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PROFESSOR: In the last two
lectures, we saw how periodic
9
00:00:58,360 --> 00:01:01,300
and non periodic signals could
be represented as linear
10
00:01:01,300 --> 00:01:04,440
combinations of complex
exponentials.
11
00:01:04,440 --> 00:01:08,570
And this led to the Fourier
series representation, in the
12
00:01:08,570 --> 00:01:12,280
periodic case, and it led
to the Fourier transform
13
00:01:12,280 --> 00:01:14,950
representation in the
aperiodic case.
14
00:01:14,950 --> 00:01:19,340
And then, in fact, what we did
was to incorporate the Fourier
15
00:01:19,340 --> 00:01:25,480
series within the framework
of the Fourier transform.
16
00:01:25,480 --> 00:01:29,350
What I'd like to do in today's
lecture is look at the Fourier
17
00:01:29,350 --> 00:01:33,620
transform more closely, in
particular with regard to some
18
00:01:33,620 --> 00:01:35,050
of its properties.
19
00:01:35,050 --> 00:01:39,460
So let me begin by reminding
you of the analysis and
20
00:01:39,460 --> 00:01:43,020
synthesis equations for the
Fourier transform, as I've
21
00:01:43,020 --> 00:01:45,350
summarized them here.
22
00:01:45,350 --> 00:01:49,550
The synthesis equation being an
equation the tells us how
23
00:01:49,550 --> 00:01:53,970
to build the time function out
of, in essence, a linear
24
00:01:53,970 --> 00:01:57,060
combination of complex
exponentials.
25
00:01:57,060 --> 00:02:01,390
And the analysis equation
telling us how to get the
26
00:02:01,390 --> 00:02:05,820
amplitudes of those complex
exponentials from the
27
00:02:05,820 --> 00:02:07,970
associated time function.
28
00:02:07,970 --> 00:02:12,360
So essentially, in the
decomposition of x of t as a
29
00:02:12,360 --> 00:02:16,270
linear combination of complex
exponentials, the complex
30
00:02:16,270 --> 00:02:20,560
amplitudes of those are, in
effect, the Fourier transform
31
00:02:20,560 --> 00:02:26,670
scaled by the differential and
scaled by 1 over 2 pi.
32
00:02:26,670 --> 00:02:33,730
As I indicated last time, the
Fourier transform is a complex
33
00:02:33,730 --> 00:02:35,810
function of frequency.
34
00:02:35,810 --> 00:02:40,490
And in particular, the complex
function of frequency has an
35
00:02:40,490 --> 00:02:44,770
important and very useful
symmetry property.
36
00:02:44,770 --> 00:02:49,030
The symmetry of the Fourier
transform, when x of t is
37
00:02:49,030 --> 00:02:53,450
real, is what is referred to
as conjugate symmetric.
38
00:02:53,450 --> 00:02:57,940
In other words, if we take the
complex function, f of omega,
39
00:02:57,940 --> 00:03:01,970
and its complex conjugate,
that's equivalent to replacing
40
00:03:01,970 --> 00:03:04,930
omega by minus omega.
41
00:03:04,930 --> 00:03:10,300
And a consequence of that, if we
think in terms of the real
42
00:03:10,300 --> 00:03:14,890
part of the Fourier transform,
the real part is an even
43
00:03:14,890 --> 00:03:19,730
function of frequency, and the
magnitude is an even function
44
00:03:19,730 --> 00:03:21,370
of frequency.
45
00:03:21,370 --> 00:03:26,550
Whereas the imaginary part is an
odd function of frequency,
46
00:03:26,550 --> 00:03:31,170
and the phase angle is an odd
function of frequency.
47
00:03:31,170 --> 00:03:35,450
So we have this symmetry
relationship that, for x of t
48
00:03:35,450 --> 00:03:39,390
real, if we think of either the
real part or the magnitude
49
00:03:39,390 --> 00:03:42,880
of the Fourier transform,
it's even symmetric.
50
00:03:42,880 --> 00:03:48,100
And the imaginary part, or the
phase angle, either one, is
51
00:03:48,100 --> 00:03:49,310
odd symmetric.
52
00:03:49,310 --> 00:03:53,740
In other words, if we flip it,
we multiply by a minus sign.
53
00:03:53,740 --> 00:03:58,260
Let's look at an example, in the
context of an example that
54
00:03:58,260 --> 00:04:02,870
we worked last time for
the Fourier transform.
55
00:04:02,870 --> 00:04:08,320
We took the case of a real
exponential of the form e to
56
00:04:08,320 --> 00:04:11,550
the minus at times the step.
57
00:04:11,550 --> 00:04:17,120
And the Fourier transform, as we
found, was of the algebraic
58
00:04:17,120 --> 00:04:20,089
form 1 over a plus j omega.
59
00:04:20,089 --> 00:04:23,580
And, incidentally, the Fourier
transform integral only
60
00:04:23,580 --> 00:04:26,790
converged for a greater
than 0.
61
00:04:26,790 --> 00:04:31,250
In other words, for this
exponential decaying.
62
00:04:31,250 --> 00:04:34,810
And I illustrated the magnitude
and angle, and what
63
00:04:34,810 --> 00:04:40,460
we see is that e to the minus
at is a real time function,
64
00:04:40,460 --> 00:04:44,110
therefore its magnitude should
be an even function of
65
00:04:44,110 --> 00:04:46,630
frequency, and indeed it is.
66
00:04:46,630 --> 00:04:50,800
And its phase angle,
shown below, is an
67
00:04:50,800 --> 00:04:54,050
odd function of frequency.
68
00:04:54,050 --> 00:05:02,170
So in fact, although I stressed
last time that the
69
00:05:02,170 --> 00:05:04,800
complex exponentials is required
to build a time
70
00:05:04,800 --> 00:05:09,000
function require exponentials of
both positive and negative
71
00:05:09,000 --> 00:05:14,160
frequencies, for x of t real
what we see is that, because
72
00:05:14,160 --> 00:05:16,270
of these symmetry properties,
either for the real and
73
00:05:16,270 --> 00:05:20,960
imaginary or magnitude and
angle, we can specify the
74
00:05:20,960 --> 00:05:24,630
Fourier transform for, let's say
only positive frequencies,
75
00:05:24,630 --> 00:05:28,470
and the symmetry, then, implies,
or tells us, what the
76
00:05:28,470 --> 00:05:30,850
Fourier transform, then, would
be for the negative
77
00:05:30,850 --> 00:05:32,100
frequencies.
78
00:05:34,560 --> 00:05:39,480
This same example, the
decaying exponential,
79
00:05:39,480 --> 00:05:44,320
demonstrates another important
and often useful property of
80
00:05:44,320 --> 00:05:47,370
the Fourier transform.
81
00:05:47,370 --> 00:05:52,410
Specifically, let's rewrite,
algebraically, this example as
82
00:05:52,410 --> 00:05:54,350
I indicate here.
83
00:05:54,350 --> 00:05:57,720
So we have, again, the
exponential, whose Fourier
84
00:05:57,720 --> 00:06:01,800
transform is 1 over
a plus j omega.
85
00:06:01,800 --> 00:06:06,020
And if I just simply divide
numerator and denominator by
86
00:06:06,020 --> 00:06:10,370
a, I can rewrite it in
the form shown here.
87
00:06:10,370 --> 00:06:15,800
And what we notice is that in
the time function we have a
88
00:06:15,800 --> 00:06:20,020
term of the form a times t, and
in the frequency function
89
00:06:20,020 --> 00:06:24,630
we have a term of the form
omega divided by a.
90
00:06:24,630 --> 00:06:30,390
Or equivalently, we could think
of the time function,
91
00:06:30,390 --> 00:06:37,280
which I show here, and as the
parameter a gets smaller, the
92
00:06:37,280 --> 00:06:41,300
exponential gets spread
out in time.
93
00:06:41,300 --> 00:06:45,880
Whereas its Fourier transform,
or the magnitude of its
94
00:06:45,880 --> 00:06:50,080
Fourier transform, has the
inverse property that as a
95
00:06:50,080 --> 00:06:56,840
gets smaller, in fact, this
scales down in frequency.
