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[MUSIC PLAYING]
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PROFESSOR: Over the last several
lectures, we developed
10
00:00:57,900 --> 00:01:03,320
the Fourier representation for
continuous-time signals.
11
00:01:03,320 --> 00:01:06,760
What I'd now like to do
is develop a similar
12
00:01:06,760 --> 00:01:09,230
representation for
discrete-time.
13
00:01:09,230 --> 00:01:12,940
And let me begin the discussion
by reminding you of
14
00:01:12,940 --> 00:01:15,280
what our basic motivation was.
15
00:01:15,280 --> 00:01:19,780
The idea is that what we wanted
to do was exploit the
16
00:01:19,780 --> 00:01:23,640
properties of linearity and
time invariance for linear
17
00:01:23,640 --> 00:01:26,330
time-invariant systems.
18
00:01:26,330 --> 00:01:31,860
So in the case of linear
time-invariant systems, the
19
00:01:31,860 --> 00:01:38,650
basic idea was to consider
decomposing the input as a
20
00:01:38,650 --> 00:01:42,730
linear combination
of basic inputs.
21
00:01:42,730 --> 00:01:46,620
And then, because of linearity,
the output could be
22
00:01:46,620 --> 00:01:49,960
expressed as a linear
combination of corresponding
23
00:01:49,960 --> 00:01:59,610
outputs where psi sub i is the
output due to phi sub i.
24
00:01:59,610 --> 00:02:03,440
So basically, what we attempted
to do was decompose
25
00:02:03,440 --> 00:02:07,710
the input, and then reconstruct
the output through
26
00:02:07,710 --> 00:02:13,620
a linear combination of the
outputs to those basic inputs.
27
00:02:13,620 --> 00:02:19,020
We then focused on the notion
of choosing the basic inputs
28
00:02:19,020 --> 00:02:20,890
with two criteria in mind.
29
00:02:20,890 --> 00:02:25,900
One was to choose them so that a
broad class of signals could
30
00:02:25,900 --> 00:02:30,020
be constructed out of
those basic inputs.
31
00:02:30,020 --> 00:02:35,000
And the second was to choose the
basic inputs, so that the
32
00:02:35,000 --> 00:02:38,800
response to those was
easy to compute.
33
00:02:38,800 --> 00:02:41,570
And as you recall, one
representation that we ended
34
00:02:41,570 --> 00:02:47,050
up with, with those basic
criteria in mind, was the
35
00:02:47,050 --> 00:02:49,500
representation through
convolution.
36
00:02:49,500 --> 00:02:52,610
And then in beginning the
discussion of the Fourier
37
00:02:52,610 --> 00:02:55,980
representation of
continuous-time signals, we
38
00:02:55,980 --> 00:03:01,880
chose as another set of basic
inputs complex exponentials.
39
00:03:01,880 --> 00:03:07,860
So for continuous-time, we chose
a set of basic inputs
40
00:03:07,860 --> 00:03:10,380
which were complex
exponentials.
41
00:03:10,380 --> 00:03:15,470
The motivation there was the
fact that the complex
42
00:03:15,470 --> 00:03:18,950
exponentials have what
we refer to as the
43
00:03:18,950 --> 00:03:20,940
eigenfunction property.
44
00:03:20,940 --> 00:03:24,190
Namely, if we put complex
exponentials into our
45
00:03:24,190 --> 00:03:29,540
continuous-time systems, then
the output is a complex
46
00:03:29,540 --> 00:03:34,830
exponential of the same
form with only a
47
00:03:34,830 --> 00:03:37,980
change in complex amplitude.
48
00:03:37,980 --> 00:03:42,450
And that change in complex
amplitude is what we referred
49
00:03:42,450 --> 00:03:48,930
to as the frequency response
of the system.
50
00:03:48,930 --> 00:03:52,040
And also, by the way, as it
developed later, that
51
00:03:52,040 --> 00:03:55,470
frequency response, as you
should now recognize from this
52
00:03:55,470 --> 00:03:59,380
expression, is in fact the
Fourier transform, the
53
00:03:59,380 --> 00:04:03,120
continuous-time Fourier
transform of the system
54
00:04:03,120 --> 00:04:06,170
impulse response.
