VLSI Implementa

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VLSI Implementation of LMS Adaptive Filter

By

Ravikumar.C

(Reg. No. 15803044)

A PROJECT REPORT

Submitted to the department of

ELECTRONICS AND COMMUNICATION

in theFACULTY OF ENGINEERING & TECHNOLOGY

in partial fulfillment of the requirementfor the award of the degree

ofMASTER OF TECHNOLOGY

IN

VLSI DESIGN

S.R.M ENGINEERING COLLEGES.R.M INSTITUTE OF SCIENCE AND TECHNOLOGY

(DEEMED UNIVERSITY)

MAY 2005


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BONAFIDE CERTIFICATE

Certified that this project report titled VLSI Implementation of LMS Adaptive

Filter is the bonafide work of Mr.Ravikumar who carried out the research under my

supervision. Certified further, that to the best of my knowledge the reported herein does

not form any part of any other project report or dissertation on the basis of which a

degree or award was conferred on an earlier occasion on this or any other certificate.

Signature of the Guide Signature of HOD

[Mr.T.VIGNESWARAN] [Dr.S. JAYASHRI]

ABSTRACT

In speech processing noise cancellation and echo cancellation are very much

important. In this project I am going to design adaptive filter which is very applicable in

the above mentioned applications. Adaptive signal processing (a new field in DSP) is a


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relatively new area in which application are increasing rapidly. Adaptive signal

processing evolved from the techniques developed to enable the adaptive control of

time-varying systems.

Since, there is no dedicated IC for adaptive filter. I am going to design the filter

using VHDL code. For designing the adaptive filter I am going to use LMS algorithm.

The LMS algorithm is the most popular for real-time adaptive system implementations.

It is employed in many areas such as modelling, controlling, beamforming and

equalization.

ACKNOWLEDGEMENT

It is my great pleasure to thank our chairman, Thiru T.R.Pachamuthu, and our

director Dr.T.P.Ganesan.


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I take this opportunity to thank our beloved principal Prof. R.Venkatramanai.

I am indebted to our Head of the Department Prof. S. Jayashree who had been

a source of inspiration for all my activities.

I wish to convey my deepest sense of gratitude to my guide

Mr.T.Vigneswaran, M.E., for his valuable guidance.

I also thank our department faculty members for their great support in all my

activities.

Signature of the Student

(C.RAVIKUMAR)

TABLE OF CONTENTS

CHAPTER TITLE PAGE

NO NO

ABSTRACT iiiLIST OF FIGURES vii

LIST OF ABBREVATIONS viii

1. INTRODUCTION 1

2. DIGITAL FILTERS 3

3. STRUCTURE OF DIGITAL FILTERS 5

3.1 ADAVANTAGES OF DIGITAL FILTERS OVERANALOG FILTERS 6


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12.2 FLOATING POINT MULTIPLIER 4512.3 FLOATING POINT ADDER 48

13. REFERENCES 51

LIST OF FIGURES

CHAPTER TITLE PAGE

NO NO

5.1 Transversal Structure 11

5.2 Symmetric Transversal Structure 12

5.3 Lattice Structure 14

8.1 Architecture adaptive filter 20

8.2 General Form Adaptive Filters 21

10.1 A strong narrowband interference N(f) in a wideband signal S(f) 30

10.2 Block diagram for the adaptive filter problem 31

10.3 a. Identification 33

10.3 b. inverse modeling 33

10.3 c. prediction 34

10.3 d. interference cancellation 34

10.4 Example cross section of an error-performance 37

. surface for a two tap filter


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CHAPTER 1

INTRODUCTION

In recent years, a growing field of research in adaptive systems has resulted in a

variety of applications such as communications, radar, sonar, seismology, mechanical

design, navigation systems and biomedical electronics. Adaptive noise canceling has

been applied to areas such as speech communications, electrocardiograph, and seismic

signal processing. Adaptive noise canceling is in fact applicable to a wide variety of

signal enhancement situations, because noise characteristics are not often stationary inreal-world situations.

An adaptive filter is a system whose structure is adjustable in such a way

that its performance improves in accordance with its environment. A simple example of

an adaptive system is the automatic gain control (AGC) used in radio and television

receivers. The function of this circuit is to adjust the sensitivity of the receiver inversely

as the average incoming signal strength. The receiver is thus able to adapt to a wide

range of inputs levels and to produce a much narrower range of inputs intensities. These

systems usually have many of the following characteristics:

1. They can automatically adapt (self-optimize) in the face changing(nonstationary) environment and changing system requirements.

