Now this is where the whole story starts and
perhaps is the most fundamental inspiration
for the scores and wavelets and multirate
digital signal cross I first talk about where
wavelets come from Well four year transforms
deal with waves sign waves to be more precise
We recognize the merits of sign waves Sign
waves have many nice properties for one they
occur naturally in many different circumstances
For example an electrical engineer recognizes
the sign waves as naturally emerging from
an electricity generation system when there
is electromagnetic induction If there is a
perfectly circular rotating device in a magnetic
field and if all is perfect in the generating
system we would be generating a perfect sign
wave from the brushes So sign wave is a good
idealization to work with for an electrical
engineer but that’s not the only point Sign
waves are in some sense the most analytic
the most smoothest possible periodic functions
They also have the power of being able to
express many other wave forms that means they
form a good basis from which many other wave
forms can be generated They have many other
nice mathematical properties
If I take two sign waves with the same frequency
possibly different amplitudes and phases I
would get back a sign wave of the same frequency
of course of a third amplitude and phase If
I differentiate a sign wave I get back a sign
of the same frequency and naturally if I make
a combination of these operations mainly differentiation
or even integration for that matter and linear
combinations and if I restrict myself to sign
waves of a particular frequency I remain within
the domain of sign waves of that particular
frequency This is something beautiful about
sign waves It is not easy to find sign wave
forms which obey this And as I said before
sign waves form a good basis so they form
good building blocks for being able to express
a wide variety of signals
For all these reasons the sign waves has been
very popular In a first course on signals
systems discreet time signal processing and
what have you communication But as I said
right in the beginning of this lecture one
of the reasons why we are not so happy with
sign waves is that they need to last forever
Beginning from minus infinity and go all the
way to plus infinity otherwise if you trinket
a sign wave if it is one sided for example
and if you look at what happens to an electric
system when you apply a one sided sign wave
By one sided sign wave I mean a sign wave
which starts from some point zero up to then
and starts from some point and continues afterwards
the sign wave The response is very different
from what would be the response for a sign
wave that started at minus infinity in general
There would be transience which are not periodic
and then all these beautiful properties of
sign waves and their responses go away So
if I really wish to be able to apply the basic
principles of signals systems and discreet
time signal processing that I learnt in the
basic course I need something unrealistic
I need a sign which lasts forever 
How can I be more realistic in my demands
By accepting that I cannot deal with waves
but its more appropriate to deal with wavelets
so that’s where the word wavelet comes from
small waves Waves that don’t last forever
Functions that are not predominant forever
They are significant in a certain range of
time perhaps only exist in a certain range
of time and insignificant outside So we have
a certain support over which one might want
to use them one might want to consider them
to exist and so on
A much more realistic assumption and that
really is what we call a wavelet Not a wave
but a wavelet For example you could if you
wish think of truncating a sign wave to a
rectangular region that means suppose we take
a sign wave to last from 0 to 1 millisecond
as an example It could be an example of wavelet
later we will see this is not a very good
example but yes in principal a wavelet a wave
that doesn’t last forever A simplistic explanation
of what wavelets means
But that’s not the whole story Our whole
objective was to talk about other domain too
so going back to the example of the audio
signal If I thought of the audio signal as
comprising of many sign waves to come together
to form an audio piece then I wish to be able
to do something simultaneously in two domains
and that is the key idea here So for example
to put it in plain language I should be able
to say well there was this two seconds audio
clip out of which there were 5 notes being
played Each note was played for different
intervals of time
Maybe the first note was played for 0.4 seconds
the second note was played for 0.7 seconds
the third note only for 0.2 seconds and so
on So I need to be able to segment in time
but when I am talking about being able to
identify notes I am also talking about being
able to segment in frequency and lo and behold
that is where the conflict arises A very basic
principle in nature says if I wish to be able
to segment in time and frequency simultaneously
I am going to run into trouble Nature does
not allow it beyond a point And that is something
very fundamental
It pops up in many different manifestation
in different subjects In modern physics they
call it uncertainty the uncertainty of position
and momentum In signal processing we call
it uncertainty the uncertainty of the time
and frequency domain So to put it simply though
not very accurately the shorter you play a
note the more difficult it is to identify
it Not very far from intuition
If you play a note for a long time and listen
to it for a long time you are likely to be
able to identify better Common sense tell
us that but what common sense does not tell
us is that you can never quiet go down to
identifying one particular frequency precisely
So if I wish to be able to come down to a
point on the time axis then I need to spread
all over the frequency axis and if I wish
to be able to come down to a point on the
frequency axis I need to spread all over the
time axis That is of course the strong version
of this restriction but there is a weaker
and a little more subtle version and that
is as follows Even if I am not quiet interested
in coming down to a point on the time axis
I am content with being in a certain region
as I said in the first 0.4 seconds out of
the two seconds clip I was playing note number
1 that means some frequency number 1 I would
be able to say this at least to a certain
degree of accuracy that is what I am trying
to point out here What the principle of uncertainty
tells us is that this can be done to a certain
degree of accuracy You can identify that note
to a certain degree of accuracy Well what
uncertainty also tell us in a more subtle
form is that if I even chose to relax to a
certain region of time so I say well in this
region of time tell me the region of frequencies
which were predominant even then there is
a restriction on simultaneous length of measure
of the time and frequency regions
And of course they have a tussle with one
another The smaller I make that time region
the larger that frequency region becomes that
means the more I want to focus in time the
less I am able to do so in frequency This
is indeed something that rouses a lot of thought
It may seem something far from our interests
at first glance but when we look at it carefully
we realize it is something very fundamental
to what we often desire That is what I am
now going to explain to you with a couple