55
00:04:06,170 --> 00:04:10,790
And the notion of decomposing
a signal as a linear
56
00:04:10,790 --> 00:04:15,960
combination of these complex
exponentials is what, first
57
00:04:15,960 --> 00:04:19,260
the Fourier series
representation, and then later
58
00:04:19,260 --> 00:04:21,130
the Fourier transform
representation
59
00:04:21,130 --> 00:04:22,680
corresponded to.
60
00:04:22,680 --> 00:04:28,390
And finally, to remind you of
one additional point, the fact
61
00:04:28,390 --> 00:04:30,370
is that because of
the eigenfunction
62
00:04:30,370 --> 00:04:34,640
property, the response--
63
00:04:34,640 --> 00:04:37,580
once we have decomposed the
input as a linear combination
64
00:04:37,580 --> 00:04:40,920
of complex exponentials, the
response to that linear
65
00:04:40,920 --> 00:04:44,820
combination is straightforward
to compute once we know the
66
00:04:44,820 --> 00:04:50,110
frequency response because of
the eigenfunction property.
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00:04:50,110 --> 00:04:54,790
Now, basically the same strategy
and many of the same
68
00:04:54,790 --> 00:05:00,120
ideas work in discrete-time,
paralleling almost exactly
69
00:05:00,120 --> 00:05:02,390
what happened in
continuous-time.
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00:05:02,390 --> 00:05:05,180
So the similarities between
discrete-time and
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00:05:05,180 --> 00:05:07,690
continuous-time are
very strong.
72
00:05:07,690 --> 00:05:11,710
Although as we'll see, there are
a number of differences.
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00:05:11,710 --> 00:05:15,460
And it's important as we go
through the discussion to
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00:05:15,460 --> 00:05:20,210
illuminate not only the
similarities, but obviously
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00:05:20,210 --> 00:05:22,160
also the differences.
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00:05:22,160 --> 00:05:26,800
Well, let's begin with the
eigenfunction property, and
77
00:05:26,800 --> 00:05:31,900
let me just state that just as
in continuous-time, if we
78
00:05:31,900 --> 00:05:36,580
consider a set of basic signals,
which are complex
79
00:05:36,580 --> 00:05:40,770
exponential rules, then
discrete-time linear time-m
80
00:05:40,770 --> 00:05:45,390
invariant systems have the
eigenfunction property.
81
00:05:45,390 --> 00:05:50,200
Namely, if we put a complex
exponential into the system,
82
00:05:50,200 --> 00:05:53,970
the response is a complex
exponential at the same
83
00:05:53,970 --> 00:06:00,270
complex frequency, and simply
multiplied by an appropriate
84
00:06:00,270 --> 00:06:03,620
complex factor, or constant.
85
00:06:03,620 --> 00:06:11,510
And just as we did in
continuous-time, we will be
86
00:06:11,510 --> 00:06:15,980
referring to this complex
constant, which is a function,
87
00:06:15,980 --> 00:06:19,910
of course, the frequency of the
complex exponential input.
88
00:06:19,910 --> 00:06:24,400
We'll be referring to this as
the frequency response.
89
00:06:24,400 --> 00:06:27,980
And although it's not
particularly evident at this
90
00:06:27,980 --> 00:06:32,020
point, as the discussion
develops through this lecture,
91
00:06:32,020 --> 00:06:36,620
what in fact will happen is
very much paralleling
92
00:06:36,620 --> 00:06:38,340
continuous-time.
93
00:06:38,340 --> 00:06:43,990
This particular expression, in
fact, will correspond to what
94
00:06:43,990 --> 00:06:48,550
we'll refer to as the Fourier
transform, the discrete-time
95
00:06:48,550 --> 00:06:52,030
Fourier transform of the system
impulse response.
96
00:06:52,030 --> 00:06:55,190
So there, of course, there's a
very strong parallel between
97
00:06:55,190 --> 00:06:58,330
continuous time and
discrete time.
98
00:06:58,330 --> 00:07:02,750
Now, just as we did in
continuous-time, let's begin
99
00:07:02,750 --> 00:07:07,720
the discussion by first
concentrating on periodic--
100
00:07:07,720 --> 00:07:11,010
the representation through
complex exponentials of
101
00:07:11,010 --> 00:07:16,030
periodic sequences, and then
we'll generalize that
102
00:07:16,030 --> 00:07:18,270
discussion to the
representation
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00:07:18,270 --> 00:07:20,570
of aperiodic signals.
