1
00:00:00,040 --> 00:00:02,470
The following content is
provided under a Creative
2
00:00:02,470 --> 00:00:03,880
Commons license.
3
00:00:03,880 --> 00:00:06,920
Your support will help MIT
OpenCourseWare continue to
4
00:00:06,920 --> 00:00:10,570
offer high quality educational
resources for free.
5
00:00:10,570 --> 00:00:13,470
To make a donation or view
additional materials from
6
00:00:13,470 --> 00:00:19,290
hundreds of MIT courses, visit
MIT OpenCourseWare at
7
00:00:19,290 --> 00:00:20,540
ocw.mit.edu.
8
00:00:56,260 --> 00:00:58,910
PROFESSOR: Last time, we began
to address the issue of
9
00:00:58,910 --> 00:01:02,610
building continuous time
signals out of a linear
10
00:01:02,610 --> 00:01:05,560
combination of complex
exponentials.
11
00:01:05,560 --> 00:01:10,110
And for the class of periodic
signals specifically, what
12
00:01:10,110 --> 00:01:13,350
this led to was the Fourier
series representation for
13
00:01:13,350 --> 00:01:15,020
periodic signals.
14
00:01:15,020 --> 00:01:17,270
Let me just summarize
the results that we
15
00:01:17,270 --> 00:01:19,110
developed last time.
16
00:01:19,110 --> 00:01:23,880
For periodic signals, we had
the continuous-time Fourier
17
00:01:23,880 --> 00:01:29,220
series, where we built the
periodic signal out of a
18
00:01:29,220 --> 00:01:33,250
linear combination of
harmonically related complex
19
00:01:33,250 --> 00:01:34,750
exponentials.
20
00:01:34,750 --> 00:01:39,030
And what that led to was what
we referred to as the
21
00:01:39,030 --> 00:01:41,050
synthesis equation.
22
00:01:41,050 --> 00:01:45,900
And we briefly addressed the
issue of when this in fact
23
00:01:45,900 --> 00:01:50,360
builds, when this in fact is a
complete representation of the
24
00:01:50,360 --> 00:01:55,970
periodic signal, and in essence,
what we presented was
25
00:01:55,970 --> 00:02:00,540
conditions either for x of t
being square integrable or x
26
00:02:00,540 --> 00:02:03,090
of t being absolutely
integrable.
27
00:02:03,090 --> 00:02:06,880
Then, the other side of the
Fourier series is what I
28
00:02:06,880 --> 00:02:10,380
referred to as the analysis
equation.
29
00:02:10,380 --> 00:02:14,120
And the analysis equation was
the equation that told us how
30
00:02:14,120 --> 00:02:19,110
we get the Fourier series
coefficients from x of t.
31
00:02:19,110 --> 00:02:23,430
And so this equation together
with the synthesis equation
32
00:02:23,430 --> 00:02:26,600
represent the Fourier
series description
33
00:02:26,600 --> 00:02:29,470
for periodic signals.
34
00:02:29,470 --> 00:02:34,580
Now what we'd like to do is
extend this idea to provide a
35
00:02:34,580 --> 00:02:39,460
mechanism for building
non-periodic signals also out
36
00:02:39,460 --> 00:02:42,500
of a linear combination of
complex exponentials.
37
00:02:42,500 --> 00:02:47,310
And the basic idea behind doing
this is very simple and
38
00:02:47,310 --> 00:02:50,680
also very clever as I
indicated last time.
39
00:02:50,680 --> 00:02:54,850
Essentially, the thought is the
following, if we have a
40
00:02:54,850 --> 00:02:59,220
non-periodic signal or aperiodic
signal, we can think
41
00:02:59,220 --> 00:03:04,720
of constructing a periodic
signal by simply periodically
42
00:03:04,720 --> 00:03:08,230
replicating that aperiodic
signal.
43
00:03:08,230 --> 00:03:12,520
So for example, if I have an
aperiodic signal as I've
44
00:03:12,520 --> 00:03:18,870
indicated here, I can consider
building a periodic signal,
45
00:03:18,870 --> 00:03:23,700
where I simply take this
original signal and repeat it
46
00:03:23,700 --> 00:03:27,910
at multiples of some
period t 0.
47
00:03:27,910 --> 00:03:30,380
Now, two things to recognize
about this.
48
00:03:30,380 --> 00:03:36,155
One is that the periodic
signal is equal to the
49
00:03:36,155 --> 00:03:39,150
aperiodic signal over
one period.
50
00:03:39,150 --> 00:03:45,020
And the second is that as the
period goes to infinity then,
51
00:03:45,020 --> 00:03:52,490
in fact, the periodic signal
goes to the aperiodic signal.
52
00:03:52,490 --> 00:03:59,360
So the basic idea then is to
use the Fourier series to
53
00:03:59,360 --> 00:04:03,920
represent the periodic signal,
and then examine the Fourier
54
00:04:03,920 --> 00:04:08,880
series expression as we let
the period go to infinity.
55
00:04:08,880 --> 00:04:14,150
Well, let's quickly see how this
develops in terms of the
56
00:04:14,150 --> 00:04:16,130
associated equations.
57
00:04:16,130 --> 00:04:22,079
Here, again, we have the
periodic signal.
58
00:04:22,079 --> 00:04:26,920
And what we want to inquire into
is what happens to the
59
00:04:26,920 --> 00:04:30,690
Fourier series expression for
this as we let the period go
60
00:04:30,690 --> 00:04:31,750
to infinity.
61
00:04:31,750 --> 00:04:35,140
As that happens, whatever
Fourier series representation
62
00:04:35,140 --> 00:04:40,260
we end up with will correspond
also to a representation for
63
00:04:40,260 --> 00:04:42,850
this aperiodic signal.
