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ocw.mit.edu. we he doesn't
he's got n e t at you
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PROFESSOR STRANG: OK, so.
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These are our two topics.
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And, Thanksgiving is
coming up, of course.
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With convolutions,
where are we?
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I'm probably starting
a little early.
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Oh, I am.
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Is that right?
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Yeah, OK.
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So, with convolutions, I
feel we've got a whole
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lot of formulas.
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We practiced on some specific
examples, but we didn't
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see the reason for them.
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We didn't say the use of them.
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And I'm unwilling to let a
whole topic, important topic
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like convolutions, go
on with just formulas.
24
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So I want to talk about
signal processing.
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So that's a nice, perfect
application of convolutions.
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And you'll see it.
27
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You'll see the point, and
it's something that you're
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going to end up doing.
29
00:01:23 --> 00:01:27
And it's quite a simple
idea, once you understand
30
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convolutions you've got it.
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OK, but then I do want to go on
to the Fourier integral part.
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00:01:34 --> 00:01:38
And basically, today I'll
just give the formulas.
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So you've got the Fourier
integral formulas that
34
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take from a function f,
defined for all x now.
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It's on the whole line, like
some bell-shaped curve or
36
00:01:50 --> 00:01:53
some exponential decaying.
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And then you get its transform,
I could call that c(k), but a
38
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more familiar notation
is f hat of k.
39
00:02:04 --> 00:02:09
So that will be the Fourier
integral transform, or just for
40
00:02:09 --> 00:02:13
short, Fourier transform of
f(x), and it'll involve
41
00:02:13 --> 00:02:15
all frequencies.
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00:02:15 --> 00:02:19
So we are getting away from the
periodic case, and integer
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frequencies to the whole
line case with a whole
44
00:02:22 --> 00:02:24
line of frequencies.
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OK, so is that alright?
46
00:02:26 --> 00:02:31
Finishing, I'll have more to
say about 4.4 convolutions,
47
00:02:31 --> 00:02:33
but I want to say this much.
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OK, let's move to an example.
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So here's a typical
block diagram.
50
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In comes the signal.
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Vector x, values x_k.
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Often, in engineering and
EE, people tend to write
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that x[k], these days.
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OK, so, as being
easier to type.
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And sort of better.
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But I'll stay with
the subscript.
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So that signal comes in.
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And it goes through a filter.
59
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And I'm taking the simplest
filter I can think of, the one
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that just averages the current
value with the previous value.
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And we want to see what's
the effect of doing that.
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So we want to see, we want to
understand the outputs, the
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y_k's, which are just the
current value and the
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previous value averaged.
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We want to see that as
a convolution, and
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see what it's doing.
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00:03:39 --> 00:03:50
OK, so that's the - and I guess
that here, it's a frequent
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convention in signal processing
to basically pretend that the
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signal is infinitely long.
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That there's no start
and no finish.
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Of course, in reality there has
to be a start and a finish.
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But if it's a long, you know
if it's a cd or something,
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and you're sampling it
every, thousands and
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thousands of times.
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And the input signal is so long
and you're not really caring
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about the very start
and the very end.
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It's simpler to just
pretend that you've got
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numbers for all of k.
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OK, then let's see what
kind of a filter this is.
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What does it do to the signal?
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OK.
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00:04:43 --> 00:04:49
Well, one way filters are -
well, first of all, let's
83
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see it as a convolution.
84
00:04:51 --> 00:04:56
So I want to see that this
formula is a convolution that
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00:04:56 --> 00:05:02
the output y, that output of
all the averages y, is the
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00:05:02 --> 00:05:08
convolution of some filter,
some, and I'll give it a proper
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name, but with the input.
88
00:05:14 --> 00:05:17
And notice that, again, there's
no circle around here.
89
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We're not doing the cyclic
case, we're just pretending
90
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infinitely long.
91
00:05:21 --> 00:05:25
So OK, now I just want to ask
you what is the h, I want to
92
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recognize this as a filter.
93
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So that convolution, let's
remember the notation.
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00:05:32 --> 00:05:36
And then we can
match this to that.
95
00:05:36 --> 00:05:42
So remember the notation
for that would be, for a
96
00:05:42 --> 00:05:46
convolution is, I take a sum.
97
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Of all x, of h_l.
98
00:05:50 --> 00:05:53
99
00:05:53 --> 00:05:56
x_(k-l), right?
100
00:05:56 --> 00:05:58
That's what convolution is.
101
00:05:58 --> 00:06:02
This is our famous, and
it would be in principle
102
00:06:02 --> 00:06:04
the sum over f.
103
00:06:04 --> 00:06:06
Sum over l's.
104
00:06:06 --> 00:06:12
So I the filter is defined
by these numbers h, these
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h's, h_0, h_1, so on.
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00:06:19 --> 00:06:26
Those numbers h multiply the
x's, where this familiar rule
107
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to give the k'th output.
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00:06:30 --> 00:06:36
OK, let's have no mystery here.
109
00:06:36 --> 00:06:41
What are the h's if
this is the output?
110
00:06:41 --> 00:06:44
Well, I want to match
that with that.
111
00:06:44 --> 00:06:51
So here I see that this takes
x_k, multiplies by half.
112
00:06:51 --> 00:06:56
So that tells me that when
l is zero, I'm getting h_0
113
00:06:56 --> 00:06:58
times x_k, so what's h_0?
114
00:07:00 --> 00:07:01
So what's the h_0?
115
00:07:03 --> 00:07:05
It's 1/2.
116
00:07:05 --> 00:07:09
That's what this
formula is saying.
117
00:07:09 --> 00:07:17
When l is zero, so l is zero,
take h_0, 1/2 times x_k, ta-da.
118
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Now, what's the other h
that's showing up here?
119
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This is a very,
very short filter.
120
00:07:22 --> 00:07:27
It's only going to have two
coefficients, h_0 and h what?
121
00:07:27 --> 00:07:28
One.
122
00:07:28 --> 00:07:30
And what's the coefficient h_1?
123
00:07:31 --> 00:07:33
What's the number h_1?
124
00:07:35 --> 00:07:37
Now you've told me
everything about h when
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you tell me that number.
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00:07:39 --> 00:07:41
Everybody sees it.
127
00:07:41 --> 00:07:43
It's also 1/2, right?
128
00:07:43 --> 00:07:46
Because I'm taking
1/2 of x_(k-1).
129
00:07:48 --> 00:07:55
So when l is that one, we have
h, the h is 1/2, times x_(k-1).
130
00:07:56 --> 00:08:00
Do you see that that simple
averaging, running average,
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00:08:00 --> 00:08:01
you could call it.
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Running average, it's the most,
the first thing you would think
133
00:08:04 --> 00:08:09
of to - why would you
do such a thing?
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00:08:09 --> 00:08:13
Why is filtering done?
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00:08:13 --> 00:08:18
This filter, this averaging
filter, would smooth the data.
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00:08:18 --> 00:08:22
So the data comes with
noise, of course.
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00:08:22 --> 00:08:27
And what you'd like, so noise
is high frequency stuff.
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00:08:27 --> 00:08:31
So what you want to do is like
damp those high frequencies a
139
00:08:31 --> 00:08:35
little bit, because much of it
is not, it hasn't got
140
00:08:35 --> 00:08:36
information in it.
141
00:08:36 --> 00:08:42
It's just noise, but you
want to keep the signal.
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00:08:42 --> 00:08:45
So it's always this
signal to noise ratio.
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00:08:45 --> 00:08:48
That's the key - SNR.
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00:08:48 --> 00:08:49
PSNR.
145
00:08:49 --> 00:08:55
That's the constant expression,
signal to noise ratio.