104
00:07:20,570 --> 00:07:26,190
So let's consider first a
periodic signal, or in
105
00:07:26,190 --> 00:07:28,900
general, signals which
are periodic.
106
00:07:28,900 --> 00:07:33,530
Period denoted by capital N.
And then, of course, the
107
00:07:33,530 --> 00:07:39,610
fundamental frequency is 2
pi divided by capital N.
108
00:07:39,610 --> 00:07:45,200
Now, we can consider
exponentials which have this
109
00:07:45,200 --> 00:07:50,390
as a fundamental frequency, or
which are harmonics of that,
110
00:07:50,390 --> 00:07:53,880
and that would correspond
to the class of complex
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00:07:53,880 --> 00:07:59,290
exponentials of the form
e to the jk omega 0 n.
112
00:07:59,290 --> 00:08:06,610
So these complex exponentials
then, as k varies, are complex
113
00:08:06,610 --> 00:08:11,930
exponentials that are
harmonically related, all of
114
00:08:11,930 --> 00:08:16,730
which are periodic with the
same period capital N.
115
00:08:16,730 --> 00:08:19,680
Although the fundamental
period is
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00:08:19,680 --> 00:08:20,960
different for each of these.
117
00:08:20,960 --> 00:08:26,150
Each of them being related
by an integer amount.
118
00:08:26,150 --> 00:08:29,830
Now, again, just as we did in
continuous time, we can
119
00:08:29,830 --> 00:08:34,390
consider attempting to build our
periodic signal out of a
120
00:08:34,390 --> 00:08:36,780
linear combination of these.
121
00:08:36,780 --> 00:08:43,280
And so we consider a periodic
signal, which is a weighted
122
00:08:43,280 --> 00:08:48,060
sum of these complex
exponentials.
123
00:08:48,060 --> 00:08:50,980
And, of course, this
periodic signal--
124
00:08:50,980 --> 00:08:52,420
this is a periodic signal.
125
00:08:52,420 --> 00:08:56,460
This can be verified, more or
less, in a straightforward way
126
00:08:56,460 --> 00:08:58,150
by substitution.
127
00:08:58,150 --> 00:09:00,950
And, of course, one of the
things that we'll want to
128
00:09:00,950 --> 00:09:04,930
address shortly is how broad a
class of signals, again, can
129
00:09:04,930 --> 00:09:07,310
be represented by this sum?
130
00:09:07,310 --> 00:09:10,730
And another question obviously
will be, how do we determine
131
00:09:10,730 --> 00:09:13,860
the coefficients a sub k?
132
00:09:13,860 --> 00:09:20,010
However, before we do that, let
me focus on an important
133
00:09:20,010 --> 00:09:23,210
distinction between
continuous-time and
134
00:09:23,210 --> 00:09:26,950
discrete-time in the context of
these complex exponentials
135
00:09:26,950 --> 00:09:30,420
and this representation.
136
00:09:30,420 --> 00:09:34,140
When we talked about complex
exponentials and sinusoids
137
00:09:34,140 --> 00:09:38,100
early in the course, one of the
differences that we saw
138
00:09:38,100 --> 00:09:42,380
between continuous-time and
discrete-time is that in
139
00:09:42,380 --> 00:09:46,990
continuous-time, as we vary the
frequency variable, we see
140
00:09:46,990 --> 00:09:52,100
different complex exponentials
as omega varies.
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00:09:52,100 --> 00:09:56,580
Whereas, in discrete-time, we
saw, in fact, that there was a
142
00:09:56,580 --> 00:09:57,750
periodicity.
143
00:09:57,750 --> 00:10:03,670
Or said another way, it's
straightforward to verify that
144
00:10:03,670 --> 00:10:08,880
if we think of this class
of complex exponentials.
145
00:10:08,880 --> 00:10:17,490
That, in fact, if we consider
varying k by adding to it
146
00:10:17,490 --> 00:10:23,670
capital N, where capital N is
the period of the fundamental
147
00:10:23,670 --> 00:10:25,690
complex exponential.
148
00:10:25,690 --> 00:10:33,700
Then in fact, if we replace k by
k plus capital N, we'll see
149
00:10:33,700 --> 00:10:38,810
exactly the same complex
exponentials over again.
