Channel Equalisation

CHANNEL EQUALISATION
BY- AJIT KUMAR PANDA
POONAN SAHOO
SAYANTAN DAS
SURAJ CHOUDHURY

THREATS IN DIGITAL
COMMUNICATION

•There are four main threats in the process of
digital communication

Inter Symbol Interference (ISI)

Multipath Propagation

Co-channel Interference

Presence of noise in the channel

INTER SYMBOL INTERFERENCE:
 Inter Symbol Interference
in Digital Transmission

 Inter-symbol interference
(ISI) arises when the data
transmitted through the
channel is dispersive, in
which each received pulse
is affected somewhat by
adjacent pulses and due to
which interference occurs
in the transmitted signals.

 It is diﬃcult to recover the
original data from one
channel sample

CO-CHANNEL INTERFERENCE:
 Co-channel Interference (CCI) and
Adjacent Channel Interference
(ACI) occur in communication
systems due to multiple access
techniques using space, frequency or
time.

 CCI occurs in cellular radio and dual-
polarized microwave radio, for
efficient utilization of the allocated
channels frequencies by reusing the
frequencies in different cells.

MULTI-PATH PROPAGATION:
 Within telecommunication channels multiple paths of propagation commonly occur.
In practical terms this is equivalent to transmitting the same signal through a number
of separate channels, each having a different attenuation and delay.
 Consider an open-air radio transmission channel that has three propagation
paths, as illustrated in Fig. These could be
- Direct
- Earth Bound
- Sky Bound
 Fig1.2b describes how a receiver picks up the
transmitted data. The direct signal is received
first whilst the earth and sky bound are delayed.
 All three of the signals are attenuated with the sky path suffering the most.
 Multipath interference between consecutively transmitted signals will take place if
one signal is received whilst the previous signal is still being detected.
 In Fig1.2 this would occur if the symbol transmission rate is greater than 1/τ where, τ
represents transmission delay. Because bandwidth efficiency leads to high data
rates, multi-path interference commonly occurs.

EQUALIZER
 Equalization is the process to remove ISI and noise effects from the
channel
 It is located at the receiver end of the channel

 It is an inverse filter placed at the front end of the receiver
 The transfer function of the equalizer is just inverse of the transfer
function of the channel
 Equalization is an iterative process of reducing the mean square
error the difference between desired response and output of filter
used in equalizer

TYPES OF EQUALIZERS:

 Equalizers are of two types

LINEAR
EQUALIZERS
NON LINEAR

 Linear equalizers aim at reducing ISI in linear channels
using various algorithms like Least Mean
Square(LMS), Recursive Least Square(RLS) and
normalized LMS
 Non linear equalizers equalize non-linear channels. They
mainly use Neural Networks(NN) and Multilayer
Perception(MLP) algorithms for equalization

Linear Adaptive Filters:

 An adaptive filter is a computational device that
attempts to model the relationship between two signals
in real time in an iterative manner
 Here the output is compared to the desired signal and
accordingly the parameters of adaptive filter are varied
and so it is known as self designing filter.

Applications of Adaptive Filters:
Identification
 Used to provide a linear model of an unknown plant

 Parameters
 u=input of adaptive filter=input to plant
 y=output of adaptive filter
 d=desired response=output of plant
 ---> e=d-y=estimation error
 Applications:
 System identification

Applications of Adaptive Filters:
Inverse Modeling
 Used to provide an inverse model of an unknown plant

 Parameters
 u=input of adaptive filter=output to plant
 y=output of adaptive filter
 d=desired response=delayed system input
 e=d-y=estimation error
 Applications:
 Channel Equalization

The channel equalization
model:

Stochastic Gradient
Approach:
 Most commonly used type of Adaptive Filters
 Define cost function as mean-squared error
 Difference between filter output and desired response
 Based on the method of steepest descent
 Move towards the minimum on the error surface to get to minimum
 Requires the gradient of the error surface to be known
 Most popular adaptation algorithm is LMS
 Derived from steepest descent
 Doesn’t require gradient to be know: it is estimated at every iteration
 Least-Mean-Square (LMS) Algorithm

update value old value learning - tap
error
of tap - weigth of tap - weight rate input
signal
vector vector parameter vector

LMS algorithm
• Introduced by Widrow & Hoff in 1959
• Simple, no matrices calculation involved in the adaptation
• In the family of stochastic gradient algorithms
• Approximation of the steepest – descent method
• Based on the MMSE criterion.(Minimum Mean square Error)
• Adaptive process containing two important signals:
• 1.) Filtering process, producing output signal.
• 2.) Desired signal (Training sequence)
• Adaptive process: recursive adjustment of filter tap
weights

Least-Mean-Square (LMS)
Algorithm continued....
 The LMS Algorithm consists of two basic processes that is
followed in the adaptive equalization processes:
 Training : It refers to adapting to the training sequence
 Tracking: keeps track of the changing characteristics of the
channel.

LMS Algorithm Steps:
M 1
 Filter output zn *
u n k wk n
k 0

 Estimation error en dn zn

wk n 1 wk n un k e* n
 Tap-weight adaptation

Derivation of the LMS MSE
expression:
 Error=E=(x(n)-x(n)’)
 Square error=E=(x(n)-x(n)’)2
 Using minimum mean square error criterion , we differentiate the
expression
dE/dw=d/dw((x(n)-x(n)’)2)
 Applying chain rule and substitution of x(n)’ ,we get
dE/dw=2(x(n)-x(n)’)*d/dw(x(n)- Ʃw*s(n-i))
dE/dw=2(e(n))(s(n-i))
 From this we can derive an update equation for every new
sample n using steepest descent and gradient method as
w(n+1)= -u*(dE/dw)
so,w(n+1)=2*u*e(n)*s(n-i)
for i=0,1,2,3...........