64
00:04:42,850 --> 00:04:44,220
Well, let's see.
65
00:04:44,220 --> 00:04:48,040
The Fourier series synthesis
expression for the periodic
66
00:04:48,040 --> 00:04:53,650
signal expresses x tilde of t,
the periodic signal as a
67
00:04:53,650 --> 00:04:57,230
linear combination of
harmonically related complex
68
00:04:57,230 --> 00:05:01,510
exponentials with the
fundamental frequency omega 0
69
00:05:01,510 --> 00:05:04,370
equaled to 2 pi divided
by the period.
70
00:05:04,370 --> 00:05:11,530
And the analysis equation tells
us what the relationship
71
00:05:11,530 --> 00:05:16,180
is for the coefficients in terms
of the periodic signal.
72
00:05:16,180 --> 00:05:20,215
Now, I indicated that the
periodic signal and the
73
00:05:20,215 --> 00:05:23,370
aperiodic signal are equal
over one period.
74
00:05:23,370 --> 00:05:27,640
We recognize that this
integration, in fact, only
75
00:05:27,640 --> 00:05:29,460
occurs over one period.
76
00:05:29,460 --> 00:05:33,840
And so we can re-express this
in terms of our original
77
00:05:33,840 --> 00:05:35,510
aperiodic signal.
78
00:05:35,510 --> 00:05:38,730
So this tells us the Fourier
series coefficients in
79
00:05:38,730 --> 00:05:40,012
terms of x of t.
80
00:05:43,320 --> 00:05:49,360
Now, if we look at this
expression, which is the
81
00:05:49,360 --> 00:05:53,020
expression for the Fourier
coefficients of the aperiodic
82
00:05:53,020 --> 00:05:58,690
signal, one of the things to
recognize is that in effect
83
00:05:58,690 --> 00:06:06,420
what this represents are samples
of an integral, where
84
00:06:06,420 --> 00:06:12,350
we can think of the variable
omega taking on values that
85
00:06:12,350 --> 00:06:14,830
are integer multiples
of omega 0.
86
00:06:14,830 --> 00:06:18,600
Said another way, let's define
a function, as I've indicated
87
00:06:18,600 --> 00:06:24,540
here, which is this integral,
where we may think of omega as
88
00:06:24,540 --> 00:06:27,930
being a continuous variable and
then the Fourier series
89
00:06:27,930 --> 00:06:35,890
coefficients correspond
to substituting for
90
00:06:35,890 --> 00:06:38,660
omega k omega 0.
91
00:06:38,660 --> 00:06:42,710
Now, one reason for doing that,
as we'll see, is that in
92
00:06:42,710 --> 00:06:47,480
fact, this will turn out to
provide us with a mechanism
93
00:06:47,480 --> 00:06:51,250
for a Fourier representation
of x of t.
94
00:06:51,250 --> 00:06:56,360
And this, in fact, then, is an
envelope of the Fourier series
95
00:06:56,360 --> 00:06:57,340
coefficients.
96
00:06:57,340 --> 00:07:03,530
In other words, t 0 the period
times the coefficients is
97
00:07:03,530 --> 00:07:07,850
equal to this integral add
integer multiples of omega 0.
98
00:07:10,540 --> 00:07:14,360
So this, in effect, tells us how
to get the Fourier series
99
00:07:14,360 --> 00:07:19,490
coefficients of the periodic
signal in terms of samples of
100
00:07:19,490 --> 00:07:20,140
an envelope.
101
00:07:20,140 --> 00:07:23,770
And that will become a very
important notion shortly.
102
00:07:23,770 --> 00:07:28,370
And that, in effect, will
correspond to an analysis
103
00:07:28,370 --> 00:07:32,280
equation to represent the
aperiodic signal.
104
00:07:32,280 --> 00:07:35,770
Now, let's look at the
synthesis equation.
105
00:07:35,770 --> 00:07:40,200
Recall that in the synthesis
our strategy is to build a
106
00:07:40,200 --> 00:07:44,190
periodic signal and let the
period go to infinity.
107
00:07:44,190 --> 00:07:50,160
Well, here is the expression
for the synthesis of the
108
00:07:50,160 --> 00:07:56,510
periodic signal now expressed
in terms of samples of this
109
00:07:56,510 --> 00:08:01,840
envelope function, and where
I've simply used the fact or
110
00:08:01,840 --> 00:08:06,870
the substitution that t 0 is 2
pi over omega 0, and so I have
111
00:08:06,870 --> 00:08:10,830
an omega 0 here and
a 1 over 2 pi.
112
00:08:10,830 --> 00:08:14,280
And the reason for doing that,
as we'll see in a minute, is
113
00:08:14,280 --> 00:08:17,440
that this then turns
into an integral.
114
00:08:17,440 --> 00:08:20,880
Specifically, then, the
synthesis equation that we
115
00:08:20,880 --> 00:08:26,690
have is what I've
indicated here.
116
00:08:26,690 --> 00:08:30,300
We would now want to examine
this as the period goes to
117
00:08:30,300 --> 00:08:33,990
infinity, which means
that omega 0 becomes
118
00:08:33,990 --> 00:08:36,140
infinitesimally small.
119
00:08:36,140 --> 00:08:39,820
And without dwelling on the
details, and with my
120
00:08:39,820 --> 00:08:43,440
suggesting that you give this
a fair amount of reflection,
121
00:08:43,440 --> 00:08:47,830
in fact, what happens as the
period goes to infinity is
122
00:08:47,830 --> 00:08:53,910
that this summation approaches
an integral over omega, where
123
00:08:53,910 --> 00:08:57,650
omega 0 becomes the differential
in omega, and the
124
00:08:57,650 --> 00:09:00,020
periodic signal, of course,
approach is
125
00:09:00,020 --> 00:09:02,740
the aperiodic signal.
