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1. INTRODUCTION

One of the most important advantages of the digital transmission systems for voice,

data and video communications is their higher reliability in noise environment in comparison

with that of their analog counterparts. Unfortunately most often the digital transmission of

information is accompanied with a phenomenon known as intersymbol interference (ISI) .

Briefly this means that the transmitted pulses are smeared out so that pulses that correspond

to different symbols are not separable. Depending on the transmission media the main causes

for ISI are:

• cable lines – the fact that they are band limited;

• cellular communications – multipath propagation

Obviously for a reliable digital transmission system it is crucial to reduce the effects of ISI

and it is where the equalizers come on the scene. The need for equalizers arises from the fact

that the channel has amplitude and phase dispersion which results in the interference of the

transmitted signals with one another. The design of the transmitters and receivers depends on

the assumption of the channel transfer function is known. But, in most of the digital

communications applications, the channel transfer function is not known at enough level to

incorporate filters to remove the channel effect at the transmitters and receivers. For example,

in circuit switching communications, the channel transfer function is usually constant, but, it

changes for every different path from the transmitter to the receiver. But, there are also non

stationary channels like wireless communications. These channels’ transfer functions vary

with time, so that it is not possible to use an optimum filter for these types of channels. So, In

order to solve this problem equalizers are designed. Equalizer is meant to work in such a way

that BER (Bit Error Rate) should be low and SNR (Signal-to-Noise Ratio) should be high.

Equalizer gives the inverse of channel to the Received signal and combination of channel and

equalizer gives a flat frequency response and linear phase.

The static equalizer is cheap in implementation but its noise performance is not very good

As it is told before, most of the time, the channels and, consequently, the transmission

system’s transfer functions are not known. Also, the channel’s impulse response may vary

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with time. The result of this is that the equalizer cannot be designed. So, mostly preferred

scheme is to exploit adaptive equalizers. An adaptive equalizer is an equalization filter that

automatically adapts to time-varying properties of the communication channel. It is a filter

that self-adjusts its transfer function according to an optimizing algorithm.

Efficient use of the available bandwidth in communication systems requires methods

such as crowding of frequency-division multiplexed signals, reusing frequencies in cellular

systems, and increasing data rates Use of these, or other methods, to improve spectral

efficiency often leads to more adjacent-channel interference (ACI), co channel interference

(CCI), and intersymbol interference (ISI). ACI is the interference due to signals with different

carrier frequencies which are close enough to cause mutual overlaps in the spectra. CCI is the

interference due to signals with similar carrier frequencies. IS1 is the interference among the

data of interest. It is necessary to find efficient techniques to simultaneously reduce the

harmful effects of ACI, CCI, and IS1 in digital communication systems. The use of

equalizers is shown to be such a technique.

This paper is concerned with the situation occurring when the signal carrying the data

of interest (containing ISI) and the signals of the interferers (ACI and CCI) are linearly

modulated by their respective data, where all systems use the same symbol rate, and where

the receivers make decisions at that symbol rate. Therefore some of the analytical subtleties

that arise in other continuous-time contexts do not arise here.

To show the effectiveness of equalizers in suppressing ACI, CCI, and IS1 we attempt to

evaluate their performance in a framework which employs all of the possibilities of

interference presence, bandwidths, and antenna diversity. The method is used to evaluate the

effectiveness of the equalizers. The method is the analysis of the number of degrees of

freedom and number of constraints for generalized zero-forcing linear and decision- feedback

equalizers employing multiple antennas which try to suppress all ACI, CCI, and ISI. This

analysis is based on the assumption that the values of the frequency responses of the channels

give a system of linearly independent equations. The analysis is a generalization of the linear

equalizer case of CCI and IS1 with one antenna.

Four results are obtained and presented in this paper. The results show some potential and

limits of using fractionally spaced equalizers and linear combiners to suppress ISI, ACI, and

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CCI. First, with one antenna and a linear equalizer, arbitrarily large receiver bandwidths

allow for marginal improvements in spectral efficiency through decreased carrier spacing,

because the carrier spacing cannot be reduced to a value below the symbol rate without

incurring insuppressible interference. Second, large receiver bandwidths assist multiple

antennas in improving the spectral efficiency in that carrier spacing values may go below the

symbol rate, even in the presence of CCI. Third, the use of equalizers and linear combiners,

together with large receiver bandwidths, allows large transmitter bandwidths to be used. This

may allow system design flexibility, e.g., constant or near-constant envelope modulation.

Fourth, for CCI and ISI, the number of interferers that may be suppressible by a generalized

zero-forcing linear equalizer increases linearly with the product of the number of antennas

and the truncated integer ratio of the total bandwidth to the symbol rate.

