The document discusses adaptive filters, which can automatically adjust their parameters to filter signals whose exact frequency response is unknown. It defines adaptive filters as having an input signal, filter structure, adjustable parameters, and adaptive algorithm. The goal of adaptive filtering is to minimize the error between the filter's output and a desired response. It describes common adaptive filtering problems and solutions like using gradient descent algorithms and the mean squared error cost function to adjust the filter parameters over time and minimize error.

1. ADAPTIVE FILTER
A Brief Discussion of
The Problem and The Solutions
Sivaranjan Goswami, B. Tech. 7th sem.
Electronics and Communication Engineering
Don Bosco College of Engineering and Technology
Air Port Road, Azara, Guwahati 781017
Contact: sivgos@gmail.com

2. INTRODUCTION
• In many practical scenario it is observed that
we are required to filter a signal whose exact
frequency response is not known.
• A solution to such problem is an adaptive
filter.
• An adaptive filter is one which can
automatically design itself and can detect
system variation in time.
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3. Defining an Adaptive Filter
An adaptive filter is defined by four aspects:
1. The signals being processed by the filter
2. The structure that defines how the output signal
of the filter is computed from its input signal
3. The parameters within this structure that can be
iteratively changed to alter the filter’s input-
output relationship
4. The adaptive algorithm that describes how the
parameters are adjusted from one time instant to
the next
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4. Block Diagram of Adaptive Filtering
Problem
x(n) = input digital signal
y(n) = output digital signal
d(n) = desired response
e(n) = error signal
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5. Adaptive Filtering Problem
• The error signal e(n) is calculated from the
desired response as shown in block diagram.
• The error signal is fed into a procedure which
alters or adapts the parameters of the filter from
time n to time (n +1) in a well-defined manner.
• Thus as time increases the output signal or actual
response y(n) is hoped to become better and
better match to the desired response d(n).
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6. Adaptive Filter Structure
• An adaptive filter is usually a linear one which
can be represented as:
Where,
X(n)=[x(n),x(n-1),….,x(n-L+1)] is the input vector
W(n)=[w0(n),w1(n),….,wL-1(n)]T is the parameter or co-efficient vector
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7. Practical Adaptive Filtering Problem 1
• So far we are focusing on the desired
response d(n). However, it is quite obvious
that in many practical situations d(n) is not
available.
• To solve this problem d(n) must be estimated
from whatever signal is available to the input.
• The fact that such schemes even work is a
tribute both to the ingenuity of the
developers of the algorithms and to the
technological maturity of the adaptive filtering
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8. Practical Adaptive Filtering Problem 2
• It should also be recognized that the
relationship between x(n) and d(n) can vary
with time.
• In this situation the adaptive filter must
continuously change its parameter values to
adapt the change.
• This behavior is commonly referred to as
tracking.
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9. Gradient- Based Adaptive Filtering
Algorithms
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10. The Mean-Squared Error Cost
Function
• The form of G (.) depends on the cost function
chosen for the given adaptive filtering task.
• We now consider one particular cost function
that yields a popular adaptive algorithm.
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11. The MSE Cost Function (contd.)
• The MSE Adaptive filter is useful for adaptive
FIR Filter because:
– JMSE(n) has a well-defined minimum with respect to
the parameters in W(n)
– The parameters at this minimum minimizes the
power of the error signal e(n), indicating that y(n)
has approached d(n).
– JMSE(n) is a smooth function of each parameter of
W(n), and differentiable w. r. t. each of these
parameters.
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12. The Wiener Solution
• WMSE(n) can be found using the relation:
• The solution of this equation is
Where,
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13. The Method of Steepest Descent
• This procedure adjusts each parameter of the
system according to
• For FIR Adaptive Filter this relation reduces to:
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14. Other Implementation
Where
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15. DISCUSSION
• There are various other methods also for
implementation of Adaptive Filter.
• The hardware or software implementations supporting
floating point arithmetic are less severe compared to
those supporting fixed point arithmetic.
• The LMS Algorithm is well known for its robust
performance in the presence of finite precision error.
• Therefore LMS algorithm can be easily implemented in
dedicated hardware using the general form of
implementation given by-
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16. Reference
Chapter 18 “Introduction to Adaptive Filters” of
Douglas, S.C. “Digital Signal Processing Handbook”
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
Available at
http://www.dsp-book.narod.ru/DSPMW/18.PDF
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17. THANK YOU
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