150
00:10:38,810 --> 00:10:39,690
Now, what does that say?
151
00:10:39,690 --> 00:10:43,430
What it says is that if I
consider this class of complex
152
00:10:43,430 --> 00:10:52,790
exponentials, as k varies from
0 through capital N minus 1,
153
00:10:52,790 --> 00:10:56,050
we will see all of the ones
that there are to see.
154
00:10:56,050 --> 00:10:58,060
There aren't anymore.
155
00:10:58,060 --> 00:11:03,160
And so, in fact, if we can
build x of n out of this
156
00:11:03,160 --> 00:11:08,600
linear combination, then we
better be able to do it as k
157
00:11:08,600 --> 00:11:16,230
varies from 0 up to N minus 1.
158
00:11:16,230 --> 00:11:20,990
Because beyond that, we'll
simply see the same complex
159
00:11:20,990 --> 00:11:22,870
exponentials over again.
160
00:11:22,870 --> 00:11:27,670
So, for example, if k takes on
the value capital N, that will
161
00:11:27,670 --> 00:11:31,630
be exactly the same complex
exponential as if
162
00:11:31,630 --> 00:11:34,300
k is equal to 0.
163
00:11:34,300 --> 00:11:41,070
So in fact, this sum ranges
only over capital N of the
164
00:11:41,070 --> 00:11:43,430
distinct complex exponentials.
165
00:11:43,430 --> 00:11:47,100
Let's say, for example, from
0 to capital N minus 1 .
166
00:11:47,100 --> 00:11:52,130
Although, in fact, since these
complex exponentials repeat in
167
00:11:52,130 --> 00:11:57,450
k, I could actually consider
instead of from 0 to N minus
168
00:11:57,450 --> 00:12:01,240
1, I could consider from
1 to N, or from 2 to
169
00:12:01,240 --> 00:12:03,800
N plus 1, or whatever.
170
00:12:03,800 --> 00:12:09,260
Or said another way, in this
representation, I could
171
00:12:09,260 --> 00:12:16,310
alternatively choose k outside
this range, thinking of these
172
00:12:16,310 --> 00:12:20,430
coefficients simply as
periodically repeating in k
173
00:12:20,430 --> 00:12:24,660
because of the fact that these
complex exponentials
174
00:12:24,660 --> 00:12:27,580
periodically repeat in k.
175
00:12:27,580 --> 00:12:31,130
So, in fact, in place of this
expression, it will be common
176
00:12:31,130 --> 00:12:35,080
in writing the Fourier series
expression to write it as I've
177
00:12:35,080 --> 00:12:38,910
indicated here, where the
implication is that these
178
00:12:38,910 --> 00:12:44,260
Fourier coefficients
periodically repeat as k
179
00:12:44,260 --> 00:12:47,350
continues to repeat outside
the interval from
180
00:12:47,350 --> 00:12:49,370
0 to N minus 1.
181
00:12:49,370 --> 00:12:54,520
And so this notation, in fact,
says that what we're going to
182
00:12:54,520 --> 00:13:02,270
use is k ranging over one
period of this periodic
183
00:13:02,270 --> 00:13:04,590
sequence, which is the Fourier
series coefficients.
184
00:13:07,730 --> 00:13:13,330
So the expression that we have
then for the Fourier series
185
00:13:13,330 --> 00:13:15,280
I've repeated here.
186
00:13:15,280 --> 00:13:18,360
And the implication
now is that the
187
00:13:18,360 --> 00:13:20,650
a sub k's are periodic.
188
00:13:20,650 --> 00:13:22,820
They periodically repeat
because, of course, these
189
00:13:22,820 --> 00:13:25,150
exponentials periodically
repeat.
190
00:13:25,150 --> 00:13:29,490
This indicates that we only
use them over one period.
191
00:13:29,490 --> 00:13:33,930
And now we can inquire as
to how we determine the
192
00:13:33,930 --> 00:13:36,120
coefficients a sub k.
193
00:13:36,120 --> 00:13:39,480
Well, we can formally go through
this much as we did in
194
00:13:39,480 --> 00:13:41,550
the continuous-time case.
195
00:13:41,550 --> 00:13:45,070
And we do, in fact, do that in
the text, which involves
196
00:13:45,070 --> 00:13:48,310
substituting some sums and
interchanging the orders of
197
00:13:48,310 --> 00:13:49,790
summation, et cetera.