146
00:08:55 --> 00:08:59
And we're sort of expecting
here that the signal to
147
00:08:59 --> 00:09:01
noise ratio is pretty good.
148
00:09:01 --> 00:09:02
High.
149
00:09:02 --> 00:09:03
Mostly signal.
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00:09:03 --> 00:09:06
But there's some noise.
151
00:09:06 --> 00:09:10
This is a very simple,
extremely short, filter.
152
00:09:10 --> 00:09:16
So this vector h, it's
a proper convolution.
153
00:09:16 --> 00:09:18
You could say h has
infinitely many components.
154
00:09:18 --> 00:09:21
But they're all zero,
except for those two.
155
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Right, do you see it?
156
00:09:23 --> 00:09:28
Another way, just at the end
of last time, I asked you
157
00:09:28 --> 00:09:33
to think of a matrix that's
doing the same thing.
158
00:09:33 --> 00:09:36
Why do I bring a matrix in?
159
00:09:36 --> 00:09:40
Because anytime I see
something linear, and that's
160
00:09:40 --> 00:09:42
incredibly linear, right?
161
00:09:42 --> 00:09:45
I think OK, there's
a matrix doing it.
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00:09:45 --> 00:09:50
So these y's, like
y_k, y_(k+1), all the
163
00:09:50 --> 00:09:52
y's are coming out.
164
00:09:52 --> 00:10:01
The x's are going in, x_k,
x_(k+1), x_(k-1), bunch of x's.
165
00:10:01 --> 00:10:04
And there's a matrix
doing exactly that.
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00:10:04 --> 00:10:06
And what does that matrix have?
167
00:10:06 --> 00:10:10
Well, it has 1/2
on the diagonal.
168
00:10:10 --> 00:10:14
So that y_k will have a 1/2
of x_k, and what's the
169
00:10:14 --> 00:10:18
other entry in that row?
170
00:10:18 --> 00:10:23
I want 1/2 of x_k, and I
want 1/2 of x_(k-1), right?
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00:10:23 --> 00:10:26
So I just put 1/2 next to it.
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00:10:26 --> 00:10:29
So there is the main diagonal
of the halves, and there is
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00:10:29 --> 00:10:32
the sub-diagonal of the half.
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00:10:32 --> 00:10:37
So it's just constant diagonal.
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00:10:37 --> 00:10:41
Now, yeah, let me
tell you the word.
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00:10:41 --> 00:10:45
When an engineer, electrical
engineer looks at this, first
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00:10:45 --> 00:10:47
thing, or this, any of these.
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00:10:47 --> 00:10:51
First letters he uses is
linear time invariant.
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00:10:51 --> 00:10:54
LTI.
180
00:10:54 --> 00:10:58
So linear we understand, right?
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00:10:58 --> 00:11:01
What does this time
invariant mean?
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00:11:01 --> 00:11:05
Time invariant means that
you're not changing the filter
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00:11:05 --> 00:11:07
as the signal comes through.
184
00:11:07 --> 00:11:10
You're keeping a
half and a half.
185
00:11:10 --> 00:11:12
You're keeping the formula.
186
00:11:12 --> 00:11:16
The formula doesn't depend
on k, the numbers are
187
00:11:16 --> 00:11:18
just 1/2 and 1/2.
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00:11:18 --> 00:11:22
They're the same, so if I shift
the whole signal by a thousand,
189
00:11:22 --> 00:11:25
the output shifts by
a thousand, right?
190
00:11:25 --> 00:11:29
If I take the whole signal
and delay it, delay it
191
00:11:29 --> 00:11:31
by a thousand times?
192
00:11:31 --> 00:11:34
Clock times?
193
00:11:34 --> 00:11:39
Then the same output will come
a thousand clock times delayed.
194
00:11:39 --> 00:11:40
So linear time invariant.
195
00:11:40 --> 00:11:44
That would be - I mean, linear
time invariant is just
196
00:11:44 --> 00:11:48
talking convolution.
197
00:11:48 --> 00:11:52
I mean, that's what it comes to
if we're in discrete problems.
198
00:11:52 --> 00:11:54
It's just that, for some h.
199
00:11:54 --> 00:12:03
Now, our h deserves like,
the other initials you see.
200
00:12:03 --> 00:12:04
OK, that was linear
time invariant.
201
00:12:04 --> 00:12:10
Now, the next initials
you'll see will be F-I-R.
202
00:12:10 --> 00:12:11
It's an FIR filter.
203
00:12:11 --> 00:12:19
So that's finite
impulse response.
204
00:12:19 --> 00:12:21
What's the impulse
response mean?
205
00:12:21 --> 00:12:23
It means the h.
206
00:12:23 --> 00:12:26
The vector h is the
impulse response.
207
00:12:26 --> 00:12:30
The vector h is what you
get if I put an impulse
208
00:12:30 --> 00:12:32
in, what comes out?
209
00:12:32 --> 00:12:33
Just tell me, what
happens here.
210
00:12:33 --> 00:12:38
Suppose an impulse, by an
impulse I mean I stick a one in
211
00:12:38 --> 00:12:39
one position and all zeroes.
212
00:12:39 --> 00:12:41
Our usual delta.
213
00:12:41 --> 00:12:46
Our impulse, our spike, is
just, suppose the x's have
214
00:12:46 --> 00:12:51
a one here, otherwise all
zero, what comes out?
215
00:12:51 --> 00:12:55
Well, suppose there's
just x_0 is one.
216
00:12:55 --> 00:12:57
What is y?
217
00:12:57 --> 00:13:01
Suppose the only input is
boom, you know a bell
218
00:13:01 --> 00:13:03
sounds at time zero.
219
00:13:03 --> 00:13:06
What comes out from the filter?
220
00:13:06 --> 00:13:11
Well, y_0 will be what?
221
00:13:11 --> 00:13:17
If the input has just a single
x, and it's one, then at that
222
00:13:17 --> 00:13:20
time zero, so x_0 is one.
223
00:13:20 --> 00:13:22
Then y_0 will be?
224
00:13:22 --> 00:13:27
1/2, and what will be y_1?
225
00:13:29 --> 00:13:30
Also 1/2, right?
226
00:13:30 --> 00:13:34
Because of y_1 will take
x_1, that's already
227
00:13:34 --> 00:13:36
dropped back to zero.
228
00:13:36 --> 00:13:38
Plus x_0, that's the bell.
229
00:13:38 --> 00:13:39
It'll be 1/2.
230
00:13:39 --> 00:13:42
In other words, the
output is this.
231
00:13:42 --> 00:13:45
No big deal.
232
00:13:45 --> 00:13:49
The impulse responses
is exactly h.
233
00:13:49 --> 00:13:51
So you can say they've
created a long word for
234
00:13:51 --> 00:13:55
a small idea, true.
235
00:13:55 --> 00:14:00
And the idea, the word finite
is the important number.
236
00:14:00 --> 00:14:04
Finite, meaning that it's
finite length, I only have a
237
00:14:04 --> 00:14:08
finite number of h's, and
in this case two h's.
238
00:14:08 --> 00:14:14
So that's - part of
every subject is just
239
00:14:14 --> 00:14:16
learning the language.
240
00:14:16 --> 00:14:21
So LTI, it means you've
got a convolution.
241
00:14:21 --> 00:14:26
FIR means that the convolution
has finite length.
242
00:14:26 --> 00:14:29
OK, now for the question.
243
00:14:29 --> 00:14:34
What is this filter
doing to the signal?
244
00:14:34 --> 00:14:35
It's certainly averaging.
245
00:14:35 --> 00:14:38
That's clear.
246
00:14:38 --> 00:14:41
But we want to be more precise.
247
00:14:41 --> 00:14:46
Well, let me take
some examples.
248
00:14:46 --> 00:14:51
Suppose the input is all ones.