96
00:06:56,840 --> 00:07:00,430
Well, this is a general property
of the Fourier
97
00:07:00,430 --> 00:07:06,350
transform, namely the fact that
a linear scaling in time
98
00:07:06,350 --> 00:07:10,590
generates the inverse linear
scaling in frequency.
99
00:07:10,590 --> 00:07:14,190
And the general statement of
this time frequency scaling is
100
00:07:14,190 --> 00:07:18,100
what I show at the top of the
transparency, namely the
101
00:07:18,100 --> 00:07:23,570
equation that if we scale the
time function in time, then we
102
00:07:23,570 --> 00:07:26,140
apply an inverse scaling
in frequency.
103
00:07:28,720 --> 00:07:33,430
This, in fact, is probably a
result that you're already
104
00:07:33,430 --> 00:07:35,580
possibly familiar with,
in somewhat
105
00:07:35,580 --> 00:07:37,190
of a different context.
106
00:07:37,190 --> 00:07:42,250
Essentially, you could think
of this as an example, or a
107
00:07:42,250 --> 00:07:47,700
generalization, rather, of the
notion that if, let's say I
108
00:07:47,700 --> 00:07:53,610
had a signal that was recorded
on a tape player, and if I
109
00:07:53,610 --> 00:07:57,140
play the tape back at, let's
say, twice the speed, which
110
00:07:57,140 --> 00:08:00,380
means that I'm compressing
the time axis linearly
111
00:08:00,380 --> 00:08:02,240
by a factor of 2.
112
00:08:02,240 --> 00:08:05,280
Then, in fact, what happens is
that the frequencies that we
113
00:08:05,280 --> 00:08:08,930
observe get pushed up
by a factor of 2.
114
00:08:08,930 --> 00:08:12,450
And, in fact, let me
illustrate that.
115
00:08:12,450 --> 00:08:18,670
I have here a glockenspiel, and
anyone who loves a parade
116
00:08:18,670 --> 00:08:22,610
certainly knows what
a glockenspiel is.
117
00:08:22,610 --> 00:08:28,700
And this particular glockenspiel
has three a's,
118
00:08:28,700 --> 00:08:30,360
separated each by an octave.
119
00:08:30,360 --> 00:08:35,500
There's a middle a, a
high a, and a low a.
120
00:08:38,960 --> 00:08:46,240
And what I've done is to record
the middle a on a tape
121
00:08:46,240 --> 00:08:48,950
at 7 and 1/2 inches
per second.
122
00:08:48,950 --> 00:08:54,680
And what I'd like to demonstrate
is that as we play
123
00:08:54,680 --> 00:08:59,790
that back, either at twice or
half speed, the effective note
124
00:08:59,790 --> 00:09:03,650
gets moved down or
up by an octave.
125
00:09:03,650 --> 00:09:07,340
So let me first play the
note at the speed at
126
00:09:07,340 --> 00:09:08,760
which it was recorded.
127
00:09:08,760 --> 00:09:12,250
And so what we'll hear is the
middle a as I've recorded it.
128
00:09:12,250 --> 00:09:15,200
And let me just start
the tape player.
129
00:09:26,190 --> 00:09:30,190
Let me stop it, and hopefully
what you heard is the same
130
00:09:30,190 --> 00:09:33,930
note from the tape recorder as
the note that I played on the
131
00:09:33,930 --> 00:09:35,980
glockenspiel.
132
00:09:35,980 --> 00:09:40,740
Now I'll rewind the tape, and
we'll go back to the beginning
133
00:09:40,740 --> 00:09:43,700
of that portion.
134
00:09:43,700 --> 00:09:49,930
And now, if I change the tape
speed to half the speed, so
135
00:09:49,930 --> 00:09:53,050
from 7 and 1/2 inches per
second, I'll change the tape
136
00:09:53,050 --> 00:09:57,980
speed to 3 and 3/4 inches
per second.
137
00:09:57,980 --> 00:10:04,820
And now when I play the tape
back, because of the inverse
138
00:10:04,820 --> 00:10:08,060
relationship between time and
frequency scaling, we're now
139
00:10:08,060 --> 00:10:10,750
scaling in time by stretching
out, we would expect the
140
00:10:10,750 --> 00:10:13,770
frequencies to be lowered
by a factor of 2.
141
00:10:13,770 --> 00:10:19,170
We should now expect the taped
note to be an octave lower,
142
00:10:19,170 --> 00:10:20,390
matching this lower a.
143
00:10:20,390 --> 00:10:22,010
So let's just play that.
144
00:10:39,300 --> 00:10:43,795
Let me stop it, and, again,
we'll rewind the tape.
145
00:10:46,410 --> 00:10:50,020
Go back to the beginning,
and now we'll play this
146
00:10:50,020 --> 00:10:51,770
at twice the speed.
147
00:10:51,770 --> 00:10:57,310
So I'll change from 3 and 3/4
to 15 inches per second.
148
00:10:57,310 --> 00:11:00,680
And now when I play it, we would
expect that to match the
149
00:11:00,680 --> 00:11:01,180
upper note.
150
00:11:01,180 --> 00:11:02,430
And let's just do that.
151
00:11:15,100 --> 00:11:18,980
Although that's a result that,
intuitively, probably makes
152
00:11:18,980 --> 00:11:22,720
considerable sense, in fact,
what that is is an
153
00:11:22,720 --> 00:11:27,130
illustration of the inverse
relationship between time
154
00:11:27,130 --> 00:11:29,250
scaling and frequency scaling.
155
00:11:29,250 --> 00:11:31,970
And also, by the way, it was
finally my opportunity to play
156
00:11:31,970 --> 00:11:33,220
the glockenspiel
on television.
157
00:11:36,560 --> 00:11:42,300
In addition, there is another
very important relationship
158
00:11:42,300 --> 00:11:45,810
between the time and frequency
domains, namely what is
159
00:11:45,810 --> 00:11:50,150
referred to as a duality
relationship.
160
00:11:50,150 --> 00:11:56,660
And the duality relationship
between time and frequency
161
00:11:56,660 --> 00:12:01,220
falls out, more or less
directly, from the equations,
162
00:12:01,220 --> 00:12:03,790
the analysis and synthesis
equations.
163
00:12:03,790 --> 00:12:08,470
In particular, if we look at the
synthesis equation, which
164
00:12:08,470 --> 00:12:12,760
I repeat here, and the analysis
equation, which I
165
00:12:12,760 --> 00:12:17,510
repeat below it, what we observe
is that, in fact,
166
00:12:17,510 --> 00:12:23,420
these equations are basically
identical, except for the fact
167
00:12:23,420 --> 00:12:26,870
that in the top integral we have
things as a function of
168
00:12:26,870 --> 00:12:30,320
omega, in the bottom integral
as a function of t, and
169
00:12:30,320 --> 00:12:32,960
there's a factor of 1 over
2 pi, and, by the
170
00:12:32,960 --> 00:12:34,590
way, a minus sign.
171
00:12:34,590 --> 00:12:39,500
You can look at the algebra
more carefully at your
172
00:12:39,500 --> 00:12:43,920
leisure, but essentially what
this says is that if x of
173
00:12:43,920 --> 00:12:51,420
omega is the Fourier transform
of a time function x of t,
174
00:12:51,420 --> 00:12:58,120
then, in fact, x of t is very
much like the Fourier
175
00:12:58,120 --> 00:13:00,180
transform of x of omega.
176
00:13:00,180 --> 00:13:04,060
In fact, it's the Fourier
transform of x of minus omega
177
00:13:04,060 --> 00:13:06,060
to account for this
minus sign.
178
00:13:06,060 --> 00:13:07,800
And, by the way, there's
just an additional
179
00:13:07,800 --> 00:13:10,550
factor of 1 over 2 pi.
