DESIGN AND EVALUATION OF ADAPTIVE

FILTER USING ALGORITHMS

A project submitted in partial fulfilment of the requirement for the award of the degree of

BACHELOR OF TECHNOLOGY

IN

ELECTRONICS AND COMMUNICATION ENGINEERING

Submitted by

S.CHAITANYA PHANI P.MONIKA REDDY

(1210409158) (1210409140)

B.SREE SHASHANK ATREY

(1210409109) (1210409155)

Under the Esteemed Guidance of

Smt. D. MADHAVIAssociate Professor

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERINGGITAM Institute of Technology

GITAM UNIVERSITY(Estd. u/s 3 of UGC act 1956)

VISAKHAPTNAM-530 045

DECLARATION

We hereby declare that the project entitled “DESIGN AND EVALUATION OF ADAPTIVE

FILTER USING ALGORITHMS” is an original and authentic work done in the Department

of Electronics and Communication Engineering, GITAM Institute of Technology, GITAM

University, Visakhapatnam, submitted in partial fulfilment of the requirements for the degree of

Bachelor of Engineering.

Project members

1210409158 : S.CHAITANYA PHANI

1210409109 : B.SREE

1210409155 :

1210409140 :

SHASHANK ATREY

P.MONIKA REDDY

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

GITAM INSTITUTE OF TECHNOLOGY

GITAMUNIVERSITY(Estd. u/s 3 of UGC act 1956)

VISAKHAPATNAM-530 045

CERTIFICATE

This is to certify that the project work entitled “Evaluation and design of adaptive filters using

different algorithms” is the bonafide record of project work carried out by(S.CHAITANYA

PHANI-1210409158; B.SREE- 1210409109; SHASHANK ATREY-1210409155;

P.MONIKA REDDY 1210409140) submitted in partial fulfillment of the requirements for the

award of the degree Bachelor of Technology in Electronics and Communication Engineering

GIT, GITAM University.

Project Guide Head of the Department

Smt. D. MADHAVI Dr.V. Malleswara Rao

Associate Professor, Head Of the Department,ECE Department, ECE Department ,GITAM University, GITAM University,Visakhapatnam. Visakhapatnam.

ACKNOWLEDGEMENTS

It is a great pleasure for us to express our grateful thanks to our honorable principal Prof. D. Prasada Rao (Ph.D), GITAM Institute of Technology, GITAM University for making project study a part of our curriculum. It intern gave us scope to showcase our theoretical concepts knowledge to practical problems.

We express our deep sense of gratitude and heartfelt thanks to Dr.V.Malleswara Rao (M.E Ph.D.), Head of department, Electronics and Communication Engineering, GITAM University for his cheerful motivation and encouragement at each stage of this Endeavour. We shall always cherish our association. It gave us a scope to have freedom of thought and action.

We record our deep sense of gratitude to our beloved Project guide Smt. D.Madhavi, Associate Professor, Electronics and Communications Department, for continuous effort, stimulating guidance and motivation. We also would like to express our heartfelt thanks to our project in-charge Dr.B.T.Krishna(M.E,Phd) for his profuse assistance.

TEAM MEMBERS 1210409158 : S.Chaitanya Phani

1210409109 : B.Sree

1210409155 :

1210409140 :

Shashank Atrey

P.Monika Reddy

ABSTRACT

In various applications of signal processing and communication we are faced with necessity to remove noise and distortion from the signals. These phenomena are due to time-varying physical processes. One of these situations is transmission of signal from one point to another. The medium known as channel introduces noise and distortion due to variations of its properties. These variations may be slow varying or fast varying. Noise is also caused due to environment. Noise problems in the environment have gained attention due to the tremendous growth of technology that has led to noisy engines, heavy machinery, high speed wind buffeting and other noise sources. The problem of controlling the noise level has become the focus of a tremendous amount of research over the years.

Since the variations are unknown, it is the use of adaptive filtering that diminishes and sometimes completely eliminates the signal distortion. An adaptive filter has the property of self-modifying its frequency response to change the behavior in time domain, allowing the filter to adapt the response to the input signal characteristics change. The most common adaptive filters used during adaptation process are, finite impulse response filters (FIR). In last few years various adaptive algorithms are developed for noise cancellation.