249
00:14:51 --> 00:14:54
All x_k are one.
250
00:14:54 --> 00:14:55
So a constant input.
251
00:14:55 --> 00:14:57
Natural thing to test.
252
00:14:57 --> 00:15:02
That's the constant input,
that's zero frequency.
253
00:15:02 --> 00:15:04
Zero frequency.
254
00:15:04 --> 00:15:12
What's the output?
255
00:15:12 --> 00:15:15
From all ones going in wow,
sorry to ask you such
256
00:15:15 --> 00:15:17
a trivial question.
257
00:15:17 --> 00:15:21
You came in for some good
math here, and I'm just
258
00:15:21 --> 00:15:22
taking 1/2 and 1/2.
259
00:15:22 --> 00:15:28
So the output is all
y's equal, right?
260
00:15:28 --> 00:15:33
So, to me, that's telling, just
to introduce an appropriate
261
00:15:33 --> 00:15:38
word, low frequencies; in fact,
bottom frequencies, zero
262
00:15:38 --> 00:15:42
frequency, is passed
straight through.
263
00:15:42 --> 00:15:44
That's a low pass filter.
264
00:15:44 --> 00:15:49
That's telling me I have
a low pass filter.
265
00:15:49 --> 00:15:52
So that's an expression.
266
00:15:52 --> 00:15:56
That's so simple that you might
as well know those words.
267
00:15:56 --> 00:16:03
Low pass means that the lowest
frequencies pass through
268
00:16:03 --> 00:16:04
virtually unchanged.
269
00:16:04 --> 00:16:09
In this case, the very zero
frequency the DC term, the
270
00:16:09 --> 00:16:12
constant term, passes through
completely unchanged.
271
00:16:12 --> 00:16:17
Now, what about another input
all - well, now I want
272
00:16:17 --> 00:16:18
high frequencies.
273
00:16:18 --> 00:16:22
Top frequencies.
274
00:16:22 --> 00:16:27
What's the the highest
oscillation I can get would
275
00:16:27 --> 00:16:34
be x equal, say, it starts
one minus one, one,
276
00:16:34 --> 00:16:37
minus one, so on.
277
00:16:37 --> 00:16:38
Both directions.
278
00:16:38 --> 00:16:41
Oscillating as
fast as possible.
279
00:16:41 --> 00:16:44
I couldn't get a faster
frequency of oscillation
280
00:16:44 --> 00:16:48
in a discrete signal
than up, down, up, down.
281
00:16:48 --> 00:16:53
What's the output for that?
282
00:16:53 --> 00:16:58
So that's really oscillation.
283
00:16:58 --> 00:17:01
That's the fastest oscillation.
284
00:17:01 --> 00:17:03
What would be the output
from my averaging
285
00:17:03 --> 00:17:06
filter for this input?
286
00:17:06 --> 00:17:08
Zero.
287
00:17:08 --> 00:17:12
At every step, I'm averaging
this with the guy before
288
00:17:12 --> 00:17:13
and they add to zero.
289
00:17:13 --> 00:17:16
I'm averaging this with the guy
before, this with the guy -
290
00:17:16 --> 00:17:20
output is, y equals all zeroes.
291
00:17:20 --> 00:17:26
OK, so that confirms in my mind
that I have a low pass filter.
292
00:17:26 --> 00:17:30
The high frequencies
are getting wiped out.
293
00:17:30 --> 00:17:34
OK, so that's two examples.
294
00:17:34 --> 00:17:36
Now, what about
frequencies in between?
295
00:17:36 --> 00:17:41
Because ultimately we want
to see what's happening to
296
00:17:41 --> 00:17:43
frequencies in between.
297
00:17:43 --> 00:17:45
OK, so what's an
in-between frequency?
298
00:17:45 --> 00:17:54
So in between x_k could
be e^(ikn), let's
299
00:17:54 --> 00:17:55
say. e^(ik*omega).
300
00:17:57 --> 00:17:58
e^(ik*omega).
301
00:18:00 --> 00:18:09
Where this omega is somewhere
between minus pi and pi.
302
00:18:09 --> 00:18:13
OK, why do I say
minus pi and pi?
303
00:18:13 --> 00:18:16
If the frequency - so
that's the frequency.
304
00:18:16 --> 00:18:19
If omega is zero,
what's my signal?
305
00:18:19 --> 00:18:22
All ones, right?
306
00:18:22 --> 00:18:25
If omega is zero,
everything is my all ones.
307
00:18:25 --> 00:18:26
This is this k.
308
00:18:26 --> 00:18:29
So I now have a letter
for it, omega=0.
309
00:18:29 --> 00:18:32
310
00:18:32 --> 00:18:34
What's this top frequency?
311
00:18:34 --> 00:18:38
One, minus one, one, minus
one, what omega will give
312
00:18:38 --> 00:18:41
me alternating signs?
313
00:18:41 --> 00:18:43
Omega equal?
314
00:18:43 --> 00:18:44
Pi, right?
315
00:18:44 --> 00:18:44
Omega=pi.
316
00:18:45 --> 00:18:50
Because if this is pi, I have
e^(i*pi), which is minus one.
317
00:18:50 --> 00:19:01
So when omega=pi, my inputs are
e^(i*omega), to the k'th power.
318
00:19:01 --> 00:19:06
But this is minus one.
e^(i*pi), to the k'th power,
319
00:19:06 --> 00:19:09
and that's minus one.
320
00:19:09 --> 00:19:12
So that's the top frequency.
321
00:19:12 --> 00:19:15
And also the bottom
frequency is pi.
322
00:19:15 --> 00:19:20
And the zero frequency
is the all one.
323
00:19:20 --> 00:19:24
And this is what happens - ah.
324
00:19:24 --> 00:19:29
Now, comes the point.
325
00:19:29 --> 00:19:32
What's the output if
this is the input?
326
00:19:32 --> 00:19:34
What's the output when
this is the input?
327
00:19:34 --> 00:19:36
We can easily figure that out.
328
00:19:36 --> 00:19:38
We can take that average.
329
00:19:38 --> 00:19:44
OK, so let me do that input.
330
00:19:44 --> 00:19:49
Input x_k is e^(ik*omega).
331
00:19:49 --> 00:19:52
332
00:19:52 --> 00:20:01
And now what's the output?
y_k is the average of that.
333
00:20:01 --> 00:20:05
And the one before.
334
00:20:05 --> 00:20:09
Divided by two, right?
335
00:20:09 --> 00:20:14
OK, now you're certainly going
to factor out, anybody who
336
00:20:14 --> 00:20:18
sees this is going to factor
out e^(ik*omega), right?
337
00:20:18 --> 00:20:22
I mean that's sitting there,
that's the whole point of these
338
00:20:22 --> 00:20:25
exponentials is they factor
out of all linear stuff.
339
00:20:25 --> 00:20:33
So if I factor that out, I get
a very very important thing.
340
00:20:33 --> 00:20:37
I get, well, it's over
two, I get a one.
341
00:20:37 --> 00:20:43
And I get, what's this term?
e^(ik*omega) is here.
342
00:20:43 --> 00:20:44
So I only want e^(-i*omega).
343
00:20:44 --> 00:20:48
344
00:20:48 --> 00:20:54
OK, that is called the
frequency response.
345
00:20:54 --> 00:20:58
So that's telling me the
response of what the filter
346
00:20:58 --> 00:21:01
does to frequency omega.
347
00:21:01 --> 00:21:04
It multiplies the signal.
348
00:21:04 --> 00:21:09
If I have a signal that's
purely with frequency omega,
349
00:21:09 --> 00:21:11
that signal is getting
multiplied by that
350
00:21:11 --> 00:21:14
response factor.