2. They can be trained to perform specific filtering and decision-making tasks.Synthesis of systems having these capabilities can be accomplished

automatically through training. In a sense, adaptive systems can beprogrammed by a training process.


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3. Because of the above, adaptive systems do not require the elaborate

synthesis procedures usually needed for nonadaptive systems. Instead, they

tend to be self designing.

4. They can extrapolate a model of behavior to deal with new situations after

having been trained on a finite and often small number of training signals or

patterns.

5. To a limited extent they can repair themselves; that is they can adapt aroundcertain kinds of internal defects.

6. They can usually be described as nonlinear systems with time-varyingparameters.

CHAPTER 2

DIGITAL FILTERS

Digital filters are very important part of digital signal processing. Infact, their

extra-ordinary performance is one of the key reasons that digital signal processing has

become so popular. Digital filters can be used for:

1. Separation of signals that have been combined.

2. Restoration of signals that have been distorted in someway.

Analog filters can be used for these same tasks. Digital filters can achieve far

superior results. Signal separation is needed when a signal has been distorted with

interference, noise or other signals. For example, imagine a device for measuring the


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electrical activity of a babys heart while still in the womb. The breathing and the

heartbeat of the mother will still corrupt the raw signal. A filter might be used to

separate these signals so that they can be individually analyzed.

Signal restoration is used when a signal has been distorted in some way. For

example, an audio recording made with poor equipment may be filtered in-order to

represent the sound in a better way as it actually occurred. Another example is the

blurring of an image acquired with an improperly focused lens, or a shaky camera.

A digital filter is a numerical procedure or an algorithm that transforms a given

sequence of numbers in to a second sequence that has some desirable properties, such

as less noise or distortion. Else, it can as well be defined as a digital machine that

performs filtering process by the numerical evaluation of linear difference equation in

real time under the program control.

In Radar applications, digital filters are used to improve the detection of

airplanes. In speech processing, digital filters have been employed to reduce the

redundancy in the speech signal so as to allow more efficient transmission and for

speech recognition.

Input sequence to a digital filter can be generated in several ways. One common

method is to sample a continuous time signal at a set of equally spaced time intervals. If

the continuous time signal is denoted by X(t), then the values of the discrete time

sequence are denoted as,

X(nTs)=X(t); t= nTs ------- (2.1)

Where Ts is the sampling period.

The implementation of a digital filter depends on the application. In education

and research, a digital filter is typically implemented as a program on a general-purpose

computer. The types of computers, which can be used, vary widely from personal


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computers through the larger minicomputers to the large time-shared mainframes. In

commercial instrumentation and in industrial applications, the digital filter program is

commonly implemented with the microcomputer that may also be used for control and

monitoring purposes as well. For high speed or large volume applications, such as use

in automobiles for controlling engine operation, the digital filter may consist of special

purpose integrated circuit chips. These circuits perform the computation and storage

functions required for the digital filter operation.

CHAPTER 3

STRUCTURE OF DIGITAL FILTERS

A digital filter consists of three simple elements: adders, multipliers and delays.

The adder and multiplier are conceptually are simple components that are readily

implemented in the arithmetic logic unit of the computer. Delays are components that

allow access to future and past va lues in the sequence. Delays come in two basic

flavors: positive and negative. A memory register that stores the current values of a

sequence for one sample interval, thus making it available for future calculations,

implements a positive delays or simple delays. A negative delay, or advance delay, is

used to look ahead to the next value in the sequence. Advance delays are typically used

in applications, such as image processing, in which the entire data sequence to be

processed is available at the start of processing. So that the advance delay serves to

access the next data sample in the sequence. The availability of the advance delay will

simplify the analysis of digital filters. A digital filter design involves selecting and

interconnecting a finite number of these elements and determining the multipliers co-

efficient values.


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3.1 ADAVANTAGES OF DIGITAL FILTERS OVER ANALOG FILTERS

1. Digital filters are programmable and its design parameters can be easilymodified and implemented through a computer.

2. Temperature will not affect digital filters whereas the passive elements ofanalog filters are sensitive to temperature.

3. Digital filters can handle low frequency signals, unlike analog filters.4. Digital filters are more flexible and are vastly superior in the level of

performance compared to the analog filters.