In this project we present an implementation of LMS (Least Mean Square), NLMS (Normalized Least Mean Square), RLMS (Recursive least mean square), BLMS (Block LMS) and signed LMS algorithms on MATLAB. We simulate the adaptive filter in MATLAB with a noisy tone signal and white noise signal and analyze the performance of algorithms.

CONTENTS:

1.INTODUCTION

2.BASIC DEFINITIONS TO BE KNOWN

3.EQUALIZER

4.ADAPTIVE FILTERS

5.WIENER FILTERS

6.GRADIENT SEARCH ALGORITHM

7.STEEPEST DESCENT ALGORITHM

8.LEAST MEAN SQUARE(LMS) ALGORITHM

DESIGN AND EVALUATION OF ADAPTIVE FILTER USING DIFFERENT ALGORITHMS

Communication is nothing but passing of information from one point (source) to another point(destination).The information is nothing but a signal.To get better results like high and faster data transfer rate. But any where noise is a common problem .Noise is nothing but an unwanted signal which gets added with original data signal and this combination is received at receiver. The signal when transmitted through a channel gets corrupted resulting in a noisy output. As a consequence, the user at the output end has to attempt to reconstruct the original signal given the noisy one the channel is the place where noise enters. To remove this noise we use some circuits called filters .Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both. So we use filters to remove noise in the channel used at receiver end. The noise added in signal has unpredictable nature and is not fixed. So to overcome this problem a filter which adjusts itself to noise is needed which is possible only with adaptive filters. Mainly filters are classified as linear and non-linear

Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity. However, in several cases one cannot find an acceptable linear filter, either because the noise is non-additive or non-gaussian.

A   nonlinear filter  is a signal processing device whose output is not a linear function of its input. Nonlinear filters locate and remove data that is recognized as noise. The algorithm is 'nonlinear' because it looks at each data point and decides if that data is noise or valid signal. If the point is noise, it is simply removed and replaced by an estimate based on surrounding data points, and parts of the data that are not considered noise are not modified at all.

FIR filters: A finite Impulse Response (FIR) filter are type of digital filters and consists of weighting sequence (impulse response) among non-recursive digital filters which is finite in length. FIR filters are non recursive digital filters  has been selected for this thesis due to their good characteristics and can be used to implement in any sort of frequency response digitally.The series of multipliers, delays and adders are used for FIR filters’ implementation for filter’s output. The output of the non recursive digital filter is formed from the weighted linear combination of current input and previous value of the input.

BASIC DEFINITIONS TO BE KNOWN :

Mean value:

Mean value or expectation value at a time n of random variable x(n) having a pdf f(x(n)) is given by

Where e{.} stands for expectation operator. we can also use the realizations to obtain value to implement the frequency interpretation formula.

Where N=number of realizations.

Where xi (n) is the i th outcome at the sample index n at i th realization.

Cross correlation :

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. Cross correlation of two random variables is given by:

Autocorrelation:

In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero unless the signal is a trivial zero signal. frequency interpretation of autocorrelation is found using the formula

Standard deviation:

Standard deviation (represented by the symbol sigma, σ) shows how much variation or "dispersion" exists from the average (mean, or expected value). The standard deviation of X is the quantity

EQUALIZERS

In telecommunication, the equalizer is a device that attempts to reverse the distortion incurred by a signal transmitted through a channel.

In digital communications  its purpose is to reduce intersymbol interference to allow recovery of the transmit symbols.It is a simple linear filter or a complex algorithm.

Channel equalization is a simple way of making less severe the harmful, effects caused by a frequency-selective and/or dispersive communication link between sender and receiver. For this demonstration, all signals are assumed to have a digital baseband representation. During the training phase of channel equalization, a digital signal s[n] that is known to both the transmitter and receiver is sent by the transmitter to the receiver. The received signal x[n] contains two signals: the signal s[n] filtered by the channel impulse response, and an unknown broadband noise signal v[n]. The goal is to filter x[n] to remove the inter-symbol interference (ISI) caused by the dispersive channel and to minimize the effect of the additive noise v[n]. Ideally, the output signal would closely follow a delayed version of the transmitted signal s[n]

In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable. ISI is usually caused by multipath propagation or the inherent non-linear frequency response of a channel causing successive symbols to "blur" together. The presence of ISI in the system introduces errors in the decision device at the receiver output. Therefore, in the design of the transmitting and receiving filters, the objective is to minimize the effects of ISI, and thereby deliver the digital data to its destination with the smallest error rate possible. Ways to fight intersymbol interference include adaptive equalization and error correcting codes

