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PROFESSOR: Over the last several
lectures, we've dealt
9
00:00:57,730 --> 00:01:01,370
with the representation of
linear time-invariant systems
10
00:01:01,370 --> 00:01:03,080
through convolution.
11
00:01:03,080 --> 00:01:07,850
And just to remind you of our
basic strategy, essentially,
12
00:01:07,850 --> 00:01:13,230
the idea was to exploit the
notion of linearity by
13
00:01:13,230 --> 00:01:17,250
decomposing the input into a sum
of basic inputs and then
14
00:01:17,250 --> 00:01:20,580
using linearity to tell us
that the output can be
15
00:01:20,580 --> 00:01:23,200
represented as the corresponding
linear
16
00:01:23,200 --> 00:01:25,880
combination of the associated
outputs.
17
00:01:25,880 --> 00:01:30,420
So, if we have a linear system,
either continuous-time
18
00:01:30,420 --> 00:01:34,570
or discrete-time, for example,
with continuous time, if the
19
00:01:34,570 --> 00:01:38,690
input is decomposed as a linear
combination of basic
20
00:01:38,690 --> 00:01:42,250
inputs, with each of these basic
inputs generating an
21
00:01:42,250 --> 00:01:46,900
associated output, and if the
system is linear, then the
22
00:01:46,900 --> 00:01:51,850
output of the system is the same
linear combination of the
23
00:01:51,850 --> 00:01:53,530
associated outputs.
24
00:01:53,530 --> 00:01:57,450
And the same statement is
identical both for continuous
25
00:01:57,450 --> 00:02:01,170
time and discrete time.
26
00:02:01,170 --> 00:02:06,600
So the strategy is to decompose
the input into these
27
00:02:06,600 --> 00:02:08,440
basic inputs.
28
00:02:08,440 --> 00:02:14,120
And the inputs were chosen
also with some particular
29
00:02:14,120 --> 00:02:15,960
strategy in mind.
30
00:02:15,960 --> 00:02:19,390
In particular, for both
continuous time or discrete
31
00:02:19,390 --> 00:02:25,120
time, in this representation,
the basic inputs used in the
32
00:02:25,120 --> 00:02:30,890
decomposition are chosen, first
of all, so that a broad
33
00:02:30,890 --> 00:02:34,440
class of signals could be
represented in terms of these
34
00:02:34,440 --> 00:02:39,840
basic inputs, and second of all,
so that the response to
35
00:02:39,840 --> 00:02:45,660
these basic inputs is, in some
sense, easy to compute.
36
00:02:45,660 --> 00:02:52,420
Now, in the representation which
led us to convolution,
37
00:02:52,420 --> 00:02:56,930
the particular choice that we
made in the discrete-time case
38
00:02:56,930 --> 00:03:02,630
for our basic inputs was a
decomposition of the input in
39
00:03:02,630 --> 00:03:06,960
terms of delayed impulses.
40
00:03:06,960 --> 00:03:11,780
And the associated outputs
that that generated were
41
00:03:11,780 --> 00:03:15,580
delayed versions of the
impulse response.
42
00:03:15,580 --> 00:03:19,100
Decomposing the input into a
linear combination of these,
43
00:03:19,100 --> 00:03:21,510
the output into the
corresponding linear
44
00:03:21,510 --> 00:03:26,190
combination of these, then led
to the convolution sum in the
45
00:03:26,190 --> 00:03:28,430
discrete time case.
46
00:03:28,430 --> 00:03:33,080
And in the continuous-time
case, a similar kind of
47
00:03:33,080 --> 00:03:36,330
decomposition, in terms of
impulses, and associated
48
00:03:36,330 --> 00:03:39,740
representation of the output,
in terms of the impulse
49
00:03:39,740 --> 00:03:44,220
response, led to the convolution
integral.
50
00:03:44,220 --> 00:03:48,380
Now, in this lecture, and for
a number of the succeeding
51
00:03:48,380 --> 00:03:52,450
lectures, we'll want to turn
our attention to a very
52
00:03:52,450 --> 00:03:56,120
different set of basic
building blocks.
53
00:03:56,120 --> 00:04:01,150
And in particular, the signals
that we'll be using as the
54
00:04:01,150 --> 00:04:04,740
building blocks for our more
general signals, rather than
55
00:04:04,740 --> 00:04:09,870
impulses, as we've dealt with
before, will be, in general,
56
00:04:09,870 --> 00:04:12,230
complex exponentials.
57
00:04:12,230 --> 00:04:17,810
So, in a general sense, in the
continuous-time case, we'll be
58
00:04:17,810 --> 00:04:21,339
thinking in terms of a
decomposition of our signals
59
00:04:21,339 --> 00:04:26,160
as a linear combination of
complex exponentials,
60
00:04:26,160 --> 00:04:31,740
continuous-time, or, in the
discrete-time case, complex
61
00:04:31,740 --> 00:04:37,020
exponentials, where z_k is
complex here in discrete time
62
00:04:37,020 --> 00:04:42,500
and s sub k is complex here
in continuous time.
63
00:04:42,500 --> 00:04:47,860
Now, the basic strategy, of
course, requires that we
64
00:04:47,860 --> 00:04:52,220
choose a set of inputs, basic
building blocks, which have
65
00:04:52,220 --> 00:04:53,580
two properties.
66
00:04:53,580 --> 00:04:57,320
One is that the system response
be straightforward to
67
00:04:57,320 --> 00:05:00,150
compute, or in some sense,
easy to compute.
68
00:05:00,150 --> 00:05:03,280
And second is that it be a
fairly general set of building
69
00:05:03,280 --> 00:05:09,290
blocks so that we can build lots
of signals out of them.
70
00:05:09,290 --> 00:05:12,690
What we'll find with complex
exponentials, either
71
00:05:12,690 --> 00:05:14,740
continuous-time or
discrete-time, is that they
72
00:05:14,740 --> 00:05:17,870
very nicely have those
two properties.
73
00:05:17,870 --> 00:05:23,280
In particular, the notion that
the output of a linear
74
00:05:23,280 --> 00:05:27,320
time-invariant system is easy
to compute is tied to what's
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00:05:27,320 --> 00:05:29,250
referred to as the Eigenfunction
function
76
00:05:29,250 --> 00:05:32,280
property of complex
exponentials, which we'll
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00:05:32,280 --> 00:05:36,780
focus on shortly in a
little more detail.
78
00:05:36,780 --> 00:05:41,600
And second of all, the fact
that we can, in fact,
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00:05:41,600 --> 00:05:44,880
represent very broad classes
of signals as linear
80
00:05:44,880 --> 00:05:50,010
combinations of these will be
a topic and an issue that
81
00:05:50,010 --> 00:05:53,780
we'll develop in detail over,
in fact, the next set of
82
00:05:53,780 --> 00:05:57,860
lectures, this lecture, and
the next set of lectures.
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00:05:57,860 --> 00:06:03,960
Now, in doing this, although we
could, in fact, begin with
84
00:06:03,960 --> 00:06:10,040
our attention focused on, in
general, complex exponentials,
85
00:06:10,040 --> 00:06:14,330
what we'll choose to do is first
focus on the case in
86
00:06:14,330 --> 00:06:19,150
which the exponent in the
continuous-time case is purely
87
00:06:19,150 --> 00:06:25,420
imaginary, as I indicate here,
and in the discrete-time case,
88
00:06:25,420 --> 00:06:27,990
where the magnitude of
the complex number
89
00:06:27,990 --> 00:06:30,590
z_k is equal to 1.
90
00:06:30,590 --> 00:06:35,320
So what that corresponds to in
the continuous-time case is a
91
00:06:35,320 --> 00:06:41,280
set of building blocks of the
form e^(j omega_k t), and in
92
00:06:41,280 --> 00:06:44,800
the discrete-time case, a set of
building blocks of the form
93
00:06:44,800 --> 00:06:46,050
e^(j Omega_k n).
94
00:06:47,930 --> 00:06:52,730
What we'll see is a
representation in these terms
95
00:06:52,730 --> 00:06:57,570
leads to what's referred
to as Fourier analysis.
96
00:06:57,570 --> 00:07:00,150
And that's what will be dealing
with over the next set
97
00:07:00,150 --> 00:07:02,870
of lectures.