198
00:13:49,790 --> 00:13:54,020
But let me draw your attention
to the fact that this, in
199
00:13:54,020 --> 00:13:59,290
fact, can be thought of as
capital N equations and
200
00:13:59,290 --> 00:14:02,210
capital N unknowns.
201
00:14:02,210 --> 00:14:08,100
In other words, we know x of n
over a period, and so we know
202
00:14:08,100 --> 00:14:12,280
what the left-hand side of this
is for capital N values.
203
00:14:12,280 --> 00:14:16,070
And we'd like to determine
these constants a sub k.
204
00:14:16,070 --> 00:14:18,860
Well, it turns out that there
is a nice convenient
205
00:14:18,860 --> 00:14:21,050
closed-form expression
for that.
206
00:14:21,050 --> 00:14:26,060
And, in fact, if we evaluate
the closed-form expression
207
00:14:26,060 --> 00:14:29,860
through any of a variety of
algebraic manipulations, we
208
00:14:29,860 --> 00:14:35,290
end up then with the
analysis equation.
209
00:14:35,290 --> 00:14:39,150
And the analysis equation, which
tells us how to get the
210
00:14:39,150 --> 00:14:47,080
coefficients a sub k from x of n
is what I've indicated here.
211
00:14:47,080 --> 00:14:53,020
And so this tells us how from
x of n to get the a sub k's.
212
00:14:53,020 --> 00:14:58,140
And, of course, the first
equation tells us how x of n
213
00:14:58,140 --> 00:15:01,120
is built up out of
the a sub k.
214
00:15:01,120 --> 00:15:05,520
Notice incidentally that there
is a strong duality between
215
00:15:05,520 --> 00:15:07,050
these two equations.
216
00:15:07,050 --> 00:15:11,590
And that's a duality that we'll
return to, actually
217
00:15:11,590 --> 00:15:13,750
toward the end of the
next lecture.
218
00:15:13,750 --> 00:15:16,340
Now, there is a real difference
between the way
219
00:15:16,340 --> 00:15:19,630
those equations look and the
way the continuous-time
220
00:15:19,630 --> 00:15:21,540
Fourier series looked.
221
00:15:21,540 --> 00:15:26,650
In the continuous-time case,
let me remind you that it
222
00:15:26,650 --> 00:15:31,630
required an infinite number of
coefficients to build up this
223
00:15:31,630 --> 00:15:33,800
continuous-time function.
224
00:15:33,800 --> 00:15:37,720
And so this was not simply a
matter of identifying how to
225
00:15:37,720 --> 00:15:41,180
invert capital N or a finite
number of equations and a
226
00:15:41,180 --> 00:15:44,280
finite number of unknowns.
227
00:15:44,280 --> 00:15:51,210
And the analysis equation was
an integration as opposed to
228
00:15:51,210 --> 00:15:55,510
the synthesis equation,
which is a summation.
229
00:15:55,510 --> 00:15:59,400
So there is a real difference
there between the
230
00:15:59,400 --> 00:16:01,960
continuous-time and
discrete-time cases.
231
00:16:01,960 --> 00:16:06,560
And the difference arises, to
a large extent, because of
232
00:16:06,560 --> 00:16:09,480
this notion that in
discrete-time, the complex
233
00:16:09,480 --> 00:16:16,050
exponentials are periodic
in their frequency.
234
00:16:16,050 --> 00:16:21,470
So we have then to summarize the
synthesis equation and the
235
00:16:21,470 --> 00:16:27,210
analysis equation for the
discrete-time Fourier series.
236
00:16:27,210 --> 00:16:32,250
Again, x of n, our original
signal is periodic.
237
00:16:32,250 --> 00:16:35,400
And, of course, the complex
exponentials
238
00:16:35,400 --> 00:16:38,230
involved are periodic.
239
00:16:38,230 --> 00:16:40,680
They're periodic
obviously in n.
240
00:16:40,680 --> 00:16:43,530
But in contrast to
continuous-time,
241
00:16:43,530 --> 00:16:46,370
these repeat in k.
242
00:16:46,370 --> 00:16:51,070
In other words, as k omega 0
goes outside a range that
243
00:16:51,070 --> 00:16:53,500
covers a 2 pi interval.