180
00:13:10,550 --> 00:13:14,070
So the duality relationship
which follows from these two
181
00:13:14,070 --> 00:13:21,450
equations, in fact, says that if
x of t and x of omega are a
182
00:13:21,450 --> 00:13:26,360
Fourier transform pair, if x and
X are a Fourier transform
183
00:13:26,360 --> 00:13:32,150
pair, then X, in fact, has a
Fourier transform which is
184
00:13:32,150 --> 00:13:35,910
proportional to x
turned around.
185
00:13:35,910 --> 00:13:40,060
This duality in the continuous
time Fourier
186
00:13:40,060 --> 00:13:42,560
transform is very important.
187
00:13:42,560 --> 00:13:44,110
It's very useful.
188
00:13:44,110 --> 00:13:48,590
It, by the way, is not a duality
that surfaced in the
189
00:13:48,590 --> 00:13:52,620
Fourier series, because, as you
recall, the Fourier series
190
00:13:52,620 --> 00:13:58,250
begins with a continuous time
function and in the frequency
191
00:13:58,250 --> 00:14:02,470
domain generates a sequence,
which would just naturally
192
00:14:02,470 --> 00:14:06,630
have problems associated with
it if we attempted to
193
00:14:06,630 --> 00:14:08,580
interpret a duality.
194
00:14:08,580 --> 00:14:12,130
And we'll see, also, that in the
discrete time case, one of
195
00:14:12,130 --> 00:14:14,560
the important differences
between continuous time and
196
00:14:14,560 --> 00:14:18,250
discrete time Fourier transforms
is the fact that in
197
00:14:18,250 --> 00:14:23,950
continuous time we have duality,
in the discrete time
198
00:14:23,950 --> 00:14:27,330
Fourier transform we don't.
199
00:14:27,330 --> 00:14:32,400
Let's illustrate this
with an example.
200
00:14:32,400 --> 00:14:37,000
Here are, in fact, two examples
of Fourier transform
201
00:14:37,000 --> 00:14:40,460
pairs taken from examples
in the text.
202
00:14:40,460 --> 00:14:46,690
The top one being example 4.11
from the text, and it's a time
203
00:14:46,690 --> 00:14:52,310
function which is a sine x
over x type of function.
204
00:14:52,310 --> 00:14:59,710
And its Fourier transform
corresponds to a rectangular
205
00:14:59,710 --> 00:15:03,830
shape in the frequency domain.
206
00:15:03,830 --> 00:15:07,320
There's also another example in
the text, the example that
207
00:15:07,320 --> 00:15:11,590
precedes this one, which
is example 4.10.
208
00:15:11,590 --> 00:15:17,460
And in example 4.10, we begin
with a rectangle, and its
209
00:15:17,460 --> 00:15:21,340
Fourier transform is
of the form of a
210
00:15:21,340 --> 00:15:24,140
sine x over x function.
211
00:15:24,140 --> 00:15:28,980
So in fact, if we look at these
two examples together,
212
00:15:28,980 --> 00:15:33,260
what we see is the duality
very evident.
213
00:15:33,260 --> 00:15:38,390
In other words, if we take this
time function and instead
214
00:15:38,390 --> 00:15:42,260
think of a frequency function
that has the same form, then
215
00:15:42,260 --> 00:15:45,360
we simply interchange the roles
of time and frequency in
216
00:15:45,360 --> 00:15:46,630
the other domains.
217
00:15:46,630 --> 00:15:50,810
So the fact that these two
correspond means that these
218
00:15:50,810 --> 00:15:52,330
two correspond.
219
00:15:52,330 --> 00:15:57,910
Of course in this particular
example, because of the fact
220
00:15:57,910 --> 00:16:01,290
that we picked a symmetric
function, an even function, in
221
00:16:01,290 --> 00:16:05,530
fact, the additional twist of
the time axis being reversed
222
00:16:05,530 --> 00:16:10,830
didn't show up in duality
with this example.
223
00:16:10,830 --> 00:16:13,930
One thing this says, of course,
is that essentially
224
00:16:13,930 --> 00:16:19,390
any time you've calculated the
Fourier transform of one time
225
00:16:19,390 --> 00:16:23,290
function, then you've actually
calculated the Fourier
226
00:16:23,290 --> 00:16:25,990
transform of two
time functions.
227
00:16:25,990 --> 00:16:28,970
Another one being the dual
example to the one that you
228
00:16:28,970 --> 00:16:30,220
just calculated.
229
00:16:32,370 --> 00:16:37,805
Also somewhat related to duality
is what is referred to
230
00:16:37,805 --> 00:16:41,650
as Parseval's relation
for the continuous
231
00:16:41,650 --> 00:16:43,620
time Fourier transform.
232
00:16:43,620 --> 00:16:47,880
And essentially, what Parseval's
relationship says,
233
00:16:47,880 --> 00:16:55,800
as a summary of it, says that
the energy in a time function
234
00:16:55,800 --> 00:17:00,260
and the energy in its Fourier
transform are proportional,
235
00:17:00,260 --> 00:17:04,430
the proportionality factor
being a factor of 2 pi.
236
00:17:04,430 --> 00:17:06,770
That's summarized here.
237
00:17:06,770 --> 00:17:10,190
What's meant by the energy is,
of course, the integral of the
238
00:17:10,190 --> 00:17:13,010
magnitude squared of x of t.
239
00:17:13,010 --> 00:17:16,819
And the statement of Parseval's
relation is that
240
00:17:16,819 --> 00:17:21,670
that integral, the energy in x
of t, is proportional to this
241
00:17:21,670 --> 00:17:25,770
integral, which is the
energy in x of omega.
242
00:17:25,770 --> 00:17:31,660
Although we've incorporated the
Fourier series within a
243
00:17:31,660 --> 00:17:36,460
framework of the Fourier
transform, Parseval's relation
244
00:17:36,460 --> 00:17:40,830
needs to be modified slightly
for Fourier series, because of
245
00:17:40,830 --> 00:17:43,840
the fact that a periodic signal
has an infinite amount
246
00:17:43,840 --> 00:17:46,930
of energy in it, and,
essentially, that form of
247
00:17:46,930 --> 00:17:50,380
Parseval's relationship for the
periodic case would say
248
00:17:50,380 --> 00:17:53,560
infinity equals infinity,
which isn't too useful.
249
00:17:53,560 --> 00:17:57,320
However, it can be modified so
that Parseval's relationship
250
00:17:57,320 --> 00:18:02,760
to the periodic case says,
essentially, that the energy
251
00:18:02,760 --> 00:18:09,740
in one period of the periodic
time function is proportional
252
00:18:09,740 --> 00:18:12,160
with, this is the
proportionality factor,
253
00:18:12,160 --> 00:18:15,870
proportional to the sum of the
magnitude squared of the
254
00:18:15,870 --> 00:18:17,020
coefficients.
255
00:18:17,020 --> 00:18:21,300
In other words, the energy in
one period is proportional to
256
00:18:21,300 --> 00:18:25,580
the energy in the sequence that
represents the Fourier
257
00:18:25,580 --> 00:18:26,830
series coefficients.
258
00:18:29,660 --> 00:18:34,070
There are lots of other
properties, and they're
259
00:18:34,070 --> 00:18:37,180
developed in the text and
in the study guide.
260
00:18:37,180 --> 00:18:42,170
A number of properties that we
want to make particular use of
261
00:18:42,170 --> 00:18:46,100
during this lecture, and in
later lectures, are ones that
262
00:18:46,100 --> 00:18:48,080
I summarize here.
263
00:18:48,080 --> 00:18:54,200
And I won't demonstrate the
proofs, but principally focus
264
00:18:54,200 --> 00:18:58,630
on some of the interpretation
as the lecture goes on.
265
00:18:58,630 --> 00:19:01,760
The first property that I have
listed here is what's referred
266
00:19:01,760 --> 00:19:04,600
to as the time shifting
property.
267
00:19:04,600 --> 00:19:08,470
And the time shifting property
says, if I have a time
268
00:19:08,470 --> 00:19:13,610
function with a Fourier
transform x of omega, if I
269
00:19:13,610 --> 00:19:19,950
shift that time function in
time, then that corresponds to
270
00:19:19,950 --> 00:19:24,440
multiplying the Fourier
transform by this factor.