126
00:09:02,740 --> 00:09:08,240
So the resulting equation that
we get out of the original
127
00:09:08,240 --> 00:09:14,680
Fourier series synthesis
equation is the equation that
128
00:09:14,680 --> 00:09:21,730
I indicate down here, x of t
synthesized in terms of this
129
00:09:21,730 --> 00:09:25,720
integral, which is what the
Fourier series approaches as
130
00:09:25,720 --> 00:09:28,780
omega 0 goes to 0.
131
00:09:28,780 --> 00:09:34,400
And we had previously that
x of omega was in fact an
132
00:09:34,400 --> 00:09:36,900
envelope function.
133
00:09:36,900 --> 00:09:39,660
And we have then the
corresponding Fourier
134
00:09:39,660 --> 00:09:45,240
transform analysis equation,
which tells us how we arrive
135
00:09:45,240 --> 00:09:49,930
at that envelope in
terms of x of t.
136
00:09:49,930 --> 00:09:54,410
So we now have an analysis
equation and a synthesis
137
00:09:54,410 --> 00:09:59,150
equation, which in effect
expresses for us how to build
138
00:09:59,150 --> 00:10:03,910
x of t in terms of
infinitesimally finely spaced
139
00:10:03,910 --> 00:10:06,750
complex exponentials.
140
00:10:06,750 --> 00:10:11,690
The strategy to review it, and
which I'd like to illustrate
141
00:10:11,690 --> 00:10:21,090
with a succession of overlays,
was to begin with our
142
00:10:21,090 --> 00:10:28,780
aperiodic signal, as I indicate
here, and then we
143
00:10:28,780 --> 00:10:34,160
constructed from that
a periodic signal.
144
00:10:34,160 --> 00:10:39,880
And this periodic signal has a
Fourier series, and we express
145
00:10:39,880 --> 00:10:43,550
the Fourier series coefficients
of this as
146
00:10:43,550 --> 00:10:46,630
samples of an envelope
function.
147
00:10:46,630 --> 00:10:51,520
The envelope function is what I
indicate on the curve below.
148
00:10:51,520 --> 00:10:55,390
So this is the envelope of the
Fourier series coefficients.
149
00:10:55,390 --> 00:11:02,640
For example, if the period t 0
was four times t1, then the
150
00:11:02,640 --> 00:11:06,270
Fourier series coefficients that
we would end up with is
151
00:11:06,270 --> 00:11:10,480
this set of samples
of the envelope.
152
00:11:10,480 --> 00:11:15,500
If instead we doubled that
period, then the Fourier
153
00:11:15,500 --> 00:11:18,470
series coefficients that
we end up with
154
00:11:18,470 --> 00:11:22,180
are more finely spaced.
155
00:11:22,180 --> 00:11:28,470
And as t 0 continues to
increase, we get more and more
156
00:11:28,470 --> 00:11:33,290
finely spaced samples of this
envelope function, and as t 0
157
00:11:33,290 --> 00:11:37,030
goes to infinity in fact, what
we get is every single point
158
00:11:37,030 --> 00:11:41,080
on the envelope, and that
provides us with the
159
00:11:41,080 --> 00:11:44,780
representation for the
aperiodic signal.
160
00:11:44,780 --> 00:11:50,530
Let me, just to really emphasize
the point, show this
161
00:11:50,530 --> 00:11:52,520
example once again.
162
00:11:52,520 --> 00:11:55,800
But now, let's look at
it dynamically on
163
00:11:55,800 --> 00:11:58,440
the computer display.
164
00:11:58,440 --> 00:12:01,220
So here we have the square
wave, and below it, the
165
00:12:01,220 --> 00:12:03,010
Fourier series coefficients.
166
00:12:03,010 --> 00:12:06,490
And we now want to look at the
Fourier series coefficients as
167
00:12:06,490 --> 00:12:09,550
the period of the square wave
starts to increase.
168
00:12:17,760 --> 00:12:21,260
And what we see is that
these look like
169
00:12:21,260 --> 00:12:24,100
samples of an envelope.
170
00:12:24,100 --> 00:12:26,970
And in fact, the envelope
of the Fourier series
171
00:12:26,970 --> 00:12:30,090
coefficients is shown in
the bottom [? trace, ?]
172
00:12:30,090 --> 00:12:33,300
and to emphasize in fact that
it is the envelope let's
173
00:12:33,300 --> 00:12:37,400
superimpose it on top of the
Fourier series coefficients
174
00:12:37,400 --> 00:12:38,840
that we've generated so far.
175
00:12:48,340 --> 00:12:48,710
OK.
176
00:12:48,710 --> 00:12:54,020
Now, let's increase the period
even further, and we'll see
177
00:12:54,020 --> 00:12:57,910
the Fourier series coefficients
fill in under
178
00:12:57,910 --> 00:13:02,780
that envelope function
even more.
179
00:13:02,780 --> 00:13:07,980
And in fact, as the period gets
large enough, what we
180
00:13:07,980 --> 00:13:13,570
begin to get a sense of is that
we're sampling more and
181
00:13:13,570 --> 00:13:15,940
more finely this envelope.
182
00:13:15,940 --> 00:13:19,060
And in fact, in the limit,
as the period goes off to
183
00:13:19,060 --> 00:13:23,160
infinity, the samples basically
will represent every
184
00:13:23,160 --> 00:13:25,050
single point on the envelope.
185
00:13:25,050 --> 00:13:28,030
Well, this is about as
far as we want to go.
186
00:13:28,030 --> 00:13:33,130
Let's once again, plot the
envelope function, and again,
187
00:13:33,130 --> 00:13:36,160
to emphasize that we've
generated samples of that,
188
00:13:36,160 --> 00:13:39,175
let's superimpose that on the
Fourier series coefficients.