98
00:07:02,870 --> 00:07:06,290
We'll then be exploiting this
representation actually
99
00:07:06,290 --> 00:07:07,710
through most of the course.
100
00:07:07,710 --> 00:07:11,700
And then toward the end of the
course, we'll return to
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00:07:11,700 --> 00:07:16,110
generalizing the Fourier
representation to a discussion
102
00:07:16,110 --> 00:07:19,210
Laplace transforms
and Z-transforms.
103
00:07:19,210 --> 00:07:22,900
So for now, we want to restrict
ourselves to complex
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00:07:22,900 --> 00:07:27,100
exponentials of a particular
form, and in fact, also
105
00:07:27,100 --> 00:07:30,980
initially to continuous-time
signals and systems.
106
00:07:30,980 --> 00:07:34,410
So let's begin with the
continuous-time case and the
107
00:07:34,410 --> 00:07:39,020
complex exponentials that we
want to deal with and focus,
108
00:07:39,020 --> 00:07:43,050
first of all, on what I refer
to as the Eigenfunction
109
00:07:43,050 --> 00:07:48,000
property of this particular
set of building blocks.
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00:07:48,000 --> 00:07:51,313
We're talking about basic
signals of the form e^(j
111
00:07:51,313 --> 00:07:52,563
omega_k t).
112
00:07:54,020 --> 00:07:58,370
And the statement is that for
a linear time-invariant
113
00:07:58,370 --> 00:08:03,770
system, the response to one of
these is of exactly the same
114
00:08:03,770 --> 00:08:09,340
form, just simply multiplied
by a complex factor, that
115
00:08:09,340 --> 00:08:12,680
complex factor depending
on what the
116
00:08:12,680 --> 00:08:16,020
frequency, omega_k, is.
117
00:08:16,020 --> 00:08:18,820
Now more or less, the
justification for this, or the
118
00:08:18,820 --> 00:08:24,170
proof, follows by simply looking
at the response to a
119
00:08:24,170 --> 00:08:28,140
complex exponential, using
the convolution integral.
120
00:08:28,140 --> 00:08:33,750
So if we put a complex
exponentials into a linear
121
00:08:33,750 --> 00:08:39,409
time-invariant system with
impulse response h(t), then we
122
00:08:39,409 --> 00:08:42,780
can express the response
as I've indicated here.
123
00:08:42,780 --> 00:08:46,320
We can then recognize that this
complex exponentials can
124
00:08:46,320 --> 00:08:48,680
be factored into two terms.
125
00:08:48,680 --> 00:08:53,180
And so we can rewrite
this complex
126
00:08:53,180 --> 00:08:56,080
exponential as this product.
127
00:08:56,080 --> 00:09:02,280
Second, recognize that this term
can be taken outside the
128
00:09:02,280 --> 00:09:07,120
integral, over here, because
of the fact that it depends
129
00:09:07,120 --> 00:09:10,360
only on t and not on Tau.
130
00:09:10,360 --> 00:09:14,640
And so what we're left with,
when we track this through, is
131
00:09:14,640 --> 00:09:19,070
that, with a complex exponential
input, we get an
132
00:09:19,070 --> 00:09:23,120
output which is the same complex
exponential, namely
133
00:09:23,120 --> 00:09:26,950
this factor, times
this integral.
134
00:09:26,950 --> 00:09:33,620
And this integral is what I
refer to above as H(omega_k).
135
00:09:37,040 --> 00:09:42,200
And so, in fact, we put in a
complex exponential, we get
136
00:09:42,200 --> 00:09:46,680
out a complex exponentials of
the same frequency, multiplied
137
00:09:46,680 --> 00:09:48,650
by a complex constant.
138
00:09:48,650 --> 00:09:54,030
And that is what's referred to
as the Eigenfunction property,
139
00:09:54,030 --> 00:09:57,840
Eigenfunction meaning that an
Eigenfunction of a system, or
140
00:09:57,840 --> 00:10:01,370
mathematical expression, is a
function which, when you put
141
00:10:01,370 --> 00:10:04,980
it through the system, comes out
looking exactly the same
142
00:10:04,980 --> 00:10:07,950
except for a change in
amplitude, the change in
143
00:10:07,950 --> 00:10:11,040
amplitude being the
Eigenvalue.
144
00:10:11,040 --> 00:10:15,440
So in fact, this function
is the Eigenfunction.
145
00:10:17,980 --> 00:10:23,800
And this value is
the Eigenvalue.
146
00:10:27,540 --> 00:10:31,920
OK, now it's because of the
Eigenfunction property that
147
00:10:31,920 --> 00:10:34,800
complex exponentials are
particularly convenient as
148
00:10:34,800 --> 00:10:35,950
building blocks.
149
00:10:35,950 --> 00:10:38,900
Namely you put it through the
system, they come out with the
150
00:10:38,900 --> 00:10:42,980
same form and simply scale.
151
00:10:42,980 --> 00:10:45,780
The other part to the question,
related to the
152
00:10:45,780 --> 00:10:51,480
strategy that we've been
pursuing, is to hope that
153
00:10:51,480 --> 00:10:57,370
these signals can be used as
building blocks to represent a
154
00:10:57,370 --> 00:11:01,030
very broad class of signals
through a linear combination.
155
00:11:01,030 --> 00:11:04,230
And in fact, that turns out to
be the case with complex
156
00:11:04,230 --> 00:11:06,820
exponentials.
157
00:11:06,820 --> 00:11:10,420
As we work our way through that,
we'll first consider the
158
00:11:10,420 --> 00:11:13,950
case of periodic signals.
159
00:11:13,950 --> 00:11:17,710
And what that leads to is a
representation of periodic
160
00:11:17,710 --> 00:11:22,360
signals through what's called
the Fourier series.
161
00:11:22,360 --> 00:11:26,690
Following that, we'll turn our
attention to non-periodic, or
162
00:11:26,690 --> 00:11:29,770
as I refer to it, aperiodic
signals.
163
00:11:29,770 --> 00:11:34,830
And the representation that's
developed in terms of linear
164
00:11:34,830 --> 00:11:38,270
combinations of complex
exponentials is what's
165
00:11:38,270 --> 00:11:41,240
referred to as the Fourier
transform.
166
00:11:41,240 --> 00:11:45,550
So the first thing we want to
deal with are periodic signals
167
00:11:45,550 --> 00:11:47,090
and the Fourier series.
168
00:11:49,690 --> 00:11:54,190
So what we're talking about then
is the continuous-time
169
00:11:54,190 --> 00:11:57,110
Fourier series.
170
00:11:57,110 --> 00:12:02,180
And the Fourier series is a
representation for periodic
171
00:12:02,180 --> 00:12:05,090
continuous-time signals.
172
00:12:05,090 --> 00:12:08,730
We have a signal, then,
which is periodic.
173
00:12:08,730 --> 00:12:13,310
And we're choosing T_0
to denote the period.
174
00:12:13,310 --> 00:12:19,380
So it's T_0 that corresponds
to the period of
175
00:12:19,380 --> 00:12:21,660
our periodic signal.
176
00:12:21,660 --> 00:12:29,550
omega_0 is 2 pi / T_0, as you
recall from our discussion pf
177
00:12:29,550 --> 00:12:32,410
periodic signals and
sinusoids before.
178
00:12:32,410 --> 00:12:34,500
And that's 2 pi f_0.
179
00:12:34,500 --> 00:12:36,930
So this is the fundamental
frequency.
180
00:12:39,640 --> 00:12:45,240
Now let's examine, first of all,
complex exponentials, and
181
00:12:45,240 --> 00:12:49,860
recognize, first of all, that
there is a complex exponential
182
00:12:49,860 --> 00:12:54,060
that has exactly the same
period and fundamental
183
00:12:54,060 --> 00:12:58,680
frequency as our more general
periodic signal, namely the
184
00:12:58,680 --> 00:13:05,550
complex exponential e^(j omega_0
t), where omega_0 is 2
185
00:13:05,550 --> 00:13:12,640
pi / T_0, or equivalently,
T_0 is 2 pi / omega_0.
186
00:13:12,640 --> 00:13:17,820
Now that's the complex
exponential which has T_0 as
187
00:13:17,820 --> 00:13:19,760
the fundamental period.
