1
00:00:25,380 --> 00:00:40,940
So now let us see how f2 t minus f1 t would
look It is very easy to see that f2 t minus
2
00:00:40,940 --> 00:00:55,750
f1 t has an appearance like this Let me flash
them before you f2 t and f1 t just for a second
3
00:00:55,750 --> 00:01:01,949
here so that you get a feel This is f2 and
this is f1 and visualize subtracting this
4
00:01:01,949 --> 00:01:08,010
from this what would you get a function which
looks something like this So I have the time
5
00:01:08,010 --> 00:01:21,630
axis here so if I mark the intervals of size
T there something like this maybe this has
6
00:01:21,630 --> 00:01:25,400
height H1 and this has height H2
7
00:01:25,400 --> 00:01:35,990
Let me mark H1 and H2 on this diagram too
so this is H1 and this is H2 of course so
8
00:01:35,990 --> 00:01:43,960
is this Simple enough now if we look carefully
we can construct all of this by using just
9
00:01:43,960 --> 00:01:50,499
one function and what is that function
10
00:01:50,499 --> 00:02:01,869
Suppose I were to visualize a function like
this 1 over the interval of 0 to one half
11
00:02:01,869 --> 00:02:13,890
and minus1 over the next half interval this
is a point half this is a point 1 point 0
12
00:02:13,890 --> 00:02:24,840
1 here and minus1 And let us give this function
a name let us call it psi of t In fact this
13
00:02:24,840 --> 00:02:32,000
is indeed what is called the Haar wavelet
Haar again the name of the mathematician It
14
00:02:32,000 --> 00:02:39,250
is very easy to see that using this function
I can construct any such f2 t minus f1 t Indeed
15
00:02:39,250 --> 00:02:46,050
if I were to take this function stretch it
or compress whatever might be the case depending
16
00:02:46,050 --> 00:02:51,070
on the value of capital T dilate dilate is
the more general word
17
00:02:51,070 --> 00:02:56,200
So if I were to dilate this function to occupy
an interval of T and bring it to this particular
18
00:02:56,200 --> 00:03:04,120
interval of T so I dilate that function psi
t and bring it to this interval of T And then
19
00:03:04,120 --> 00:03:11,600
I multiply psi t so dilated by the constant
H1 of course H1 should be is an algebraic
20
00:03:11,600 --> 00:03:19,380
constant it should be given a sign Here for
example H1 should be given a negative value
21
00:03:19,380 --> 00:03:24,410
because we started psi t with a positive minus1
here Similarly H2 has a positive value here
22
00:03:24,410 --> 00:03:31,590
So in other words this segment of f2 t minus
f1 t is of the following form
23
00:03:31,590 --> 00:03:40,910
Some H1 times psi of t dilated by T plus H2
times psi of t minus T dilated by T So here
24
00:03:40,910 --> 00:03:49,460
this is both dilated and translated all right
In other words in general when we start from
25
00:03:49,460 --> 00:04:11,430
the function psi of t we are constructing
functions of the form psi t minus tao by S
26
00:04:11,430 --> 00:04:18,620
where of course S is a positive real number
and tao is real This is a general function
27
00:04:18,620 --> 00:04:22,490
that we are using as a building block different
values of tao and different values of S Of
28
00:04:22,490 --> 00:04:29,210
course here at a particular resolution at
a particular level of detail the value of
29
00:04:29,210 --> 00:04:31,419
S is only one
30
00:04:31,419 --> 00:04:38,220
For example when we are representing the function
on intervals of size T we take S equal to
31
00:04:38,220 --> 00:04:42,935
T If we were to represent the function on
intervals of size T by 2 then S would become
32
00:04:42,935 --> 00:04:52,370
T by 2 and so on Then what we are doing in
effect is dilating and translating now we
33
00:04:52,370 --> 00:05:22,850
introduce those terms Tao is called a translation
index or translation variant and S is called
34
00:05:22,850 --> 00:05:27,110
the dilation index or dilation variant and
we are dilating and translating or we are
35
00:05:27,110 --> 00:05:29,139
constructing dilates and translates of a basic
function
36
00:05:29,139 --> 00:05:43,670
Dilates and translates of psi t capture the
additional information in f2 t minus f1 t
37
00:05:43,670 --> 00:06:02,360
now let us spend a minute in reflecting about
why this is so important What have we done
38
00:06:02,360 --> 00:06:16,590
so far just looks like very simple function
analysis or just a very simple transformation
39
00:06:16,590 --> 00:06:32,740
or algebra of functions What is so striking
in what we have just said What is striking
40
00:06:32,740 --> 00:06:46,940
is that what we have done to go from T to
T by 2 can also be done to go from T by 2