271
00:19:24,440 --> 00:19:29,310
As you examine this factor, what
you can see is that this
272
00:19:29,310 --> 00:19:34,300
factor has magnitude unity and
it has a phase, which is
273
00:19:34,300 --> 00:19:39,740
linear with frequency, and
a slope of minus t0.
274
00:19:39,740 --> 00:19:43,730
So a statement to remember, that
will come up many times
275
00:19:43,730 --> 00:19:49,010
throughout the course, is
that a time shift, or a
276
00:19:49,010 --> 00:19:58,190
displacement in time,
corresponds to a linear change
277
00:19:58,190 --> 00:20:01,110
in phase and frequency.
278
00:20:01,110 --> 00:20:06,020
Another property and, in fact,
a pair of properties that
279
00:20:06,020 --> 00:20:09,840
we'll make reference to as we
turn our attention toward the
280
00:20:09,840 --> 00:20:14,660
end of this lecture to solving
differential equations using
281
00:20:14,660 --> 00:20:18,920
the Fourier transform, is what's
referred to as the
282
00:20:18,920 --> 00:20:23,790
differentiation property and
its companion, which is the
283
00:20:23,790 --> 00:20:26,600
integration property.
284
00:20:26,600 --> 00:20:30,700
The differentiation property
says, again, if we have a time
285
00:20:30,700 --> 00:20:34,950
function with Fourier transform
x of omega, the
286
00:20:34,950 --> 00:20:41,470
Fourier transform of the time
derivative of that corresponds
287
00:20:41,470 --> 00:20:45,950
to multiplying the Fourier
transform by a linear function
288
00:20:45,950 --> 00:20:46,940
of frequency.
289
00:20:46,940 --> 00:20:51,460
So here it's a linear amplitude
change that
290
00:20:51,460 --> 00:20:55,470
corresponds to differentiation.
291
00:20:55,470 --> 00:20:59,500
At first glance, what you
might think is that the
292
00:20:59,500 --> 00:21:02,230
integration property is just
the reverse of that.
293
00:21:02,230 --> 00:21:05,030
If for the differentiation
property you multiply by j
294
00:21:05,030 --> 00:21:09,150
omega, then for integration you
must divide by j omega.
295
00:21:09,150 --> 00:21:12,460
And that's almost correct,
except not quite.
296
00:21:12,460 --> 00:21:16,930
And the reason for the not quite
is that recall that if
297
00:21:16,930 --> 00:21:20,400
you differentiate, what happens,
of course, is that
298
00:21:20,400 --> 00:21:21,970
you lose a constant.
299
00:21:21,970 --> 00:21:25,450
And if we have a time function
that's some finite energy
300
00:21:25,450 --> 00:21:29,520
signal plus a constant,
differentiating will destroy
301
00:21:29,520 --> 00:21:32,250
the constant.
302
00:21:32,250 --> 00:21:34,000
The integration property,
in essence, tries
303
00:21:34,000 --> 00:21:35,300
to bring that back.
304
00:21:35,300 --> 00:21:39,730
So the integration property,
which is the inverse of the
305
00:21:39,730 --> 00:21:44,450
differentiation property, says
that we divide the transform
306
00:21:44,450 --> 00:21:50,610
by j omega, and then if, in
fact, there was a constant
307
00:21:50,610 --> 00:21:54,600
added to x of t, we have to
account for that by inserting
308
00:21:54,600 --> 00:21:58,220
an impulse into the
Fourier transform.
309
00:21:58,220 --> 00:22:03,370
And the final property that I
want to draw your attention to
310
00:22:03,370 --> 00:22:08,090
on this view graph is the
linearity property, which is
311
00:22:08,090 --> 00:22:11,320
very straightforward to
demonstrate from the analysis
312
00:22:11,320 --> 00:22:17,010
and synthesis equations, which
simply says if x1 of omega is
313
00:22:17,010 --> 00:22:21,120
the Fourier transform x1 of t,
and x2 of omega is the Fourier
314
00:22:21,120 --> 00:22:25,080
transform of x2 of t, then the
Fourier transform of a linear
315
00:22:25,080 --> 00:22:29,370
combination is a linear
combination of the Fourier
316
00:22:29,370 --> 00:22:30,620
transforms.
317
00:22:32,700 --> 00:22:36,370
Let me emphasize, also, that
these properties, for the most
318
00:22:36,370 --> 00:22:40,310
part, apply both to Fourier
series and Fourier transforms
319
00:22:40,310 --> 00:22:44,530
because, in fact, what we've
done is to incorporate the
320
00:22:44,530 --> 00:22:48,260
Fourier series within
the framework
321
00:22:48,260 --> 00:22:49,510
of the Fourier transform.
322
00:22:51,900 --> 00:22:54,970
We'll be using a number of these
properties shortly, when
323
00:22:54,970 --> 00:22:57,740
we turn our attention to linear
constant coefficient
324
00:22:57,740 --> 00:22:59,350
differential equations.
325
00:22:59,350 --> 00:23:03,670
However, before we do that
I'd like to focus on two
326
00:23:03,670 --> 00:23:10,010
additional major properties, and
these are what I refer to
327
00:23:10,010 --> 00:23:13,780
as the convolution property and
the modulation property.
328
00:23:13,780 --> 00:23:17,610
And in fact, the convolution
property, as I'm about to
329
00:23:17,610 --> 00:23:22,090
introduce it, forms the
mathematical and conceptual
330
00:23:22,090 --> 00:23:27,250
basis for the whole notion of
filtering, which, in fact,
331
00:23:27,250 --> 00:23:32,810
will be a topic by itself in a
set of lectures, and, in fact,
332
00:23:32,810 --> 00:23:36,700
is a chapter by itself
in the textbook.
333
00:23:36,700 --> 00:23:40,220
Similarly, what I'll refer to
as the modulation property,
334
00:23:40,220 --> 00:23:45,130
again, will occupy its own set
of lectures as we go through
335
00:23:45,130 --> 00:23:49,060
the course, and, in
fact, has its own
336
00:23:49,060 --> 00:23:52,160
chapter in the textbook.
337
00:23:52,160 --> 00:23:58,450
Let me just indicate what the
convolution property is.
338
00:23:58,450 --> 00:24:08,870
And what the convolution
property tells us is that the
339
00:24:08,870 --> 00:24:13,050
Fourier transform of the
convolution of two time
340
00:24:13,050 --> 00:24:19,620
functions is the product of
their Fourier transforms.
341
00:24:19,620 --> 00:24:25,340
So it says, for example, that
if I have a linear time
342
00:24:25,340 --> 00:24:30,500
invariant system, and I have
an input x of t, an impulse
343
00:24:30,500 --> 00:24:33,550
response h of t, and the output,
of course, being the
344
00:24:33,550 --> 00:24:38,280
convolution, then, in fact,
if I look at this in the
345
00:24:38,280 --> 00:24:45,200
frequency domain, the Fourier
transform of the output is the
346
00:24:45,200 --> 00:24:49,920
Fourier transform of the input
times the Fourier transform of
347
00:24:49,920 --> 00:24:53,190
the impulse response.
348
00:24:53,190 --> 00:24:58,340
You can demonstrate this
property algebraically by
349
00:24:58,340 --> 00:25:00,990
essentially taking the
convolution integral and
350
00:25:00,990 --> 00:25:04,270
applying the Fourier transform
and doing the appropriate
351
00:25:04,270 --> 00:25:08,510
interchanging of the order of
integration, et cetera.
352
00:25:08,510 --> 00:25:13,000
But what I'd like to draw your
attention to is a somewhat
353
00:25:13,000 --> 00:25:16,560
more intuitive interpretation
of the property.
354
00:25:16,560 --> 00:25:20,070
And the intuitive interpretation
stems from the
355
00:25:20,070 --> 00:25:23,720
relationship between the Fourier
transform of the
356
00:25:23,720 --> 00:25:26,890
impulse response and what
we've referred to as the
357
00:25:26,890 --> 00:25:28,840
frequency response.