188
00:13:19,760 --> 00:13:24,950
But there are harmonically
related complex exponentials
189
00:13:24,950 --> 00:13:30,130
that also have T_0 as a period,
although in fact,
190
00:13:30,130 --> 00:13:32,520
their fundamental period
is shorter.
191
00:13:32,520 --> 00:13:37,017
So we can also look at complex
exponentials of the form e^(j
192
00:13:37,017 --> 00:13:38,267
k omega_0 t).
193
00:13:39,710 --> 00:13:46,590
These likewise are periodic
with a period of T_0.
194
00:13:46,590 --> 00:13:52,060
Although, in fact, their
fundamental period is T_0 / k,
195
00:13:52,060 --> 00:13:55,480
or equivalently, 2 pi divided
by their fundamental
196
00:13:55,480 --> 00:13:58,520
frequency, k omega_0.
197
00:13:58,520 --> 00:14:04,070
So as k, an integer, varies,
these correspond to
198
00:14:04,070 --> 00:14:09,140
harmonically related complex
exponentials.
199
00:14:09,140 --> 00:14:14,060
Now what the Fourier series
says, and we'll justify this
200
00:14:14,060 --> 00:14:18,170
bit by bit as the discussion
goes on, what the Fourier
201
00:14:18,170 --> 00:14:23,030
series says, and in fact, what
Fourier said, which was
202
00:14:23,030 --> 00:14:27,150
essentially his brilliant
insight, is that, if I have a
203
00:14:27,150 --> 00:14:31,670
very general periodic signal, I
can represent it as a linear
204
00:14:31,670 --> 00:14:34,940
combination of these
harmonically-related complex
205
00:14:34,940 --> 00:14:36,900
exponentials.
206
00:14:36,900 --> 00:14:41,950
So that representation is what
I've indicated here.
207
00:14:41,950 --> 00:14:49,580
And this summation is what will
be referred to as the
208
00:14:49,580 --> 00:14:52,890
Fourier series.
209
00:14:55,990 --> 00:14:59,320
And as we proceed with the
discussion, there are two
210
00:14:59,320 --> 00:15:02,010
issues that will develop.
211
00:15:02,010 --> 00:15:06,670
One is, assuming that our
periodic signal can be
212
00:15:06,670 --> 00:15:11,040
represented this way, how do
we determine the Fourier
213
00:15:11,040 --> 00:15:14,820
series coefficients, as they're
referred to, a_k.
214
00:15:14,820 --> 00:15:16,720
That's one question.
215
00:15:16,720 --> 00:15:20,580
And the second question will
be how broad a class of
216
00:15:20,580 --> 00:15:24,070
signals, in fact, can be
represented this way.
217
00:15:24,070 --> 00:15:27,340
And that's another question
that we'll deal with
218
00:15:27,340 --> 00:15:28,590
separately.
219
00:15:30,120 --> 00:15:35,430
Now just focusing on this
representation for a minute,
220
00:15:35,430 --> 00:15:43,230
this representation of the
Fourier series, which I've
221
00:15:43,230 --> 00:15:52,000
repeated again here, is what's
referred to as the complex
222
00:15:52,000 --> 00:15:55,910
exponential form of the
Fourier series.
223
00:15:55,910 --> 00:15:59,740
And it's important to note,
incidentally, that the
224
00:15:59,740 --> 00:16:05,740
summation involves frequencies,
k omega_0, that
225
00:16:05,740 --> 00:16:07,800
are both positive
and negative.
226
00:16:07,800 --> 00:16:12,420
In other words, this index k
runs over limits that include
227
00:16:12,420 --> 00:16:16,700
both negative values and
positive values.
228
00:16:16,700 --> 00:16:21,710
Now that complex exponential
form is one representation for
229
00:16:21,710 --> 00:16:23,190
the Fourier series.
230
00:16:23,190 --> 00:16:27,080
And in fact, it's the one that
we will be principally relying
231
00:16:27,080 --> 00:16:29,300
on in this course.
232
00:16:29,300 --> 00:16:32,090
There is another representation
that perhaps
233
00:16:32,090 --> 00:16:37,030
you've come across previously
and that in a variety of other
234
00:16:37,030 --> 00:16:40,520
contexts is typically used,
which is called the
235
00:16:40,520 --> 00:16:44,010
trigonometric form for
the Fourier series.
236
00:16:44,010 --> 00:16:47,240
Without really tracking
through the algebra,
237
00:16:47,240 --> 00:16:50,770
essentially we can get to the
trigonometric form from the
238
00:16:50,770 --> 00:16:56,890
complex exponential form by
recognizing that if we express
239
00:16:56,890 --> 00:17:02,290
the complex coefficient in polar
form or in rectangular
240
00:17:02,290 --> 00:17:09,150
form and expand the complex
exponential term out in terms
241
00:17:09,150 --> 00:17:14,099
of cosine plus j sine, using
just simply Euler's relation,
242
00:17:14,099 --> 00:17:21,220
then we will end up with a
representation for the
243
00:17:21,220 --> 00:17:26,230
periodic signal, or a
re-expression of the Fourier
244
00:17:26,230 --> 00:17:31,140
series expression that we had
previously, either in the form
245
00:17:31,140 --> 00:17:35,660
that I indicate here, where
now the periodic signal is
246
00:17:35,660 --> 00:17:41,090
expressed in terms of a
summation of cosines with
247
00:17:41,090 --> 00:17:45,100
appropriate amplitude
and phase.
248
00:17:45,100 --> 00:17:50,020
Or another equivalent
trigonometric form involves
249
00:17:50,020 --> 00:17:54,450
rearranging this in terms
of a combination
250
00:17:54,450 --> 00:17:57,910
of cosines and sines.
251
00:17:57,910 --> 00:18:02,160
Now in this representation,
the frequencies of the
252
00:18:02,160 --> 00:18:08,790
sinusoids vary only over
positive frequencies.
253
00:18:08,790 --> 00:18:12,860
And typically one thinks of
periodic signals as having
254
00:18:12,860 --> 00:18:16,590
positive frequencies associated
with them.
255
00:18:16,590 --> 00:18:23,910
However, let's look back and the
complex exponential form
256
00:18:23,910 --> 00:18:26,600
for the Fourier series at
the top of the board.
257
00:18:26,600 --> 00:18:30,110
And in that representation,
when we use this
258
00:18:30,110 --> 00:18:35,180
representation, we'll find it
convenient to refer to both
259
00:18:35,180 --> 00:18:38,700
positive frequencies and
negative frequencies.
260
00:18:38,700 --> 00:18:42,800
So the representation that we
will most typically be using
261
00:18:42,800 --> 00:18:45,310
is the complex exponential
form.
262
00:18:45,310 --> 00:18:49,480
And in that form, what we'll
find as we think of
263
00:18:49,480 --> 00:18:55,160
decomposing a periodic signal
into its components at
264
00:18:55,160 --> 00:18:57,880
different frequencies, it will
involve both positive
265
00:18:57,880 --> 00:18:59,610
frequencies and negative
frequencies.
266
00:19:03,710 --> 00:19:09,850
Okay, now we have the Fourier
series representation, as I've
267
00:19:09,850 --> 00:19:11,320
indicated here.
268
00:19:11,320 --> 00:19:16,050
Again, so far I've sidestepped
the issue as to whether this
269
00:19:16,050 --> 00:19:18,040
in fact represents all
the signals that
270
00:19:18,040 --> 00:19:20,340
we'd like to represent.
271
00:19:20,340 --> 00:19:24,730
Let's first address the issue
of how we determine these
272
00:19:24,730 --> 00:19:28,350
coefficients a_k, assuming
that, in fact, this
273
00:19:28,350 --> 00:19:30,450
representation is valid.
274
00:19:30,450 --> 00:19:32,930
And again, I'll kind
of move through the
275
00:19:32,930 --> 00:19:36,490
algebra fairly quickly.
276
00:19:36,490 --> 00:19:40,460
The algebraic steps are ones
that you can pursue more
277
00:19:40,460 --> 00:19:42,940
leisurely just to kind
of verify them and
278
00:19:42,940 --> 00:19:44,660
step through them.