189
00:13:46,300 --> 00:13:52,000
So what we have then is now
our Fourier transform
190
00:13:52,000 --> 00:13:57,530
representation, the continuous
time Fourier transform with
191
00:13:57,530 --> 00:14:01,960
the synthesis equation expressed
as an integral, as
192
00:14:01,960 --> 00:14:06,870
I've indicated here, and this
integral is what the Fourier
193
00:14:06,870 --> 00:14:12,820
series sum went to as we let the
period go to infinity or
194
00:14:12,820 --> 00:14:14,720
the frequency go to zero.
195
00:14:14,720 --> 00:14:19,080
The corresponding analysis
equation, which we have here,
196
00:14:19,080 --> 00:14:23,670
the analysis equation being
the expression for the
197
00:14:23,670 --> 00:14:26,970
envelope of the Fourier series
coefficients for the
198
00:14:26,970 --> 00:14:30,620
periodically replicated
signal.
199
00:14:30,620 --> 00:14:33,690
And in shorthand notation,
we would think
200
00:14:33,690 --> 00:14:35,370
of x of t and [? its ?]
201
00:14:35,370 --> 00:14:40,380
Fourier transform as a pair,
as I've indicated here.
202
00:14:40,380 --> 00:14:45,830
And the Fourier transform, as
we'll emphasize in several
203
00:14:45,830 --> 00:14:49,800
examples, and certainly as is
consistent with the Fourier
204
00:14:49,800 --> 00:14:54,590
Series, is a complex valued
function even
205
00:14:54,590 --> 00:14:55,730
when x of t is real.
206
00:14:55,730 --> 00:14:59,690
So with x of t real, we end up
with a Fourier transform,
207
00:14:59,690 --> 00:15:01,770
which is a complex function.
208
00:15:01,770 --> 00:15:06,060
Just as the Fourier series
coefficients were complex for
209
00:15:06,060 --> 00:15:08,250
a real value time function.
210
00:15:08,250 --> 00:15:12,420
So we could alternatively, as
with the Fourier series,
211
00:15:12,420 --> 00:15:18,150
express the Fourier transform in
terms of it's real part and
212
00:15:18,150 --> 00:15:23,530
imaginary part, or
alternatively, in terms of its
213
00:15:23,530 --> 00:15:25,520
magnitude and its angle.
214
00:15:28,270 --> 00:15:33,560
All right, now let's look at an
example of a time function
215
00:15:33,560 --> 00:15:35,750
in its Fourier transform.
216
00:15:35,750 --> 00:15:39,680
And so let's consider an
example, which in fact is an
217
00:15:39,680 --> 00:15:41,350
example worked out
in the text.
218
00:15:41,350 --> 00:15:44,500
It's example 4.7 in the text.
219
00:15:44,500 --> 00:15:49,520
And this is our old familiar
friend the exponential.
220
00:15:49,520 --> 00:15:53,230
It's Fourier transform is the
integral from minus infinity
221
00:15:53,230 --> 00:15:57,950
to plus infinity, x of t, e
to the minus j omega t dt.
222
00:15:57,950 --> 00:16:03,960
And so, if we substitute in x
of t and combine these two
223
00:16:03,960 --> 00:16:09,200
exponentials together, these two
exponentials combined are
224
00:16:09,200 --> 00:16:15,130
e to the minus t times
a plus j omega.
225
00:16:15,130 --> 00:16:21,070
And if we carry out the
integration of this, we end up
226
00:16:21,070 --> 00:16:25,790
with the expression indicated
here and provided now, and
227
00:16:25,790 --> 00:16:32,830
this is important, provided that
a is greater than 0, then
228
00:16:32,830 --> 00:16:36,310
at the upper limit, this
exponential becomes 0.
229
00:16:36,310 --> 00:16:38,870
At the lower limit, of
course, it's one.
230
00:16:38,870 --> 00:16:44,460
And so what we have finally is
for the Fourier transform
231
00:16:44,460 --> 00:16:48,780
expression 1 over
a plus j omega.
232
00:16:52,700 --> 00:16:56,970
Now, this Fourier transform as
I indicated is a complex
233
00:16:56,970 --> 00:16:58,380
valued function.
234
00:16:58,380 --> 00:17:04,210
Let's just take a look at what
it looks like graphically.
235
00:17:04,210 --> 00:17:10,640
We have the expression for the
Fourier transform pair, e to
236
00:17:10,640 --> 00:17:11,910
the minus a t times [? a ?]
237
00:17:11,910 --> 00:17:12,839
[? step ?].
238
00:17:12,839 --> 00:17:16,980
And its Fourier transform is
1 over a plus j omega.
239
00:17:16,980 --> 00:17:22,609
And I indicated that that's
true for a greater than 0.
240
00:17:22,609 --> 00:17:26,710
Now, in the expression that we
just worked out, if a is less
241
00:17:26,710 --> 00:17:33,360
than 0, in fact, the expression
doesn't converge e
242
00:17:33,360 --> 00:17:38,800
to the minus a t for a
negative as t goes to
243
00:17:38,800 --> 00:17:44,170
infinity, blows up, and so in
fact the Fourier transform
244
00:17:44,170 --> 00:17:48,990
doesn't converge except for the
case where a is greater
245
00:17:48,990 --> 00:17:52,930
than 0 And in fact, there is a
more detailed discussion of
246
00:17:52,930 --> 00:17:54,990
convergence issues
in the text.
247
00:17:54,990 --> 00:17:58,000
The convergence issues are very
much the same for the
248
00:17:58,000 --> 00:18:00,860
Fourier transform as they are
for the Fourier series.
249
00:18:00,860 --> 00:18:03,510
And in fact, that's not
surprising, because we
250
00:18:03,510 --> 00:18:05,360
developed the Fourier
transform out of a
251
00:18:05,360 --> 00:18:07,410
consideration of the
Fourier series.