358
00:25:28,840 --> 00:25:34,610
Recall that one of the things
that led us to use complex
359
00:25:34,610 --> 00:25:37,910
exponentials as building blocks
was the fact that
360
00:25:37,910 --> 00:25:39,610
they're eigenfunctions
of linear
361
00:25:39,610 --> 00:25:41,000
time and variant systems.
362
00:25:41,000 --> 00:25:45,640
In other words, if we have a
linear time invariant system,
363
00:25:45,640 --> 00:25:48,990
and I have an input which is
a complex exponential, the
364
00:25:48,990 --> 00:25:51,770
output is a complex exponential
of the same
365
00:25:51,770 --> 00:25:55,790
frequency multiplied
by what we call
366
00:25:55,790 --> 00:25:57,510
the frequency response.
367
00:25:57,510 --> 00:26:02,410
And, in fact, the expression for
the frequency response is
368
00:26:02,410 --> 00:26:05,970
identical to the expression for
the Fourier transform of
369
00:26:05,970 --> 00:26:07,710
the impulse response.
370
00:26:07,710 --> 00:26:11,670
In other words, the frequency
response is the Fourier
371
00:26:11,670 --> 00:26:15,750
transform of the impulse
response.
372
00:26:15,750 --> 00:26:20,350
Now in that context, how
can we interpret
373
00:26:20,350 --> 00:26:22,260
the convolution property?
374
00:26:22,260 --> 00:26:26,630
Well, remember what I said at
the beginning of the lecture,
375
00:26:26,630 --> 00:26:30,850
when I pointed to the synthesis
equation and I said,
376
00:26:30,850 --> 00:26:35,010
in essence, the synthesis
equation tells us how to
377
00:26:35,010 --> 00:26:38,490
decompose x of t as a linear
combination of complex
378
00:26:38,490 --> 00:26:40,740
exponentials.
379
00:26:40,740 --> 00:26:43,460
What are the complex amplitudes
of those complex
380
00:26:43,460 --> 00:26:45,020
exponentials?
381
00:26:45,020 --> 00:26:50,250
In terms of our notation here,
the complex amplitude of those
382
00:26:50,250 --> 00:26:55,590
complex exponentials is x of
omega, or proportional to x of
383
00:26:55,590 --> 00:26:58,650
omega, in particular it's x of
omega, d omega, and then a
384
00:26:58,650 --> 00:27:01,160
factor of 2 pi.
385
00:27:01,160 --> 00:27:06,480
As this signal goes through
this linear time invariant
386
00:27:06,480 --> 00:27:12,100
system, what happens to each of
those exponential is each
387
00:27:12,100 --> 00:27:15,610
one gets multiplied by the
frequency response at the
388
00:27:15,610 --> 00:27:18,090
associated frequency.
389
00:27:18,090 --> 00:27:22,770
What comes out is the amplitude
of the complex
390
00:27:22,770 --> 00:27:28,610
exponentials that are used
to build the output.
391
00:27:28,610 --> 00:27:35,990
So in fact, the convolution
property simply is telling us
392
00:27:35,990 --> 00:27:39,730
that, in terms of the
decomposition of the signal,
393
00:27:39,730 --> 00:27:42,940
in terms of complex
exponentials, as we push that
394
00:27:42,940 --> 00:27:46,230
signal through a linear time
invariant system, we're
395
00:27:46,230 --> 00:27:52,780
separately multiplying by the
frequency response, the
396
00:27:52,780 --> 00:27:56,590
amplitudes of the exponential
components used
397
00:27:56,590 --> 00:27:58,100
to build the input.
398
00:27:58,100 --> 00:28:03,860
And that sum, in turn, is the
decomposition of the output in
399
00:28:03,860 --> 00:28:05,270
terms of complex exponentials.
400
00:28:08,630 --> 00:28:12,310
I understand that going through
that involves a little
401
00:28:12,310 --> 00:28:16,640
bit of sorting out, and I
strongly encourage you to try
402
00:28:16,640 --> 00:28:20,940
to understand and interpret the
convolution property in
403
00:28:20,940 --> 00:28:25,400
those conceptual terms, rather
than simply by applying the
404
00:28:25,400 --> 00:28:29,050
mathematics to the convolution
integral and seeing the terms
405
00:28:29,050 --> 00:28:30,300
match up on both sides.
406
00:28:33,930 --> 00:28:38,190
As I indicated, the convolution
property forms the
407
00:28:38,190 --> 00:28:41,580
basis for what's referred
to as filtering.
408
00:28:41,580 --> 00:28:46,700
And this is a topic that we'll
be treating in a considerable
409
00:28:46,700 --> 00:28:51,330
amount of detail after we've
also gone through a discussion
410
00:28:51,330 --> 00:28:54,400
of the discrete time Fourier
transform in the
411
00:28:54,400 --> 00:28:56,050
next several lectures.
412
00:28:56,050 --> 00:28:59,600
However, what I'd like to do
is just indicate, now, a
413
00:28:59,600 --> 00:29:04,560
little bit of the conceptual
ideas involved.
414
00:29:04,560 --> 00:29:08,990
Essentially, conceptually,
what filtering, as it's
415
00:29:08,990 --> 00:29:15,250
typically referred to,
corresponds to is modifying
416
00:29:15,250 --> 00:29:18,220
separately the individual
frequency
417
00:29:18,220 --> 00:29:20,030
components in a signal.
418
00:29:20,030 --> 00:29:25,360
The convolution property told
us that if we look at the
419
00:29:25,360 --> 00:29:28,330
individual frequency components,
they get
420
00:29:28,330 --> 00:29:30,970
multiplied by the frequency
response, and so what that
421
00:29:30,970 --> 00:29:34,710
says is that we can amplify
or attenuate any of those
422
00:29:34,710 --> 00:29:39,570
components separately using a
linear time invariant system.
423
00:29:39,570 --> 00:29:43,050
For example, what I've
illustrated here is the
424
00:29:43,050 --> 00:29:47,320
frequency response of what is
commonly referred to as an
425
00:29:47,320 --> 00:29:50,220
ideal low pass filter.
426
00:29:50,220 --> 00:29:55,990
What an ideal low pass filter
does is to pass exactly
427
00:29:55,990 --> 00:30:01,200
frequencies in one frequency
range and eliminate totally
428
00:30:01,200 --> 00:30:04,350
frequencies outside
that range.
429
00:30:04,350 --> 00:30:08,890
Another filter which is not so
ideal might, for example,
430
00:30:08,890 --> 00:30:15,980
attenuate components in
this band but not
431
00:30:15,980 --> 00:30:17,230
totally eliminate them.
432
00:30:19,650 --> 00:30:22,190
In terms of filtering, we
can think back to the
433
00:30:22,190 --> 00:30:27,390
differentiation property
and, in fact, interpret
434
00:30:27,390 --> 00:30:29,680
differentiator as a filter.
435
00:30:29,680 --> 00:30:32,710
Recall that the differentiation
property said
436
00:30:32,710 --> 00:30:39,330
that the Fourier transform of
the differentiated signal is
437
00:30:39,330 --> 00:30:41,180
the Fourier transform of
the original signal
438
00:30:41,180 --> 00:30:43,600
multiplied by j omega.
439
00:30:43,600 --> 00:30:47,060
So what that says, then,
is that if we have a
440
00:30:47,060 --> 00:30:52,490
differentiator, the frequency
response of that is j omega.
441
00:30:52,490 --> 00:30:55,640
In other words, the Fourier
transform of the output is j
442
00:30:55,640 --> 00:31:00,120
omega times the Fourier
transform of the input.
443
00:31:00,120 --> 00:31:04,640
And so the frequency response
of the differentiator looks
444
00:31:04,640 --> 00:31:07,140
like this, in terms
of its magnitude.
445
00:31:07,140 --> 00:31:10,700
And what it does, of course,
is it amplifies high
446
00:31:10,700 --> 00:31:14,950
frequencies and attenuates
low frequencies.
447
00:31:17,670 --> 00:31:23,520
Let me just, to cement some of
these ideas, illustrate them
448
00:31:23,520 --> 00:31:29,140
in the context of one kind of
signal, namely a signal which,
449
00:31:29,140 --> 00:31:34,170
in fact, is a spatial signal
rather than a time signal.