279
00:19:44,660 --> 00:19:48,460
But essentially, the algebra
develops out of the
280
00:19:48,460 --> 00:19:53,580
recognition that if we
integrate a complex
281
00:19:53,580 --> 00:19:58,140
exponential over one
period, T_0--
282
00:19:58,140 --> 00:20:02,610
and I mean by this notation that
this is an integral over
283
00:20:02,610 --> 00:20:05,550
a period, where I don't
particularly care where the
284
00:20:05,550 --> 00:20:09,340
period starts and where the
period stops, in other words,
285
00:20:09,340 --> 00:20:12,040
exactly what period I picked--
286
00:20:12,040 --> 00:20:17,140
that this integral is equal to
T_0 when m is equal to 0.
287
00:20:17,140 --> 00:20:21,490
And it's equal to 0 if
m is not equal to 0.
288
00:20:21,490 --> 00:20:26,250
That follows simply from the
fact that if we substitute in
289
00:20:26,250 --> 00:20:29,640
for using or Euler's relation,
so that we have the integral
290
00:20:29,640 --> 00:20:36,540
of a cosine plus j times the
sine, if m is not equal to 0,
291
00:20:36,540 --> 00:20:41,470
then both of these integrals
over a period are 0.
292
00:20:41,470 --> 00:20:45,320
The integral of a of a periodic
of a sinusoid, cosine
293
00:20:45,320 --> 00:20:49,440
or sine, over an integral
number of periods is 0.
294
00:20:49,440 --> 00:20:55,660
Whereas, if m is equal to 0,
this integral will be equal to
295
00:20:55,660 --> 00:20:58,350
T_0, the integral
of the cosine.
296
00:20:58,350 --> 00:21:01,840
And the integral of the
sine is equal to 0.
297
00:21:04,400 --> 00:21:07,480
Okay, well, the next step in
developing the expression for
298
00:21:07,480 --> 00:21:13,180
the coefficient a_k is to refer
back to the Fourier
299
00:21:13,180 --> 00:21:18,170
series expression, which was
that x(t) is equal to the sum
300
00:21:18,170 --> 00:21:19,420
of a_k e^(j k omega_0 t).
301
00:21:22,310 --> 00:21:28,370
If we multiply both sides of
that by e^(-j n omega_0 t),
302
00:21:28,370 --> 00:21:34,135
and integrate that
over a period--
303
00:21:36,810 --> 00:21:41,340
both sides of the equation
integrated over a period, so
304
00:21:41,340 --> 00:21:44,160
these two equations
are equal--
305
00:21:44,160 --> 00:21:47,680
and then in essence, interchange
the summation and
306
00:21:47,680 --> 00:21:52,390
the integration so that this
part of the expression comes
307
00:21:52,390 --> 00:21:57,030
outside the sum, and then we
combine these two complex
308
00:21:57,030 --> 00:22:02,540
exponentials together, where we
come out is the expression
309
00:22:02,540 --> 00:22:04,840
that I've indicated here.
310
00:22:04,840 --> 00:22:07,930
And then essentially what
happens at this point,
311
00:22:07,930 --> 00:22:12,060
algebraically, is that we use
the result that we just
312
00:22:12,060 --> 00:22:16,140
developed to evaluate
this integral.
313
00:22:16,140 --> 00:22:19,800
So multiplying both sides of
the Fourier series and then
314
00:22:19,800 --> 00:22:23,850
doing the integration leads
us, after the appropriate
315
00:22:23,850 --> 00:22:27,670
manipulation, to the expression
316
00:22:27,670 --> 00:22:31,140
that I have up here.
317
00:22:31,140 --> 00:22:42,550
And this integral is equal to
T_0 if k is equal to n,
318
00:22:42,550 --> 00:22:44,920
corresponding to 0 up here.
319
00:22:44,920 --> 00:22:49,020
And it's 0 otherwise, which is
what we had demonstrated or
320
00:22:49,020 --> 00:22:50,840
argued previously.
321
00:22:50,840 --> 00:22:53,970
And the upshot of all that,
then, is that the right hand
322
00:22:53,970 --> 00:22:58,620
side of this expression
disappears except for the term
323
00:22:58,620 --> 00:23:01,020
when k is equal to n.
324
00:23:01,020 --> 00:23:07,400
And so finally, we have what I
indicate here, taking T_0 and
325
00:23:07,400 --> 00:23:12,050
moving it over to the other
side of the equation, that
326
00:23:12,050 --> 00:23:16,580
then tells us how we determine
the Fourier series
327
00:23:16,580 --> 00:23:20,540
coefficients a_n, or a_k.
328
00:23:20,540 --> 00:23:25,810
So that, in effect, then is
what we refer to as the
329
00:23:25,810 --> 00:23:30,040
analysis equation, the equation
that begins with x(t)
330
00:23:30,040 --> 00:23:34,260
and tells us how to get the
Fourier series coefficients.
331
00:23:34,260 --> 00:23:38,620
What I'll refer to as the
Fourier series synthesis
332
00:23:38,620 --> 00:23:45,330
equation is the equation that
tells us how to build x(t) out
333
00:23:45,330 --> 00:23:48,340
of these complex exponentials.
334
00:23:48,340 --> 00:23:51,680
So we have the synthesis
equation, which is the one we
335
00:23:51,680 --> 00:23:53,070
started from.
336
00:23:53,070 --> 00:23:58,250
We have the analysis equation,
which is the equation that we
337
00:23:58,250 --> 00:23:59,500
just developed.
338
00:24:02,210 --> 00:24:07,950
So we in effect have gone
through the issue of, assuming
339
00:24:07,950 --> 00:24:10,580
that a Fourier series
representation is in fact
340
00:24:10,580 --> 00:24:14,980
valid, how we get the
coefficients.
341
00:24:14,980 --> 00:24:18,440
We'll want to address somewhat
the question of how broad a
342
00:24:18,440 --> 00:24:21,680
class of signals are
we talking about.
343
00:24:21,680 --> 00:24:26,090
And what's in fact amazing,
and was Fourier's amazing
344
00:24:26,090 --> 00:24:29,250
insight, was that it's a very
broad class of signals.
345
00:24:29,250 --> 00:24:34,690
But let's first look at just
some examples in which we take
346
00:24:34,690 --> 00:24:38,170
a signal, assume that it
has the Fourier series
347
00:24:38,170 --> 00:24:41,750
representation, and see what
the Fourier series
348
00:24:41,750 --> 00:24:44,080
coefficients look like.
349
00:24:44,080 --> 00:24:50,710
So we'll begin with what I refer
to as an antisymmetric
350
00:24:50,710 --> 00:24:53,110
periodic square wave--
351
00:24:53,110 --> 00:24:56,010
periodic of course, because
we're talking about periodic
352
00:24:56,010 --> 00:24:59,950
signals; square wave referring
to its shape; and
353
00:24:59,950 --> 00:25:03,660
antisymmetric referring
to the fact that it
354
00:25:03,660 --> 00:25:05,740
is an odd time function.
355
00:25:05,740 --> 00:25:07,440
In other words, it is
356
00:25:07,440 --> 00:25:12,030
antisymmetric about the origin.
357
00:25:12,030 --> 00:25:16,650
Now the expression for the
Fourier series coefficients
358
00:25:16,650 --> 00:25:23,450
tells us that we determine a_k
by taking 1 / T0 times the
359
00:25:23,450 --> 00:25:26,140
integral over a period of x(t),
e^(-j k omega_0 t) dt.
360
00:25:30,490 --> 00:25:35,560
The most convenient thing in
this case is to choose a
361
00:25:35,560 --> 00:25:41,550
period, which let's say goes
from -T_0 / 2 to +T_0 / 2.
362
00:25:41,550 --> 00:25:44,370
So here x(t) is -1.
363
00:25:44,370 --> 00:25:47,110
Here x(t) is +1.
364
00:25:47,110 --> 00:25:52,060
And so I've expressed the
Fourier series coefficients as
365
00:25:52,060 --> 00:25:55,950
this integral, that's
from -T_0 / 2 to 0.
366
00:25:55,950 --> 00:26:02,010
And then added to that is the
positive part of the cycle.
367
00:26:02,010 --> 00:26:05,550
And so we have these
two integrals.
368
00:26:05,550 --> 00:26:10,150
Now, I don't want to track
through the details of the
369
00:26:10,150 --> 00:26:11,490
algebra again.
370
00:26:11,490 --> 00:26:13,980
I guess I've decided that that's
much more fun for you
371
00:26:13,980 --> 00:26:15,620
to do on your own.