2. EQUALIZER AND ITS APPLICATION

For a reliable digital transmission system it is crucial to reduce the effects of ISI and it is

where the equalizers come on the scene. The need for equalizers arises from the fact that the

channel has amplitude and phase dispersion which results in the interference of the

transmitted signals with one another. The design of the transmitters and receivers depends on

the assumption of the channel transfer function is known. But, in most of the digital

communications applications, the channel transfer function is not known at enough level to

incorporate filters to remove the channel effect at the transmitters and receivers. For example,

in circuit switching communications, the channel transfer function is usually constant, but, it

changes for every different path from the transmitter to the receiver. But, there are also non

stationary channels like wireless communications. These channels’ transfer functions vary

with time, so that it is not possible to use an optimum filter for these types of channels. So, In

order to solve this problem equalizers are designed. Equalizer is meant to work in such a way

that BER (Bit Error Rate) should be low and SNR (Signal-to-Noise Ratio) should be high.

Equalizer gives the inverse of channel to the Received signal and combination of channel and

equalizer gives a flat frequency response and linear phase.

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Fig 1. Concept of equalizer

2.1.Interference

Interference is anything which alters, modifies, or disrupts a signal as it travels along a

channel between a source and a receiver.

The term typically refers to the addition of unwanted signals to a useful signal.

Common examples are:

1. Adjacent-channel interference (ACI)

2. Co-channel interference (CCI), also known as crosstalk

3. Intersymbol interference (ISI)

2.1.1 Adjacent-channel interference

Adjacent-channel interference or ACI is interference caused by extraneous power from a

signal in an adjacent channel.

ACI is the interference due to signals with different carrier frequencies which are close

enough to cause mutual overlaps in the spectra.

ACI may be caused by inadequate filtering, improper tuning, or poor frequency control

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2.1.2 Co channel interference

CCI is the interference due to signals with similar carrier frequencies.

CCI may be caused by Frequency reuse phenomenon

Cellular Mobile Networks:

In cellular mobile communication frequency spectrum is a precious resource which is

divided into non-overlapping spectrum bands which are assigned to different cells (In cellular

communications, a cell refers to the hexagonal/circular area around the base station antenna).

However, after certain geographical distance, the frequency bands are re-used, i.e. the same

spectrum bands are re-assigned to other distant cells. The co-channel interference arises in the

cellular mobile networks owing to this phenomenon of Frequency reuse. Thus, besides the

intended signal from within the cell, signals at the same frequencies (co-channel signals)

arrive at the receiver from the undesired transmitters located (far away) in some other cells

and lead to deterioration in receiver performance.

Fig 2. Adjacent-channel interference in 3 FM stations

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Fig.3. Co channel interference in cellular network

2.1.3.Intersymbol interference

In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in

which one symbol interferes with subsequent symbols. One of the most important advantages

of the digital transmission systems for voice, data and video communications is their higher

reliability in noise environment in comparison with that of their analog counterparts.

Unfortunately most often the digital transmission of information is accompanied with a

phenomenon known as intersymbol interference (ISI).Briefly this means that the transmitted

pulses are smeared out so that pulses that correspond to different symbols are not separable.

Depending on the transmission media the main causes for ISI are:

• Cable lines – the fact that they are band limited

One of the causes of intersymbol interference is the transmission of a signal through a band

limited channel, i.e., one where the frequency response is zero above a certain frequency (the

cut-off frequency). Passing a signal through such a channel results in the removal of

frequency components above this cut-off frequency; in addition, the amplitude of the

frequency components below the cut-off frequency may also be attenuated by the channel.

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This filtering of the transmitted signal affects the shape of the pulse that arrives at the

receiver. The effects of filtering a rectangular pulse; not only change the shape of the pulse

within the first symbol period, but it is also spread out over the subsequent symbol periods.

When a message is transmitted through such a channel, the spread pulse of each individual

symbol will interfere with following symbols

Fig 4.Spreading of pulse in a band-limited channel

Fig 5.Overlapping of pulses in a time variant channel

• Cellular communications – multipath propagation

Another cause 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 different lengths - plus some of these

effects will also slow the signal down - this results in the different versions of the signal

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arriving at different times. This delay means 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.

Fig 6. Multipath propagation

When a radio signal is transmitted, it can propagate along many paths to reach the receiver.

At the receiver antenna, the received signal can be viewed as a complex sum of vectors with

independent gains and phases. Multipath interference is the effect of multiple versions of a

transmitted signal arriving at a radio receiver and combining in a way that produces distortion

of the original signal. Figure bellow shows how multipath interference is produced by

reflections from buildings or other objects.

Fig 7.Multipath propagation in a moving system

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When multiple versions of the same transmitted signal arrive simultaneously, an interference

pattern is formed with the various signals combining according to their amplitudes and

phases. As the mobile receiver moves, the relationship between the amplitudes and phases of

the signals from the various paths changes, causing undulations in both the composite

amplitude and the composite phase.

2.2. INTERSYMBOL INTERFERENCE & ITS EFFECTS

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.

EFFECTS:-

Pulse spreading:-

Assume polar NRZ line code. The channel outputs are shown as spreaded (width Tb becomes

2Tb) pulses shown (Spreading due to bandlimited channel characteristics).