252
00:18:07,410 --> 00:18:10,970
So the convergence conditions as
you'll see as you refer in
253
00:18:10,970 --> 00:18:15,310
detail to the text relate to
whether the time function is
254
00:18:15,310 --> 00:18:18,400
absolutely integrable under
one set of conditions and
255
00:18:18,400 --> 00:18:22,290
square integrable under another
set of conditions.
256
00:18:22,290 --> 00:18:28,110
OK, now, if we plot the Fourier
transform, let's first
257
00:18:28,110 --> 00:18:30,870
consider the shape of
the time function.
258
00:18:30,870 --> 00:18:34,580
And as I indicated, we're
restricting the time function
259
00:18:34,580 --> 00:18:37,630
so that the exponential
factor a is positive.
260
00:18:37,630 --> 00:18:40,730
In other words, e to the
minus a t decays
261
00:18:40,730 --> 00:18:42,920
as t goes to infinity.
262
00:18:42,920 --> 00:18:48,830
The magnitude of the Fourier
transform is as I indicate
263
00:18:48,830 --> 00:18:53,970
here and the phase below it.
264
00:18:53,970 --> 00:18:59,350
And there are a number of things
we can see about the
265
00:18:59,350 --> 00:19:02,630
magnitude and phase of the
Fourier transform for this
266
00:19:02,630 --> 00:19:07,990
example, which in fact we'll
see in the next lecture are
267
00:19:07,990 --> 00:19:10,210
properties that apply
more generally.
268
00:19:10,210 --> 00:19:13,810
For example, the fact that the
Fourier transform magnitude is
269
00:19:13,810 --> 00:19:17,440
an even function of frequency,
and the phase is an odd
270
00:19:17,440 --> 00:19:19,870
function of frequency.
271
00:19:19,870 --> 00:19:26,810
Now, let me also draw your
attention to the fact that on
272
00:19:26,810 --> 00:19:31,220
this curve we have both positive
frequencies and
273
00:19:31,220 --> 00:19:32,830
negative frequencies.
274
00:19:32,830 --> 00:19:35,950
In other words, in our
expression for the Fourier
275
00:19:35,950 --> 00:19:40,220
transform, it requires
both omega
276
00:19:40,220 --> 00:19:43,920
positive and omega negative.
277
00:19:43,920 --> 00:19:46,700
This, of course, was exactly
the same in the case of the
278
00:19:46,700 --> 00:19:48,160
Fourier series.
279
00:19:48,160 --> 00:19:53,530
And the reason you should recall
and keep in mind is
280
00:19:53,530 --> 00:19:56,390
related to the fact that we're
building our signals out of
281
00:19:56,390 --> 00:20:01,010
complex exponentials, which
require both positive values
282
00:20:01,010 --> 00:20:03,900
of omega and negative
values of omega.
283
00:20:03,900 --> 00:20:07,000
Alternatively, if we chosen
other representation, which
284
00:20:07,000 --> 00:20:10,060
turns out notationally to be
much more difficult, namely
285
00:20:10,060 --> 00:20:13,300
sines and cosines, then we would
in fact only consider
286
00:20:13,300 --> 00:20:14,700
positive frequencies.
287
00:20:14,700 --> 00:20:19,100
So it's important to keep in
mind that, in our case, both
288
00:20:19,100 --> 00:20:23,050
with the Fourier series and the
Fourier transform, we deal
289
00:20:23,050 --> 00:20:26,400
and require both positive and
negative frequencies in order
290
00:20:26,400 --> 00:20:27,650
to build our signals.
291
00:20:29,910 --> 00:20:34,220
Now, in the graphical
representation that I've shown
292
00:20:34,220 --> 00:20:39,210
here, I've chosen a linear
amplitude scale and a linear
293
00:20:39,210 --> 00:20:40,360
frequency scale.
294
00:20:40,360 --> 00:20:44,070
And that's one graphical
representation for the Fourier
295
00:20:44,070 --> 00:20:47,550
transform that we'll
typically use.
296
00:20:47,550 --> 00:20:54,110
There's another one that very
commonly arises, which I'll
297
00:20:54,110 --> 00:20:57,450
just briefly indicate
for this example.
298
00:20:57,450 --> 00:21:04,240
And that is what's referred to
as a bode plot in which the
299
00:21:04,240 --> 00:21:07,640
magnitude is displayed on
a log amplitude and log
300
00:21:07,640 --> 00:21:09,160
frequency scale.
301
00:21:09,160 --> 00:21:11,940
And the phase is displayed
on a log frequency scale.
302
00:21:11,940 --> 00:21:14,140
Let me show you what I mean.
303
00:21:14,140 --> 00:21:17,560
Here is the general expression
for the bode plot.
304
00:21:17,560 --> 00:21:23,080
The bode plot expresses for us
the amplitude in terms of the
305
00:21:23,080 --> 00:21:26,430
logarithm to the base
10 of the magnitude.
306
00:21:26,430 --> 00:21:33,020
And it also expresses the angle
in both cases expressed
307
00:21:33,020 --> 00:21:37,810
as a function of a logarithmic
frequency axis.
308
00:21:37,810 --> 00:21:45,240
So here is the amplitude
as I've displayed it.
309
00:21:45,240 --> 00:21:50,110
And this is a log magnitude
scale, a logarithmic frequency
310
00:21:50,110 --> 00:21:54,280
scale as indicated by the fact
that as we move in equal
311
00:21:54,280 --> 00:21:58,190
increments along this axis,
we change frequency
312
00:21:58,190 --> 00:22:00,050
by a factor of 10.