372
00:26:15,620 --> 00:26:19,270
But the way it comes out when
you go through it is the
373
00:26:19,270 --> 00:26:24,100
expression that I finally
indicate after suggesting that
374
00:26:24,100 --> 00:26:26,860
there are few more
steps to follow.
375
00:26:26,860 --> 00:26:31,970
And what develops is that those
two integrals together,
376
00:26:31,970 --> 00:26:38,040
for k not equal to 0, come
out to this expression.
377
00:26:38,040 --> 00:26:42,830
And that expression is
not valid for k = 0.
378
00:26:42,830 --> 00:26:48,160
For k equal to 0, we can go back
to the basic expression
379
00:26:48,160 --> 00:26:53,340
for the Fourier series, which is
1 / T_0, the integral over
380
00:26:53,340 --> 00:26:58,770
a period, x(t) e^(-j
k omega_0 t) dt.
381
00:26:58,770 --> 00:27:04,170
For k = 0, of course this term
just simply becomes 1.
382
00:27:04,170 --> 00:27:10,520
And so the zeroth coefficient is
1 / T_0 times the integral
383
00:27:10,520 --> 00:27:13,730
of x(t) over a period.
384
00:27:13,730 --> 00:27:18,460
Now, going back to the original
function that we
385
00:27:18,460 --> 00:27:23,460
have, what we're saying then is
that the zeroth coefficient
386
00:27:23,460 --> 00:27:29,780
is 1 / T_0 times the integral
over one period, which is, in
387
00:27:29,780 --> 00:27:31,950
effect, the average value.
388
00:27:31,950 --> 00:27:35,060
And it's straightforward to
verify for this case that
389
00:27:35,060 --> 00:27:37,210
average value is equal to 0.
390
00:27:40,050 --> 00:27:45,540
Now let's look at these Fourier
series coefficients on
391
00:27:45,540 --> 00:27:47,840
a bar graph.
392
00:27:47,840 --> 00:27:50,610
And I've indicated that here.
393
00:27:50,610 --> 00:27:53,090
The expression for the
Fourier series
394
00:27:53,090 --> 00:27:54,980
coefficients we just developed.
395
00:27:54,980 --> 00:27:58,110
And it involves--
396
00:27:58,110 --> 00:28:02,270
it's 0 for k = 0, it's
a factor of this
397
00:28:02,270 --> 00:28:04,990
form for k =/= 0.
398
00:28:04,990 --> 00:28:10,330
Plotted on a bar graph, then we
see values like this, 0 at
399
00:28:10,330 --> 00:28:13,920
k = 0 and then associated
values.
400
00:28:13,920 --> 00:28:16,820
And there are a number of things
to focus on when you
401
00:28:16,820 --> 00:28:19,890
look at this.
402
00:28:19,890 --> 00:28:24,210
One is the fact that the Fourier
series coefficients
403
00:28:24,210 --> 00:28:29,870
for this example are
purely imaginary.
404
00:28:29,870 --> 00:28:33,160
A second is that the Fourier
series coefficients for this
405
00:28:33,160 --> 00:28:36,200
example are an odd sequence.
406
00:28:36,200 --> 00:28:39,160
In other words, if you look at
this sequence, what you see
407
00:28:39,160 --> 00:28:44,460
are these values for
-k flipped over.
408
00:28:44,460 --> 00:28:48,660
So they're imaginary and odd.
409
00:28:48,660 --> 00:28:57,430
And what that results in, when
you look at the trigonometric
410
00:28:57,430 --> 00:29:02,300
form of the Fourier series,
is that in fact, those
411
00:29:02,300 --> 00:29:06,060
conditions, if you put the terms
all together, lead you
412
00:29:06,060 --> 00:29:10,590
to a trigonometric
representation, which involves
413
00:29:10,590 --> 00:29:13,020
only sine terms--
414
00:29:13,020 --> 00:29:15,280
in other words, no
cosine terms.
415
00:29:15,280 --> 00:29:18,470
Let me just draw your attention
to the fact that,
416
00:29:18,470 --> 00:29:23,760
since a_k's are imaginary, this
j takes care of that fact
417
00:29:23,760 --> 00:29:27,530
so that these coefficients
are in fact real.
418
00:29:27,530 --> 00:29:33,350
So what this says is that for
the antisymmetric square wave,
419
00:29:33,350 --> 00:29:36,845
in effect, the Fourier series
is a sine series.
420
00:29:36,845 --> 00:29:40,910
The antisymmetric square wave
is an odd function.
421
00:29:40,910 --> 00:29:43,450
Sinusoids are odd functions.
422
00:29:43,450 --> 00:29:45,590
And so this is all kind of
reasonable, that we're
423
00:29:45,590 --> 00:29:50,760
building an odd function
out of odd functions.
424
00:29:50,760 --> 00:29:56,320
As an additional aside, which
I won't exploit or refer to
425
00:29:56,320 --> 00:29:59,740
any further here, but just draw
your attention to, is
426
00:29:59,740 --> 00:30:04,530
that another aspect of this
periodic square wave, the
427
00:30:04,530 --> 00:30:08,890
particular one that we chose, is
that it is what's referred
428
00:30:08,890 --> 00:30:11,720
to as an odd harmonic
function.
429
00:30:11,720 --> 00:30:17,480
In other words, for even values
of k, the Fourier
430
00:30:17,480 --> 00:30:19,660
series coefficients are 0.
431
00:30:19,660 --> 00:30:22,734
They're are only non-zero
for odd values of k.
432
00:30:25,340 --> 00:30:27,970
Now let's look at
another example.
433
00:30:27,970 --> 00:30:31,430
Another example is
the symmetric
434
00:30:31,430 --> 00:30:33,830
periodic square wave.
435
00:30:33,830 --> 00:30:40,240
And this is in fact example 4.5,
worked out in more detail
436
00:30:40,240 --> 00:30:40,910
in the text.
437
00:30:40,910 --> 00:30:46,150
Then I won't bother to work
this out in detail here,
438
00:30:46,150 --> 00:30:50,630
except to draw your attention
to several points.
439
00:30:50,630 --> 00:30:54,330
Here is the symmetric periodic
square wave.
440
00:30:54,330 --> 00:30:58,470
And what I mean by symmetric
is that it's
441
00:30:58,470 --> 00:31:00,750
an even time function.
442
00:31:00,750 --> 00:31:06,430
Now just kind of extrapolating
your intuition, what you
443
00:31:06,430 --> 00:31:10,120
should expect is that if it's
only an even time function, it
444
00:31:10,120 --> 00:31:13,820
should be built up or buildable,
if it's buildable
445
00:31:13,820 --> 00:31:18,110
at all, out of only
even sinusoids.
446
00:31:18,110 --> 00:31:20,630
And in fact, that's the case.
447
00:31:20,630 --> 00:31:25,720
So if we look at the Fourier
series coefficients for this,
448
00:31:25,720 --> 00:31:29,700
is zeroth coefficient, again, is
the average value, which in
449
00:31:29,700 --> 00:31:31,350
this case, is 1/2.
450
00:31:31,350 --> 00:31:35,550
Here I've plotted pi times the
Fourier series coefficients.
451
00:31:35,550 --> 00:31:39,590
So the zeroth value is pi / 2.
452
00:31:39,590 --> 00:31:45,280
The coefficients are now an even
sequence, in other words,
453
00:31:45,280 --> 00:31:48,000
symmetric about k = 0.
454
00:31:48,000 --> 00:31:51,760
And the consequence of that is
that when you take these
455
00:31:51,760 --> 00:31:57,080
coefficients and put together
the equivalent trigonometric
456
00:31:57,080 --> 00:32:05,290
form, the trigonometric form
involves only cosines and no
457
00:32:05,290 --> 00:32:08,040
sine terms.
458
00:32:08,040 --> 00:32:12,530
Now you'll see this in other
examples, not that we'll do in
459
00:32:12,530 --> 00:32:15,090
the lecture, but examples in
the text and in the video
460
00:32:15,090 --> 00:32:19,150
manual, if in fact the square
wave was neither symmetric or
461
00:32:19,150 --> 00:32:22,590
antisymmetric, then the
trigonometric form would
462
00:32:22,590 --> 00:32:25,480
involve both sines
and cosines.