351
00:21:14 --> 00:21:14
1+e^(i*omega).
352
00:21:15 --> 00:21:21
When omega is zero,
what is this quantity?
353
00:21:21 --> 00:21:24
So let me call this
cap h(omega).
354
00:21:24 --> 00:21:27
355
00:21:27 --> 00:21:27
what.
356
00:21:27 --> 00:21:31
Is this factor, if
omega is zero?
357
00:21:31 --> 00:21:35
Then h(omega)=0 is?
358
00:21:35 --> 00:21:37
One.
359
00:21:37 --> 00:21:42
That's telling me again that
at zero frequency the output
360
00:21:42 --> 00:21:44
is the same as the input.
361
00:21:44 --> 00:21:45
Multiplied by one.
362
00:21:45 --> 00:21:52
And at omega equal to pi, what
is this frequency response?
363
00:21:52 --> 00:21:53
Zero, right.
364
00:21:53 --> 00:21:58
At omega=pi, this is
minus one so I get zero.
365
00:21:58 --> 00:22:02
And it's telling me again
that this is the response.
366
00:22:02 --> 00:22:08
And now it's also telling me
what the response factor is for
367
00:22:08 --> 00:22:10
the frequencies in between.
368
00:22:10 --> 00:22:15
And everybody would draw a
graph of the darn thing, right?
369
00:22:15 --> 00:22:19
So this was simple, let me
do its graph over here.
370
00:22:19 --> 00:22:21
So I'm going to graph h(omega).
371
00:22:23 --> 00:22:27
Well, I've a little problem.
h(omega)'s a complex number.
372
00:22:27 --> 00:22:31
I'll graph the
magnitude response.
373
00:22:31 --> 00:22:36
So here I'm going to do a
graph from minus pi to pi.
374
00:22:36 --> 00:22:38
This is the picture.
375
00:22:38 --> 00:22:41
This is the picture
people look at.
376
00:22:41 --> 00:22:46
This is the picture of
what the filter is doing.
377
00:22:46 --> 00:22:50
All the information about
the filter is in here.
378
00:22:50 --> 00:22:53
All the information
is in there.
379
00:22:53 --> 00:22:56
So if I graph that, I know
what the filter's doing.
380
00:22:56 --> 00:23:01
So you said at omega=0,
I get a value of one.
381
00:23:01 --> 00:23:03
At omega=pi, I get
a value of zero.
382
00:23:03 --> 00:23:06
At omega equal minus pi,
I get a value of zero.
383
00:23:06 --> 00:23:10
And I think if you figure
out the magnitude,
384
00:23:10 --> 00:23:12
it's just a cosine.
385
00:23:12 --> 00:23:15
It's just an arc of a cosine.
386
00:23:15 --> 00:23:19
OK, for that really,
really simple filter.
387
00:23:19 --> 00:23:23
So any engineer, any signal
processing person, looks at
388
00:23:23 --> 00:23:31
this graph of h(omega) and says
that is a very fuzzy filter.
389
00:23:31 --> 00:23:37
A good, an ideal filter, an
ideal low pass filter, would
390
00:23:37 --> 00:23:38
do something like this.
391
00:23:38 --> 00:23:44
An ideal filter would
stay at one up to some
392
00:23:44 --> 00:23:47
frequency - say, pi/2.
393
00:23:48 --> 00:23:50
And drop instantly to zero.
394
00:23:50 --> 00:23:52
There is a really good filter.
395
00:23:52 --> 00:23:55
I mean, people would pay
money for that filter.
396
00:23:55 --> 00:23:58
Because what happens when
you send a signal through
397
00:23:58 --> 00:24:00
that ideal filter?
398
00:24:00 --> 00:24:03
It completely wipes out
the top frequencies.
399
00:24:03 --> 00:24:06
Let's say, up after pi/2.
400
00:24:06 --> 00:24:09
And it completely saves
the in-between ones.
401
00:24:09 --> 00:24:11
So that's really
a sharp filter.
402
00:24:11 --> 00:24:16
Actually, what people would
like to do would be to have
403
00:24:16 --> 00:24:18
that filter available.
404
00:24:18 --> 00:24:22
And then also to have a perfect
ideal high pass filter.
405
00:24:22 --> 00:24:26
What would be an ideal
high pass filter?
406
00:24:26 --> 00:24:29
Yeah, let's talk about high
pass filters just a moment.
407
00:24:29 --> 00:24:34
Because this is, you're seeing
the reality of what people do,
408
00:24:34 --> 00:24:39
and how they - and that little
easy bit of math they do.
409
00:24:39 --> 00:24:41
Do you want to suggest a
high pass filter, let
410
00:24:41 --> 00:24:43
me come back to this?
411
00:24:43 --> 00:24:46
And just change it a little?
412
00:24:46 --> 00:24:53
So I plan to do not - I'm now
going to do a different filter.
413
00:24:53 --> 00:24:56
That's going to be a
high pass filter.
414
00:24:56 --> 00:24:58
And what do I mean by that?
415
00:24:58 --> 00:25:04
A high pass filter
will kill the x_k=1.
416
00:25:05 --> 00:25:07
I now want the output from
- this is now going to
417
00:25:07 --> 00:25:09
be, I'm going to change.
418
00:25:09 --> 00:25:11
Can I just erase, change
a lot of things?
419
00:25:11 --> 00:25:17
I'm now going to produce
a high pass filter.
420
00:25:17 --> 00:25:19
Sorry, pi.
421
00:25:19 --> 00:25:21
And what's the difference?
422
00:25:21 --> 00:25:26
When all x's are one, the
output is going to be?
423
00:25:26 --> 00:25:27
Zero.
424
00:25:27 --> 00:25:31
And when I have the highest
frequency, the output
425
00:25:31 --> 00:25:32
is going to be?
426
00:25:32 --> 00:25:37
The input.
427
00:25:37 --> 00:25:40
And what am I - and then
in between, I'll do
428
00:25:40 --> 00:25:42
something in between.
429
00:25:42 --> 00:25:47
OK, what do you think would
be a high pass filter, like
430
00:25:47 --> 00:25:52
the simplest high pass
filter we can think of?
431
00:25:52 --> 00:25:53
Anybody think of it?
432
00:25:53 --> 00:25:58
You're only getting, like, 15
seconds to think in this class.
433
00:25:58 --> 00:26:01
That's a small
drawback, 15 seconds.
434
00:26:01 --> 00:26:06
But, the high pass filter
that I think of first
435
00:26:06 --> 00:26:12
is, take the difference.
436
00:26:12 --> 00:26:13
Take the difference.
437
00:26:13 --> 00:26:18
Put minus 1/2s on
the sub-diagonal.
438
00:26:18 --> 00:26:22
This is the same, this is
also a convolution, but now
439
00:26:22 --> 00:26:24
what h_0 is still a half.
440
00:26:24 --> 00:26:27
But now h_1 is?
441
00:26:27 --> 00:26:29
Minus 1/2.
442
00:26:29 --> 00:26:31
We're still convolving.
443
00:26:31 --> 00:26:34
We're still convolving with
- it's still linear time
444
00:26:34 --> 00:26:37
invariant, that just means
it's a convolution.
445
00:26:37 --> 00:26:39
It's still a finite
impulse response.
446
00:26:39 --> 00:26:46
But the response, the impulse
response is now 1/2 minus 1/2.
447
00:26:46 --> 00:26:50
So what happens if I, in my
picture over here, if I send
448
00:26:50 --> 00:26:55
in any pure frequency, I'm
now doing minus 1/2 here.
449
00:26:55 --> 00:26:57
So I'll just keep the plus.
450
00:26:57 --> 00:27:02
But I'll also add in the minus.
451
00:27:02 --> 00:27:04
So now I'm looking at
1-e^(-1*omega/2).