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Example 1

Fig 8. Spreading of pulse by channel

Example 2

Fig 9. Superposition of bit

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If the rectangular multilevel pulses are filtered improperly as they pass through a

communications system, they will spread in time, and the pulse for each symbol may be

smeared into adjacent time slots and cause Intersymbol Interference.

Example 3

Fig 10. Interference of symbol in receiver block

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The amount of ISI can be seen on an oscilloscope using an Eye Diagram or Eye pattern.

Fig 11. Eye diagram for ISI

3. EQUALIZATION TECHNIQUE

Channel equalization can be done by using different types of equalization techniques.

By using this technique the weights of the filter are adjusted so as to have the characteristics

of the equalizer inverse to that of the channel. If the transfer function or characteristics of the

channel is known then the equalizer design is easy. But in practice the channel characteristics

is unknown and time varying. In this case it is difficult to design an equalizer. So different

adaptive equalization techniques are used to design a practical equalizer.

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3.1. TYPES OF EQUALIZER

3.1.1. Static equalizer:-

The type of equalizers in which the transfer function of the system is known and time

invariant is called as static equalizer.

The static equalizer is cheap in implementation but its noise performance is not very good.

But most of the time, the channels and, consequently, the transmission system’s transfer

functions are not known. Also, the channel’s impulse response may vary with time. The

result of this is that the equalizer cannot be designed. So, mostly preferred scheme is to

exploit adaptive equalizers.

3.1.2. Adaptive equalizer:-

An adaptive equalizer is an equalization filter that automatically adapts to time-varying

properties of the communication channel. It is a filter that self-adjusts its transfer function

according to an optimizing algorithm.

Types of adaptive equalizer:-

a. Linear equalizer

b. Non linear equalizer

c. Blind equalizer

a. Linear equalizer:-

A linear equalizer is a linear filter wk that is designed to reduce the noise and ISI according

to some criterion of optimality

xk = wk * yk

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X(z) = W (z) Y (z)

xk = Input signal to the channel

yk = Output signal from the channel

Types of linear equalizer:-

1.Zero forcing equalizer:-

The Zero Forcing (ZF)-equalizer is designed using the peak-distortion criterion. Ideally

eliminates all ISI. The names zero forcing corresponds to bringing down to intersymbol

interference to zero in a noise free case. This will be useful when ISI is significant compared

to noise .Zero-forcing equalizers ignore the additive noise and may significantly amplify

noise for channels with spectral nulls. The limitations of ZFE are:

1. Even though the channel impulse response has finite length, the impulse response

of the equalizer needs to be infinitely long.

2. At some frequencies the received signal may be weak. To compensate the

magnitude of the zero forcing equalizer grows very large. As a consequence any noise added

after the channel gets boosted by a large factor and destroys the overall SNR. Further more

the channel may have zeros in its frequency response that cannot be inverted at all.

2. MMSE equalizer:-

The Minimum Mean-Square Error (MMSE) equalizer minimizes the mean of the square of

the error (noise+ISI) after the equalizer.

Minimum-mean-square error (MMSE) equalizers minimize the mean-square error between

the output of the equalizer and the transmitted symbol. They require knowledge of some auto

and cross-correlation functions, which in practice can be estimated by transmitting a known

signal over the channel.

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3. Symbol spaced equalizer:-

A symbol-spaced linear equalizer consists of a tapped delay line that stores samples from the

input signal. Once per symbol period, the equalizer outputs a weighted sum of the values in

the delay line and updates the weights to prepare for the next symbol period. This class of

equalizer is called symbol-spaced because the sample rates of the input and output are equal.

In this configuration the equalizer is attempting to synthesize the inverse of the folded

channel spectrum which arises due to the symbol rate sampling (aliasing) of the input. In

typical application, the equalizer begins in training mode together information about the

channel, and later switches to decision –directed mode. Here, the newest of weights depends

on these quantities:

The current set of weights

The input signal

The output signal

For adaptive algorithms other than CMA, a reference signal, d, whose characteristics

depend on the operation mode of the equalizer.

Fig 12. Symbol spaced equalizer

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b. Non linear Equalizer

Decision feedback equalizer:-

The basic limitation of a linear equalizer, such as transversal filter, is the poor perform on the

channel having spectral nulls. A decision feedback equalizer (DFE) is a nonlinear equalizer

that uses previous detector decision to eliminate the ISI on pulses that are currently being

demodulated. In other words, the distortion on a current pulse that was caused by previous

pulses is subtracted. Figure 7shows a simplified block diagram of a DFE where the forward

filter and the feedback filter can each be a linear filter, such as transversal filter. The

nonlinearity of the DFE stems from the nonlinear characteristic of the detector that provides

an input to the feedback filter. The basic idea of a DFE is that if the values of the symbols

previously detected are known, then ISI contributed by these symbols can be cancelled out

exactly at the output of the forward filter by subtracting past symbol Values with appropriate

weighting. The forward and feedback tap weights can be adjusted simultaneously to fulfil a

criterion such as minimizing the MSE. The DFE structure is particularly useful for

equalization of channels with severe amplitude distortion, and is also less sensitive to

sampling phase offset. The improved performance comes about since the addition of the

feedback filter allows more freedom in the selection of feed forward coefficients. The exact

inverse of the channel response need not be synthesized in the feed forward filter, therefore

excessive noise enhancement is avoided and sensitivity to sampler phase is decreased.