463
00:32:25,480 --> 00:32:30,660
And that is, of course,
the more general case.
464
00:32:30,660 --> 00:32:35,250
Furthermore, in the two examples
I've shown here, in
465
00:32:35,250 --> 00:32:40,310
both cases, the signal
is odd harmonic.
466
00:32:40,310 --> 00:32:44,720
In other words, for even values
of k, the coefficients
467
00:32:44,720 --> 00:32:46,710
are equal to 0.
468
00:32:46,710 --> 00:32:49,260
Although I won't justify that
here, that's a consequence of
469
00:32:49,260 --> 00:32:54,570
the fact that this symmetry is
exactly about half a period.
470
00:32:54,570 --> 00:32:58,570
And if you made the on time of
the square wave different in
471
00:32:58,570 --> 00:33:00,980
relation to the off
time, then that
472
00:33:00,980 --> 00:33:02,575
property would also disappear.
473
00:33:06,400 --> 00:33:13,240
Now what's kind of amazing,
actually, is that if we take a
474
00:33:13,240 --> 00:33:17,500
square wave, like I have
here or as I had in the
475
00:33:17,500 --> 00:33:21,940
antisymmetric case, the
implication is that I can
476
00:33:21,940 --> 00:33:26,000
build that square wave
by adding up
477
00:33:26,000 --> 00:33:30,110
enough sines or cosines.
478
00:33:30,110 --> 00:33:33,690
And it really seems kind of
amazing because the square
479
00:33:33,690 --> 00:33:37,700
wave, after all, is a very
discontinuous function.
480
00:33:37,700 --> 00:33:40,080
Sinusoids are very continuous.
481
00:33:40,080 --> 00:33:43,760
And it seems puzzling that
in fact you can do that.
482
00:33:43,760 --> 00:33:48,690
Well let's look in a little
bit of detail how the
483
00:33:48,690 --> 00:33:55,280
sinusoidal terms add up to
build a square wave.
484
00:33:55,280 --> 00:34:01,750
And to do that, let's first
define what I refer to as a
485
00:34:01,750 --> 00:34:03,480
partial sum.
486
00:34:03,480 --> 00:34:10,670
So here we have the expression
which is the synthesis
487
00:34:10,670 --> 00:34:15,510
equation, telling us how x(t)
could be represented as
488
00:34:15,510 --> 00:34:18,179
complex exponentials
if it can be.
489
00:34:18,179 --> 00:34:21,280
And let's consider just
a finite number of
490
00:34:21,280 --> 00:34:22,770
terms in this sum.
491
00:34:22,770 --> 00:34:27,630
And so x_n(t), of course, as n
goes to infinity, approaches
492
00:34:27,630 --> 00:34:30,469
the infinite sum that
we're talking about.
493
00:34:30,469 --> 00:34:32,889
And although we could do this
more generally, let's not.
494
00:34:32,889 --> 00:34:36,760
Let's focus on the symmetric
square wave case, where
495
00:34:36,760 --> 00:34:39,880
because of the symmetry of these
coefficients, namely
496
00:34:39,880 --> 00:34:46,810
that a_k is equal to a_(-k), we
can rewrite these terms as
497
00:34:46,810 --> 00:34:48,389
cosine terms.
498
00:34:48,389 --> 00:34:52,880
And so this partial sum can
be expressed the way I'm
499
00:34:52,880 --> 00:34:55,900
expressing it here.
500
00:34:55,900 --> 00:34:58,990
Well let's look at a
few of these terms.
501
00:34:58,990 --> 00:35:05,930
On the graph, I have, first
of all, x(t), which is our
502
00:35:05,930 --> 00:35:09,330
original square wave.
503
00:35:09,330 --> 00:35:14,400
The term that I indicate here
is the factor of 1/2,
504
00:35:14,400 --> 00:35:19,530
which is this term.
505
00:35:19,530 --> 00:35:22,080
With n = 1, that would
correspond to adding one
506
00:35:22,080 --> 00:35:23,940
cosine term to that.
507
00:35:23,940 --> 00:35:28,420
And so the sum of those two
would be this, which looks a
508
00:35:28,420 --> 00:35:32,680
little closer to the square
wave, but certainly not very
509
00:35:32,680 --> 00:35:35,050
close to it at all.
510
00:35:35,050 --> 00:35:39,570
And in fact, it's somewhat hard
to imagine without seeing
511
00:35:39,570 --> 00:35:43,380
the terms build up how in fact,
by adding more and more
512
00:35:43,380 --> 00:35:47,470
terms, we can generate something
that is essentially
513
00:35:47,470 --> 00:35:50,230
flat, except at the
discontinuities.
514
00:35:50,230 --> 00:35:52,340
So let's look at this example.
515
00:35:52,340 --> 00:35:57,420
And what I'd like to show is
this example, but now as we
516
00:35:57,420 --> 00:35:59,580
add many more terms to it.
517
00:35:59,580 --> 00:36:04,640
And let's see in fact how these
individual terms add up
518
00:36:04,640 --> 00:36:07,150
to build up the square wave.
519
00:36:07,150 --> 00:36:11,190
So this is the square wave
that we want to build up
520
00:36:11,190 --> 00:36:15,280
through the Fourier series
as a sum of sinusoids.
521
00:36:15,280 --> 00:36:19,590
And the term for k = 0 will be
a constant which represents
522
00:36:19,590 --> 00:36:21,550
the DC value of this.
523
00:36:21,550 --> 00:36:27,380
And so in the partial sum, as we
develop it, the first thing
524
00:36:27,380 --> 00:36:31,450
that we'll show is just
the term for k = 0.
525
00:36:31,450 --> 00:36:38,020
Now for k = 1, we would add to
that one sinusoidal term.
526
00:36:38,020 --> 00:36:43,410
And so the sum of the term
for k = 1 and k = 0
527
00:36:43,410 --> 00:36:44,800
is represented here.
528
00:36:48,350 --> 00:36:52,250
Now when we go to k = 2, because
of the fact that this
529
00:36:52,250 --> 00:36:58,450
is an odd harmonic function,
in fact, the term for k = 2
530
00:36:58,450 --> 00:37:03,230
will have zero amplitude and
so this won't change.
531
00:37:03,230 --> 00:37:08,480
Here we show the Fourier
series with k = 2
532
00:37:08,480 --> 00:37:10,390
and there's no change.
533
00:37:10,390 --> 00:37:14,110
And then we will go to k = 3.
534
00:37:14,110 --> 00:37:16,650
And we will be adding,
then, one
535
00:37:16,650 --> 00:37:18,570
additional sinusoidal term.
536
00:37:18,570 --> 00:37:20,620
Here is k = 3.
537
00:37:20,620 --> 00:37:25,120
When we go to k = 4, again,
there won't be any change.
538
00:37:25,120 --> 00:37:31,420
But there will be another term
that's added at k = 5 here.
539
00:37:31,420 --> 00:37:35,560
Then k = 6, again, because it's
odd harmonic, no change.
540
00:37:35,560 --> 00:37:41,350
And finally k = 7
is shown here.
541
00:37:41,350 --> 00:37:45,040
And we can begin to see that
this starts to look somewhat
542
00:37:45,040 --> 00:37:47,160
like the square wave.
543
00:37:47,160 --> 00:37:52,500
But now to really emphasize how
this builds up, let's more
544
00:37:52,500 --> 00:37:56,680
rapidly add many more terms,
and in fact increase the
545
00:37:56,680 --> 00:38:00,790
number of terms up to about
100, recognizing that the
546
00:38:00,790 --> 00:38:05,560
shape will only change on the
inclusion of the odd-numbered
547
00:38:05,560 --> 00:38:08,370
terms, not the even-numbered
terms, because it's an odd
548
00:38:08,370 --> 00:38:10,970
harmonic function.
549
00:38:10,970 --> 00:38:16,620
So now we're increasing and
we're building up toward k =
550
00:38:16,620 --> 00:38:18,530
100, 100 terms.
551
00:38:18,530 --> 00:38:24,970
And notice that it is the
higher-order terms that tend
552
00:38:24,970 --> 00:38:30,340
to build up the discontinuity
corresponding to the notion
553
00:38:30,340 --> 00:38:34,940
that the discontinuity, or sharp
edges in a signal, in
554
00:38:34,940 --> 00:38:39,340
fact, are represented through
the higher frequencies in the
555
00:38:39,340 --> 00:38:41,550
Fourier series.