452
00:27:08 --> 00:27:11
And again, let's plot a
few points for that guy.
453
00:27:11 --> 00:27:15
So what, at x at omega
- so this is omega
454
00:27:15 --> 00:27:16
in this direction.
455
00:27:16 --> 00:27:19
And this is h in
this direction.
456
00:27:19 --> 00:27:24
So at omega=0, what's
my high pass guy?
457
00:27:24 --> 00:27:29
When I send in a zero
frequency, constant,
458
00:27:29 --> 00:27:32
I get what output?
459
00:27:32 --> 00:27:37
Zeroes, because now - I I'll
call it a differencing filter.
460
00:27:37 --> 00:27:44
So I'll just, instead of
averaging I'm differencing.
461
00:27:44 --> 00:27:50
OK, so now for this one, maybe
I'll put an x to indicate I'm
462
00:27:50 --> 00:27:53
now doing, I'll do x's
for the high pass.
463
00:27:53 --> 00:27:57
So this now, the high pass guy,
kills the low frequency and
464
00:27:57 --> 00:28:00
preserves the high frequency.
465
00:28:00 --> 00:28:05
And you won't be surprised to
find it's some cosine or
466
00:28:05 --> 00:28:12
something that, well yeah,
it's got so sorry that's
467
00:28:12 --> 00:28:15
not much of a cosine.
468
00:28:15 --> 00:28:26
It's the mirror image
of the low pass guy.
469
00:28:26 --> 00:28:29
And maybe the sum of
squares adds to one
470
00:28:29 --> 00:28:30
or two or something.
471
00:28:30 --> 00:28:32
One, probably.
472
00:28:32 --> 00:28:34
The sum of squares
probably adds to one.
473
00:28:34 --> 00:28:39
And they're kind of
complementary filters.
474
00:28:39 --> 00:28:41
But they're very poor.
475
00:28:41 --> 00:28:47
Very crude, I mean that's so
far from the ideal filter.
476
00:28:47 --> 00:28:52
So how would we create a
closer to ideal filter?
477
00:28:52 --> 00:28:57
Well, we need more h's.
478
00:28:57 --> 00:29:00
With two h's, we're doing
the best we can, with
479
00:29:00 --> 00:29:01
just h_0 and h_1.
480
00:29:02 --> 00:29:05
With a longer filter, for which
we're going to have to pay a
481
00:29:05 --> 00:29:08
little more to use, but
we'll get a lot more.
482
00:29:08 --> 00:29:12
We'll get something, we could
get a filter that stays
483
00:29:12 --> 00:29:14
pretty close to this,
drops pretty fast.
484
00:29:14 --> 00:29:15
There's a whole world.
485
00:29:15 --> 00:29:22
Bell Labs had a little
team of filter experts.
486
00:29:22 --> 00:29:27
Creating, and now MATLAB will
create it for you, The
487
00:29:27 --> 00:29:30
coefficients, h, that would
give you a response, a
488
00:29:30 --> 00:29:34
frequency response, it'll stay
up toward, up close to
489
00:29:34 --> 00:29:36
one as long as possible.
490
00:29:36 --> 00:29:41
And drop as fast as possible,
and bounce around there.
491
00:29:41 --> 00:29:45
So next week, if I come back
to that topic, I can say a
492
00:29:45 --> 00:29:50
little more about these
really good filters.
493
00:29:50 --> 00:29:53
What was I trying to do today?
494
00:29:53 --> 00:29:59
Trying to see how
convolution is used.
495
00:29:59 --> 00:30:02
And this is a use you
will really make.
496
00:30:02 --> 00:30:04
So now I just have, I think,
about two more things to
497
00:30:04 --> 00:30:06
say about this example.
498
00:30:06 --> 00:30:08
Let's see, what are they?
499
00:30:08 --> 00:30:11
Well, first, so all
the information is
500
00:30:11 --> 00:30:12
in this h(omega).
501
00:30:12 --> 00:30:15
502
00:30:15 --> 00:30:18
Oh, yeah.
503
00:30:18 --> 00:30:24
This simple example gives us a
way to visualize convolution.
504
00:30:24 --> 00:30:26
And I think we need that.
505
00:30:26 --> 00:30:27
Right?
506
00:30:27 --> 00:30:30
Because up to now, convolution
has been a formula.
507
00:30:30 --> 00:30:31
Right?
508
00:30:31 --> 00:30:34
It's been this formula.
509
00:30:34 --> 00:30:36
That's the formula for
convolution, and how
510
00:30:36 --> 00:30:39
do I visualize that?
511
00:30:39 --> 00:30:43
Just think of, may I
try to visualize that?
512
00:30:43 --> 00:30:50
Here I have, this
is the time line.
513
00:30:50 --> 00:30:54
The different k's. k equals
zero, one, two, minus one.
514
00:30:54 --> 00:30:57
515
00:30:57 --> 00:30:58
And I have x_-1,
x_0, x_1, x_2, x_3.
516
00:30:58 --> 00:31:03
517
00:31:03 --> 00:31:08
So that would be a little
bouncy up and down.
518
00:31:08 --> 00:31:11
And the averaging filter,
let me go back to
519
00:31:11 --> 00:31:12
the averaging one.
520
00:31:12 --> 00:31:17
The averaging filter would
smooth out the bumps.
521
00:31:17 --> 00:31:23
Because it would take the,
like, average neighbors.
522
00:31:23 --> 00:31:26
And that's a smoothing process.
523
00:31:26 --> 00:31:32
As we see here, it's a process
that kills high frequency.
524
00:31:32 --> 00:31:37
Now, what is this visualization
I want you to think of?
525
00:31:37 --> 00:31:41
I want you to just think
of, like, a moving window.
526
00:31:41 --> 00:31:44
So here is the input.
527
00:31:44 --> 00:31:47
Now, I move a window along.
528
00:31:47 --> 00:31:51
And that window, so let's
say here's the window,
529
00:31:51 --> 00:31:54
I should have another.
530
00:31:54 --> 00:31:55
So that's the window.
531
00:31:55 --> 00:32:00
When the window is there, it
takes the average of those two.
532
00:32:00 --> 00:32:02
That gives me the new output.
533
00:32:02 --> 00:32:06
Now, think of the window as
moving along here, taking
534
00:32:06 --> 00:32:08
the average of these.
535
00:32:08 --> 00:32:10
Move the window along, take
the average of these.
536
00:32:10 --> 00:32:12
Move the window along.
537
00:32:12 --> 00:32:18
Do you see the sort of, this is
what a convolution is doing.
538
00:32:18 --> 00:32:24
This is a picture of my
formula, sum of h_k*x_(l-k).
539
00:32:24 --> 00:32:28
540
00:32:28 --> 00:32:33
So the window is the h's, is
the width of the h's, and as
541
00:32:33 --> 00:32:35
that window moves along.
542
00:32:35 --> 00:32:41
I mean, you could write, you
could create, design a little
543
00:32:41 --> 00:32:44
circuit that would
do exactly this.
544
00:32:44 --> 00:32:46
That would do the convolution.
545
00:32:46 --> 00:32:51
You just have to put together
some multipliers, because
546
00:32:51 --> 00:32:54
you have these h's,
these, like, halves.
547
00:32:54 --> 00:32:59
And you have to put in an
adder, that'll add the pieces.
548
00:32:59 --> 00:33:07
And those are the essential
little electronic pieces
549
00:33:07 --> 00:33:12
of an actual filter.
550
00:33:12 --> 00:33:13
Then you just move it along.
551
00:33:13 --> 00:33:16
So it needs a delay.
552
00:33:16 --> 00:33:18
That's about the
content of a filter.