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Fig 13.decision feedback equalizer

The advantage of a DFE implementation is the feedback filter, which is additionally working

to remove ISI, operates on noiseless quantized levels, and thus its output is free of channel

noise. One drawback to the DFE structure surfaces when an incorrect decision is applied to

the feedback filter. The DFE output reflects this error during the next few symbols as the

incorrect decision propagates through the feedback filter. Under this condition, there is a

greater likelihood for more incorrect decisions following the first one, producing a condition

known as error propagation. On most channels of interest the error rate is low enough that the

overall performance degradation is slight.

c. Blind equalizer:-

Blind equalization of the transmission channel has drawn achieving equalization without

the need of transmitting a training signal. Blind equalization algorithms that have been

proposed are the constant modulus algorithm (CMA) and multimodal’s algorithm (MMA).

This reduces the mean square error (MSE) to acceptable levels. Without the aid of training

sequences, a blind equalization is used as an adaptive equalization in communication

systems. In the blind equalization algorithm, the output of the equalizer is quantized and the

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quantized output is used to update the coefficients of the equalizer. Then, for the complex

signal case, an advanced algorithm was presented in. However, the convergence property of

this algorithm is relatively poor. In an effort to overcome the limitations of decision directed

equalization, the desired signal is estimated at the receiver using a statistical measure based

on a priori symbol properties; this technique is referred to as blind equalization. In the Blind

Error is selected as the basis for the filter coefficient update. In general, blind equalization

directs the coefficient adaptation process towards the optimal filter parameters even when the

initial error rate is large. For best results the error calculation is switched to decision directed

method after an initial period of equalization, we call this the shift blind method. Referring to,

the Reference Selector selects the Decision Device Output as the input to the error calculation

and the Error Selector selects the Standard Error as the basis for the filter coefficient update.

Fig 14. Blind equalizer

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Fig 15. A typical blind equalizer

The constant modulus algorithm (CMA) is the most popular adaptive blind equalizer used

today because of its relative simplicity and its good global convergence properties. It is very

effective in equalizing signals that are of constant modulus. However, for non-constant

modulus signals, CMA equalization can suffer from a considerable amount of residual ISI,

and produce non-minimal convergence.

Barbarossa and Scaglione propose an alphabet-matched algorithm (AMA), which is able to

provide better equalization of non-constant modulus signals, and they introduce a scheme that

uses CMA to initialize the alphabet-matched algorithm (AMA). The AMA cost function tries

to force the equalizer output to belong to the signal constellation of interest, and it is shown

that this method is able to lower the residual ISI and improve the convergence rate over that

of CMA alone. However, the AMA equalizer works well only when the initialization is good.

An issue here is how to automatically switch from the initialization step using CMA, to the

AMA. Using too few iterations of CMA will lead to ineffective equalization overall, while

using too many iterations increases the runtime without improving performance.

A new adaptive channel equalization scheme based on combining the two well-defined

cost functions of the constant modulus algorithm (CMA) and the alphabet-matched algorithm

(AMA) into a single cost function.

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3.2. ADAPTATION ALGORITHMS

The main adaptation algorithm used in equalizer is

3.2.1. Least Mean Square (LMS)

The Least Mean Square (LMS) algorithm, introduced by Widrow and Hoff in 1959 is an

adaptive algorithm, which uses a gradient-based method of steepest decent. LMS algorithm

uses the estimates of the gradient vector from the available data. LMS incorporates an

iterative procedure that makes successive corrections to the weight vector in the direction of

the negative of the gradient vector which eventually leads to the minimum mean square error.

Compared to other algorithms LMS algorithm is relatively simple; it does not require

correlation function calculation nor does it require matrix inversions.

Least mean squares (LMS) algorithms are class of adaptive filter used to mimic a

desired filter by finding the filter coefficients that relate to producing the least mean squares

of the error signal (difference between the desired and the actual signal). It is a stochastic

gradient descent method in that the filter is only adapted based on the error at the current

time. LMS filter is built around a transversal (i.e. tapped delay line) structure. Two practical

features, simple to design, yet highly effective in performance have made it highly popular in

various application. LMS filter employ, small step size statistical theory, which provides a

fairly accurate description of the transient behavior. It also includes H8 theory which

provides the mathematical basis for the deterministic robustness of the LMS filters. As

mentioned before LMS algorithm is built around a transversal filter, which is responsible for

performing the filtering process. A weight control mechanism responsible for performing the

adaptive control process on the tape weight of the transversal filter.