Causes for ISI are:

1.Multipath propagation:

One of the causes of intersymbol interference is what is known as multipath propagation in which a wireless signal from a transmitter reaches the receiver via many different paths. The causes of this include reflection (for instance, the signal may bounce off buildings), refraction (such as through the foliage of a tree) and atmospheric effects such as atmospheric ducting and ionospheric reflection. Since all of these paths are of different lengths, this results in the different versions of the signal arriving at the receiver at different times. These delays mean that part or all of a given symbol will be spread into the subsequent symbols, thereby interfering with the correct detection of those symbols. Additionally, the various paths often distort the amplitude and/or phase of the signal thereby causing further interference with the received signal.

2.Bandlimitted channels:

Another cause of intersymbol interference is the transmission of a signal through a bandlimited channel, i.e., one where the frequency response is zero above a certain frequency (the cutoff frequency). Passing a signal through such a channel results in the removal of frequency components above this cutoff frequency; in addition, the amplitude of the frequency components below the cutoff frequency may also be attenuated by the channel.

APPLICATIONS OF EQUALIZERS:

1. To reduce the higher order harmonics

2. To make the original signal faithful to next block of the system

3. Pre-equalization is done at the transmission end so that the signal sent is of correct

specification that receiver needs

4. Post equalization is done at the receiver end so that the signal received is of correct specifications

ADAPTIVE FILTERS:

The usage of adaptive filters is one of the most popular proposed solutions to reduce the signal corruption caused by predictable and unpredictable noise added to the source signal. An adaptive filter has the property of self-modifying its frequency response to change the behavior in time domain, allowing the filter to adapt the response to the input signal characteristics change .The purpose of adaptive filter is noise cancellation is to remove noise from the signal and to improve signal to noise ratio. Mostly used filters are FIR filters. These are preferable because they are stable, and no special adjustments are needed for implementation. The selection of FIR filter is due to coefficient sensitivity, round off noise, stability and suitable for high speed applications.

An adaptive filter is defined by four aspects:

1. The signals being processed by the filter.

2. The structure that defines how the output signal of the filter is computed from its input signal.

3. The parameters within this structure that can be iteratively changed to alter the filter’s input-output relationship.

4. The adaptive algorithm that describes how the parameters are adjusted from one time.

By choosing a particular adaptive filter structure ,one specifies the number and type of parameters that can be adjusted. The adaptive algorithm used to update the parameter values of the system can take on a myriad of forms and is often derived as a form of optimization procedure that minimizes an error criterion that is useful for the task at hand.

Adaptive filters are composed of three basic modules:

1. Filtering structure ◮ Determines the output of the filter given its input samples. ◮ Its weights are periodically adjusted by the adaptive algorithm. ◮ Can be linear or nonlinear, depending on the application. ◮ Linear filters can be FIR or IIR

2. Performance criterion ◮ Defined according to application and mathematical tractability. ◮ Is used to derive the adaptive algorithm. ◮ Its value at each iteration affects the adaptive weight updates.

3. Adaptive algorithm ◮ Uses the performance criterion value and the current signals. ◮ Modifies the adaptive weights to improve performance. ◮ Its form and complexity are function of the structure and of the performance criterion.

Properties They can operate satisfactorily in unknown and possibly time-varying environments

without user intervention. They improve their performance during operation by learning statistical characteristics

from current signal observations. They can track variations in the signal operating environment (SOE).

Applications:

Because of its capability, overall performance and construction facility the adaptive filters have been used in various applications like:

1. Telephone echo cancellation 2. Radar signal processing3. Navigation systems4. Communication channel equalization and5. Biometric signal processing.

Now we design this adaptive filter which changes its filter coefficients following different algorithms.

Following is the block diagram of adaptive noise cancellation system:

Above Figure shows Adaptive Noise Cancellation (ANC) system . The discrete adaptive filter processed the reference signal x(n) to produce the output signal y(n) by a convolution with filter’s weights w(n).The filter output y(n) is subtracted from d(n) to obtain an estimation error e(n). The primary sensor receives noise x1(n) which has correlation with noise x(n) in an unknown way. The objective here is to minimize the error signal e(n). This error signal is used to incrementally adjust the filter’s weights for the next time instant. We use algorithms for minimizing these errors.

y(n)=x(n)*w(n); e(n)=y(n)-d(n);

In this project we use different algorithms like LMS, NLMS, RLMS to design the Adaptive filter and compare output in each algorithm case i.e, ‘equalizer’ application.