CHAPTER 4

DIGITAL FILTER DESIGN

The design technique will be forcing the digital filter to behave very closely

with some reference analog filter. The analog filter is analyzed by standard technique to

obtain analog representation in terms of the differential equation. The equation can be

solved easily by analytic means for the unknown voltage, when the input voltage is a

sinusoidal function of time, or an exponential function of time, or a step function of

time. Digital filters allow digital signal processors to work in real time. When the input

signal is a more complicated function of time, though it is possible to obtain an

appropriate solution by numerical methods. The numerical methods provide the basics

of digital filtering.


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Digital filters use a digital processor to perform numerical calculations on

sampled values of the signal. The processor in this case may be specialized with a

digital signal processor chip. They are highly reliable and predictable since they work

with the binary states of zero and one. They are very flexible. The most basic step being

the selection of the filter type based on parameters such as linear phase, efficiency and

stability. Then the filter specifications need to be specified along with the number of

co-efficients involved. The realization structure involved also is analyzed.

4.1 TYPES OF DIGITAL FILTERS

Digital filters are classified as either recursive or nonrecursive. Depending on

whether the filter is recursive or nonrecursive, the digital can be categorized as finite or

Infinite Impulse Response filters.

A nonrecursive filters that has a finite time duration impulse response is known

as Infinite Impulse Response filter or FIR.A recursive filter with an infinite time

duration impulse response is known as Infinite Impulse Response filter or IIR.

4.1.1 FIR FILTER

Finite Impulse Response (FIR) filter is one in which the impulse response h(n)

is limited to a finite number of samples defined over the range (0, N-1)

For an N- tap FIR filter with co-efficients h(k), whose output is described by:

Y(n)= h(0) x(n) +h(1) x(n-1)+h(2) x(n-2)+. . . . . . . .+h(N-1) x(n-N-1) ------- (4.1),

The filters Z transform is,

H(Z) = h(0) Z+ h(1) Z + h(2) Z +. . . . . . . .+ h(N-1) Z ------- (4.2),


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CHAPTER 5

FILTER STRUCTURE

Several types of filter structures can be implemented for the adaptive filters such

as IIR or FIR. The FIR filter has only adjustable zeroes and hence it is free of stability

problems associated with adaptive. IIR filters have adjustable poles as well as zeroes.

An adaptive IIR filter with poles as well as zeroes, makes it possible to offer the same

filter characteristics as the FIR filter but with higher filter complexity. An adaptive FIR

filter can be realized using transversal and lattice structures.

5.1TRANSVERSAL STRUCTURE

The most common implementation of the adaptive filter is the transversal

structure (tapped delay line). Figure 5.1 shows the structure of a transversal FIR filter

with N tap weights (adjustable during the adaptation process).

The filter output signal y(n) is given as,

y(n) = WT(n) U(n) = ? wi(n)u(n-i) ------- (5.1)

where

U(n) = [u(n), u(n-1),..........., u(n-N+1)]Tis the input vector,

W(n) = [wo(n), w1(n)...........WN-1(n)]Tis the weight vector


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u(n)

u( -1) u( -2) u( -N+1)

wo(n) w1(n) w2(n)wN-1(n)

+

Y(n)

Figure 5.1 Transversal Structure

5.2 SYMMETRIC TRANSVERSAL STRUCTURE


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? ?

+

u(n)

u( -1) u( -2) u( -N+1))

+ +

wo(n) w1(n) wN/2-1(n)

y(n)

A transversal filter with symmetric impulse response (weight values) about the

center weight has a linear phase response. The characteristic of linear phase response in

filter is sometimes desirable because it allows a system to reject or shape energy bands

of the spectrum and still maintain the basic pulse integrity with a constant filter group

delay. Image and digital communication are examples where this characteristic is

desirable. The adaptive symmetric transversal structure is shown in the figure 5.2.

Figure 5.2 Symmetric Transversal Structure

An FIR filter with time domain symmetry, such as


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wo(n) = wN-1(n) w1(n) = wN-2(n) ------- (5.2)

has a linear phase response in the frequency domain. Consequently, the number of

weights is reduced by half in a transversal structure, as shown in the figure 5.2 with

even n tap weights.

The tap-input vector becomes

u(n) = [u(n) + u(n-N+1), 8(n-10 + u(n-N+2),........., u(n-N/2+1) + u(n-N/2)]T ------- (5.3)

As a result, output y(n) becomes,

y(n) =? [wi(n) u(n- i) + (n- N + i + 1)] ------- (5.4)

5.3 LATTICE STRUCTURE

The lattice structure offers several advantages over the transversal structure:

1.