450
00:31:34,170 --> 00:31:39,270
And this also gives me an
opportunity to introduce you
451
00:31:39,270 --> 00:31:43,080
to our colleague, J.
B. J. Fourier.
452
00:31:43,080 --> 00:31:47,790
So if we could look at our
colleague, Mr. Fourier, who,
453
00:31:47,790 --> 00:31:53,500
by the way, is not only a person
who had tremendously
454
00:31:53,500 --> 00:31:59,140
brilliant insights, and his
insights, in fact, have led to
455
00:31:59,140 --> 00:32:02,670
forming the foundation of the
developments that is the basis
456
00:32:02,670 --> 00:32:04,040
for this course.
457
00:32:04,040 --> 00:32:05,730
He was also a very
458
00:32:05,730 --> 00:32:07,560
interesting, fascinating person.
459
00:32:07,560 --> 00:32:11,830
And there's a certain amount of
historical discussion about
460
00:32:11,830 --> 00:32:15,300
Fourier and his background,
which you might enjoy reading
461
00:32:15,300 --> 00:32:17,120
in the text.
462
00:32:17,120 --> 00:32:18,740
In any case, what you're
looking at, of
463
00:32:18,740 --> 00:32:20,500
course, is a signal.
464
00:32:20,500 --> 00:32:25,160
And the signal is a spatial
signal, and it has high
465
00:32:25,160 --> 00:32:26,880
frequencies and low
frequencies.
466
00:32:26,880 --> 00:32:31,030
High frequencies corresponding
to things that are varying
467
00:32:31,030 --> 00:32:34,310
rapidly spatially, and
low frequencies
468
00:32:34,310 --> 00:32:37,810
varying slowly spatially.
469
00:32:37,810 --> 00:32:43,250
And so, for example, we could
low pass filter this picture
470
00:32:43,250 --> 00:32:48,090
simply by asking the video crew
if they could be slightly
471
00:32:48,090 --> 00:32:50,270
defocus it.
472
00:32:50,270 --> 00:32:53,540
And what you see as the picture
is defocused, if you
473
00:32:53,540 --> 00:32:58,560
could hold it there, is that
we've lost edges, which is the
474
00:32:58,560 --> 00:33:00,230
rapid variation.
475
00:33:00,230 --> 00:33:05,020
And what we've retained is the
broader, slow variation.
476
00:33:05,020 --> 00:33:09,170
And now let's take out the
defocusing low pass filter and
477
00:33:09,170 --> 00:33:11,550
go back to a focused image.
478
00:33:19,480 --> 00:33:23,260
What we can also consider is
what would happen if we looked
479
00:33:23,260 --> 00:33:25,090
at the differentiated image.
480
00:33:25,090 --> 00:33:28,210
And there are several ways
we can think about this.
481
00:33:28,210 --> 00:33:30,580
One is that a differentiator--
482
00:33:30,580 --> 00:33:35,400
Of course the output of a
differentiator is larger where
483
00:33:35,400 --> 00:33:40,460
the discontinuity, or where
the variation, is faster.
484
00:33:40,460 --> 00:33:44,890
And so we would expect the edges
to be enhanced if, in
485
00:33:44,890 --> 00:33:47,490
fact, we differentiated
the image.
486
00:33:47,490 --> 00:33:51,200
Or if we interpret
differentiation in the context
487
00:33:51,200 --> 00:33:59,190
of our filter, then what we're
saying is that, in effect,
488
00:33:59,190 --> 00:34:04,730
what's happening is that the
differentiator is accentuating
489
00:34:04,730 --> 00:34:08,130
the high frequencies because of
the frequency shape of the
490
00:34:08,130 --> 00:34:09,790
differentiator.
491
00:34:09,790 --> 00:34:13,940
Recall that this all fits
together as a nice package.
492
00:34:13,940 --> 00:34:16,120
We expect intuitively
that differentiation
493
00:34:16,120 --> 00:34:18,020
will enhance edges.
494
00:34:18,020 --> 00:34:21,250
When we talked about square
waves and we saw how the
495
00:34:21,250 --> 00:34:25,380
Fourier series built up a square
wave, we saw that it
496
00:34:25,380 --> 00:34:29,420
was the high frequencies that
were required in order to
497
00:34:29,420 --> 00:34:32,610
build up the sharp edges.
498
00:34:32,610 --> 00:34:36,250
And so either viewed as a
filter, or viewed intuitively,
499
00:34:36,250 --> 00:34:40,070
we would expect that the
differentiated image would, in
500
00:34:40,070 --> 00:34:46,560
fact, attenuate this slowly
varying background and amplify
501
00:34:46,560 --> 00:34:48,320
the rapidly varying edges.
502
00:34:48,320 --> 00:34:52,170
So let's look again at our
original image, just to remind
503
00:34:52,170 --> 00:34:55,829
you of the fact that there are
edges, of course, and there is
504
00:34:55,829 --> 00:34:59,480
a more slowly varying
background.
505
00:34:59,480 --> 00:35:05,860
And now let's look at the result
of passing that through
506
00:35:05,860 --> 00:35:07,730
a differentiator.
507
00:35:07,730 --> 00:35:11,790
And, as I think is very evident
in the resulting
508
00:35:11,790 --> 00:35:17,610
image, clearly it's the edges
that are retained and the
509
00:35:17,610 --> 00:35:21,590
slower background variations
are destroyed, which is
510
00:35:21,590 --> 00:35:23,220
consistent with everything
that we've said.
511
00:35:27,270 --> 00:35:31,790
I've emphasized that we'll be
returning to a much broader
512
00:35:31,790 --> 00:35:35,540
discussion of filtering at a
later point in the course.
513
00:35:35,540 --> 00:35:40,690
I'd now like to comment on
another property, which is
514
00:35:40,690 --> 00:35:44,110
also, as I indicated, a topic
in its own right, and which
515
00:35:44,110 --> 00:35:48,740
really is the dual property to
the convolution property, and,
516
00:35:48,740 --> 00:35:53,360
in fact, could be argued
directly from duality.
517
00:35:53,360 --> 00:35:59,230
And that is what's referred to
as the modulation property.
518
00:35:59,230 --> 00:36:04,210
The convolution property told us
that if we convolve in the
519
00:36:04,210 --> 00:36:08,940
time domain, we multiply in
the frequency domain.
520
00:36:08,940 --> 00:36:10,920
And we know that time and
frequency domains are
521
00:36:10,920 --> 00:36:13,920
interchangeable because of
duality, so what that would
522
00:36:13,920 --> 00:36:19,860
suggest is that if we multiply
in the time domain, that would
523
00:36:19,860 --> 00:36:23,140
correspond to convolution
in the frequency domain.
524
00:36:23,140 --> 00:36:25,430
And, in fact, that is
exactly what the
525
00:36:25,430 --> 00:36:27,610
modulation property is.
526
00:36:27,610 --> 00:36:31,940
I have it summarized here that
if we multiply a time function
527
00:36:31,940 --> 00:36:34,340
by another time function,
then in the
528
00:36:34,340 --> 00:36:37,880
frequency domain we convolve.
529
00:36:37,880 --> 00:36:40,850
Whereas the convolution property
is just the dual of
530
00:36:40,850 --> 00:36:45,870
that, namely convolving in the
time domain corresponds to
531
00:36:45,870 --> 00:36:49,040
multiplication in the
frequency domain.
532
00:36:49,040 --> 00:36:52,170
The convolution property
is the basis, as I
533
00:36:52,170 --> 00:36:54,390
indicated, for filtering.
534
00:36:54,390 --> 00:36:58,620
The modulation property, as I've
summarized it here, in
535
00:36:58,620 --> 00:37:06,000
fact, is the entire basis for
amplitude modulation systems
536
00:37:06,000 --> 00:37:10,080
as used almost universally
in communications.