244
00:16:53,500 --> 00:16:57,570
And because of that, we're
imposing, in a sense, the
245
00:16:57,570 --> 00:17:00,450
interpretation that
the a sub k's are
246
00:17:00,450 --> 00:17:02,820
likewise a periodic sequence.
247
00:17:02,820 --> 00:17:07,890
And in fact, if we look at the
analysis equation, as we let k
248
00:17:07,890 --> 00:17:14,010
vary outside the range from 0
to N minus 1, what you can
249
00:17:14,010 --> 00:17:18,680
easily verify by substitution in
here is that this sequence
250
00:17:18,680 --> 00:17:22,030
will, in fact, periodically
repeat.
251
00:17:22,030 --> 00:17:26,300
So to underscore the difference
between the
252
00:17:26,300 --> 00:17:30,320
continuous-time and
discrete-time cases, we have
253
00:17:30,320 --> 00:17:33,730
this periodicity in the time
domain, and that's a
254
00:17:33,730 --> 00:17:38,200
periodicity that is, of course,
true in discrete-time
255
00:17:38,200 --> 00:17:43,520
and it's also true in
continuous-time if we replace
256
00:17:43,520 --> 00:17:48,820
the integer variable by the
discrete-time time variable.
257
00:17:48,820 --> 00:17:53,620
And we also, in discrete-time,
have this periodicity in k, or
258
00:17:53,620 --> 00:17:55,430
in k omega 0.
259
00:17:55,430 --> 00:17:58,700
And correspondingly, a
periodicity in the Fourier
260
00:17:58,700 --> 00:18:00,240
coefficients.
261
00:18:00,240 --> 00:18:06,980
And that is a set of properties
that does not
262
00:18:06,980 --> 00:18:10,300
happen in continuous-time.
263
00:18:10,300 --> 00:18:15,520
And it is that that essentially
leads to all of
264
00:18:15,520 --> 00:18:18,650
the important differences
between discrete-time Fourier
265
00:18:18,650 --> 00:18:21,390
representations and
continuous-time Fourier
266
00:18:21,390 --> 00:18:24,250
representations.
267
00:18:24,250 --> 00:18:29,430
Now, just quickly, let me draw
your attention to the issue of
268
00:18:29,430 --> 00:18:33,400
convergence and when a sequence
can and can't be
269
00:18:33,400 --> 00:18:35,370
represented, et cetera.
270
00:18:35,370 --> 00:18:39,620
And recall that in the
continuous-time case, we
271
00:18:39,620 --> 00:18:44,530
focused on convergence in the
context either of conditions,
272
00:18:44,530 --> 00:18:48,310
which I referred to as square
integrability, or another set
273
00:18:48,310 --> 00:18:51,510
of conditions, which were the
Dirichlet conditions.
274
00:18:51,510 --> 00:18:55,310
And there was this issue about
when the signal does and
275
00:18:55,310 --> 00:18:58,480
doesn't converge at
discontinuities, et cetera.
276
00:18:58,480 --> 00:19:02,330
Let me just simply draw your
attention to the fact that in
277
00:19:02,330 --> 00:19:07,470
the discrete-time case, what we
have is the representation
278
00:19:07,470 --> 00:19:12,620
of the periodic signal
as a sum of a
279
00:19:12,620 --> 00:19:14,750
finite number of terms.
280
00:19:14,750 --> 00:19:17,350
This represents capital
N equations
281
00:19:17,350 --> 00:19:19,850
and capital N unknowns.
282
00:19:19,850 --> 00:19:24,980
If we consider earth the partial
sum, namely taking a
283
00:19:24,980 --> 00:19:31,640
smaller number of terms, then
simply what happens is as the
284
00:19:31,640 --> 00:19:36,480
number of terms increases to the
finite number required to
285
00:19:36,480 --> 00:19:41,620
represent x of n, we simply end
up with the partial sum
286
00:19:41,620 --> 00:19:44,160
representing the finite
[? length ?] sequence.
287
00:19:44,160 --> 00:19:48,600
What all that boils down to
is the statement that in
288
00:19:48,600 --> 00:19:53,090
discrete-time there really are
no convergence issues as there
289
00:19:53,090 --> 00:19:55,670
were in continuous-time.