In all algorithms the basic algorithm is LMS algorithm. To understand this algorithm in designing adaptive filter we have to know about wiener filter and steepest descent algorithm.

WEINER FILTERS

Wiener filter is a linear optimum discrete time filters. These filters are optimum in sense of minimizing appropriate function of the error, known as the cost function. The cost function that is commonly used in filter design optimization is the mean square error (MSE).Wiener filter is

not an adaptive filter because the theory behind this filter assumes that the inputs are stationary. To minimize this MSE we use wiener filter. Minimizing MSE involves only 2nd order statistics.

Wiener filters are characterized by the following:[3]

1. Assumption: signal and (additive) noise are stationary linear stochastic processes with known spectral characteristics or known autocorrelation and cross-correlation

2. Requirement: the filter must be physically realizable/causal (this requirement can be dropped, resulting in a non-causal solution)

3. Performance criterion: minimum mean-square error (MMSE)

v(n) d(n)

d(n) x(n) d^(n) e(n)

In the process of transmission of information from the source to receiver side in all the channels, noise from the surroundings automatically gets added to the signal. In above diagram d(n) is desired signal, d^(n) is estimated signal that the filter produces, v(n) is noise added, x(n) is received signal at receiver, w(n) represents the filter coefficients and e(n) is the error signal.

According to error the filter coefficients are varied according to algorithm.x(n)=d(n)+v(n);e(n)=d(n)-d^(n);

d^(n)=x(n)*w(n) --------------(1)

=wTx(n)-------convolution

where wT= [w1 w2 ……….wM-1]T ; x(n)=[x(n) x(n-1)………… x(n-M+1)]T

Here the MSE is given by

J(n)=E{e2(n)} ; where E{x(n)} is mean value or expected value

MSE= J(w)=E{e2(n)}

filter w(n)+

=E{[d(n)-d^(n)]2}

=E{[d(n)-wTx(n)]2} ------from (1)

=E{[d(n)- wTx(n)] [ d(n)- wTx(n)]T}

J(w) =E{[d2(n)-2d(n) wTx(n)+ (wT)2 x2(n)]}

= E{d2(n)}-2wT E{d(n) x(n)}+ (wT)2 E{x2(n)}

=σd2 -2wTPdx+ wTE[xT(n) x(n)]w

J(w)= σd2 -2wTPdx+wTRxw

Where

Pdx = [Pdx (0) Pdx (1) Pdx (2) Pdx (3) ………. Pdx (M-1)]T

= cross correlation between x(n),d(n)

Rx=correlation between xT(n),

x(n)=matrix of size n*n with below contents

Here rx(0)=E{x(n)x(n)}, rx(1)=E{x(n)x(n-1)}… rx(-M+1)=E{x(n-M+1)x(n)}, rx(0)=E{x(n-M+1)x(n-M+1)}.

The below shown is the schematic representation of wiener filtering

Now to find optimum Wiener filter coefficients desired signal d^(n) is to be known.

MSE surface is found by inserting different values of w0,w1 in J(w).

Values of coefficients that correspond to bottom of the surface are optimum wiener coefficients .

The vertical distance from w0-w1 plane to the bottom of the surface is known as minimum error Jmin and corresponds to the optimum wiener coefficients.

The below shown is the MSE surface.

WIENER SOLUTION

Using above MSE equation J(w)= σd2 -2wTPdx+wTRxw , now for a 2coefficient filter we get

J(w0 ,w1).To get a minimum value we use mathematical analysis that for an expression/equation to have a minimum value its first derivative(partial derivative) is zero and its second derivative(partial derivative) must be greater zero.The same is given below

∂ J(w0,w1) =0 ; ∂ J(w0,w1) =0 ∂w0 ∂w1

∂ 2 J(w 0,w1) >0 ; ∂ 2 J(w 0,w1) >0 ∂w0

2 ∂w12

In the process of finding MSE we came across the below equation from which we start our analysis.