The lattice structure has good numerical round-off characteristics that makesit less sensitive than the transversal structure to round-off errors and

parameter variations.

2. The lattice structure orthoganalises the input signal stage-by-stage, whichleads to faster convergence and efficient tracking capabilities when used in

an adaptive environment.

3. The various stages are decoupled from each other, so it is relatively easy to

increase the prediction order if required.

5. The lattice structure is order recursive, which allows adding or deleting ofstages from the lattice without affecting the existing stages.


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6. The lattice filter (predictor) can be interpreted as wave propagation in astratified medium. This can represent an acoustical tube model of the

human vocal tract, which is extremely useful in Digital Signal Processing.]

The lattice filter has a modular structure with cascaded identical stages, as shown

below,

Figure 5.3 Lattice Structure

CHAPTER 6

SAMPLING THEORY

Sampling has a great importance in DSP. The transformation of a signal from

digital to analog and from analog to digital is vital in Digital signal processing. High

sampling rate provides a good frequency response. Sampling theory defines how a

variant quantity can be captured in an instant of time. A signal can be sampled at a set

of points spaced at equal intervals of time. Sampling is the process in which the

continuous time signal is converted into discrete time signal.

Method of sampling is based on:

STAGE

1

STAGE

2

STAGE

m

fo(n)

bo(n)

u(n)

f1(n)

b1(n)

fm(n)

bm(n)


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1. Establishing the rate of sampling.2. Changing an analog signal to a digital one.

3. Changing the digital number to an analog signal.

Nyquist theory is also known as the sampling theory, and it states that in-order

to obtain all information in a signal of one-sided bandwidth B, it must be sampled at a

rate greater than 2B.

CHAPTER 7

HISTORY OF ADATPIVE FILTERS

Until the mid 1960's, telephone channel equalizers were fixed equalizers that

caused fixed performance degradation or they were manually adjustable equalizers that

were cumbersome to adjust.

In 1965, Lucky introduced the Zero-Forcing algorithm for automatic adjustment

of the equalizer weights. This algorithm minimizes a certain distortion, which has the

effect of forcing the intersymbol interference to zero. This breakthrough by Lucky

inspired the other researchers to investigate different aspects of adaptive equalization

problem, leading to new improved solutions.

The gain in popularity of adaptive signal processing is primarily due to

advances in the digital technology that have increased computing capabilities and

therefore broadened the scope of Digital Signal Processing as a whole.


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If the signal is deterministic with known spectrum and this spectrum does not

overlap with that of the noise, then the signal can be recovered by the conventional

filtering techniques. However, this situation is very rare. Instead, we are often faced

with the problem of estimating an unknown random signal in the presence of noise.

This is usually accomplished so as to minimize the error in the estimation according to

a certain criterion. This leads to the area of adaptive filtering.

For example, when a telephone call is initiated, the transfer function of the

telephone channel is unknown, and the connection may involve signal reflections that

produce undesirable echoes. This problem can be corrected with a linear filter once the

transfer function is measured. The measurement can be made, however, only after the

system is running. Therefore, the correction filter must be designed by the system itself,

and in a reasonably short time. This can be done by using adaptive filters.

8.1 OPERATION

For designing the adaptive filter, we have used Least Mean Square (LMS)

algorithm. The LMS adaptive algorithm has found many applications in noise

cancellation, line enhancing, etc., to design our filter, we used conventional Look-ahead

technique that uses serial LMS algorithm. There is also another technique called

Relaxed Look-ahead technique that uses delayed LMS algorithm. The Relaxed Look-

ahead technique does not maintain input-output mapping and does not result in a final

unique architecture. Convergence characteristics would also differ for different

combinations. These disadvantages are overcomes in Conventional Look-ahead

technique.


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We designed a fourth order adaptive filter. Designing the adaptive filter does

not require any other frequency response information or specification. To define the

self learning process, the filter uses the adaptive algorithm, used to reduce the error

between the output signal y(k) and the desired signal d(k). [Initially the weight values

are initialized to zero].

When the LMS performance criteria for e(k) have achieved its minimum value

through the iterations of the adaptive algorithm, the adaptive filter has finished its

weight updation process and its coefficients have converged to a solution. Now the

output from the adaptive filter matches closely the desired signal d(k).