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Fig.16. Block diagram of adaptive transversal filter employing LMS algorithm

The LMS algorithm in general, consists of two basics procedure:

Filtering process-which involve, computing the output of a linear filter in response to the

input signal and generating an estimation error by comparing this output with a desired

response as follows:

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

y(n) is filter output and is the desired response at time n

Adaptive process - which involves the automatics adjustment of the parameter of the filter in

accordance with the estimation error.

푤(푛 + 1) = 푤(푛) + 휇(푛)푒∗(푛)

Where

휇 is the step size

(n+1) is estimate of tape weight vector at time (n+1)

If prior knowledge of the tape weight vector n is not available set n=0

The combination of these two processes working together constitutes a feedback loop, as

illustrated in the block diagram. First, we have a transversal filter, around which the LMS

algorithm is built; this component is responsible for performing the filtering process. Second,

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we have a mechanism for performing the adaptive control process on the tap weight of the

transversal filter- hence the designated “adaptive weight –control mechanism” in the figure.

Characteristics of LMS algorithm:-

i. LMS is the most well-known adaptive algorithms by a value that is proportional to the

product of input to the equalizer and output error.

ii. LMS algorithms execute quickly but converge slowly, and its complexity grows linearly

with the no of weights.

iii. Computational simplicity

iv. In which channel parameter don’t vary very rapidly

3.2.1.1.LMS algorithm formulation

From the method of steepest descent, the weight vector equation

Where µ is the step-size parameter and controls the convergence characteristics of the LMS

algorithm; e2(n) is the mean square error between the beam former output y(n) and the

reference signal which is given by

The gradient vector in the above weight update equation can be computed as

In the method of steepest descent the biggest problem is the computation involved in finding

the values r and R matrices in real time. The LMS algorithm on the other hand simplifies this

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by using the instantaneous values of covariance matrices r and R instead of their actual values

i.e.

Therefore the weight update can be given by the following equation,

The LMS algorithm is initiated with an arbitrary value w(0) for the weight vector at n=0. The

successive corrections of the weight vector eventually leads to the minimum value of the

mean squared error.

Therefore the LMS algorithm can be summarized in following equations;

Output, y(n)=푤 푥(푛)

Error, 푒(푛) = 푑∗(푛)− 푦(푛)Weight, w(n+1) =w(n)+µx(n)e*(n)

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Description

Fig 17. N-tap transversal adaptive filter

We assume that the signals involved are real-valued.

The LMS algorithm changes (adapts) the filter tap weights so that e(n) is minimized

in the mean-square sense. When the processes x(n) & d(n) are jointly stationary, this

algorithm converges to a set of tap-weights which, on average, are equal to the

Wiener-Hopf solution.

The LMS algorithm is a practical scheme for realizing Wiener filters, without

explicitly solving the Wiener-Hopf equation.

The conventional LMS algorithm is a stochastic implementation of the steepest

descent algorithm. It simply replaces the cost function ξ = E[e2(n)] by its

instantaneous coarse estimate ξˆ = e2 (n) .

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Substituting ξˆ = e2 (n) for ξ in the steepest descent recursion, we obtain

푊 (푛 + 1) = 푊 (푛) − 휇훻푒 (푛)

Where

Note that the i-th element of the gradient vector ∇e2 (n) is

Then

Where x(n)= [x(n) x(n-1) x(n-2)푥(푛 − 푁 + 1)]

Finally we obtain

3.2.1.2.Summary of the LMS algorithm

Input :-

Tap-weight vector, 푊(푛)

input vector, 푥̅(푛)

desired output, d(n)

Output :-

Filter output, y(n)

Tap-weight vector update, 푊(푁 + 1)

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3.2.1.3.Convergence and Stability of the LMS algorithm

The LMS algorithm initiated with some arbitrary value for the weight vector is seen to

converge and stay stable for

0 < μ < 1/λmax

Where λmax is the largest eigenvalue of the correlation matrix R. The convergence of the

algorithm is inversely proportional to the eigenvalue spread of the correlation matrix R.

When the Eigen values of R are widespread, convergence may be slow. The eigenvalue

spread of the correlation matrix is estimated by computing the ratio of the largest eigenvalue

to the smallest eigenvalue of the matrix. If µ is chosen to be very small then the algorithm

converges very slowly. A large value of µ may lead to a faster convergence but may be less

stable around the minimum value. One of the literatures also provides an upper bound for µ

based on several approximations as

µ <= 1/(3trace(R)).

3.2.1.4.Normalised least mean squares filter (NLMS)

The main drawback of the "pure" LMS algorithm is that it is sensitive to the scaling of its

input x(n). This makes it very hard (if not impossible) to choose a learning rate µ that

guarantees stability of the algorithm (Haykin 2002). The Normalised least mean squares filter

(NLMS) is a variant of the LMS algorithm that solves this problem by normalising with the

power of the input. The NLMS algorithm can be summarised as:

Parameters: p = filter order µ = step size

Initialization: ℎ(0)=0

Computation: For n = 0,1,2,..