J(w) =E{[d2(n)-2d(n) wTx(n)+ (wT)2 x2(n)]}

For a two coefficient filter above equation gets modified as

J(w0,w1)= w02 rx(0)+2w0w1 rx(1)+ w1

2 rx(0)- 2w0rdx(0)-2w1 rdx(1)+ σd2

Here rdx(1) is nothing but cross correlation between d(n),x(n).Now applying first partial differential equations we get

2wo0 rx(0)+ 2wo

1rx(1)- 2rdx(0)=0;

2wo1 rx(0)+ 2wo

0rx(1)- 2rdx(0)=0;

Writing above obtained in matrix form using Wiener Hopf equation the modified equation is

Rxwo=Pdx

where‘o’ represents optimum wiener solution for filter

Rx(a)=E{x(n)x(n-a)},

Pdx=E{d(n)x(n)},

M is filter order

If matrix is invertible wo=Rx-1Pdx

Rx is a M*M matrix;

wo,P are M*1 vectors .

Now if second partial derivation is also applied we get

2rx(0)=0 for both the equations. The rx(0)=E[{x(m)x(m)}]= σx2>0,hence it is the minimum value

notation and the surface is concave upward. Therefore ,the extreme is minimum point of the surface.

We obtain the minimum error in the mean-square sense

Jmin = σd2 - woPdx T

= σd2 - Pdx T Rx

-1Pdx

=> The minimum point of error surface is at a distance Jmin above the w-plane.

If no correlation between d(n) and d^(n) i.e,Pdx=0,error= σd2

According to the geometric properties of error surface now the cost function J can be written using J= σd

2 -2wTPdx+wTRxw as

wTRxw-2pTw-(J- σd2)=0

If we set J>Jmin ,the w plane will cut the second order surface of a two coefficient filter along a line whose projection on the w-plane are ellipses.If we apply shift and rotation transformations we obtain the relation

J(w)=Jmin+(w- wo)T Rx(w- wo)

STEEPEST DESCENT METHOD OR NEWTON’S METHOD

The adaptive algorithms are nothing but iterative search algorithms derived from minimizing a cost function with the true statistics replaced by their estimates.One such is this Steepest Descent Algorithm.

The one-coefficient MSE surface is given from above as

J(w)=Jmin+rx(0)(w- wo)2

If we calculate now first and second derivatives of above equation we get

∂ J(w) =2 rx(0)(w- wo); ∂ 2 J(w) =2rx(0)>0∂w ∂w2

At w=w0 the first derivative becomes zero and the second derivative is greater than zero .Hence we get a global minima at the surface and hence it is concave upward

To find the optimum value of w,we use iterative approach.We start with an arbitrary value w(0) and measure the slope of curve J(w) at w(0).Next,we set w(1) to be equal to w(0) plus negative increment of an increment proportional to the slope of J(w) at w(0).Proceeding with an iteration we will eventually find the minimum value w0.The values w(0),w(1),w(2)…….are known as gradient estimates.

GRADIENT SEARCH ALGORITHM:

According to above method the next filter coefficient,filter coefficient at iteration n ,w(n), is found using the relation

w(n+1)=w(n)+µ[-∂/∂w J(w)]|w=w(n)

=w(n)+ µ[-∇ J(w(n))]

w(n+1)=w(n)-2 µrx(0)[w(n)-w0]

Where µ is a constant to be determined called step size factor

Now rearrange the above equation as follows

w(n+1)=w(n)-2 µrx(0)w(n)- 2 µrx(0)w(n)w0

w(n+1)=w(n)[1-2 µrx(0)]+ 2µrx(0) w0

The solution of above difference equation using iteration approach is

w(n)= w0+[1-2 µrx(0)]n [w(0)-w0]

w(n)= wo +(1- 2µ rx(0))n (w(0)- wo) -----(a)

The above equation gives w(n)explicitly at any iteration in search procedure. This is the solution to the gradient search algorithm.If we had initially guessed w(0) the first wiener filter coefficient as optimum value w0 we will make that w(1)= w0 which gives optimum value in single step

To have a convergence value of w(n), we must impose the condition - |1- 2µ rx(0)|<1

The above inequality defines the range of the step size constant µ so that the algorithm will converge.

-1<1- 2µ rx(0)<1 (or ) 0<2 µ rx(0)<2 (or) 0<µ <1/ rx(0)

If µ >1/ rx(0) process is unstable and no convergence takes place.

If the filter co-efficient has a value w(n) at nth iteration, then the MSE surface is

J(w)=Jmin+rx(0)(w(n)- wo)2

Substituting the quantity (w(n)- wo) of (a) we get

J(w)=Jmin+rx(0)(w(0)- wo)2 (1- 2µ rx(0))2n

The above equation show that w(n) wo as n tends to infinity, and the MSE undergoes a G.P towards Jmin and the plot is the performance J(n) vs iteration number n is known as the learning curve.