Stability of LMS:
 The LMS algorithm is convergent in the mean square if and only
if the step-size parameter satisfy

1
0
m ax

Here max is the largest eigen value of the correlation matrix of
the input data.
1
 More practical test for stability is 0
input signal power

 The value of step size has to be a trade off between fast
convergence rates and less steady state misadjustment.
 Larger values for step size
 Increases adaptation rate (faster adaptation)
 Increases residual mean-squared error

LMS-Pros & cons:

 LMS – Advantage:
• Simplicity of implementation
• Not neglecting the noise like Zero forcing equalizer
• Stable and robust performance against different signal
conditions

 LMS – Disadvantage:
 Slow Convergence

 Demands using of training sequence as reference ,thus
decreasing the communication BW.

NLMS-Normalised LMS
algorithm
 Is mainly required to provide better performance than LMS
as LMS performance is slow
 Uses normalization technique to provide a variable step size
as step size ‘u’ is divided by instantaneous signal power thus
providing more stability and faster convergence.
 Is equivalent to running the LMS recursion for a new sample of
inputs every time recursion or the NLMS operation is carried out.
W(n+1) = w(n) + (1/xT(n)x(n)) * e(n) x(n)
 The step size value for the input vector is calculated
µ (n) = 1/xT(n)x(n)
 The filter tap weights are updated in preparation for the next
iteration
W(n+1) = w(n) + 2*µ (n) * e(n) * x(n)

Results for LMS algorithm:

 Convergence is faster with increased step size .
Plot is for noise=30 dB

Results for NLMS algorithm:

 Convergence is faster in case of NLMS
algorithm
 It provides a more stable output.

NON-LINEAR CHANNEL
EQUALISATION

Need For Non-Linear
Equalizer:
 Linear Equalizers do not perform well on channels
having deep spectral nulls in the pass band.

 To compensate distortion linear equalizer places too
much gain in the vicinity of spectral nulls thereby
enhancing the noise present in these frequencies.

 BER is better in Non-linear channel equalizer

 Linear equalizer-inverse problem
Non-linear equalizer-pattern classification

Non-Linear Channel
Equalizer:
 t k denotes a sequence of
T spaced complex symbols
of an BPSK
constellation, where 1/T
denotes the symbol rate and
k denotes the discrete time
index.
 A widely used model for a
linear dispersive channel is
an FIR filter whose output at
th
the kN instant is given by
h-1
Schematic Diagram of a Non-Linear
Wireless Digital Communication system
∑
ak= i=0 hi * t k-i with channel equalizer

Continued…
 where
hi- denotes the FIR filter weights
Nh- denotes the FIR order.

 Considering the channel to be a nonlinear one the NL block introduces
channel nonlinearity to the filter output.

 The transmitted signal t k after being passed through the nonlinear
channel and added with the additive noise arrives at the receiver, which
is denoted by r k .The received signal at the kth time instant is given by r k.

 The purpose of equalizer attached at the receiver front end is to recover
the transmitted sequence t k or its delayed version t k-1 ,where t is the
propagation delay associated with the physical channel.

Neural Network:
 Started in 1800s as an effort to
describe how human mind
performs.

 It is applied to computational
models with Turing ‘s B-type
machine and Perceptron

 A neural network is a massively
parallel distributed processor made
up of simple processing units, which
has a natural propensity for storing
experimental knowledge and making
it available for use.

Continued:
 Today in general form a neural network is a
machine that is designed by using electronic
components or is simulated in software on
a digital computer.

 To achieve good performance, neural
networks employ a massive interconnection of
simple computing cells referred to as
“Neurons” or “processing units”

 The procedure is called a learning
algorithm, the function of which is to modify the
synaptic weights of the network in an orderly
fashion to attain a desired design objective.
McCulloch and Pitts have developed the neural
networks for different computing machines.


Artificial Neural Network:
 Artificial Neural Network (ANN)
have become a powerful tool for
many complex applications including
functional approximation, nonlinear
system identification, motor
control, pattern recognition, adaptive
channel equalization and
optimization.

 ANN is capable of performing
nonlinear mapping between the
input and output space due to its
large parallel interconnection
between different layers and the
nonlinear processing characteristics.

Continued:
 An artificial neuron basically consists of a
computing element that performs the
weighted sum of the input signal and the
connecting weight. The weighted sum is
added with the bias called threshold and
the resultant signal is passed through a
nonlinear activation function. Common
types of activation functions are sigmoid and
hyperbolic tangent.
 Each neuron is associated with three
parameters whose learning can be adjusted.
These are the connecting weights, the bias
and the slope of the nonlinear function.
 For the structural point of view a NN may
be single layer or it may be multilayer

Multi-layer Perceptron:
 The perceptron is a single level
connection of McCulloch-Pitts
neurons is called as Single-layer
feed forward networks.
 The network is capable of
linearly separating the input
vectors into pattern of classes by
a hyper plane. Similarly many
perceptrons can be connected in
layers
 To provide a MLP network, the
input signal propagates through
the network in a forward
direction, on a layer-by-layer
basis. This network has been
applied successfully to solve
diverse problems.

Continued…
 Generally MLP is trained using popular error back-
propagation algorithm.
 The scheme of MLP using four layers is shown. Si
represent the inputs s1, s2, ….. , sn to the network, and yk
represents the output of the final layer of the neural
network.
 The connecting weights between the input to the first
hidden layer, first to second hidden layer and the second
hidden layer to the output layers are represented by W i
,W ji ,W kj respectively.
 The final output layer of the MLP may be expressed as

Channel Equalisation

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