When you change the input data characteristics, sometimes called the filter

environment by generating a new set of coefficients for the new data. Notice that when

e(k) goes to zero and remains these, we achieve perfect adaptation, but not likely in the

real world[2].


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XXX

+ + +

XX

+++ +

XX

W1(n W2(n)

0

WN(n)

y(n)

u(n)

F-BLOCK

d(n)

e(n)

m

WUD BLOCK

Figurer 8.1 architecture adaptive filter


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8.2 GENERAL FORM

An adaptive filter is a filter containing the co-efficients that are updated by an

adaptive algorithm to optimize the filter's response to a desired performance criterion.

Figure 8.2 General Form Adaptive Filters

In general, adaptive filters consists of two distinct parts: a filter, whose structure

is designed to perform a desired processing function: and an adaptive algorithm, for

adjusting the co-efficients of that filter to improve its performance, as illustrated in the

figure 8.2

The incoming signal, u(n), is weighted in a digital filter to produce an output

y(n). The adaptive algorithm adjusts the weights in the filter to minimize the error, e(n),

between the filter output, y(n) and the desired response of the filter, d(n). Because of

their robust performance in the unknown and time-variant environment, adaptive filters

have been widely used from telecommunications to control applications.

X(n)

Y(n)

+

-

_

en =xn-xn

Adaptive

Filter

Adaptive

Algorithm


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An adaptive filter can eliminate phase errors that arise from distortion in the

propagating medium or from the signal processing system. Suppose that a distortion of

a test impulse u(t) = ? (t), which has a uniform amplitude as a function of frequency

when it passes through a transmission medium, is H(? ), where

H(? ) = A(? ) exp [-j? (? )] ------- (8.1)

and A(? ) is a real function.

If a compensating matched filter is constructed with a response H*(? ), the

resultant output, when a signal U(? ) is passed through the medium, will be

Y(? ) = U(? ) H(? ) H*(? ) = A2(? )U(? ) ------- (8.2)

All phase errors due to transmission medium are removed by this process, and

the output of a test impulse, after passing through the distortion medium and the

matched filter, has no phase variation with frequency. The transform of this output will

be y(t), and the combined response of the transmission system and the equalizer will be

the autocorrelation function of h(t) convolved with u(t).

By removing the phase errors an output at t=0 corresponds to the peak of the

autocorrelation function of h(t) and can be obtained from an impulse input

[u(t) = ? (t)]. If H(? ) H*(? ) is constant in amplitude over a finite frequency range ? , so

that H(? ) = ? [(? -? o) / ? ], the output with an impulse input will be a sinc function of

time. If there are major peaks in the frequency spectrum, the time response will exhibit

service ringing, but provided that the input spectrum is reasonably smooth (ideally, a

Gaussian function).


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The time domain response corresponding to H(? )H*(? ) should be reasonably

compact with no severe ringing. Suppose that a reference impulse is sent through a

distorting medium. If a distorted pulse h(t) stored in the storage correlator, phase

distortion caused by the medium can be removed from it. Now any other signal u(t),

sent along the same path into the correlator will be correlated with the stored reference

H*(? ) or h(-t) and will have its phase distortion removed.

8.3 CHARACTERISTICS

An adaptive system is one that is designed primarily for the purpose of adaptive

control and adaptive signal processing. Such a system usually has some or all of the

following characteristics:

1. Firstly they can automatically adapt in the face of changing environmentsand changing system requirements.

2. Secondly they can be trained to perform specific filtering and decision-

making tasks, i.e., they can be programmed by a training process. Because

of this adaptive systems do not require the elaborate synthesis procedures

usually needed for non-adaptive systems. Instead, they tend to be self

designing.

3. Thirdly, they can extrapolate a model of behaviour to deal with new

situations after having been trained on a finite small number of training

signals or patterns.


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4. Fourthly, to a limited extent they can repair themselves, i.e., adapt around

certain kinds of internal defects.

5. Finally, they are more complex and difficult to analyze than non-adaptive

systems, but they offer the possibility of substantially increased system

performance when input signal characteristics are unknown or time varying.

By an adaptive, we mean a self-designing device that has the following

characteristics:

1. It contains a set of adjustable filter co-efficients.2. The co-efficients are updated in accordance with an algorithm.3. The algorithm operates with arbitrary initial conditions; each time new

samples are received for the input signal and the desired response,

appropriate corrections are made to the previous values of the filter co-

efficients.