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3.2.1.5.Advantages & disadvantages of LMS algorithm:

(1) Simplicity in implementation

(2) Stable and robust performance against different signal conditions

(3) slow convergence ( due to Eigen value spread )

3.2.2. Blind Adaptive algorithm

The scheme that we propose combines the conventional CMA with a constellation matched

error (CME) term, similar to the modified CMA (MCMA) scheme presented by He et. al. The

authors show that for QAM signals, this new scheme improves the convergence rate of the

adaptive equalizer and is more robust than the conventional CMA for low signal-to-noise

ratios (SNRs). The cost function presented is a weighted sum of two individual cost

functions, one is CMA and the other is a constellation matched error (CME) term. The

authors state that the CME term should satisfy certain desirable properties namely uniformity,

symmetry, and zero/maximum penalties at the zero/maximum distance from the QAM

constellation points. Uniformity implies that the CME function does not favour any one data

symbol in the alphabet over another, while symmetry implies that the function is symmetric

around each constellation symbol. It is also desirable that the CME assigns the

zero/maximum penalty at the zero/maximum distance from a constellation point. A zero

penalty occurs when an equalized symbol lies directly on a constellation point and the

maximum penalty occurs midway between two adjacent constellation points. In this work we

propose the use of AMA, presented by Barbarossa and Scaglione as the CME term. AMA

satisfies both the uniformity and symmetry properties while satisfying a minimum/maximum

penalty assignment at the zero/maximum deviation points.

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3.2.2.1.Constant Modulation algorithm(CMA)

For both the CMA and the AMA, the cost function J(w) to be minimized is a function of the

difference between the equalizer output, and the known constellation

points. The cost function for CMA is given by

JCMA(w) = 퐸{(|푦(푛) − 푅 |) } (10)

Where R2= { ( ) }{| ( ) |}

, and c(i)=1,2…..M are the known constellation points. M is the number

of constellation points. The notation E{⋅} denotes the expected value, taken over the entire

block of equalizer output symbols. This cost function attempts to restore the shape of the

constellation by taking into account the distance between the equalizer output symbols and

the computed constant modulus of the known constellation symbols. CMA does not take the

phase of the constellation points into consideration. CMA is well studied and is the most

popular algorithm used for blind adaptive equalization.

3.2.2.2. Alphabet matched algorithm(AMA)

The AMA cost function, proposed by Babarossa and Scaglione is given by

JAMA(w)=퐸 1 −∑ 푒| ( ) ( )|

(11)

The AMA cost function uses the same notation as above and a parameter, σ, is used to

control the width of the nulls that the function places around each constellation point. Its

value is chosen so that these nulls do not overlap for adjacent constellation points. The AMA

cost function attempts to restore the shape of the constellation by considering the distance

between the equalizer output symbol and each of the known constellation symbols. It assigns

an appropriate penalty based on this distance. When an equalized symbol is sufficiently close

to a constellation point the cost function is minimized. The opposite is true when an equalized

symbol is far from the constellation points. AMA is known to depend on good initialization

for optimization of the cost function.

29

In our equalization scheme we combine the two cost functions described above into a

single cost function but modified as given below:

J(w)=JCMA(w)+ßJAMA(w) (12)

This combined cost function attempts to restore the constellation shape by taking into account

both the amplitude and phase of the equalized symbol. The parameter β is a fixed scaling

factor between the CMA and AMA terms. Making the substitution of the CMA and AMA

expressions into equation (12), we obtain

J(w)= 퐸{(|푦(푛) − 푅 |) }+ß 퐸 1 −∑ 푒| ( ) ( )|

(13)

The update for the equalizer weights in this proposed scheme is based on the following

stochastic block gradient descent rule:

w n+1 = wn -µ 훻 J(w) (14)

where µ is the algorithm step-size. The equalizer output is given as for

equalizer weight vector,w and a received data vector, xn, which is a function of time n. The

gradient of the cost function is given in equation (17) as the linear combination of the

gradients of the CMA and AMA cost functions shown in equations (15) and (16)

respectively:

훻 JCMA(w)= 퐸{(4|푦(푛) − 푅 |) |푦(푛)|푥∗} (15)

훻 JAMA(w)=퐸{ ∑ 푥 ∗ 푦(푛)− 푐(푖) 푒| ( ) ( )|

} (16)

Note that the * operator denotes the complex conjugate. Thus the gradient expression for the

combined scheme is:

훻 J(w)= 훻 JCMA(w) +ß 훻 JAMA(w) (17)

30

and it should be noted that the averaged block gradient is computed at each iteration.

The cost function given in (12) combines the previously described CMA and AMA

equalizers into a single equalizer. Note that if β is zero, we have the pure CMA equalizer and

if β is very large, (12) is essentially controlled by the AMA equalizer. For a fixed β, it should

also be noted that when the equalizer is far from convergence, the AMA cost function

remains fairly flat, and the evolution of the equalizer is controlled by CMA. Once CMA has

converged, the evolution can then be controlled by the AMA. Thus the combined cost

function eliminates the need to monitor CMA for convergence. Further, it provides some

additional gains in the region where CMA has not yet converged, and the AMA initialization

is poor.