537
00:37:10,080 --> 00:37:13,110
And what the modulation
property, as we'll see when we
538
00:37:13,110 --> 00:37:18,300
explore it in more detail, tells
us is that if we have a
539
00:37:18,300 --> 00:37:22,560
signal with a certain spectrum,
and we multiply by a
540
00:37:22,560 --> 00:37:27,020
sinusoidal signal whose Fourier
transform is a set of
541
00:37:27,020 --> 00:37:31,500
impulses, then in a frequency
domain we convolve.
542
00:37:31,500 --> 00:37:35,570
And that corresponds to taking
the original spectrum and
543
00:37:35,570 --> 00:37:41,160
translating it, shifting it in
frequency up to the frequency
544
00:37:41,160 --> 00:37:45,060
of the carrier, namely the
sinusoidal signal.
545
00:37:45,060 --> 00:37:49,330
And as I said, we'll come to
that in much more detail in a
546
00:37:49,330 --> 00:37:50,580
number of lectures.
547
00:37:52,960 --> 00:37:56,780
We've seen a number of
properties, and I indicated
548
00:37:56,780 --> 00:38:00,100
sometime earlier when we talked
about differential
549
00:38:00,100 --> 00:38:03,350
equations, that, in fact, it's
the properties of the Fourier
550
00:38:03,350 --> 00:38:08,730
transform that provide us with
a very useful and important
551
00:38:08,730 --> 00:38:13,420
mechanism for solving linear
constant coefficient
552
00:38:13,420 --> 00:38:15,350
differential equations.
553
00:38:15,350 --> 00:38:21,340
And what I'd like to do now is
illustrate the procedure, the
554
00:38:21,340 --> 00:38:24,780
basis for that, and I think what
we'll do is illustrate it
555
00:38:24,780 --> 00:38:26,985
simply in the context
of several examples.
556
00:38:31,120 --> 00:38:39,380
What I've indicated is a
system with an impulse
557
00:38:39,380 --> 00:38:43,550
response h of t, or frequency
response h of omega.
558
00:38:43,550 --> 00:38:46,630
And, of course, we know that
in the time domain it's
559
00:38:46,630 --> 00:38:50,880
described through convolution,
in the frequency domain it's
560
00:38:50,880 --> 00:38:54,020
described through
multiplication.
561
00:38:54,020 --> 00:38:59,040
And I, in essence, am assuming
that we're talking about a
562
00:38:59,040 --> 00:39:01,400
linear time invariant system.
563
00:39:01,400 --> 00:39:05,560
And we're also going to assume
then it's characterized by a
564
00:39:05,560 --> 00:39:09,540
linear constant coefficient
differential equation, where
565
00:39:09,540 --> 00:39:13,730
we're going to impose the
condition that it's causal
566
00:39:13,730 --> 00:39:17,760
linear and time invariant, or
equivalently that the initial
567
00:39:17,760 --> 00:39:22,320
conditions are consistent
with the initial rest.
568
00:39:22,320 --> 00:39:26,160
And it's because of the fact
that we're assuming that it's
569
00:39:26,160 --> 00:39:30,380
a linear time invariant system
that we can describe it in the
570
00:39:30,380 --> 00:39:34,560
frequency domain through the
convolution property, and we
571
00:39:34,560 --> 00:39:38,940
can use the properties of
the Fourier transform.
572
00:39:38,940 --> 00:39:43,190
So let's take, as our example,
a first order differential
573
00:39:43,190 --> 00:39:46,670
equation as I indicate here.
574
00:39:46,670 --> 00:39:49,560
So the derivative of the output
plus a times the output
575
00:39:49,560 --> 00:39:52,330
is equal to the input.
576
00:39:52,330 --> 00:39:57,080
And now we can use the
differentiation property.
577
00:39:57,080 --> 00:40:01,250
If we Fourier transform this
entire expression, the
578
00:40:01,250 --> 00:40:06,690
differentiation property tells
us that the Fourier transform
579
00:40:06,690 --> 00:40:10,290
of the derivative of the
output is the Fourier
580
00:40:10,290 --> 00:40:14,560
transform of the output
multiplied by j omega.
581
00:40:14,560 --> 00:40:20,330
And linearity will let us write
the Fourier transform of
582
00:40:20,330 --> 00:40:22,810
this as a times y of omega.
583
00:40:22,810 --> 00:40:25,690
And since these are added
together, and since we have
584
00:40:25,690 --> 00:40:29,330
the linearity property, these
are added together.
585
00:40:29,330 --> 00:40:34,600
And x of omega is the Fourier
transform of x of t.
586
00:40:34,600 --> 00:40:40,630
So what we've used is the
differentiation property, and
587
00:40:40,630 --> 00:40:43,185
we've used the linearity
property.
588
00:40:48,290 --> 00:40:52,980
We can solve this equation
for y of omega.
589
00:40:52,980 --> 00:40:57,420
The Fourier transform of the
output in terms of x of omega,
590
00:40:57,420 --> 00:41:01,970
the Fourier transform of the
input, and a simple algebraic
591
00:41:01,970 --> 00:41:05,430
step gets us to this
expression.
592
00:41:05,430 --> 00:41:08,550
So the Fourier transform of the
output is 1 over j omega
593
00:41:08,550 --> 00:41:12,110
plus a times the Fourier
transform of the input.
594
00:41:12,110 --> 00:41:16,220
And I've just simply repeated
that equation up here.
595
00:41:19,980 --> 00:41:25,470
So far this is algebra, and the
question is, now, how do
596
00:41:25,470 --> 00:41:27,540
we interpret this?
597
00:41:27,540 --> 00:41:32,880
Well, we know that the Fourier
transform of the output is the
598
00:41:32,880 --> 00:41:38,060
Fourier transform of the input
times the Fourier transform of
599
00:41:38,060 --> 00:41:40,120
the impulse response of
the system, namely
600
00:41:40,120 --> 00:41:41,830
the frequency response.
601
00:41:41,830 --> 00:41:47,170
So, in fact, if we think of
h of t and h of omega as a
602
00:41:47,170 --> 00:41:51,980
Fourier transform pair, it's the
convolution property that
603
00:41:51,980 --> 00:41:58,295
lets us equate this term
with h of omega.
604
00:41:58,295 --> 00:42:02,230
So here we're using the
convolution property.
605
00:42:06,170 --> 00:42:10,750
So we know what the Fourier
transform of the impulse
606
00:42:10,750 --> 00:42:14,030
response is, namely 1
over j omega plus a.
607
00:42:17,270 --> 00:42:20,870
We may have, for example, wanted
in our problem, instead
608
00:42:20,870 --> 00:42:23,420
of getting the frequency
response, to
609
00:42:23,420 --> 00:42:25,790
get the impulse response.
610
00:42:25,790 --> 00:42:28,690
And there are a variety of
ways that we can do this.
611
00:42:28,690 --> 00:42:32,400
We can attempt to go through the
inverse Fourier transform
612
00:42:32,400 --> 00:42:33,570
expression.
613
00:42:33,570 --> 00:42:38,780
But in fact, one of the most
useful ways is formally called
614
00:42:38,780 --> 00:42:40,490
the inspection method.
615
00:42:40,490 --> 00:42:46,190
Informally it's called, if you
worked it out going one way,
616
00:42:46,190 --> 00:42:48,630
then you ought to remember the
answer so that you know how to
617
00:42:48,630 --> 00:42:51,000
get back method.
618
00:42:51,000 --> 00:42:56,140
So what that says is, remember
that we worked an example, and
619
00:42:56,140 --> 00:42:58,520
in fact I showed you the
example earlier in the
620
00:42:58,520 --> 00:43:03,540
lecture, that the Fourier
transform of e to the minus at
621
00:43:03,540 --> 00:43:07,460
times the step is 1 over
j omega plus a?
622
00:43:07,460 --> 00:43:11,550
So what is the inverse Fourier
transfer of 1 over
623
00:43:11,550 --> 00:43:13,130
j omega plus a?
624
00:43:13,130 --> 00:43:19,890
Well, it's e to the minus
at times a unit step.
625
00:43:19,890 --> 00:43:22,840
And that's just simply
remembering, in essence, this
626
00:43:22,840 --> 00:43:24,090
particular transform pair.