553
00:33:18 --> 00:33:26
Is multipliers that will
multiply by the h's, so in come
554
00:33:26 --> 00:33:35
the x, multiply by the h's, do
the addition, and do a shift
555
00:33:35 --> 00:33:37
to get onto the next one.
556
00:33:37 --> 00:33:39
You see how a filter works?
557
00:33:39 --> 00:33:43
I think that image, or
convoluted is a little
558
00:33:43 --> 00:33:44
bit vague, maybe?
559
00:33:44 --> 00:33:46
This window moving along?
560
00:33:46 --> 00:33:51
But it's quite meaningful.
561
00:33:51 --> 00:33:56
And then the final thing I'll
say about filters is this.
562
00:33:56 --> 00:34:02
That, what's the connection
between h(omega) and h(k)?
563
00:34:03 --> 00:34:05
Or h_k, let me call it h_k?
564
00:34:06 --> 00:34:11
What's the connection
between the numbers
565
00:34:11 --> 00:34:14
in impulse response?
566
00:34:14 --> 00:34:19
Just, which were the h's,
and the function, which is
567
00:34:19 --> 00:34:21
the frequency response?
568
00:34:21 --> 00:34:25
Which tells me what happens
to a particular frequency?
569
00:34:25 --> 00:34:30
Each frequency, e to the - you
notice how the frequency that
570
00:34:30 --> 00:34:34
went in is the frequency
that comes out.
571
00:34:34 --> 00:34:39
It's just amplified or
diminished by this
572
00:34:39 --> 00:34:41
h(omega) factor.
573
00:34:41 --> 00:34:46
So you see the h's are the
coefficients here of h(omega).
574
00:34:47 --> 00:34:51
In other words, h(omega)
is the sum of the
575
00:34:51 --> 00:34:53
h_k's, e to the, e^-ik.
576
00:34:53 --> 00:34:56
577
00:34:56 --> 00:35:00
Here's the beautiful formula.
578
00:35:00 --> 00:35:05
That's obvious, right?
579
00:35:05 --> 00:35:08
Here you're seeing the formula
in the simplest case, with
580
00:35:08 --> 00:35:10
just an h_0 and an h_1.
581
00:35:11 --> 00:35:15
But of course, it would have
worked if I had several h's.
582
00:35:15 --> 00:35:19
So this h(omega), this
factor that comes out,
583
00:35:19 --> 00:35:22
is just this guy.
584
00:35:22 --> 00:35:30
Now, if I look at that,
what am I saying?
585
00:35:30 --> 00:35:37
I've seen things that connect a
function of omega with a number
586
00:35:37 --> 00:35:41
of filter coefficients.
587
00:35:41 --> 00:35:45
I saw that in Section
4.1, in Fourier series.
588
00:35:45 --> 00:35:50
This is the Fourier series
for that function.
589
00:35:50 --> 00:35:52
Right?
590
00:35:52 --> 00:35:55
You might say, OK,
why that minus?
591
00:35:55 --> 00:35:57
I say, it's there
because the electrical
592
00:35:57 --> 00:35:59
engineers put it there.
593
00:35:59 --> 00:36:00
They liked it.
594
00:36:00 --> 00:36:04
And the rest of the world
has to live with it.
595
00:36:04 --> 00:36:08
So, you notice I
don't concede on i.
596
00:36:08 --> 00:36:11
I refuse to write j.
597
00:36:11 --> 00:36:14
But they all would.
598
00:36:14 --> 00:36:16
I speak about they,
but probably some
599
00:36:16 --> 00:36:17
you would write j.
600
00:36:17 --> 00:36:22
So I'm hoping it's OK if I
write i. i is for imaginary.
601
00:36:22 --> 00:36:25
I don't see how you could
say the word imaginary
602
00:36:25 --> 00:36:29
starting with a j.
603
00:36:29 --> 00:36:33
And what was the matter
with i, anyway?
604
00:36:33 --> 00:36:33
Current.
605
00:36:33 --> 00:36:35
Well, current used to be i.
606
00:36:35 --> 00:36:39
I mean who, is it still?
607
00:36:39 --> 00:36:41
Well, let's just accept it.
608
00:36:41 --> 00:36:44
OK, they can call the current
i, and the square root of minus
609
00:36:44 --> 00:36:48
one j, but not in 18.085.
610
00:36:48 --> 00:36:52
So OK, here we are.
611
00:36:52 --> 00:36:57
So my point is just that
we have a Fourier series.
612
00:36:57 --> 00:37:01
Here we a 2pi
periodic function.
613
00:37:01 --> 00:37:04
Here we have its
Fourier coefficients.
614
00:37:04 --> 00:37:10
The only difference is that we
started with the coefficients.
615
00:37:10 --> 00:37:12
And created the function.
616
00:37:12 --> 00:37:19
But otherwise, we're back to
Section 4.1, Fourier series.
617
00:37:19 --> 00:37:22
But that fact that we started
with the coefficients and built
618
00:37:22 --> 00:37:28
the function, sometimes you
could say, OK that sounds a
619
00:37:28 --> 00:37:30
little different from the
regular Fourier series,
620
00:37:30 --> 00:37:32
where you go the other way.
621
00:37:32 --> 00:37:36
So people give it the
name discrete time
622
00:37:36 --> 00:37:37
Fourier transform.
623
00:37:37 --> 00:37:40
You might see those
letters sometime.
624
00:37:40 --> 00:37:43
The discrete time Fourier
transform goes from the
625
00:37:43 --> 00:37:47
coefficients to the function.
626
00:37:47 --> 00:37:50
Where the standard Fourier
series starts with a function,
627
00:37:50 --> 00:37:51
goes to the coefficients.
628
00:37:51 --> 00:37:54
But really, it doesn't matter.
629
00:37:54 --> 00:37:56
The point is, yeah.
630
00:37:56 --> 00:38:01
So you could say, maybe we
have now a force transform.
631
00:38:01 --> 00:38:03
Like the first transform
was Fourier series.
632
00:38:03 --> 00:38:05
The second one was
the discrete.
633
00:38:05 --> 00:38:07
The third one is the
Fourier integral that's
634
00:38:07 --> 00:38:09
coming in one minute.
635
00:38:09 --> 00:38:12
And the fourth is this one.
636
00:38:12 --> 00:38:20
But hey, it's just the
coefficients and the function
637
00:38:20 --> 00:38:23
have switched places.
638
00:38:23 --> 00:38:30
In the, which one is the start
and which one is at the end.
639
00:38:30 --> 00:38:32
OK, let me pause a minute
because that's everything
640
00:38:32 --> 00:38:36
I wanted to say about
simple filters.
641
00:38:36 --> 00:38:41
And you can see that this
is a very simple filter,
642
00:38:41 --> 00:38:44
and could be improved.
643
00:38:44 --> 00:38:47
Better numbers would
give, I mean what would
644
00:38:47 --> 00:38:48
be better numbers?
645
00:38:48 --> 00:38:54
I suppose that 1/4, 1/2, 1/4
would probably be better.
646
00:38:54 --> 00:38:58
If I took those numbers,
I'm pretty sure that this
647
00:38:58 --> 00:39:06
thing would be closer to
ideal by quite a bit.
648
00:39:06 --> 00:39:11
If I plotted - so what do I
mean by, those are the h's.
649
00:39:11 --> 00:39:14
So I would take 1/4
plus 1/2 e^(-i*omega).
650
00:39:15 --> 00:39:18
Plus 1/4 e^(-2i*omega).
651
00:39:19 --> 00:39:22
This would be my
better h(omega).
652
00:39:23 --> 00:39:27
This would be my frequency
response to a better averaging
653
00:39:27 --> 00:39:32
filter, sort of this is like
averaged averaged, right?