4. The adaptation is continued until the operating point of the filter on the error

performance surface moves close enough to the minimum points.

CHAPTER 9

ALGORITHMS

Two types of adaptive algorithms are discussed in this section:

1. Recursive least square algorithm (RLS)


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2. Least mean square algorithm (LMS)RLS algorithm provides faster convergence but has more computational

complexity. LMS algorithms based on gradient-type search for tracking time-varying

signal characteristics.

9.1 RLS ALGORITHM

In the Recursive Least Square Algorithm, the order of operations are

1. Compute the filter output2. Find the error signal3. Compute the kalman gain vector4. Update the inverse of the correlation matrix5. Update the weights

9.2 LMS ALGORITHM

LMS algorithm has simplicity, which is its major advantages, and hence it is

widely used in many applications. The LMS algorithm is described as,

W(n+1)=W(n)+m*e(n)*U(n) ------- (9.1)

e(n) = d(n)-WT(n)*U(n) ------- (9.2)


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where

W(n) = {w1),w2(n), . . . . . . . .WN(n)}T

U(n) = {u(n),u(n-1), . . . . . . . .,u(n-N+1)}T

m is the adaptation constant ,

d(n) is the desired output ,

e(n) is the error obtained during every iteration.

As we specify m smaller, the correction to filter weights gets smaller for each

sample and the LMS error falls more slowly. Larger m changes the weights more each

step and hence the error falls more rapidly, but the resulting error does not approach the

ideal solution closely.

9.2.1 MSE CRITERION

The adaption algorithm uses the error signal,

e(n) = d(n) y(n) ------- (9.3)

where

d(n) is the desired signal

y(n) is the filter output

The input vector u(n) and e(n) are used to update the adaptive filter co-efficients

according to a criterion. The criterion employed in this section is the Mean Square

Error (MSE) e:

e = E[e2 (n)] ------- (9.4)

we know that,


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approximate solutions. In this algorithm, the next weight vector W(n+1) is increased by

a change proportional to the negative gradient of mean-square error performance.

W(n+1) = W(n) m? (n) ------- (9.10)

Where m is the adaptation step size that controls the stability and convergence

rate. For the LMS algorithm, the gradient at the nth iteration, ? (n), is estimated by

assuming squared error e2(n) as an estimate of the MSE in equation (9.3). Thus the

expression for the gradient estimate can be simplified to,

? (n) = d[e2 (n)] / dW(n) ------- (9.11)

= -2e(n) U(n) ------- (9.12)

Substitution of this instantaneous gradient estimate into (9.10) yields the

Windrow- Hoff LMS algorithm,

W(n+1) = W(n) + 2me(n)U(n) ------- (9.13)

where 2m in the above equation is replaced by m in practical implementation.

m should be selected such that,

0


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CHAPTER 10

ADAPTIVE FILTER OVERVIEW

Adaptive filters learn the statistics of their operating environment and

continually adjust their parameters accordingly. This chapter presents the theory of the

algorithms needed to train the filters.

10.1 INTRODUCTION

In practice, signals of interest often become contaminated by noise or other

signals occupying the same band of frequency. When the signal of interest and the

noise reside in separate frequency bands, conventional linear filters are able to extract

the desired signal .However, when there is spectral overlap between the signal and

noise, or the signal or interfering signals statistics change with time, fixed coefficient

filters are inappropriate.

.


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Figure 10.2 Block diagram for the adaptive filter problem.

The discrete adaptive filter accepts an input u(n) and Produces an output y(n) by

a convolution with the filters weights, w(k). A desired reference signal, d(n), is

compared to the output to obtain an estimation error e(n). This error signal is used to

incrementally adjust the filters weights for the next time instant. Several algorithms

exist for the weight adjustment, such as the Least-Mean-Square (LMS) and the

Recursive Least-Squares (RLS) algorithms .The choice of training algorithm is

dependent upon needed convergence time and the computational complexity available,

as statistics of the operating environment.

10.3 APPLICATIONS

Because of the ir ability to perform well in unknown environments and track

statistical time-variations, adaptive filters have been employed in a wide range of fields.

However, there are essentially four basic classes of applications for adaptive filters.

These are: Identification, inverse modeling, prediction, and interference cancellation,

with the main difference between them being the manner in which the desired response

is extracted. These are presented in figure 10.3 a, b, c, and d, respectively. The

adjustable parameters that are dependent upon the applications at hand are the number

of filter taps, choice of FIR or IIR, choice of training algorithm, and the learning rate.

