So we are going to build up the whole idea
of wavelengths by starting from what is called
the har multi resolution analysis Har incidentally
is a name of a mathematician Call him a mathematician
call him a scientist what you will but one
of the beautiful things that this gentleman
proposed was what is called the dual of the
idea of ferial analysis What do we do in ferial
analysis
We allow even discontinuous wave forms we
allow non smooth wave forms and we convert
them into a sum or a linear combination of
extremely smooth functions namely the sign
waves Har said can we do exactly the dual
Can we in principle take smooth functions
and convert them into a linear combination
of effectively jagget of discontinuous functions
Why on earth would one like to do something
like that?
Again lets deflect for a minute a few years
before this might have seem silly to do but
today it is not What are you doing when you
are doing digital communication?
You are transmitting audio you are transmitting
pictures and you are doing all this actually
with a large level of discontinuity How does
one record digital audio ? One firstly samples
so one takes values of the audio signal at
different points in time One digitizes them
and one then records those digital values
as a stream of bits
All of these are highly discontinuous operations
You are forcibly introducing discontinuity
in time and on top of that you are introducing
discontinuity in amplitude by quantization
So wanting to represent the beautiful smooth
audio in terms of very discontinuous bit streams
is very-very beneficial to digital communication
And in fact none of complain when we have
a good audio digital recording sometimes we
even say that digital recording is better
than an analogue recording that we had in
the past
So going from smooth to non smooth has its
place in modern communication and signal crossing
And when Har proposed that one should look
at the whole philosophy and the whole principle
of being able to go from smooth to non smooth
perhaps he was looking into the future When
this would be absolutely essential What we
are going to do in the very first few lectures
immediately following this is to look at one
whole angle of wavelengths and multi rate
digital processing based on the principles
that Har propounded
So we are going to look at what is called
the Har multi resolution analysis And in fact
if we understand the Har multi resolution
analysis in depth we actually end up understanding
many principles of wavelength Many of the
essentials of multi rate processing specifically
what is called 2 band processing very well
So we shall draw upon the Har mutli resolution
analysis to understand some of the basic concepts
that underlie this course and of course build
upon them further later on
From the Har we shall slightly progress to
better multi resolution analysis From the
better in what sense we understand and there
are many such different families from this
better multi resolution analysis one of them
being what is called as the Dobash family
Dobash again is the name of a mathematician
scientist who proposed that family of multi
resolution analysis As I said at a certain
point in the course immediately following
this we shall then look at the uncertainty
principle
Fundamentally and in terms of its implications
From there we shall move to the continuous
wavelength transform so in the Har multi resolution
analysis we have a certain dicretization in
the variables associated with the wavelength
transform Later on we shall go to what is
called the continuous wavelength transform
where the variables the independent variables
are associated with the wavelength transform
all become continuous
Following that we shall look at some of the
generalizations of the ideas that we have
build up earlier in this course And towards
the last phase of the course we shall look
in depth at some of the important applications
to which wavelengths at multiple digital processing
provide great advantages
Now I would like to spend a little while in
this lecture on building up in parallel Some
of the developments that took place to introduce
the subject of multi rate digital processing
What is multi rate?
What rate are we talking about here and why
do we need to talk about multi rate?
Why is it connected with wavelengths?
Let's go back to the audio example Or may
be let's first go to the biomedical example
In the biomedical example we said we would
have quicker parts in the response and slower
parts in the response The slower parts of
the response are likely to last for a longer
region in time The quicker parts in the response
are likely to last for smaller regions in
time So here other than the concept of being
able to localize in a certain region of time
and of course correspondingly on frequency
There is also a distinction between what kind
of localization is required for higher frequencies
and lower frequencies If we spend a little
bit of thought and time in understanding these
2 kinds of components we'll realize that most
of the time when we are talking about the
slower parts of the response or lower frequencies
we are talking about compromising on what
is called time resolution So I bring in the
idea of resolution here
Resolution means the ability to resolve The
ability to be able to identify the specific
components So for example frequency resolutions
relates to being able to identify specific
frequency components And going further and
being a little more pin pointed when I am
talking about frequency resolution what I
am saying in effect is suppose I have 2 sign
waves whose frequency start coming closer
and closer together Over what region of time
do I need to observe them so I can actually
identify the 2 frequencies separately?