41
00:06:46,940 --> 00:06:50,419
to T by 4
42
00:06:50,419 --> 00:07:07,950
Not only that what we have done to go from
T to T by 2 in other words for intervals of
43
00:07:07,950 --> 00:07:28,161
length T to intervals of length T by 2 all
over the time axis can be done all over the
44
00:07:28,161 --> 00:07:38,360
time axis to go from intervals of size T by
2 to intervals of size T by 4 And then you
45
00:07:38,360 --> 00:07:48,479
could go from intervals of size T by 4 to
intervals of size T by 8 T by 16 T by 32 T
46
00:07:48,479 --> 00:07:55,790
by 64 and what have you to as small an interval
as you desire each time what you add in terms
47
00:07:55,790 --> 00:08:01,949
of information is going to get captured by
these dilates and translates of the single
48
00:08:01,949 --> 00:08:02,949
function psi t
49
00:08:02,949 --> 00:08:06,030
A very serious statement if we think about
it deeply enough That one single function
50
00:08:06,030 --> 00:08:12,410
psi t allows you to bring in resolutions step-by-step
to any level of detail In fact in formal language
51
00:08:12,410 --> 00:08:21,460
in functional analysis we would put it something
like this You know in mathematics in these
52
00:08:21,460 --> 00:08:49,560
arguments of limits on continuity and so on
or in some of these proofs related to convergence
53
00:08:49,560 --> 00:09:02,290
there is this notion of this adversary and
the defendant So here the defendant is trying
54
00:09:02,290 --> 00:09:10,030
to show the one makes the proposition is trying
to show that by this process you can go arbitrary
55
00:09:10,030 --> 00:09:15,030
close to a continuous function as close as
you desire
56
00:09:15,030 --> 00:09:22,610
Now as close in what sense Well it could be
in terms of what is called the mean square
57
00:09:22,610 --> 00:09:27,120
error or the square error So let us formulate
that adversary-proponent kind of argument
58
00:09:27,120 --> 00:09:34,620
here So what we are saying is the proponent
says we can go arbitrarily close to xt to
59
00:09:34,620 --> 00:09:43,020
a continuous function xt by this mechanism
Arbitrarily close in what sense In the sense
60
00:09:43,020 --> 00:09:50,670
if xa is the approximation approximation at
a particular resolution and if xt is the original
61
00:09:50,670 --> 00:09:58,730
function then if we take what is called the
square error so we look at xe t that is xt
62
00:09:58,730 --> 00:10:06,180
minus x a t and integrate x e t the whole
square the modulus whole square actually over
63
00:10:06,180 --> 00:10:08,620
all t we call this the squared error script
e
64
00:10:08,620 --> 00:10:14,230
Then adversary or opponent says bring e to
the small value let us say e0 and the proponent
65
00:10:14,230 --> 00:10:35,940
says certainly here is the m such that T by
2 raised to the power of m is okay So that
66
00:10:35,940 --> 00:10:46,100
is the idea of proponent and opponent here
The adversary or the opponent gives you a
67
00:10:46,100 --> 00:10:55,810
target He says I want the square error to
be less than this number e0 and the proponent
68
00:10:55,810 --> 00:10:59,560
says well here you are if you make that that
interval of size T by 2 to the power m lo
69
00:10:59,560 --> 00:11:02,880
and behold your error is going to be less
than or equal to e0
70
00:11:02,880 --> 00:11:15,080
And what is striking in this whole discussion
is no matter how small we make that e0 the
71
00:11:15,080 --> 00:11:25,730
proponent is always able to come out with
an error such that T by 2 raised to the power
72
00:11:25,730 --> 00:11:38,459
of m I mean piecewise constant approximation
for intervals of size T to the power of 30
73
00:11:38,459 --> 00:11:46,060
by 2 raised to the power of n would give you
an approximation close enough for that small
74
00:11:46,060 --> 00:11:54,649
e0 We need to spend a minute or 2 to reflect
on this it is a serious thing we are saying
75
00:11:54,649 --> 00:12:06,220
In fact let us for a moment think on how this
is dual to the idea of representation of a
76
00:12:06,220 --> 00:12:11,570
function in terms of its Fourier series for
example
77
00:12:11,570 --> 00:12:21,580
In the Fourier series presentation what do
we do we say give me a periodic function or
78
00:12:21,580 --> 00:12:31,290
for that matter give me a function on certain
interval of time let us say an interval of
79
00:12:31,290 --> 00:12:52,300
T size T If I simply periodically extend that
function that means I take this basic function
80
00:12:52,300 --> 00:13:26,290
on the interval of T I repeat it on every
such interval of T translated from the original