556
00:38:41,550 --> 00:38:44,900
And here we have a
not-too-unreasonable
557
00:38:44,900 --> 00:38:48,150
approximation to the original
square wave.
558
00:38:48,150 --> 00:38:51,890
There is the artifact of the
ripples at the discontinuity.
559
00:38:51,890 --> 00:38:56,710
And in fact, that rippling
behavior at the discontinuity
560
00:38:56,710 --> 00:39:00,180
is referred to the
Gibbs phenomenon.
561
00:39:00,180 --> 00:39:03,650
And it's an inherent part
of the Fourier series
562
00:39:03,650 --> 00:39:06,380
representation at
discontinuities.
563
00:39:06,380 --> 00:39:10,700
Now to emphasize this, let's
decrease the number
564
00:39:10,700 --> 00:39:13,450
of terms back down.
565
00:39:13,450 --> 00:39:21,600
And we will carry this down to
k = 1, again to emphasize how
566
00:39:21,600 --> 00:39:26,370
the sinusoids are building
up the square wave.
567
00:39:26,370 --> 00:39:29,100
Here we are back at k = 1.
568
00:39:29,100 --> 00:39:33,490
And then finally, we will add
back in the sinusoids
569
00:39:33,490 --> 00:39:34,640
that we took out.
570
00:39:34,640 --> 00:39:39,550
And let's build this back up
to 100 terms, showing the
571
00:39:39,550 --> 00:39:43,380
approximation that we generated
with 100 terms to
572
00:39:43,380 --> 00:39:44,630
the square wave.
573
00:40:00,900 --> 00:40:05,650
Okay, so what you saw is that,
in fact, we got awfully close
574
00:40:05,650 --> 00:40:06,990
to a square wave.
575
00:40:06,990 --> 00:40:09,810
And the other thing that was
kind of interesting about it
576
00:40:09,810 --> 00:40:14,610
as it went along was the
fact that, with the low
577
00:40:14,610 --> 00:40:17,650
frequencies, what we were
tending to build was the
578
00:40:17,650 --> 00:40:19,460
general behavior.
579
00:40:19,460 --> 00:40:26,490
And as the higher frequencies
came in, that tended
580
00:40:26,490 --> 00:40:28,590
contribute to the
discontinuity.
581
00:40:28,590 --> 00:40:34,030
And in fact, something that will
stand out more and more
582
00:40:34,030 --> 00:40:37,150
as we go through our discussion
of Fourier series
583
00:40:37,150 --> 00:40:41,430
and Fourier transforms, is that
general statement, that
584
00:40:41,430 --> 00:40:48,750
it's the low-frequency terms
that represent the broad time
585
00:40:48,750 --> 00:40:52,860
behavior, and it's the
high-frequency terms that are
586
00:40:52,860 --> 00:40:54,940
used to build up the sharp
587
00:40:54,940 --> 00:40:56,660
transitions in the time domain.
588
00:41:00,060 --> 00:41:03,670
Now we need to get a little
more precise about the
589
00:41:03,670 --> 00:41:10,210
question of how, in fact, the
Fourier series, or when the
590
00:41:10,210 --> 00:41:13,730
Fourier series represents the
functions that we're talking
591
00:41:13,730 --> 00:41:17,140
about and in what sense
they represent them.
592
00:41:17,140 --> 00:41:26,840
And so if we look again at the
synthesis equation, what we
593
00:41:26,840 --> 00:41:31,750
really want to ask is, if we add
up enough of these terms,
594
00:41:31,750 --> 00:41:37,700
in what sense does this sum
represent this time function?
595
00:41:37,700 --> 00:41:42,660
Well, let's again use the notion
of our partial sum.
596
00:41:42,660 --> 00:41:47,000
So we have the partial
sum down here.
597
00:41:47,000 --> 00:41:51,120
And we can think of the
difference between this
598
00:41:51,120 --> 00:41:59,300
partial sum and the original
time function as the error.
599
00:41:59,300 --> 00:42:02,210
And I've defined
the error here.
600
00:42:02,210 --> 00:42:07,540
And what we would like to know
is does this error decrease as
601
00:42:07,540 --> 00:42:10,150
we add more and more terms?
602
00:42:10,150 --> 00:42:13,660
And in fact, in what sense, if
the error does decrease, in
603
00:42:13,660 --> 00:42:16,640
what sense does it decrease?
604
00:42:16,640 --> 00:42:20,420
Now in detail this is
a fairly complicated
605
00:42:20,420 --> 00:42:21,810
and elaborate topic.
606
00:42:21,810 --> 00:42:23,930
I don't mean to make that
sound frightening.
607
00:42:23,930 --> 00:42:27,130
It's mainly a statement
that I don't want to
608
00:42:27,130 --> 00:42:29,770
explore in a lot of detail.
609
00:42:29,770 --> 00:42:34,030
But it relates to what it's
referred to as the issue of
610
00:42:34,030 --> 00:42:37,010
convergence of the
Fourier series.
611
00:42:37,010 --> 00:42:41,310
And the convergence of the
Fourier series, the bottom
612
00:42:41,310 --> 00:42:45,220
line on it, the kind of end
statement, can be made in
613
00:42:45,220 --> 00:42:47,710
several ways.
614
00:42:47,710 --> 00:42:50,090
One statement related to the
convergence of the Fourier
615
00:42:50,090 --> 00:42:52,590
series is the following.
616
00:42:52,590 --> 00:42:58,590
If I have a time function, which
is what is referred to
617
00:42:58,590 --> 00:43:01,400
as square integrable, namely
its integral, over
618
00:43:01,400 --> 00:43:04,350
a period, is finite.
619
00:43:04,350 --> 00:43:10,060
Then what you can show, kind
of amazingly, is that the
620
00:43:10,060 --> 00:43:14,440
energy in that error, in other
words, the energy and the
621
00:43:14,440 --> 00:43:17,160
difference between the original
function and the
622
00:43:17,160 --> 00:43:21,370
partial sum, the energy
in that, goes to 0
623
00:43:21,370 --> 00:43:22,770
as n goes to infinity.
624
00:43:25,430 --> 00:43:30,120
A somewhat tighter condition is
a condition referred to as
625
00:43:30,120 --> 00:43:37,590
it Dirichlet conditions, which
says that if the time function
626
00:43:37,590 --> 00:43:42,170
is absolutely integrable, not
square integrable, but
627
00:43:42,170 --> 00:43:44,170
absolutely integrable--
628
00:43:44,170 --> 00:43:47,510
and I've kind of hedged the
issue by just simply referring
629
00:43:47,510 --> 00:43:50,800
to x(t) as being
well behaved--
630
00:43:50,800 --> 00:43:56,170
then the statement is that the
error in fact goes to 0 as n
631
00:43:56,170 --> 00:44:00,090
increases, except at the
discontinuities.
632
00:44:00,090 --> 00:44:04,690
And what well behaved means in
that statement is that, as
633
00:44:04,690 --> 00:44:07,580
discussed in the book, there are
a finite number of maxima
634
00:44:07,580 --> 00:44:11,310
and minima in any period and
a finite number of finite
635
00:44:11,310 --> 00:44:15,500
discontinuities, which is,
essentially, always the case.
636
00:44:15,500 --> 00:44:19,890
So under square integrability
what we have is the statement
637
00:44:19,890 --> 00:44:23,240
not that the partial sum goes
to the right value at every
638
00:44:23,240 --> 00:44:28,250
point, but that the energy
in the error goes to 0.
639
00:44:28,250 --> 00:44:30,790
Under the Dirichlet conditions,
it says that, in
640
00:44:30,790 --> 00:44:35,800
fact, the signal goes to the
right value at every time
641
00:44:35,800 --> 00:44:40,560
instant except at the
discontinuities.
642
00:44:40,560 --> 00:44:47,770
So going back to the square
wave, the square wave
643
00:44:47,770 --> 00:44:49,880
satisfies either one of
those conditions.
644
00:44:49,880 --> 00:44:55,180
And so what the consequence is
is that, with the square wave,
645
00:44:55,180 --> 00:44:59,680
if we looked at the error, then
in fact what we would
646
00:44:59,680 --> 00:45:04,690
find is that the energy in the
error would go to zero as we
647
00:45:04,690 --> 00:45:07,810
add more and more terms
in the partial sum.