4. CHANNEL EQUALIZATION & SYSTEM MODEL

As mentioned in the introduction the intersymbol interference imposes the main obstacles

to achieving increased digital transmission rates with the required accuracy. ISI problem is

resolved by channel equalization in which the aim is to construct an equalizer such that the

impulse response of the channel/equalizer combination is as close to z–Δ as possible, where

is a delay. Frequently the channel parameters are not known in advance and moreover they

may vary with time, in some applications significantly. Hence, it is necessary to use the

adaptive equalizers, which provide the means of tracking the channel characteristics. The

following figure shows a diagram of a channel equalization system

31

Fig 18. Digital transmission system using channel equalization

In the previous figure, s(n) is the signal that you transmit through the communication

channel, and x(n) is the distorted output signal. To compensate for the signal distortion, the

adaptive channel equalization system completes the following two modes:

Training mode - This mode helps you determine the appropriate coefficients of the

adaptive filter. When you transmit the signal s(n) to the communication channel, you

also apply a delayed version of the same signal to the adaptive filter. In the previous

figure, z is a delay function and d(n) is the delayed signal, y(n) is the output signal

from the adaptive filter and e(n) is the error signal between d(n) and y(n) .The

adaptive filter iteratively adjusts the coefficients to minimize e(n) .After the power of

e(n)converges, y(n) is almost identical to d(n),which means that you can use the

resulting adaptive filter coefficients to compensate for the signal distortion.

Decision-directed mode - After you determine the appropriate coefficients of the

adaptive filter, you can switch the adaptive channel equalization system to decision-

directed mode. In this mode, the adaptive channel equalization system decodes the

signal and y(n) produces a new signal, which is an estimation of the signal s(n)

except for a delay of Δ taps.

32

Here, Adaptive filter plays an important role. The structure of the adaptive filter is

showed in Fig.14

푑(n) d(n)

x(n)

- + e(n)

Wn

Fig 19. Adaptive filter

Fig 20.function of equalizer

SYSTEM MODEL

We consider a communications environment, which is coherent and synchronous with

single-tone signaling. Carrier timing and waveform recovery have also been achieved. The

received data block is of length N symbols and the equalizer attempts to remove the

distortions introduced by the channel. Both the channel and equalizer are constrained to be

Variable filter(Wn)

Update algorithm(n)

+

33

linear, time-invariant FIR filters. The time-invariance assumption on the channel is

reasonable because we consider short blocks of data. Noise in the channel is modeled as

zero-mean additive white Gaussian noise (AWGN). Figure 16 provides a block diagram for

the communication system.

Fig 21. Communication system

The channel is a single-input single-output (SISO) system with an input/output relationship

characterized by

where T is the number of channel zeros and {h(0) …. h(T-1)} is the channel impulse

response, s(n) is the transmitted symbol sequence, and v(n) is AWGN noise in a data block of

length N. We assume the symbol sequence in the block of data is independently and

identically distributed (i.i.d.) and is non-Gaussian. The equalizer output, y(n) at time n is

defined as

The discussion develops results for linear equalizers and decision-feedback equalizers

impaired by ISI, ACI, and CCI, which state that relatively wider bandwidths (with respect to

the symbol rate) in the combined channel, combined interferers, and equalizer, may provide

34

the flexibility to an equalizer to suppress larger numbers of interferers. This analysis also

includes the effect of receiver antenna diversity. If the number of receiver antennas is Ar it

means that in Fig. l(a) and l(b) there are Ar times the number of combined channels and

combined interferers and that receiver inputs are put together using a continuous-time

infinite-length optimal combiner. In other words, the output of each receiver antenna would

be fed to a continuous-time infinite-length linear equalizer with all equalizer outputs added

together, and the adder output would be fed to the input of the quantizer. Note that the

description of this result emphasized the work may; this is because wide relative bandwidths

do not guarantee improved interference suppression capability under all conditions.

Pathological conditions exist where wide relative bandwidths do not guarantee improved

interference suppression, such as when the signals of interest and interferers have identical

impulse responses. However, under conditions of linear independence, wide relative

bandwidths allow for substantial equalizer/combiner performance improvements over the

case where narrow relative bandwidths are used, and the improvements are compounded by

the use of multiple antennas.

In this generalization, the number of interferers L contains two components

L = Na + Nc (1)

Where Na is the number of ACI signals and Nc is the number of CCI signals present at the

receiver input. Fig. 7(a) and 7(b) introduce three parameters. The transmitter bandwidth Bt

equalizer/combiner (receiver) bandwidth Br and carrier spacing C all of which are measured

relative to the symbol rate, 1/T. The figures also show the effect of the

equalizer/combiner/receiver bandwidth. All signals are zero outside the frequency band (-

Br/T, Bt/T). All the CCI signals have carrier frequency of zero. However, although it is not

shown in this paper, it would be a slight generalization to include multiple CCI signals at

each carrier frequency. All the ACI carrier frequencies are separated by C/T with one ACI

signal per carrier frequency. Depending on Bt, Br and C, a number of ACI signals may be

present after receiver filtering.