81
00:13:26,290 --> 00:13:38,700
interval So suppose that original interval
is 0 to T then repeat whatever is between
82
00:13:38,700 --> 00:13:46,540
0 and T between T and 2T between minus T and
0 between minus 2T and minus T between 2T
83
00:13:46,540 --> 00:13:51,470
and 3T and go on doing this so you have a
periodic function
84
00:13:51,470 --> 00:14:00,779
Decompose that periodic function into its
Fourier series presentation so what am I doing
85
00:14:00,779 --> 00:14:16,381
in effect I have a sum of sinusoids sine waves
all of whose frequencies are multiples of
86
00:14:16,381 --> 00:14:23,910
the fundamental frequency What is that fundamental
frequency In angular frequency terms it is
87
00:14:23,910 --> 00:14:37,220
2 pie by T some in hertz terms it is 1 by
T So in hertz terms you have sine waves with
88
00:14:37,220 --> 00:14:49,530
frequencies which are all multiples of 1 by
T
89
00:14:49,530 --> 00:15:01,579
And an appropriate set of amplitudes and phrases
assigned to these different sinusoidal components
90
00:15:01,579 --> 00:15:40,370
with frequencies of multiples of 1 by
T when added together would go arbitrarily
91
00:15:40,370 --> 00:15:49,500
close to the original function of course the
original periodic function on the entire real
92
00:15:49,500 --> 00:16:00,870
axis or for that matter specifically on the
interval from 0 to T if you restrict yourself
93
00:16:00,870 --> 00:16:06,250
to the function from where you started
94
00:16:06,250 --> 00:16:25,519
So not only does the Fourier series allow
you to represent by using the tool of continuous
95
00:16:25,519 --> 00:16:40,470
functions analytic functions remember we talked
about sine waves in the previous lecture Sine
96
00:16:40,470 --> 00:16:54,890
waves are the most continuous in some sense
the smoothest function that you can think
97
00:16:54,890 --> 00:17:12,289
of
The derivative of a sine wave is a sine wave
98
00:17:12,289 --> 00:17:31,020
integral of a sign wave is a sine wave when
you add 2 sine waves of the same frequency
99
00:17:31,020 --> 00:17:41,110
they give you back a sine wave of the same
frequency
100
00:17:41,110 --> 00:17:54,750
So sine waves are the smoothest function that
you could deal with and even if you had somewhat
101
00:17:54,750 --> 00:18:14,480
discontinuous function on the interval from
0 to
102
00:18:14,480 --> 00:18:38,250
T and if you use this mechanism of Fourier
series decomposition you would lined up expressing
103
00:18:38,250 --> 00:19:08,980
a discontinuous function in terms of extremely
smooth analytic functions What would you be
104
00:19:08,980 --> 00:19:21,750
doing in the Haar approach that we discussed
a few minutes ago Exactly the dual
105
00:19:21,750 --> 00:19:32,500
Even if you had this continuous audio pattern
you would decompose it into highly discontinuous
106
00:19:32,500 --> 00:19:44,780
functions which are piecewise constants Constant
on intervals of size T in the resolution T
107
00:19:44,780 --> 00:20:16,120
on intervals of size T by2 at the resolution
T by 2 and so on and
108
00:20:16,120 --> 00:20:30,429
so forth now this tells in the Fourier series
presentation you have this proponent-opponent
109
00:20:30,429 --> 00:20:56,880
kind of argument that is for a given class
of functions even if they are discontinuous
110
00:20:56,880 --> 00:21:25,910
even if they have a lot of non-analytic points
and so on for a reasonably wide class of functions
111
00:21:25,910 --> 00:22:13,260
Remember in the Fourier series that wide class
of functions is captured by
112
00:22:13,260 --> 00:23:49,440
what are by what are called the Dirichlet
conditions now I
113
00:23:49,440 --> 00:24:08,350
would not go into those details here But there
are certain kinds of conditions very very
114
00:24:08,350 --> 00:24:37,270
mild conditions which a function needs to
obey before it can be decomposed into the
115
00:24:37,270 --> 00:25:46,320
Fourier series or in other words before the
Fourier series can do this job of representing
116
00:25:46,320 --> 00:26:06,440
that discontinuous function in terms of highly
continuous and analytic smooth functions
117
00:26:06,440 --> 00:26:21,900
So similar set of conditions thus exist even
for the Haar case I mean if one really wishes
118
00:26:21,900 --> 00:26:43,970
to be finicky one does need to restrict oneself
to a certain subclass of functions where again
119
00:26:43,970 --> 00:26:54,669
that restriction is not really serious in
most physical situations For the time being
120
00:26:54,669 --> 00:27:12,200
in this course we may even just ignore that
restriction