Beyond these, the underlying architecture required for realization is independent of the

application. Therefore, this thesis will focus on one particular application, namely noise

cancellation, as it is the most likely to require an embedded VLSI implementation. This

is because it is sometimes necessary to use adaptive noise cancellation incommunication systems such as handheld radios and satellite systems that are contained

on a single silicon chip, where real-time processing is required. Doing this efficiently is

important, because adaptive equalizers are a major component of receivers in modern

communications systems and can account for up to 90% of the total gate count.


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Figure 10.3 a Identification

Figure 10.3 b inverse modeling


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Figure 10.3 c prediction

Figure 10.3 d interference cancellation

.


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Also, let Prepresent the cross-correlation vector between the tap inputs and the

desired response d(n):

p = E [u(n)d *(n)], ------- (10.8)

which expanded is:

p = [p(0), p(-1),,p(1-M)]T ------- (10.9)

Since the lags in the definition of are p either zero or negative, the Wiener-Hopf

equation may be written in compact matrix form:

RwO= P, ------- (10.10)

with wO stands for the M-by-1 optimum tap-weight vector, for the transversal

filter. That is, the optimum filters coefficients will be:

wO = [wO0 , wO1, , wO, M-1 ]T-------(10.11)

This produces the optimum output in terms of the mean-square-error, however if

the signals statistics change with time then the Wiener-Hopf equation must berecalculated. This would require calculating two matrices, inverting one of them and

then multiplying them together. This computation cannot be feasibly calculated in real

time, so other algorithms that approximate the Wiener filter must be used.


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10.4.2 METHOD OF STEEPEST DESCENT

With the error-performance surface defined previously, one can use the method

of steepest-descent to converge to the optimal filter weights for a given problem. Since

the gradient of a surface (or hypersurface) points in the direction of maximum increase,

then the direction opposite the gradient ( ) will point towards the minimum point of

the surface. One can adaptively reach the minimum by updating the weights at each

time step by using the equation

w n+1 = w n+ ( n ), ------- (10.12)

where the constant is the step size parameter. The step size parameter

determines how fast the algorithm converges to the optimal weights. A necessary and

sufficient condition for the convergence or stability of the steepest descent algorithm is

for to satisfy,

0


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10.4.3 LEAST-MEAN-SQUARE ALGORITHM

The least-mean-square (LMS) algorithm is similar to the method of Steepest-

descent in that it adapts the weights by iteratively approaching the MSE minimum.

Widrow and Hoff invented this technique in 1960 for use in training neural networks.

The key is that instead of calculating the gradient at every time step, the LMS algorithm

uses a rough approximation to the gradient. The error at the output of the filter can be

expressed as,

en = dn wT

n un , ------- (10.14)

which is simply the desired output minus the actual filter output. Using this

definition for the error an approximation of the gradient is found by

Substituting this expression for the gradient into the weight update equation

from the method of steepest-descent gives

wn+1 = wn +2 .en un , ------- (10.15)


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which is the Widrow-Hoff LMS algorithm. As with the steepest-descent

algorithm, it can be shown to converge for values of less than the reciprocal of ?max,

but ?max may be time-varying, and to avoid computing it another criterion can be used.

This is

where M is the number of filter taps and Smax is the maximum value of the

power spectral density of the tap inputs u. The relatively good performance of the LMS

algorithm given its simplicity has caused it to be the most widely implemented in

practice. For an N-tap filter, the number of operations has been reduced to 2*N

multiplications and N additions per coefficient update. This is suitable for real-time

applications, and is the reason for the popularity of the LMS algorithm[4].

CHAPTER 11

CONCLUSIONS

This paper presents the development of algorithm, architecture and

implementation for speech processing using FPGAs. The VHDL code developed is

RTL compliant and works for using Xilinx tools. Adaptive filter is a filter that vary in

time, adapting their coefficients according to some reference using LMS algorithm. We

are often faced with the problem of estimating an unknown random signal in the

presence of noise. This is usually accomplished so as to minimize the error in the

estimation according to a certain criterion. This leads to the area of adaptive filtering.