35

Fig 22. (a)Magnitude response of the signals after receiver filtering separated adjacent

channels

(b) Magnitude response of the signals after receiving filtering unseparated adjacent

channels

condition for zero interference

Fig. 22(a) shows a case where the adjacent bands are separated from the signal carrying the

data of interest. In this particular case, the problem of ACI suppression is trivial; the receiver

filter

frequency response is set to zero in the adjacent interferers' bands. Of more interest in this

paper is the situation shown in Fig. 22(b) where substantial spectral overlap occurs.

Define the equalized combined channel as

(2)

36

Where the symbol * denotes convolution, rm(t) denotes the impulse response of the mth

equalizer, Ar { 1, 2, 3,. . .} is the number of antennas, and denotes the impulse response

between the desired signal transmitter and the mth antenna element. Define the equalized

combined interferers to be

The time-domain condition for zero intersymbol interference at the sampling time can be

written as

and the time-domain condition for zero interference at the sampling time can be written as

The concept of suppressing interference by putting T-spaced zero crossings in the equalized

combined interference responses is similar to the idea avoiding interference by generating

nulls in an antenna pattern. A linear equalizer/combiner which causes (4) and (5) to be

satisfied is called a generalized zero-forcing linear equalizer

Conditions (4) and (5) can be expressed in the frequency domain as

(3)

(4)

(5)

(6)

37

is the fourier transform of

is the frequency response of the equalizer at mth antenna output

This equation can only be satisfied (and suppressing all ISI, ACI, and CCI) if for any

frequency the number of unknowns equals or exceeds the number of

equations L + 1

TABLE 1: FOUR BASIC CLASSES OF ADAPTIVE FILTERING APPLICATION

Class of adaptive

Filtering

Application Purpose

Identification System

identification

Layered earth

Modelling

Given an unknown

dynamical system, the

purpose system

identification is to design

an adaptive filter that

provides an approximation

to the system.

In exploration seismology,

a layered modelled of the

earth is developed to

unravel the complexities of

the earth’s surface.

Inverse modelling Equalization Given a channel of

unknown impulse

response, the purpose of an

adaptive equalizer is to

operate on the channel

output such that the

cascade connection of the

channel and the equalizer

38

provides an approximation

to an ideal transmission

medium.

Prediction Predictive coding

Spectrum analysis

The adaptive prediction is

used to develop a model of

a signal of interest (e.g., a

speech signal); rather than

encode the signal directly,

in predictive coding the

prediction error is encoded

for transmission or storage.

Typically, the prediction

error has a smaller

variance than the original

signal-Hence the basis for

improved encoding.

In this application,

predictive modelling is

used to estimate the power

spectrum of a signal of

interest.

Interference

cancellation

Noise cancellation

The purpose of an adaptive

noise canceller is to

subtract noise from a

received signal in

adaptively controlled

manner so as to improve

the signal-to-noise ratio.

Echo cancellation,

experienced on telephone

circuits, is a special form

39

Beam forming

of noise cancellation. Noise

cancellation is also used in

Electrocardiography.

A beam forming is spatial

filter consist of an array of

antenna elements with

adjustable weights

(Coefficients). The twin

purposes of an adaptive

beam former are to

adaptively control the

weights so as to cancel

interfering signal impinging on

the array

from unknown direction

and, at the same time,

provide protection to a

target signal of interest.

4. FUTURE WORK

Recursive Least Square Algorithm (RLS)

The Recursive least squares (RLS) adaptive filter is an algorithm which recursively finds the

filter coefficients that minimize a weighted linear least squares cost function relating to the

input signals. This in contrast to other algorithms such as the least mean squares (LMS) that

aim to reduce the mean square error. In the derivation of the RLS, the input signals are

considered deterministic, while for the LMS and similar algorithm they are considered

stochastic. Compared to most of its competitors, the RLS exhibits extremely fast

convergence. However, this benefit comes at the cost of high computational complexity, and

potentially poor tracking performance when the filter to be estimated changes.

40

As illustrated in Figure the RLS algorithm has the same to procedures as LMS algorithm,

except that it provides a tracking rate sufficient for fast fading channel, moreover RLS

algorithm is known to have the stability issues due to the covariance update formula p(n) ,

which is used for automatics adjustment in accordance with the estimation error as follows:

p(0) =I

Where p is inverse correlation matrix and is regularization parameter, positive constant for

high SNR and negative constant for low SNR

For each instant of time n=1,2,3.....

Time varying gain vector

Then the priori estimation error

Fig 23.Block diagram of adaptive transversal filter using RLS algorithm

41

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42

[13] Mahmood Farhan Mosleh, Aseel Hameed AL-Nakkash, “Combination of LMS and RLS

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