627
00:43:26,940 --> 00:43:32,070
I've drawn, graphically, the
magnitude of the Fourier
628
00:43:32,070 --> 00:43:35,890
transform here.
629
00:43:35,890 --> 00:43:41,100
And below it we have the
impulse response.
630
00:43:41,100 --> 00:43:45,610
The impulse response is, as I
just indicated, e to the minus
631
00:43:45,610 --> 00:43:48,450
at times u of t.
632
00:43:48,450 --> 00:43:53,400
Now let's go back up and look
at the magnitude of the
633
00:43:53,400 --> 00:43:56,240
frequency response.
634
00:43:56,240 --> 00:44:00,030
And given just the little bit
of discussion that we had
635
00:44:00,030 --> 00:44:05,680
previously about filtering, you
should be able to infer
636
00:44:05,680 --> 00:44:09,940
something about the filtering
characteristics of this
637
00:44:09,940 --> 00:44:13,680
simple, first order differential
equation.
638
00:44:13,680 --> 00:44:17,740
In particular, if you look at
that frequency response, the
639
00:44:17,740 --> 00:44:21,550
frequency response falls off
with frequency and so what it
640
00:44:21,550 --> 00:44:27,080
tends to do is attenuate high
frequencies and retain low
641
00:44:27,080 --> 00:44:28,160
frequencies.
642
00:44:28,160 --> 00:44:32,950
So in fact, you could think of
the defocusing that we did on
643
00:44:32,950 --> 00:44:35,640
the image of Fourier, you
could think of that,
644
00:44:35,640 --> 00:44:39,390
approximately, as similar to the
kind of filtering action
645
00:44:39,390 --> 00:44:43,630
that you would get by passing a
signal through a first order
646
00:44:43,630 --> 00:44:44,880
differential equation.
647
00:44:47,620 --> 00:44:52,640
Just to illustrate one
additional step in both
648
00:44:52,640 --> 00:44:56,090
evaluating inverse transforms
and using Fourier transform
649
00:44:56,090 --> 00:44:59,690
properties to solve linear
constant coefficient
650
00:44:59,690 --> 00:45:04,250
differential equations, let's
take the same example and,
651
00:45:04,250 --> 00:45:07,830
rather than finding the impulse
response, let's find
652
00:45:07,830 --> 00:45:11,910
the response to another
exponential input.
653
00:45:11,910 --> 00:45:13,830
We could, of course,
do that using
654
00:45:13,830 --> 00:45:14,950
the convolution integral.
655
00:45:14,950 --> 00:45:17,090
We've just gotten the impulse
response, and we could put
656
00:45:17,090 --> 00:45:20,710
that through the convolution
integral to get the response
657
00:45:20,710 --> 00:45:21,930
to this input.
658
00:45:21,930 --> 00:45:25,160
But let's do it, instead,
by going back to the
659
00:45:25,160 --> 00:45:27,890
differential equation.
660
00:45:27,890 --> 00:45:32,330
And so here I'm taking a
differential equation.
661
00:45:32,330 --> 00:45:35,920
I'll choose, just to have
some numbers to work
662
00:45:35,920 --> 00:45:38,400
with, a equal to 2.
663
00:45:38,400 --> 00:45:41,690
And now I'll choose an
exponential on the right hand
664
00:45:41,690 --> 00:45:46,030
side, e to the minus
t times u of t.
665
00:45:46,030 --> 00:45:51,700
And again, we Fourier transform
the equation.
666
00:45:51,700 --> 00:45:55,510
And we can remember
this particular
667
00:45:55,510 --> 00:45:57,490
Fourier transform pair.
668
00:45:57,490 --> 00:46:00,690
It's just the one we worked
out previously, now with a
669
00:46:00,690 --> 00:46:01,650
equal to 1.
670
00:46:01,650 --> 00:46:05,430
Clearly we're getting a lot of
mileage out of that example.
671
00:46:05,430 --> 00:46:09,660
And now, if we want to determine
what the output y of
672
00:46:09,660 --> 00:46:14,920
t is, we can do that by solving
for y of omega and
673
00:46:14,920 --> 00:46:18,520
then generating the inverse
Fourier transform.
674
00:46:18,520 --> 00:46:22,850
Let's solve this algebraically
for y of omega, and that gets
675
00:46:22,850 --> 00:46:25,460
us to this expression.
676
00:46:25,460 --> 00:46:29,420
And this is not a Fourier
transform that we've worked
677
00:46:29,420 --> 00:46:30,830
out before.
678
00:46:30,830 --> 00:46:36,190
And this is the second part to
the inspection procedure.
679
00:46:36,190 --> 00:46:39,430
What we have is a Fourier
transform which is a product
680
00:46:39,430 --> 00:46:42,940
of two terms, each of which
we can recognize.
681
00:46:42,940 --> 00:46:48,660
And what we can consider doing
is expanding that out in a
682
00:46:48,660 --> 00:46:52,160
partial fraction expansion,
namely as a sum of terms.
683
00:46:52,160 --> 00:46:54,650
Because of the linearity
property associated with the
684
00:46:54,650 --> 00:46:59,040
Fourier transform, the inverse
transform is then the sum of
685
00:46:59,040 --> 00:47:01,030
the inverse transform of
each of those terms.
686
00:47:03,750 --> 00:47:08,020
So if we expand this out in a
partial fraction expansion,
687
00:47:08,020 --> 00:47:12,990
and you can just verify that if
you add these two together
688
00:47:12,990 --> 00:47:15,440
you'll get back to
where we started.
689
00:47:15,440 --> 00:47:22,670
We now have the sum of two
terms, and if we now
690
00:47:22,670 --> 00:47:26,740
recognize, by inspection, the
inverse Fourier transform of
691
00:47:26,740 --> 00:47:32,520
this, we see that it's simply
minus e to the minus 2t times
692
00:47:32,520 --> 00:47:33,720
the unit step.
693
00:47:33,720 --> 00:47:38,320
This one is plus e to the minus
t times the unit step.
694
00:47:38,320 --> 00:47:43,280
And so, in fact, it's the sum
of these two terms which are
695
00:47:43,280 --> 00:47:47,970
the inverse transforms of the
individual terms in the
696
00:47:47,970 --> 00:48:02,820
partial fraction expansion that
then give us the output.
697
00:48:02,820 --> 00:48:08,170
So this is y of t, which
is the sum of these two
698
00:48:08,170 --> 00:48:14,350
exponentials, and this is the
inverse Fourier transform of y
699
00:48:14,350 --> 00:48:16,690
of omega as we calculated
it previously.
700
00:48:21,110 --> 00:48:24,940
Hopefully you're beginning to
get some sense, now, of how
701
00:48:24,940 --> 00:48:29,190
powerful and also beautiful
the Fourier transform is.
702
00:48:29,190 --> 00:48:33,980
We've seen already a glimpse
of how it plays a role in
703
00:48:33,980 --> 00:48:38,050
filtering, modulation, how its
properties help us with linear
704
00:48:38,050 --> 00:48:39,510
constant coefficient
differential
705
00:48:39,510 --> 00:48:42,456
equations, et cetera.
706
00:48:42,456 --> 00:48:48,520
What we will do, beginning
with the next lecture, is
707
00:48:48,520 --> 00:48:52,310
develop a similar set of tools
for the discrete time case.
708
00:48:52,310 --> 00:48:56,650
And there are some very strong
similarities to what we've
709
00:48:56,650 --> 00:48:58,850
done in continuous time, also
some very important
710
00:48:58,850 --> 00:49:00,190
differences.
711
00:49:00,190 --> 00:49:04,150
And then, after we have the
continuous time and discrete
712
00:49:04,150 --> 00:49:10,270
time Fourier transforms, we'll
then see how the concepts
713
00:49:10,270 --> 00:49:14,030
involved and the properties
involved lead to very
714
00:49:14,030 --> 00:49:18,690
important and powerful notions
of filtering, modulation,
715
00:49:18,690 --> 00:49:22,250
sampling, and other signal
processing ideas.
716
00:49:22,250 --> 00:49:23,500
Thank you.