industry experience. She has three papers in her credit
which are published in refereed International Journals and
Conferences. Here as of interest include Advanced Digital
Signal Processing, VLSI Design and Image Processing.
http://www.ijettjournal.org
Page 43