Introduction
1
ECE 5650/4650 Computer Project #3
Adaptive Filter Simulation
This project is to be treated as a takehome exam, meaning each student is to due his/her own
work without consulting others. The grading for this third computer project will be handled differ
ently than the first two. A separate grade category exits for this project, thus allowing this project
to count up to 20% percent of the final grade (details given below).
The project due date is 5:00
PM Thursday, December 15, 2011 (Final Exam week)
.
The grading options are as follows: (1) (current syllabus) Computer project #3 20%, final exam
25%; (2) Computer project #3 15%, final exam 30%. The other grade distribution percentages
remain the same as the syllabus sheet discussed the first day of class.
Introduction
In this project you will be investigating adaptive noise cancellation (correlation cancellation) tech
niques using adaptive FIR and adaptive IIR filters. The input signal will be modeled as a real sinu
soid(s) in additive white noise. Specifically the systems investigated are called adaptive line
enhancers or ALEs. A rework of the ALE for adaptive interference cancellation, is also consid
ered using a real speech waveform.
Background Theory
A common estimation theory problem is to estimate a random process of interest (signal) from an
observed random process (e.g. signal plus noise). A solution is to use a causal minimum mean
square error (MMSE) filter (Wiener filter) to process the observations. A discretetime Wiener fil
ter takes the form of an FIR filter with
M
+ 1 weights.
Let
be the observations,
be the signal to estimate, and
be the filter weights. The
MMSE estimate is of the form
(1)
The weights an,
are chosen such that
(2)
is minimized. A block diagram of the filter is shown in Figure 1.
The optimal weights are found by setting
y n
x n
a
m
x
ˆ
n
a
m
y n
m
–
m
0
=
M
=
a
m
m
0 1
M
=
E e
2
n
E
x n
x
ˆ
n
–
2
=
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Background Theory
2
(3)
The solution is a result of the projection theorem or orthogonality principle
1
, which says that we
choose constants
such that the error
is orthogonal to the observations (
data
), i.e.,
(4)
The resulting system of equations
(5)
are known as the
normal equations
, or in Papoulis the YuleWalker equations. The function
is the autocorrelation sequence corresponding to
and
is the cross correla
tion sequence between
and
.
In an adaptive Wiener filter the error signal
is fed back to the filter weights to adjust
them using a
steepestdescent algorithm
. Note that the error surface generated by
over the
parameter space
is convex cup (i.e. a bowl shape) as shown in Figure 2. The filter
decorrelates the output error
so that signals in common to both
and
in a correla
tion sense are cancelled. A block diagram of an adaptive FIR filter is shown in Figure 3.
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 Fall '09
 Digital Signal Processing, Signal Processing, Finite impulse response, Adaptation Equations, IIR ALE

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