654
00:39:32 --> 00:39:35
If I do a half an average and
then I do the average again.
655
00:39:35 --> 00:39:41
In other words, if I just send
these signals y_k through that
656
00:39:41 --> 00:39:45
same averaging filter, so
average again to get a z_k.
657
00:39:47 --> 00:39:51
I think probably the
coefficients would be 1/4, 1/2,
658
00:39:51 --> 00:39:54
1/4, and it would be, I've
taken out more noise.
659
00:39:54 --> 00:39:56
Right?
660
00:39:56 --> 00:40:00
Each time I did that averaging,
I damp the high frequencies, so
661
00:40:00 --> 00:40:03
if I do it twice I
get more damping.
662
00:40:03 --> 00:40:07
But I lose signal, of course.
663
00:40:07 --> 00:40:11
I mean, presumably there's some
information in the signal
664
00:40:11 --> 00:40:12
in these frequencies.
665
00:40:12 --> 00:40:15
And I'm reducing it.
666
00:40:15 --> 00:40:18
And if I average twice
I'm reducing it further.
667
00:40:18 --> 00:40:23
So a better one would be
to get a sharp cutoff.
668
00:40:23 --> 00:40:27
OK, that's filters.
669
00:40:27 --> 00:40:33
I guess what I hope is that we
have the idea of a convolution,
670
00:40:33 --> 00:40:36
and now we see what
we can use it for.
671
00:40:36 --> 00:40:37
Right?
672
00:40:37 --> 00:40:39
And there are many others.
673
00:40:39 --> 00:40:40
So we'll have another.
674
00:40:40 --> 00:40:44
We'll come back to convolutions
and de-convolution.
675
00:40:44 --> 00:40:49
Because if you have a CT
scanner, that doing a
676
00:40:49 --> 00:40:51
little convolution.
677
00:40:51 --> 00:40:54
I mean, you're the input,
right, to the CT scanner?
678
00:40:54 --> 00:40:57
You march in, hoping
for the best.
679
00:40:57 --> 00:41:02
OK, CT scanner convolves you
with their little filter.
680
00:41:02 --> 00:41:10
And then it does a
deconvolution, to an
681
00:41:10 --> 00:41:15
approximate deconvolution, to
have a better image of you.
682
00:41:15 --> 00:41:19
OK, let's leave that.
683
00:41:19 --> 00:41:23
Can I change direction and just
write down the formulas for the
684
00:41:23 --> 00:41:27
Fourier integral transform?
685
00:41:27 --> 00:41:29
And do one example?
686
00:41:29 --> 00:41:32
OK.
687
00:41:32 --> 00:41:38
I don't know what you think
about a lecture that stops
688
00:41:38 --> 00:41:41
and starts a new topic.
689
00:41:41 --> 00:41:46
Is it - maybe it's
tough on the listener?
690
00:41:46 --> 00:41:48
Or maybe it's a break.
691
00:41:48 --> 00:41:49
I don't know.
692
00:41:49 --> 00:41:51
Let's look at it positively.
693
00:41:51 --> 00:41:53
Alright, break.
694
00:41:53 --> 00:41:58
Alright.
695
00:41:58 --> 00:42:02
So let me remember the
Fourier series formulas.
696
00:42:02 --> 00:42:05
So I'm just going to break,
and now we go to the
697
00:42:05 --> 00:42:09
integral transform.
698
00:42:09 --> 00:42:13
OK, so let me remember the
formula for the coefficients,
699
00:42:13 --> 00:42:13
which was 1/2pi.
700
00:42:15 --> 00:42:22
The integral of
f(x)e^(-ikx)dx, right?
701
00:42:22 --> 00:42:26
And then when we added it
up to get f(x) back again,
702
00:42:26 --> 00:42:31
we added up a sum of
the c_k's e^(ikx)'s.
703
00:42:31 --> 00:42:34
704
00:42:34 --> 00:42:37
That's 4.1.
705
00:42:37 --> 00:42:39
We know those formulas.
706
00:42:39 --> 00:42:40
And we notice again.
707
00:42:40 --> 00:42:42
Complex conjugate.
708
00:42:42 --> 00:42:44
One direction is the
conjugate compared to
709
00:42:44 --> 00:42:45
the other direction.
710
00:42:45 --> 00:42:51
Now, all I plan to do is
write down the formula.
711
00:42:51 --> 00:42:55
And remember, I'm going to
use f hat of k, instead
712
00:42:55 --> 00:42:57
of the coefficients.
713
00:42:57 --> 00:43:00
Because it's a function of
k, all k's and not just
714
00:43:00 --> 00:43:02
integers are allowed.
715
00:43:02 --> 00:43:05
And then I'm going
to recover f(x).
716
00:43:05 --> 00:43:08
OK, now this integral went
from minus pi to pi,
717
00:43:08 --> 00:43:10
because that was periodic.
718
00:43:10 --> 00:43:12
But now all the integrals
are going to go from minus
719
00:43:12 --> 00:43:15
infinity to infinity.
720
00:43:15 --> 00:43:17
We've got every k, every x.
721
00:43:17 --> 00:43:19
So we take, what do you
expect here? f(x)?
722
00:43:21 --> 00:43:21
e^(-ikx)?
723
00:43:23 --> 00:43:23
dx?
724
00:43:24 --> 00:43:25
Yes.
725
00:43:25 --> 00:43:27
Fine.
726
00:43:27 --> 00:43:32
Same thing, f(x) is there, but
now any k is allowed so I
727
00:43:32 --> 00:43:34
have a function of all k's.
728
00:43:34 --> 00:43:35
And now I want to recover f(x).
729
00:43:36 --> 00:43:38
So what do I do?
730
00:43:38 --> 00:43:39
You can guess.
731
00:43:39 --> 00:43:41
I've got an integral now.
732
00:43:41 --> 00:43:43
Not a sum.
733
00:43:43 --> 00:43:47
Because a sum was when I
had only integer numbers.
734
00:43:47 --> 00:43:54
Now, I've got f hat of k, and
would you like to tell me what
735
00:43:54 --> 00:44:00
the magic factor is, there
in the integral formula?
736
00:44:00 --> 00:44:02
It's just what you hope.
737
00:44:02 --> 00:44:04
It's the e^(ik*omega).
738
00:44:05 --> 00:44:08
Where d what?
739
00:44:08 --> 00:44:14
Now this is where it's easy
to make a mistake. d,
740
00:44:14 --> 00:44:16
I'm integrating here.
741
00:44:16 --> 00:44:19
I'm putting the whole,
reconstructing the function.
742
00:44:19 --> 00:44:24
I'm putting back the harmonics
with the amount f hat of k
743
00:44:24 --> 00:44:29
tells me how much e^(ik*omega)
there is in the function.
744
00:44:29 --> 00:44:32
I put them all together,
so I integrate dk.
745
00:44:34 --> 00:44:36
I'm integrating over
the frequencies.
746
00:44:36 --> 00:44:39
This was the sum over k.
747
00:44:39 --> 00:44:41
From minus infinity
to infinity.
748
00:44:41 --> 00:44:44
Now this is an integral,
because we've got,
749
00:44:44 --> 00:44:46
it's all filled in.
750
00:44:46 --> 00:44:48
And it remains to
deal with this 2pi.
751
00:44:49 --> 00:44:54
And I see in the book
that the 2pi went there.
752
00:44:54 --> 00:44:55
I don't know why.
753
00:44:55 --> 00:44:56
Anyway, there it is.
754
00:44:56 --> 00:45:00
So let's follow
that convention.
755
00:45:00 --> 00:45:01
Put the 2pi here.
756
00:45:01 --> 00:45:05
So there's the formula.
757
00:45:05 --> 00:45:07
The pair of formulas,
the twin formulas.