313
00:22:00,050 --> 00:22:07,440
And similarly, what we have is
a display for the phase again
314
00:22:07,440 --> 00:22:10,190
on a log frequency scale.
315
00:22:10,190 --> 00:22:15,000
And I indicated that there is
a symmetry to the Fourier
316
00:22:15,000 --> 00:22:19,570
transform, and so in fact, we
can infer from this particular
317
00:22:19,570 --> 00:22:23,360
picture what it looks like for
the negative frequencies as
318
00:22:23,360 --> 00:22:24,695
well as for the positive
frequencies.
319
00:22:29,560 --> 00:22:39,450
Now, what we've done so far
is to develop the Fourier
320
00:22:39,450 --> 00:22:44,285
transform on the basis, the
Fourier transform of an
321
00:22:44,285 --> 00:22:48,170
aperiodic signal on the basis
of periodically repeating it
322
00:22:48,170 --> 00:22:52,420
and recognizing that the Fourier
series coefficients
323
00:22:52,420 --> 00:22:57,200
are samples of an envelope and
that these become more finely
324
00:22:57,200 --> 00:22:59,990
spaced as frequency increases.
325
00:22:59,990 --> 00:23:08,080
And in fact, we can go back to
our original equation in which
326
00:23:08,080 --> 00:23:13,810
we developed an envelope
function, and what we had
327
00:23:13,810 --> 00:23:19,200
indicated is that the Fourier
series coefficients were
328
00:23:19,200 --> 00:23:22,920
samples of this envelope.
329
00:23:22,920 --> 00:23:31,290
We then defined this envelope
as the Fourier transform of
330
00:23:31,290 --> 00:23:33,760
this aperiodic signal.
331
00:23:33,760 --> 00:23:38,670
So that provided us with a way--
and it was a mechanism--
332
00:23:38,670 --> 00:23:44,810
for getting a representation
for an aperiodic signal.
333
00:23:44,810 --> 00:23:50,820
Now, suppose that we have
instead a periodic signal, are
334
00:23:50,820 --> 00:23:54,660
there, in fact, some statements
that we can make
335
00:23:54,660 --> 00:23:58,050
about how the Fourier series
coefficients of that are
336
00:23:58,050 --> 00:24:02,240
related to the Fourier transform
of something.
337
00:24:02,240 --> 00:24:05,780
Well, in fact, this
statement tells us
338
00:24:05,780 --> 00:24:07,390
exactly how to do that.
339
00:24:07,390 --> 00:24:11,000
What this statement says is
that, in fact, the Fourier
340
00:24:11,000 --> 00:24:16,140
series coefficients are
samples of the Fourier
341
00:24:16,140 --> 00:24:19,180
transform of one period?
342
00:24:19,180 --> 00:24:28,450
So if we now consider a periodic
signal, we can in
343
00:24:28,450 --> 00:24:33,080
fact get the Fourier series
coefficients of that periodic
344
00:24:33,080 --> 00:24:35,980
signal by considering
the Fourier
345
00:24:35,980 --> 00:24:38,630
transform of one period.
346
00:24:38,630 --> 00:24:42,400
Said another way, the Fourier
series coefficients are
347
00:24:42,400 --> 00:24:45,510
proportional to samples
of the Fourier
348
00:24:45,510 --> 00:24:47,670
transform of one period.
349
00:24:47,670 --> 00:24:53,030
So if we consider this a
periodic signal, computed as
350
00:24:53,030 --> 00:25:00,380
Fourier transform, and selected
these samples that I
351
00:25:00,380 --> 00:25:06,780
indicate here, namely samples
equally spaced in omega by
352
00:25:06,780 --> 00:25:12,260
integer multiples of omega 0,
then in fact, those would be
353
00:25:12,260 --> 00:25:14,680
the Fourier series
coefficients.
354
00:25:14,680 --> 00:25:22,890
So we can go back to our
example previously that
355
00:25:22,890 --> 00:25:27,190
involved the square wave.
356
00:25:27,190 --> 00:25:31,910
And now, in this case, we could
argue that if in fact it
357
00:25:31,910 --> 00:25:36,490
was the periodic signal that we
started with, we could get
358
00:25:36,490 --> 00:25:40,610
the Fourier series coefficients
of that by
359
00:25:40,610 --> 00:25:46,950
thinking about the Fourier
transform of one period, which
360
00:25:46,950 --> 00:25:49,590
I indicate here.
361
00:25:49,590 --> 00:25:53,920
And then the Fourier series
coefficients of the periodic
362
00:25:53,920 --> 00:26:00,070
signal, in fact, are the
appropriate set of samples of
363
00:26:00,070 --> 00:26:01,320
this envelope.
364
00:26:04,710 --> 00:26:11,400
All right, now, we have a way of
getting the Fourier series
365
00:26:11,400 --> 00:26:14,930
coefficients from the Fourier
transform of one period.
366
00:26:14,930 --> 00:26:18,100
We originally derived the
Fourier transform of one
367
00:26:18,100 --> 00:26:21,420
period from the Fourier
series.
368
00:26:21,420 --> 00:26:25,820
What would, in fact, be nice is
if we could incorporate the
369
00:26:25,820 --> 00:26:28,790
Fourier series and the
Fourier transform
370
00:26:28,790 --> 00:26:30,530
within a common framework.
371
00:26:30,530 --> 00:26:34,390
And in fact, it turns out that
there is a very convenient way
372
00:26:34,390 --> 00:26:38,040
of doing that almost
by definition.
373
00:26:38,040 --> 00:26:44,380
Essentially, if we consider
what the equation for the
374
00:26:44,380 --> 00:26:51,640
synthesis looks like in both
cases, we can in effect define
375
00:26:51,640 --> 00:26:56,820
a Fourier transform for the
periodic signal, which we know
376
00:26:56,820 --> 00:26:59,360
is represented by its Fourier
series coefficients.