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CHAPTER 12

SYNTHESIS REPORT

12.1. LMS REPORT

=============================================================Final Report

=============================================================

Final ResultsRTL Top Level Output File Name : lmsvhd.ngr

Top Level Output File Name : lmsvhdOutput Format : NGCOptimization Goal : Speed

Keep Hierarchy : NO

Design Statistics# IOs : 1

Cell Usage:

=============================================================

Device utilization summary:---------------------------

Selected Device : 2s15tq144-5

TIMING REPORT

NOTE: THESE TIMING NUMBERS ARE ONLY A SYNTHESIS ESTIMATE.

FOR ACCURATE TIMING INFORMATION PLEASE REFER TO THE TRACEREPORT

GENERATED AFTER PLACE-and-ROUTE.


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Clock Information:

------------------No clock signals found in this design

Timing Summary:

---------------Speed Grade : -5

Minimum period: No path foundMinimum input arrival time before clock: No path found

Maximum output required time after clock: No path foundMaximum combinational path delay: No path found

Timing Detail:

--------------All values displayed in nanoseconds (ns)

CPU: 56.64 / 57.50 s | Elapsed : 56.00 / 57.00 s

-->

Total memory usage is 115620 kilobytes

12.2 FLOATING POINT ADDER

=============================================================Final Report

=============================================================

Final ResultsRTL Top Level Output File Name : floatadd.ngr

Top Level Output File Name : float addOutput Format : NGCOptimization Goal : Speed

Keep Hierarchy : NO

Design Statistics : 96# IOs

Macro Statistics:# Multiplexers : 46


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# 2-to-1 multiplexer : 46# Logic shifters : 46

# 23-bit shifter logical left : 23# 24-bit shifter logical right : 23

# Adders/Subtractors : 49# 24-bit adder carry out : 1# 26-bit subtractor : 1

# 8-bit adder : 1# 8-bit subtractor : 46# Comparators : 29

# 23-bit comparator equal : 1# 23-bit comparator less : 1

# 8-bit comparator equal : 1# 8-bit comparator greater : 25# 8-bit comparator less : 1

Cell Usage:# BELS : 2 995

# GND : 1# LUT1 : 8# LUT2 : 85

# LUT3 : 652# LUT4 : 1356

# MUXCY : 431# MUXF5 : 41# VCC : 1

# XORCY : 420

# IO Buffers : 96# IBUF : 64# OBUF : 32

=============================================================

Device utilization summary:---------------------------


Number of Slices : 1305 out of 192 679% (*)Number of 4 input LUTs : 2101 out of 384 547% (*)Number of bonded IOBs : 96 out of 90 106% (*)

CPU: 80.20 / 81.20 s | Elapsed: 80.00 / 81.00 s


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12.3 FLOATING POINT MULTIPLIER

=============================================================

Final Report

Final Results

RTL Top Level Output File Name : floatmult.ngrTop Level Output File Name : floatmult

Output Format : NGCOptimization Goal : SpeedKeep Hierarchy : NO

Design Statistics

# IOs : 96

Macro Statistics:# Multiplexers : 23# 2-to-1 multiplexer : 23

# Logic shifters : 23# 23-bit shifter logical left : 23

# Adders/Subtractors : 26# 8-bit adder : 2# 8-bit subtractor : 24

# Multipliers : 1# 24x24-bit multiplier : 1


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Cell Usage:

# BELS : 3490# GND : 1# LUT1 : 55

# LUT2 : 322# LUT3 : 464

# LUT4 : 712# MULT_AND : 276# MUXCY : 775

# MUXF5 : 117# VCC : 1

# XORCY : 767# IO Buffers : 96# IBUF : 64

# OBUF : 32

Device utilization summary:


Number of Slices : 867 out of 192 451% (*)

Number of 4 input LUTs : 1553 out of 384 404% (*)Number of bonded IOBs : 96 out of 90 106% (*)

=============================================================

CPU: 35.66 / 36.51 s | Elapsed: 35.00 / 36.00 s

-->

Total memory usage is 79780 kilobytes


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Figure 12.2 synthesized FLOATMULT Architecture

CHAPTER 13

REFERENCES

1. Thomas Kailath, Adaptive Filter Theory Fourth edition, Pearson Education Asia,

2002.

2. K.Parhi. VLSI digital signal processing systems, design and implementation.

John Wiley and sons, 1999.

3. S.Palnitkar. Verilog HDL, a guide to digital design and synthesis. Prentice Hall.

4. Nabeel shirazi, Al Walters, and Peter Athanans, Quantitative Analysis of Floating

Point Arithmetic on FPGA Based Custom Computing Machines.

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