648
00:45:07,810 --> 00:45:11,890
And in fact, since the square
wave also satisfies the
649
00:45:11,890 --> 00:45:16,770
Dirichlet conditions, the actual
value of the error, the
650
00:45:16,770 --> 00:45:22,180
difference between the partial
sum and the true value, will
651
00:45:22,180 --> 00:45:23,610
actually go to 0.
652
00:45:23,610 --> 00:45:28,330
That difference will go to 0
except at the discontinuities.
653
00:45:28,330 --> 00:45:34,860
And that, in fact, is kind of
evident as we watch the
654
00:45:34,860 --> 00:45:38,170
function build up by adding
up these terms.
655
00:45:38,170 --> 00:45:44,880
And so in fact, let's go back
and see again the development
656
00:45:44,880 --> 00:45:47,690
of the partial sums
in relation to the
657
00:45:47,690 --> 00:45:49,700
original time function.
658
00:45:49,700 --> 00:45:53,000
Let's observe, this time again,
basically what we saw
659
00:45:53,000 --> 00:45:56,710
before, which is that it builds
up to the right answer.
660
00:45:56,710 --> 00:46:00,200
And furthermore what we'll
plot this time, also as a
661
00:46:00,200 --> 00:46:03,540
function of time, is the
energy in the error.
662
00:46:03,540 --> 00:46:07,300
And what we'll see is that the
energy in the error will be
663
00:46:07,300 --> 00:46:12,270
tending towards 0 as the number
of terms increases.
664
00:46:12,270 --> 00:46:15,310
So once again, we have
the square wave.
665
00:46:15,310 --> 00:46:21,010
And we want to again show the
buildup of the Fourier series,
666
00:46:21,010 --> 00:46:25,510
this time showing also how the
energy in the error decreases
667
00:46:25,510 --> 00:46:28,120
as we add more and more terms.
668
00:46:28,120 --> 00:46:32,930
Well, once again, we'll begin
with k = 0, corresponding to
669
00:46:32,930 --> 00:46:34,510
the constant term.
670
00:46:34,510 --> 00:46:38,060
And what's shown on the bottom
trace is the energy in the
671
00:46:38,060 --> 00:46:40,120
error between those two.
672
00:46:40,120 --> 00:46:46,870
And we'll then add the term k
= 1 to the DC term and we'll
673
00:46:46,870 --> 00:46:52,150
see that the energy will
decrease when we do that.
674
00:46:52,150 --> 00:46:56,160
Here we have then the sum
of k = 0 and k = 1.
675
00:46:56,160 --> 00:47:01,470
Now with k = 2, the energy won't
decrease any further
676
00:47:01,470 --> 00:47:03,685
because it's an odd
harmonic function.
677
00:47:07,310 --> 00:47:10,210
That's what we've
just added in.
678
00:47:10,210 --> 00:47:13,710
When we add in the term for k
= 3, again, we'll see the
679
00:47:13,710 --> 00:47:16,430
energy in the error decrease
as reflected
680
00:47:16,430 --> 00:47:17,680
in the bottom curve.
681
00:47:22,440 --> 00:47:25,620
So there we are at k = 3.
682
00:47:25,620 --> 00:47:30,100
When we go to k = 4,
there again is no
683
00:47:30,100 --> 00:47:32,850
change in the error.
684
00:47:32,850 --> 00:47:35,203
At k = 5, again the
error decreases.
685
00:47:39,890 --> 00:47:43,940
k = 6, there will be
no change again.
686
00:47:43,940 --> 00:47:47,500
And at k = 7, the energy
decreases.
687
00:47:47,500 --> 00:47:51,020
And now let's show how the error
decreases by building up
688
00:47:51,020 --> 00:47:53,830
the number of terms
much more rapidly.
689
00:47:53,830 --> 00:47:56,610
Already the error has gotten
somewhat small on the scale in
690
00:47:56,610 --> 00:48:00,380
which we're showing it, so let's
expand out the error
691
00:48:00,380 --> 00:48:04,090
scale, the vertical axis
displaying the energy in the
692
00:48:04,090 --> 00:48:09,220
error, so that we could watch
how the energy decreases as we
693
00:48:09,220 --> 00:48:11,090
add more and more terms.
694
00:48:11,090 --> 00:48:15,070
So here we have the vertical
scale expanded.
695
00:48:15,070 --> 00:48:20,030
And now what we'll do is
increase the number of terms
696
00:48:20,030 --> 00:48:23,490
in the Fourier series and watch
the energy in the error
697
00:48:23,490 --> 00:48:28,750
decreasing, always decreasing,
of course, on the inclusion of
698
00:48:28,750 --> 00:48:33,380
the odd-numbered terms and not
on the inclusion of the
699
00:48:33,380 --> 00:48:36,380
even-numbered terms because of
the fact that it's an odd
700
00:48:36,380 --> 00:48:38,030
harmonic function.
701
00:48:38,030 --> 00:48:41,230
Now the energy in the error
asymptotically will approach
702
00:48:41,230 --> 00:48:46,580
0, although point by point, the
Fourier series will never
703
00:48:46,580 --> 00:48:48,820
be equal to the square wave.
704
00:48:48,820 --> 00:48:53,370
It will, at every instant
of time, except at the
705
00:48:53,370 --> 00:48:58,640
discontinuities, where there
will always be some ripple
706
00:48:58,640 --> 00:49:01,370
corresponding to what's referred
to as the Gibbs
707
00:49:01,370 --> 00:49:02,620
phenomenon.
708
00:49:06,190 --> 00:49:11,870
So what we've seen, then, is
a quick look at the Fourier
709
00:49:11,870 --> 00:49:17,560
series representation
of periodic signals.
710
00:49:17,560 --> 00:49:25,370
We more broadly want to have a
more general representation of
711
00:49:25,370 --> 00:49:28,080
signals in terms of complex
exponentials.
712
00:49:28,080 --> 00:49:32,250
And so our next step will
be to move toward a
713
00:49:32,250 --> 00:49:37,440
representation of nonperiodic
or aperiodic signals.
714
00:49:37,440 --> 00:49:40,950
Now the details of this, I leave
for the next lecture.
715
00:49:40,950 --> 00:49:44,760
The only thought that I want to
introduce at this point is
716
00:49:44,760 --> 00:49:49,420
the basic strategy which is
somewhat amazing and kind of
717
00:49:49,420 --> 00:49:52,810
interesting to reflect
on in the interim.
718
00:49:52,810 --> 00:50:00,060
The basic strategy with an
aperiodic signal is to think
719
00:50:00,060 --> 00:50:05,410
of representing this aperiodic
signal as a linear combination
720
00:50:05,410 --> 00:50:12,590
of complex exponentials by the
simple trick of periodically
721
00:50:12,590 --> 00:50:18,540
replicating this signal,
generating a periodic signal
722
00:50:18,540 --> 00:50:21,590
using a Fourier series
representation for that
723
00:50:21,590 --> 00:50:27,100
periodic signal, and then simply
letting the period go
724
00:50:27,100 --> 00:50:28,650
to infinity.
725
00:50:28,650 --> 00:50:31,620
As the period goes to infinity,
that periodic signal
726
00:50:31,620 --> 00:50:35,520
becomes the original aperiodic
one that we had before.
727
00:50:35,520 --> 00:50:39,470
And the Fourier series
representation then becomes
728
00:50:39,470 --> 00:50:43,340
what we'll refer to the
Fourier transform.
729
00:50:43,340 --> 00:50:49,060
So that's just a quick look at
the basic idea and approach
730
00:50:49,060 --> 00:50:50,140
that we'll take.
731
00:50:50,140 --> 00:50:52,620
In the next lecture, we'll
develop this a little more
732
00:50:52,620 --> 00:50:55,500
carefully and more fully,
moving from the Fourier
733
00:50:55,500 --> 00:50:59,670
series, which we've used for
periodic signals, to develop
734
00:50:59,670 --> 00:51:03,740
the Fourier transform, which
will then be representation
735
00:51:03,740 --> 00:51:05,310
for aperiodic signals.
736
00:51:05,310 --> 00:51:06,560
Thank you.