377
00:26:59,360 --> 00:27:04,150
We can define a Fourier
transform, and the definition
378
00:27:04,150 --> 00:27:09,960
of the Fourier transform is as
an impulse train, where the
379
00:27:09,960 --> 00:27:14,220
coefficients in the impulse
train are proportional, with a
380
00:27:14,220 --> 00:27:17,200
proportionality factor of 2
pi for a more or less a
381
00:27:17,200 --> 00:27:20,140
bookkeeping reason, proportional
to the Fourier
382
00:27:20,140 --> 00:27:22,220
series coefficients.
383
00:27:22,220 --> 00:27:28,430
And the validity of this is,
more or less, can be seen
384
00:27:28,430 --> 00:27:30,860
essentially by substitution.
385
00:27:30,860 --> 00:27:37,210
Specifically, here is then the
synthesis equation for the
386
00:27:37,210 --> 00:27:42,030
Fourier transform if we
substitute this definition for
387
00:27:42,030 --> 00:27:45,290
the Fourier transform of the
periodic signal into this
388
00:27:45,290 --> 00:27:51,310
expression then when we do the
appropriate bookkeeping and
389
00:27:51,310 --> 00:27:54,850
interchange the order of
summation and integration the
390
00:27:54,850 --> 00:27:58,690
impulse integrates out
to the exponential
391
00:27:58,690 --> 00:28:02,530
factor that we want.
392
00:28:02,530 --> 00:28:05,820
So we have the exponential
factor.
393
00:28:05,820 --> 00:28:07,990
We have the Fourier series
coefficients.
394
00:28:07,990 --> 00:28:11,450
The 2 pis take care of each
other, and what we're left
395
00:28:11,450 --> 00:28:16,620
with is the synthesis equation
for aperiodic signal in terms
396
00:28:16,620 --> 00:28:21,160
of the Fourier transform, or in
terms of its Fourier series
397
00:28:21,160 --> 00:28:23,600
coefficients.
398
00:28:23,600 --> 00:28:30,230
Now, we can just see this in
terms of a simple example.
399
00:28:30,230 --> 00:28:36,570
If we consider the example of a
symmetric square wave, then
400
00:28:36,570 --> 00:28:40,320
in effect what we're saying is
that for this symmetric square
401
00:28:40,320 --> 00:28:45,240
wave, this has a set of Fourier
series coefficients,
402
00:28:45,240 --> 00:28:50,020
which we worked out previously
and which I indicate on this
403
00:28:50,020 --> 00:28:52,440
figure with a bar graph.
404
00:28:52,440 --> 00:28:55,930
And really all that we're saying
is that, whereas these
405
00:28:55,930 --> 00:28:59,840
Fourier series coefficients
are indexed on an integer
406
00:28:59,840 --> 00:29:04,790
variable k, and [? they're ?]
bars not impulses.
407
00:29:04,790 --> 00:29:10,020
If we simply redefine or define
the Fourier transform
408
00:29:10,020 --> 00:29:15,360
of the periodic signal as an
impulse train, where the
409
00:29:15,360 --> 00:29:20,840
weights of the impulses are 2
pi times the corresponding
410
00:29:20,840 --> 00:29:25,750
Fourier series coefficients,
then this, in fact, is what we
411
00:29:25,750 --> 00:29:29,690
would use as the Fourier
transform of
412
00:29:29,690 --> 00:29:30,940
the periodic signal.
413
00:29:34,010 --> 00:29:39,080
Now, we've kind of gone back and
forth, and maybe even it
414
00:29:39,080 --> 00:29:42,180
might seem like we've gone
around in circles.
415
00:29:42,180 --> 00:29:45,880
So let me just try to summarize
the various
416
00:29:45,880 --> 00:29:49,490
relationships and steps that
we've gone through, keeping in
417
00:29:49,490 --> 00:29:52,670
mind that one of our objectives
was first to
418
00:29:52,670 --> 00:29:56,690
develop a representation for
aperiodic signals and then
419
00:29:56,690 --> 00:30:01,910
attempt to incorporate within
one framework both periodic
420
00:30:01,910 --> 00:30:05,530
and aperiodic signals.
421
00:30:05,530 --> 00:30:11,250
We began with an aperiodic
signal.
422
00:30:11,250 --> 00:30:15,290
And the strategy was to
develop a Fourier
423
00:30:15,290 --> 00:30:21,150
representation by constructing
a periodic signal for which
424
00:30:21,150 --> 00:30:22,860
that was one period.
425
00:30:22,860 --> 00:30:25,770
And then we let the period
go to infinity,
426
00:30:25,770 --> 00:30:27,580
as I indicate here.
427
00:30:27,580 --> 00:30:32,770
So we have an aperiodic
signal.
428
00:30:32,770 --> 00:30:37,760
We construct a periodic signal,
x tilde of t for which
429
00:30:37,760 --> 00:30:41,500
one period is the aperiodic
signal.
430
00:30:41,500 --> 00:30:45,550
X tilde of t, the periodic
signal, has a Fourier series,
431
00:30:45,550 --> 00:30:50,700
and as its period increases that
approaches the aperiodic
432
00:30:50,700 --> 00:30:57,050
signal, and the Fourier series
of that approaches the Fourier
433
00:30:57,050 --> 00:31:00,720
transform of the original
aperiodic signal.
434
00:31:00,720 --> 00:31:05,700
So that was the first
step we took.
435
00:31:05,700 --> 00:31:10,940
Now, the second thing that we
recognize is that once we have
436
00:31:10,940 --> 00:31:15,280
the concept of the Fourier
transform, we can, in fact,
437
00:31:15,280 --> 00:31:19,570
relate the Fourier series
coefficients to the Fourier
438
00:31:19,570 --> 00:31:22,210
transform of one period.