STEEPEST DESCENT ALGORITHM:

The steepest descent algorithm is an old mathematical tool for numerically finding the minimum value of a function, based on the gradient of that function.   Steepest descent uses the gradient function to determine the direction in which a function is increasing or decreasing most rapidly. The method of steepest descent is a useful tool for signal processing because it can be applied iteratively. We can apply the steepest descent algorithm to the Wiener filter in such a way that at each new step we can calculate a new set of filter coefficients. Using the steepest descent

method, we can approach a minimum error value in relatively few iterations, and we can track a signal that changes in time to apply new minimum coefficients to each new version of the signal.

To find the minimum value of MSE surface,Jmin,using the steepest descent algorithm,we proceed as follows:

1.We start with the initial value w(o),usually using the null vector.

2.At the MSE surface point that corresponds to w(0),we compute the gradient vector ∇ J(w(0)).

3.We compute the value -∇J(w(0)) and add it to w(0) to obtain w(1)

4.we go to step2 and continue the procedure until we find the optimum value of the vector coefficients.

If w(n) is the filter-coefficient vector at step n(time),then it is updated value w(n+1) is given by

w(n+1)=w(n)+µ[-∇J(w(n))]

The gradient vector is equal to

∇J(w(n))=-2Pdx+2Rx w(n)

W(n+1) = w(n) + 2 µ [Pdx - Rx w(n)] = w(n) + µ’[2Pdx+2Rx w(n)]

=[I - µ’ Rx]w(n) + µ’Pdx

The below figure shows the signal flow graph representation of the steepest descent algorithm

Fig. Signal flow grow representation of the steepest descent algorithm.

Where we set µ’= 2µ,and I is the identity matrix.The value of primed step-size parameter must be much less than ½ for convergence.

To apply the method of steepest descent,we must find first estimate the autocorrelation matrix Rx

and cross-correlation vector Px from the data.This is necessary,since we do not have an ensemble of data to find Pdx and Rx

LMS ALGORITHM:

LMS was developed by Hoff and Widrow in 1960.

This algorithm is a member of stochastic gradient algorithms, and because of its robustness and low computational complexity. LMS algorithm has the following most important properties

1.It can be used to solve the Wiener-Hopf equation without finding matrix inversion.Furthermore,itdoesnot require the availability of the auto-correlation matrix of the filter input and cross correlation between the filter input and its desired signal.

2.Its form is simple as well as its implementation,yet it is capable of delivering high performance during adaptation process

3.Its iterative procedure involves:

a)computing the output of an FIR filter produced by a set of tap inputs(filter coefficients)

b)generation of an estimated error by comparing the filter output to a desired response

c)adjusting the tap weights based on the estimation error

4.The correlation term needed to find the values of the coefficients at the n+1 iteration contains the stochastic product x(n)e(n) without the expectation operation that is present in steepest descent method

5.Since the expectation operation is not present,each coefficient goes through sharp variations(noise) during the iteration process.Therefore,instead of terminating at the Wiener solution,the LMS algorithm suffers random variation around the minimum point(optimum value) of the error-performance surface

6.It includes a step-size parameter,µ,that must be selected properly to control stability and convergence speed of the algorithm

7.It is stable and robust for a variety of signal conditions

DERIVATION OF LMS ALGORITHM

Using steepest descent algorithm from above

w(n+1)=w(n)-µ∇J(w(n))

∇J(w(n))=-2Pdx+2Rx w(n)

And Rx=x(n)xT(n);Pdx=d(n)x(n) which are substituted in above equation and after it is added to 1st equation resulting

w(n+1)=w(n)+2µx(n)[d(n)-xT(n)w(n)]

where e(n)=d(n)-y(n) is the error

y(n)=wT(n)x(n) is the output

w(n)=[w0(n) w1(n)…..wM-1(n)]T is filter taps at time n

x(n)=[x(n) x(n-1) ………….x(n-M+1) input data

µ is a constant called step size factor

w(n+1)=w(n)+2µe(n)x(n)

REFERENCES:

1. http://en.wikipedia.org,

2. http://www.mathworks.in,

3.Adaptive Filtering Primer with MATLAB by Alexander D.Poularikas,Zayed M.Ramadan.