The document provides an introduction to adaptive filters, which are computational devices that model the relationship between input and output signals in real time to minimize the error between the actual and desired response. It describes the basic elements of adaptive filters including input/output signals, filter structure, coefficients, and adaptive algorithm. It also summarizes common adaptive filter structures like FIR, IIR, and linear combiners and applications such as system identification, inverse modeling, signal prediction, and interference cancellation.

1. Introduction to Adaptive Filters
 The Adaptive Filter is a computational device that
attempts to model the relationship between two signals
(input and output) in real time and in an iterative manner.
Simplified Block Diagram of the Adaptive filter
x(n): Input Signal, y(n): Output Signal, d(n): Desired Response Signal, e(n): Error Signal
n: Time index
e(n) = d(n) – y(n)
1

2. The task of the filter is to adapt or change its coefficients
iteratively from time n to time n+1 using an optimization
procedure in order to minimize the error signal between the
actual response and the desired response. This optimization
procedure is usually known as the adaptive algorithm.
 Basic Elements of the Adaptive Filter
1. The signals being processed
by the filters.
2. The structure of the filter.
3. The filter coefficients.
4. The adaptive algorithm of
the filter.
5. The performance error
function.
2

3. Adaptive Filter Realizations
A program running on
a DSP chip
A set of logic operations
implemented in an FPGA chip
 Adaptive Filter Structures
1- FIR Filter Structure (Transversal Filter Model)
2- IIR Filter Structure (Recursive Filter Model)
3- Adaptive linear Combiner
3

4. 4
The FIR Adaptive Filter Model
N: Filter length = Number of filter coefficients
Advantages:
1- Stability
2- Ease of implementation
wk(n): Filter tap weights (or coefficients) at time instant n




1
0
)()()(
N
k
k nkxnwny

5. 5
The IIR Adaptive Filter Model






1
1
1
0
)()()()()(
M
k
k
N
k
k knynbknxnany
ak(n): Forward tap weights with a total number of N,
bk(n): Feedback tap weights with a total number of M
This filter structure is
rarely used in practice
due to its complexity,
instability, and
convergence problems.

6. 6
Adaptive Linear Combiner




1
0
)()()(
N
k
kk nxnwny
N: Number of input signals
This filter model
has multiple
input signals and
is used in some
applications such
as beam-forming
or adaptive
antenna arrays.
Note: This filter can be viewed as a
special case of the FIR filter if xk(n) =
x(n-k).

7. 7
Important Features of Adaptive Filters
Classification of Adaptive Algorithms

8. 8
Mathematical Model of a Generalized FIR Adaptive Filter
T
N nwnwnwnw )](.........)()([)( 121 
)(nw : Filter coefficient vector at instant n
T
Nnxnxnxnx )]1(.........)1()([)( 
)(nx : Input signal vector
)(.)()( nxnwny T


9. 9
)()()( nyndne 
The general form of the adaptive algorithm can be written as:
))(),((.)()1( nenxfnwnw 
Or )()()1( nwnwnw 
µ : step size
The performance error function, also called the cost function, is
represented by J and can take several forms. The most popular form
is the mean squared error (MSE) function.
)]([ 2
neEJMSE 

10. 10
Numerical Example
Suppose it is required to calculate the coefficients of a two-tap FIR adaptive
filter in the next instant of time knowing that the initial values of coefficients
are 0.1 and 0.2, the input signal samples are 1 and 0.5, and assume the
desired output is 1. Use the following algorithm:
)().(.)()1( nenxnwnw 
Take the step-size of the algorithm as 0.25.

11. 11
T
nx ]5.01[)( 
T
nw ]2.01.0[)( 
)(.)()( nxnwny T

 2.0
5.0
0.1
.2.01.0 







8.02.01)()()(  nyndne
)().(.)()1( nenxnwnw 































3.0
3.0
1.0
2.0
2.0
1.0
5.0
1
8.025.0
2.0
1.0
So, the updated weights of the filter are w0=0.3 and w1=0.3.

12. 12
Applications of Adaptive Filters
1- System Identification
An example of system identification is the modeling of channels in
communication systems (channel estimation). The input signal x(n)
in this case is a training sequence.

13. 13
2- Inverse System Modeling
In this case, the filter attempts to model the inverse
characteristics of the unknown system. Typical example is
channel equalizers.

14. 14
3- Signal Prediction
In this case the input signal x(n) is to be predicted from its
samples x(n-n0), x(n-n0-1), ……..

15. 15
4- Interference Cancellation
In this application, desired signal d(n) is the corrupted signal with
the interference. The adaptive filter tries to synthesize a replica of
the interference signal that is combined with the desired signal. The
output signal of the filter is subtracted from the corrupted signal to
produce a clean signal e(n).

16. 16
Outline of the Material
1- Random Variables and Stochastic Processes
2- Wiener Filtering and the difference between Optimum Linear
Filters and Adaptive Filters
3- The Steepest-descent Algorithm
4- The Least-Mean Square (LMS) Algorithm
5- The Recursive Least–Squares (RLS) Algorithm
6- The Kalman Algorithm
References
1- B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications, John
Wiley & Sons, 1998.
2- Alexander Poularikas and Zayed Ramadan, Adaptive Filtering Primer
with MATLAB, Taylor & Francis, 2006.