439
00:31:22,210 --> 00:31:27,330
So the second statement that
we made was that if in fact
440
00:31:27,330 --> 00:31:33,070
we're trying to represent a
periodic signal, we can get
441
00:31:33,070 --> 00:31:35,900
the Fourier series coefficients
of that by
442
00:31:35,900 --> 00:31:42,340
computing the Fourier transform
of one period and
443
00:31:42,340 --> 00:31:47,380
then samples of that Fourier
transform are, in fact, the
444
00:31:47,380 --> 00:31:50,110
Fourier series coefficients
for the periodic signal.
445
00:31:53,210 --> 00:31:58,320
Then, the third step that we
took was to inquire as to
446
00:31:58,320 --> 00:32:02,340
whether there is a Fourier
transform that can
447
00:32:02,340 --> 00:32:06,420
appropriately be defined for the
periodic signal, and the
448
00:32:06,420 --> 00:32:11,260
mechanism for doing that was to
recognize that if we simply
449
00:32:11,260 --> 00:32:14,840
defined the Fourier transform
of the periodic signal as an
450
00:32:14,840 --> 00:32:19,420
impulse train, where the impulse
heights or areas were
451
00:32:19,420 --> 00:32:22,580
proportional to the Fourier
series coefficients, then, in
452
00:32:22,580 --> 00:32:29,630
fact, the Fourier transform
synthesis equation reduced to
453
00:32:29,630 --> 00:32:32,140
the Fourier series synthesis
equation.
454
00:32:32,140 --> 00:32:37,960
So the third step, then, was
with a periodic signal.
455
00:32:37,960 --> 00:32:42,280
The Fourier transform of that
periodic signal, defined as an
456
00:32:42,280 --> 00:32:44,610
impulse train, where the
heights or areas of the
457
00:32:44,610 --> 00:32:47,810
impulses are proportional
to the Fourier series
458
00:32:47,810 --> 00:32:53,040
coefficients, provides us with
a mechanism for combining it
459
00:32:53,040 --> 00:32:57,825
together the concepts or
notation of the Fourier series
460
00:32:57,825 --> 00:33:00,900
and Fourier transform.
461
00:33:00,900 --> 00:33:07,340
So if we just took a very simple
example, here is an
462
00:33:07,340 --> 00:33:12,950
example in which we have an
aperiodic signal, which is
463
00:33:12,950 --> 00:33:16,620
just an impulse,
and its Fourier
464
00:33:16,620 --> 00:33:20,670
transform is just a constant.
465
00:33:20,670 --> 00:33:24,730
We can think of a periodic
signal associated with this,
466
00:33:24,730 --> 00:33:29,820
which is this signal
periodically replicated with a
467
00:33:29,820 --> 00:33:32,300
spacing t 0.
468
00:33:32,300 --> 00:33:35,720
The Fourier transform of
this is a constant.
469
00:33:35,720 --> 00:33:38,970
And this, of course, has a
Fourier series representation.
470
00:33:38,970 --> 00:33:42,800
So the Fourier transform
of the original
471
00:33:42,800 --> 00:33:46,170
impulse is just a constant.
472
00:33:46,170 --> 00:33:51,830
The Fourier transform of the
periodic signal is an impulse
473
00:33:51,830 --> 00:33:56,170
train, where the heights of the
impulses are proportional
474
00:33:56,170 --> 00:33:58,830
to the Fourier series
coefficients.
475
00:33:58,830 --> 00:34:02,700
And, of course, we could
previously have computed the
476
00:34:02,700 --> 00:34:06,680
Fourier series coefficients for
that impulse train, and
477
00:34:06,680 --> 00:34:09,040
those Fourier series
coefficients are
478
00:34:09,040 --> 00:34:10,370
as I've shown here.
479
00:34:10,370 --> 00:34:14,290
So in both of these cases, these
in effect represent just
480
00:34:14,290 --> 00:34:16,909
a change in notation, where here
we have a bar graph, and
481
00:34:16,909 --> 00:34:19,199
here we have an impulse train.
482
00:34:19,199 --> 00:34:23,980
And both of these simply
represent samples of what we
483
00:34:23,980 --> 00:34:30,690
have above, which is the Fourier
transform of the
484
00:34:30,690 --> 00:34:31,940
original aperiodic signal.
485
00:34:35,179 --> 00:34:39,239
Once again, I suspect that kind
of moving back and forth
486
00:34:39,239 --> 00:34:42,239
and trying to straighten out
when we're talking about
487
00:34:42,239 --> 00:34:46,989
periodic and aperiodic signals
may require a little mental
488
00:34:46,989 --> 00:34:48,810
gymnastics initially.
489
00:34:48,810 --> 00:34:53,310
Basically, what we've tried to
do is incorporate within one
490
00:34:53,310 --> 00:34:58,200
framework a representation for
both aperiodic and periodic
491
00:34:58,200 --> 00:35:01,450
signals, and the Fourier
transform provides us with a
492
00:35:01,450 --> 00:35:04,410
mechanism to do that.
493
00:35:04,410 --> 00:35:07,640
In the next lecture, I'll
continue with the discussion
494
00:35:07,640 --> 00:35:11,310
of the continuous-time Fourier
transform in particular
495
00:35:11,310 --> 00:35:15,260
focusing on a number of its
properties, some of which
496
00:35:15,260 --> 00:35:18,000
we've already seen, namely
the symmetry properties.
497
00:35:18,000 --> 00:35:21,360
We'll see lots of other
properties that relate, of
498
00:35:21,360 --> 00:35:23,780
course, both to the Fourier
transform and
499
00:35:23,780 --> 00:35:24,970
to the Fourier series.
500
00:35:24,970 --> 00:35:26,220
Thank you.