of more examples
We live in an age where we use mobile telephones
in fact more fundamentally digital communication
What are we asking for in digital communication
when we look at it from a signal or system
perspective a transformed domain perspective
or time and frequency perspective Going right
down to brass tacks what we are asking for
in digital communication is I should be able
to transmit a sequence of bits binary values
0 or 1 And how do I transmit the sequence
of binary values I chose maybe 1 of 2 possible
wave forms in the simplest scheme for corresponding
to 0 I have 1 wave form corresponding to bit
1 I have a different wave form To make life
simple the 2 wave forms have the same time
interval So for example we talk about I mean
all of us hear about computer networks and
they talk about the speed of the network So
they say well this network can operate at
the speed of 1 megabit per second What does
that mean it means that in 1 second I can
transmit 10 raised the power of 6 bits so
you have 1 milli 1 micro second to allow each
bit to be transmitted
Give it a thought Here we are talking about
time what are we saying about frequency Now
let’s quote the mobile communication context
I have so many different mobile operators
obviously each operator will want its own
privacy So what is being communicated on the
network of operator 1 should not interfere
with what is being communicated on network
of operator 2 Now where is the separation
going to occur not in time After all there
are many different people simultaneously using
mobiles bought from both of the operators
so the separation can not be in time
We may argue that the separation can be in
space So in 1 region you may have mobiles
from 1 operator in another region in space
I mean mobiles from the other operator So
far so good But that's also not always true
It is very common to see mobiles purchased
from different operators operating in the
same room So there is not separation in time
no separation in space So where is the separation
then
The separation has to be in a domain which
is not so easy to see but once we have done
a course on signals and systems reasonably
easy to understand and that domain is frequency
So we say well operator 1 has this bandwidth
allocated to him operator 2 has other bandwidth
allocated to him Now when we say this bandwidth
maybe a certain region of the frequency access
of size let us say 2 mega hertz When we say
this region of 2 mega hertz is allocated to
operator 1 and another region of 2 mega hertz
is allocated to operator 2 are we not talking
about segmentation in a different domain In
fact there we are talking about simultaneous
segmentation
We have a segmentation in time because you
want to transmit different bits in different
time segments and you want to have separation
in frequency because what is transmitted by
operated 1 should not interfere with what
is transmitted by operator 2 So here is a
very common though not so obvious example
of simultaneous design of localization in
time and frequency Other than the audio example
which is of course little more obvious a little
easier to understand This example is equally
common at least in scenario today but perhaps
not so easy to understand but a little reflection
makes it very clear to us There is a desire
to localize in two domains simultaneously
Well let us go to a bio medical example Very
often when one analyses an electro cardio
graphic wave form What one wishes to indentify
are the features in the ECG signal Now I don’t
intend to go into the medical details But
there are different segments in a typical
ECG signal They are often indexed by letters
P Q and so on Without meaning to focus on
specific details of an ECG signal let us try
and understand the connection to time and
frequency localization When we talk about
an ECG signal all features are not the same
length and time some features are kind of
shorter some are longer In fact to go away
from an ECG signal bio medical engineers often
talk about what are called evoked potentials
So we provide stimulus to bio medical system
or to a bio physical system and we evoke a
response and the wave form responding to the
response is called the evoked potential It
can be studied as an electrical signal Now
the evoked potential again typically has quicker
parts in the response and slower parts in
the response Naturally we expect the slower
parts in the response would be predominantly
located if you think of the frequency domain
in the lower ranges of frequency and the quicker
parts of the evoked potential wave form will
be located in the higher ranges of frequency
Now here is an example of time frequency conflict
Suppose I wish to be able to isolate the quicker
parts Is it all right simply to isolate the
higher frequency content in a certain signal
and which comes from an evoked potential and
suppress the lower frequency part Well you
see if we try to suppress the lower frequency
part then we have already suppressed the slower
parts of the response and if we try and suppress
the higher frequencies in a bid to emphasize
the slower parts of the response we have suppressed
the quicker parts of the response So if we
think conventionally in terms of the frequency
domain maybe high pass filtering or low pass
filtering nothing works for us If we do high
pass filtering then we have effectively suppressed
the slower parts of the response and if we
do lower pass filtering we have suppressed
the quicker parts of response So we need a
different paradigm or a different perspective
on filtering
We need to identify in the different parts
of the time axis which regions of the frequency
axis are predominant and therefore in a certain
sense identify different parts of the frequency
axis to be emphasized in different time ranges
This is another perspective again on the time
frequency conflict and all this is going to
lead us in the direction of building up this
course on the wavelets We shall of course
understand some of these concepts a little
better as we progress in the lectures but
for the time being I have given you these
3 examples with the intent of bringing before
you perhaps not completely but at least in
a way to inspire your imagination the whole
idea of time frequency conflict or more generally
the conflict between 2 domains domains of
analysis and representation of a signal and
of course then going further even of a system
In a first course we understand the domains
very well we understand there is a time domain
We understand there is a frequency domain
We do well because we keep them apart
It makes life easier but what we are trying
to bring out through these 3 examples the
audio example the digital communication example
and the bio medical wave form example whether
it is the electrocardiographic wave forms
or the evoked potential wave form what we
are trying to bring out is that one normally
needs to consider the two domains together
time and frequency and when we try and do
so there is a certain very fundamental conflict
that we have to deal with That conflict called
uncertainty appears as I said in different
manifestations in different subjects and we
are going to look at that principle the uncertainty
principle as a reply to signal processing
in great depth at a certain stage in this
course but before that we are going to consider
one particular tool for analyzing signals
analyzing situations with a recognition that
we need to be local and not global
