1
00:00:17,109 --> 00:00:22,349
Today we will talking about the projection
of functions on different subspaces then how
2
00:00:22,349 --> 00:00:27,599
can we use them to get a better and better
approximation to that function We start with
3
00:00:27,599 --> 00:00:33,170
the function which is the absolute value of
2 sinusoids which is given by yt equal to
4
00:00:33,170 --> 00:00:44,780
sign of twice pie t plus sign of thrice pie
t this function is shown in this figure in
5
00:00:44,780 --> 00:00:52,229
blue Now we can get the approximation to this
function for interval of size 1 that is for
6
00:00:52,229 --> 00:00:54,619
m equal to 0
7
00:00:54,619 --> 00:01:00,180
The approximation has been shown in green
note that the function basically looks very
8
00:01:00,180 --> 00:01:04,470
very similar on the both sides of 0 because
of the original function is symmetric about
9
00:01:04,470 --> 00:01:10,290
0 So now what we do is like we add detail
function to this function that is the projection
10
00:01:10,290 --> 00:01:18,250
of original signal onto the W0 subspace and
this is shown in red So when we add the approximation
11
00:01:18,250 --> 00:01:23,830
and the detailed function what we get is a
better approximation to the original signal
12
00:01:23,830 --> 00:01:25,460
so which looks like this
13
00:01:25,460 --> 00:01:30,950
Now we can add more and more final details
to the function that is we take now the interval
14
00:01:30,950 --> 00:01:40,750
of size 2 to the power minus1 that is half
and we add the detail to the approximation
15
00:01:40,750 --> 00:01:47,830
that we just now got so which is shown in
red So when we add these 2 what we get is
16
00:01:47,830 --> 00:01:53,180
a much more better approximation to the original
signal Now we can add another finer detail
17
00:01:53,180 --> 00:02:04,560
that is the projection of signal on W2 space
and which looks like this
18
00:02:04,560 --> 00:02:09,869
When we add these 2 what we again get is like
even more better approximation to the signal
19
00:02:09,869 --> 00:02:15,910
which looks like this which is shown in green
Now in the similar way we can go for more
20
00:02:15,910 --> 00:02:21,480
and more finer approximations that is for
the interval of size 2 to the power minus3
21
00:02:21,480 --> 00:02:29,620
2 to the power minus4 that is for m equal
to 3 m equal to 4 and what we get is like
22
00:02:29,620 --> 00:02:34,530
a much more better approximation of the in
the signal that we had started This looks
23
00:02:34,530 --> 00:02:38,739
like this adding another detail
24
00:02:38,739 --> 00:02:44,110
So let us have a look at another signal function
which is a triangular function The approximation
25
00:02:44,110 --> 00:02:49,510
to this function is shown in green and the
original function is shown in blue When we
26
00:02:49,510 --> 00:02:56,099
take the projection of the blue function on
the W0 subspace it looks like this and when
27
00:02:56,099 --> 00:03:01,260
we add this detailed function to the green
approximation what we obtain is a better approximation
28
00:03:01,260 --> 00:03:03,750
to the triangular function which looks something
like this
29
00:03:03,750 --> 00:03:08,990
When we take the projection of the blue function
on the W1 subspace that is on the interval
30
00:03:08,990 --> 00:03:15,019
of size 2 to the power minus1 that is interval
of size half what we get is a function shown
31
00:03:15,019 --> 00:03:20,750
in red which is a detailed function on W1
subspace When we add these 2 functions we
32
00:03:20,750 --> 00:03:26,540
get even better approximation of the blue
function which is shown something like a staircase
33
00:03:26,540 --> 00:03:32,800
Now when we add more and more details to the
function say this we get better approximation
34
00:03:32,800 --> 00:03:39,000
to the blue function Again adding finer and
finer details results in getting even better
35
00:03:39,000 --> 00:03:44,900
approximation to the original signal So this
was another example in which we had the approximation
36
00:03:44,900 --> 00:03:52,240
of the single on different subspaces So what
we have done is essentially that we have got
37
00:03:52,240 --> 00:03:57,000
a much more better approximation by combining
more and more finer details In this way we
38
00:03:57,000 --> 00:04:01,050
can construct a much Uhh better and better
approximation of the function that we started
39
00:04:01,050 --> 00:04:03,330
with by adding more and more details thank
you