758
00:45:07 --> 00:45:12
The transform, from f to f hat,
and the inverse transform,
759
00:45:12 --> 00:45:14
from f hat back to f.
760
00:45:14 --> 00:45:22
And it's just like the one
you've seen for Fourier series.
761
00:45:22 --> 00:45:28
Well, I think the only good way
to remember those is to put in
762
00:45:28 --> 00:45:32
a function and find
its transform.
763
00:45:32 --> 00:45:38
So my final thing for today
would be take a particular
764
00:45:38 --> 00:45:39
function, f(x).
765
00:45:39 --> 00:45:42
766
00:45:42 --> 00:45:48
Here, let me take ever f(x) to
be, here's one. f to be zero
767
00:45:48 --> 00:45:52
here, and then a jump to one.
768
00:45:52 --> 00:45:54
And then an exponential decay.
769
00:45:54 --> 00:45:55
So e^-ax.
770
00:45:55 --> 00:46:00
771
00:46:00 --> 00:46:03
OK, so that's the input.
772
00:46:03 --> 00:46:06
It's not odd, it's not even.
773
00:46:06 --> 00:46:12
So I expect sort of a complex f
hat of k, which I can compute.
774
00:46:12 --> 00:46:14
So f hat of k is what?
775
00:46:14 --> 00:46:18
Now, let's just figure out
f hat of k and look at
776
00:46:18 --> 00:46:21
the decay rate and all
the other good stuff.
777
00:46:21 --> 00:46:23
So what do I do?
778
00:46:23 --> 00:46:27
I'm just doing this integral.
779
00:46:27 --> 00:46:28
For practice.
780
00:46:28 --> 00:46:31
OK, so f is zero in
the first half.
781
00:46:31 --> 00:46:34
So I really only integrate
zero to infinity.
782
00:46:34 --> 00:46:41
And in that region it's e^-ax,
and I multiply by e^(-ikx),
783
00:46:41 --> 00:46:46
and I integrate dx,
and what do I get?
784
00:46:46 --> 00:46:52
I'll get this, is an integral
we can do, and it's easy
785
00:46:52 --> 00:46:53
because this is e^-(a+ik)x.
786
00:46:53 --> 00:46:58
787
00:46:58 --> 00:47:02
You're always going to
see it that way, right?
788
00:47:02 --> 00:47:04
That we're integrating.
789
00:47:04 --> 00:47:07
And then the integral of an
exponential is the exponential
790
00:47:07 --> 00:47:14
divided by the factor that
will come down when we
791
00:47:14 --> 00:47:15
take the derivative.
792
00:47:15 --> 00:47:19
So I think we just
have this, right?
793
00:47:19 --> 00:47:23
Don't you think, to integrate
that exponential, we just get
794
00:47:23 --> 00:47:26
the exponential divided
by its little factor.
795
00:47:26 --> 00:47:29
And now we have to
stick in the limits.
796
00:47:29 --> 00:47:31
And what do I get
at the limits?
797
00:47:31 --> 00:47:36
This is like, a fun part of
Fourier integral formulas.
798
00:47:36 --> 00:47:41
What do I get at the upper
limit, x equal infinity?
799
00:47:41 --> 00:47:47
If x is very large, what
does this thing do?
800
00:47:47 --> 00:47:50
Goes to zero.
801
00:47:50 --> 00:47:52
It's gone.
802
00:47:52 --> 00:47:57
The e^(ikx) is oscillating
around, it's of size one.
803
00:47:57 --> 00:48:01
But the e^(-ax), so I needed
a to be positive here.
804
00:48:01 --> 00:48:03
That picture had to
be the right one.
805
00:48:03 --> 00:48:07
A positive.
806
00:48:07 --> 00:48:10
Then at infinity, I get zero.
807
00:48:10 --> 00:48:13
So now I just plug in this
lower limit, that comes
808
00:48:13 --> 00:48:14
with a minus sign.
809
00:48:14 --> 00:48:15
So what do I get?
810
00:48:15 --> 00:48:20
The minus sign will make
this a+ik, and what does
811
00:48:20 --> 00:48:21
it thing equal at x=0?
812
00:48:23 --> 00:48:27
One. e^0 is one.
813
00:48:27 --> 00:48:31
So there is the
Fourier transform.
814
00:48:31 --> 00:48:36
Of my one-sided exponential.
815
00:48:36 --> 00:48:39
Now, just a quick look at that
and then I'll do some more,
816
00:48:39 --> 00:48:46
this example's Example one in
Section 4.5, and we'll
817
00:48:46 --> 00:48:49
do more examples.
818
00:48:49 --> 00:48:53
But let's just
look at that one.
819
00:48:53 --> 00:48:58
I see a jump in the function.
820
00:48:58 --> 00:49:05
What do I expect in the decay
rate of the transform?
821
00:49:05 --> 00:49:10
So a jump in the function, I
expect a decay rate of 1/k.
822
00:49:12 --> 00:49:15
Decay rate, right?
823
00:49:15 --> 00:49:17
Just as for Fourier
coefficients, so for the
824
00:49:17 --> 00:49:19
integral transform.
825
00:49:19 --> 00:49:23
So a decay rate in f hat.
826
00:49:23 --> 00:49:25
And it's here.
827
00:49:25 --> 00:49:28
1/k in the denominator.
828
00:49:28 --> 00:49:29
Yeah.
829
00:49:29 --> 00:49:34
So that's a a good example.
830
00:49:34 --> 00:49:38
You might say, wait a minute,
OK that's fine but what
831
00:49:38 --> 00:49:40
about the second one?
832
00:49:40 --> 00:49:47
Could I put in 1/(a+ik)
and get back the pulse?
833
00:49:47 --> 00:49:49
The exponential pulse?
834
00:49:49 --> 00:49:52
The answer is yes, but
maybe I don't know how
835
00:49:52 --> 00:49:53
to do that integral.
836
00:49:53 --> 00:49:58
So I'm sort of fortunate that
these formulas are proved for
837
00:49:58 --> 00:50:00
any function including
this function.
838
00:50:00 --> 00:50:08
So this example shows
the decay rate.
839
00:50:08 --> 00:50:12
The possibility sometimes of
doing one integral but maybe
840
00:50:12 --> 00:50:16
the other integral going the
other direction is not so easy.
841
00:50:16 --> 00:50:17
And that's normal.
842
00:50:17 --> 00:50:20
So that's, like, Fourier
transforms and inverse
843
00:50:20 --> 00:50:26
transforms, we don't expect to
be able to do them all by hand.
844
00:50:26 --> 00:50:33
I mean, I'll just say that
actually, anybody who studies
845
00:50:33 --> 00:50:38
complex variables and residues,
I don't know if you know any of
846
00:50:38 --> 00:50:42
this, heard these words about,
there are ways to integrate.
847
00:50:42 --> 00:50:47
I could put 1/(a+ik) in here.
848
00:50:47 --> 00:50:50
And, actually, and do
this integral for minus
849
00:50:50 --> 00:50:51
infinity to infinity.
850
00:50:51 --> 00:50:54
By stuff that's in Chapter 5.
851
00:50:54 --> 00:50:59
Can I just point ahead without
any plan to discuss it.
852
00:50:59 --> 00:51:06
That some integrals can be
done by x+iy tricks, by
853
00:51:06 --> 00:51:08
using complex numbers.
854
00:51:08 --> 00:51:10
But I won't do more.
855
00:51:10 --> 00:51:11
OK, thanks.
856
00:51:11 --> 00:51:15
So that's the formulas.
857
00:51:15 --> 00:51:17
And that's one example.
858
00:51:17 --> 00:51:21
Wednesday there will be more
examples and then no review
859
00:51:21 --> 00:51:23
session Wednesday evening.