**For many types of noise filtering applications, the standard fixed
IIR or FIR filter proves to be inadequate, if the noise varies in frequency,
phase, or amplitude. Adaptive filters will process varying noise in a signal
if the noise is differentiable in some way from the signal.**

**Although more complex adaptive filter types are available, this project
examines one of the simplest filter forms, the transversal, tap weight
filter which uses the least mean squares method for the adjustment of tap
weights.**

**This type of filter requires that the noise be statistically uncorrelated
from the signal; further, the noise may not be purely random, but instead
must vary over time in a smooth way (ie, derivatives must exist).**

**We describe four applications of adaptive filters, provide the underlying
theory, then demonstrate via examples, an implementation of Type I and
Type IV filtering systems.**

*11/05/96*

**Noise cancellation is an important topic in many different fields,
ranging from satelite photo imaging, speech recognition, and telecommunications,
to noise control in industry and automobiles. With continued advances in
telecommunications and digital processing, the problem of noise elimination
will become ever more important.**

**From a theoretical viewpoint, the problem is one of separating an
information signal which has had a noise signal superimposed upon it. If
it is possible to obtain a separate noise signal, then it is possible to
subtract the noise from the information signal. Such a technique can be
found in certain kinds of noise cancelling headphones. Microphones which
are strategically placed on the headset "listen" for noise, then generate
a cancelling sound pattern which is piped into the headphones.**

**More often though, it is not possible to obtain a separate noise
signal; instead, one must devise a means for removing the noise from the
signal through the use of filters. Two kinds of noise must be dealt with.
The first is a constant type of noise, such as the static that might be
introduced in a phone line or satelite transmission. Variations, if any,
will tend to be gradual. The second type of noise is more random in nature.
An example of this type of noise is what might be found in a typical classroom:
Coughing, shuffling of papers, scraping of chair legs, and snippets of
conversation are all imposed upon the lecturer's voice in a random and
unpredictable way.**

**It turns out that extracting a signal from random noise is a very
difficult problem, a problem whose theoretical underpinnings have yet to
be fully worked out. On the other hand, techniques for dealing with constant
noise sources have been developed, and are more within the scope of an
introductory DSP course.**

**This project will concern itself with the design of "adaptive" filters
which can be used to remove constant noise. The project will proceed on
two tacks. First, it will explore and define the use of statistical methods,
such as recursive least squares, to create an adaptive filter. If time
permits, we will examine more complicated adaptive filters.**

**Second, the project will attempt to identify a real world process,
model that process within Matlab, and then apply the filter technology
derived from the theoretical examination. We will review at least two applications
for modelling: Removing noise from a communications circuit, or quieting
the noise emanating from an automobile engine.**

*10/31/96*

**Our project will cover some subset of the field of Noise Cancellation.
If we are successful, we shall have achieved a method to remove the noise
in a classroom lecture, for example, or if more appropriate, leave the
noise and remove the lecture.**

**In actuality, we understand that noise cancellation involves the
use of adaptive filters, and although we can hardly spell them right now,
we expect to have a reasonably firm grasp of their operation by the end
of the project.**