How can I resolve the 2 frequencies?
How much can I narrow down on the frequency
axis Now what we are talking about is not
so much how much we can narrow down but how
much we need to When we talk about higher
frequency content or things that vary quickly
It is often though not always the case that
we are willing there to compromise on frequency
resolution but we want time resolution So
things that take place quickly and are transient
short lived demand time resolution
And things that occupy the lower frequency
ranges which last for a long time demand frequency
resolution So very often it is true that when
one goes down on the frequency axis one demands
more frequency resolution the ability to resolve
frequency is more accurately as opposed to
time resolution The ability to resolve which
time segment it occurs And when one goes to
higher frequencies where one normally demands
more time resolution and lesser frequency
resolution
One is asking for how closely one can identify
2 segments or 2 parts of the wave forms which
vary quickly that means one is trying is narrow
down on the time axis and in doing so one
must compromise on the frequency axis So this
is what brings us to the idea of multi rate
processing You see it means that when I talk
about bands of higher frequencies I must use
smaller sampling rates in a discrete time
processing system
When I am talking about lower frequency ranges
I must use larger time sampling points or
sampling intervals Why must I do so?
To be most efficient in the processing operation
When I am talking about lower frequencies
so in an evoke potential wave form If I am
trying to look at the slower components I
should not unnecessarily sample too frequently
It only increases my data burden and does
not offer me anything special
On the other hand when I am analyzing the
quicker components It is inadequate to use
a low sampling rate I would be doing injustice
to the components For those of us who might
be exposed to the concepts of sampling analyzing
If I am not faithful in my sampling rate of
the quicker components I would introduce elaises
I would introduce purious effects which I
don’t want So on in all we recognize that
it is not a good idea to be using the same
sampling rate for all frequency components
So unlike a basic course on discrete time
signal processing where we assume all frequencies
are at the same sampling rate here we need
to deal with the same frequency rates that
are effectively at different sampling rates
in the same system That means we also need
to deal with systems that operate with different
sampling rates and that is why we talk about
multi rate discrete time signal processing
Now at a conceptual level we understand at
a very well why there is a close relationship
between multi rate discrete time signal processing
the idea of uncertainty the requirement of
the resolution And if we go further than when
we do multi rate discrete time signal processing
we also bring in a new concept of filter banks
verses filters As I said if we go back to
the bio medical example there is the effect
or there is the desire to separate components
So when I wish to separate components naturally
I wish to have all different operators all
at once So I need a system of filters which
not only have certain individual characteristics
but which also have collective characteristics
So I need to be able to analyze and then synthesize
and all this with localization included This
is what we mean by a filter bank
So a bank of filters as opposed to a single
filter in discrete time signal processing
refers to a set of filters which either have
a common input or a common point of output
or a summation output This concept of a bank
of filters in fact 2 banks of filters an analysis
filter bank and a synthesis filter bank taken
together in very central to multi rate discrete
time signal processing We shall be looking
at that concept in great depth
So we shall be building up the idea of a 2
band filter bank in reasonably great depth
in this course The concept of a 2 band filter
bank is of great importance in being able
to construct wavelengths In fact we should
see even from the Har multi resolution example
that there is an intimate relationship between
the wavelet or the Har wavelet and the 2 band
Har filter bank
So much so that if I construct a properly
designed 2 band filter bank I also construct
a multi resolution analysis that goes with
it so to speak All this is very exciting And
what we intend to do in the lectures that
follow from here is to take these concepts
one by one So in the next few lectures we
intend to talk about the Har multi resolution
analysis to build up certain basic ideas from
it
With that then we come to the end of this
first introductory lecture on the subject
of wavelets and multi rate digital processing
and proceed there with- in the next lecture
to talk about the Har multi resolution analysis
Thank you