290
00:19:55,670 --> 00:20:00,260
OK, well let's look at an
example of the Fourier series
291
00:20:00,260 --> 00:20:03,390
representation for a
particular signal.
292
00:20:03,390 --> 00:20:06,420
And the one that I've picked
here is a simple one.
293
00:20:06,420 --> 00:20:14,430
Namely, a constant, a sine
term, and a cosine term.
294
00:20:14,430 --> 00:20:17,860
Now, for this particular
example, we can expand this
295
00:20:17,860 --> 00:20:22,720
out directly in terms of complex
exponentials and
296
00:20:22,720 --> 00:20:26,710
essentially recognize this as a
sum of complex exponentials.
297
00:20:26,710 --> 00:20:33,370
It's examined in more detail
in Example 5.2 in the text.
298
00:20:33,370 --> 00:20:37,920
And if we look at the Fourier
series coefficients, we can
299
00:20:37,920 --> 00:20:41,430
either look at it in terms of
real and imaginary parts or
300
00:20:41,430 --> 00:20:44,160
magnitude and angle.
301
00:20:44,160 --> 00:20:48,260
On the left side here, I have
the real part of the Fourier
302
00:20:48,260 --> 00:20:50,150
coefficients.
303
00:20:50,150 --> 00:20:55,480
And let me draw your attention
to the fact that I've drawn
304
00:20:55,480 --> 00:21:01,330
this to specifically illuminate
the periodicity of
305
00:21:01,330 --> 00:21:04,360
the Fourier series coefficients
with a period of
306
00:21:04,360 --> 00:21:05,570
capital N.
307
00:21:05,570 --> 00:21:08,740
So here are the Fourier
coefficients.
308
00:21:08,740 --> 00:21:12,920
And, in fact, it's this line
that represents the DC, or
309
00:21:12,920 --> 00:21:17,510
constant term, and these two
lines that represent the
310
00:21:17,510 --> 00:21:19,160
cosine term.
311
00:21:19,160 --> 00:21:23,540
And of course, these are the
three terms that are required.
312
00:21:23,540 --> 00:21:27,500
Or equivalently, this one,
this one, and this one.
313
00:21:27,500 --> 00:21:29,960
And then because of the
periodicity of the Fourier
314
00:21:29,960 --> 00:21:34,140
series coefficients, this simply
periodically repeats.
315
00:21:34,140 --> 00:21:39,150
So here is the real part
and below it I show
316
00:21:39,150 --> 00:21:41,390
the imaginary part.
317
00:21:41,390 --> 00:21:45,560
And in the imaginary part,
incidentally let me draw your
318
00:21:45,560 --> 00:21:53,090
attention to the fact that it's
this term and this term
319
00:21:53,090 --> 00:21:56,760
in the imaginary part that
represent the sinusoid.
320
00:21:56,760 --> 00:22:00,280
Whereas it's the symmetric terms
in the real part the
321
00:22:00,280 --> 00:22:03,110
represent the cosine.
322
00:22:03,110 --> 00:22:07,580
OK, let's look at
another example.
323
00:22:07,580 --> 00:22:12,420
This is another example from the
text, and one that we'll
324
00:22:12,420 --> 00:22:18,000
be making frequent reference to
in this particular lecture.
325
00:22:18,000 --> 00:22:24,090
And what it is, is
a square wave.
326
00:22:24,090 --> 00:22:27,940
And I've expressed the Fourier
series coefficients, which are
327
00:22:27,940 --> 00:22:30,340
algebraically developed
in the text.
328
00:22:30,340 --> 00:22:35,000
I've expressed the Fourier
series coefficients as samples
329
00:22:35,000 --> 00:22:38,616
of an envelope function.
330
00:22:38,616 --> 00:22:43,120
And so I've expressed it as
samples of this particular
331
00:22:43,120 --> 00:22:46,120
function, which is referred
to as a sin
332
00:22:46,120 --> 00:22:49,020
nx over sin x function.
333
00:22:49,020 --> 00:22:52,850
And let me just compare it to
a continuous-time example,
334
00:22:52,850 --> 00:22:57,980
which is the continuous-time
square wave, where with the
335
00:22:57,980 --> 00:23:02,840
continuous-times square wave the
form of the Fourier series
336
00:23:02,840 --> 00:23:08,490
coefficients was as samples of
what we refer to as a sin x
337
00:23:08,490 --> 00:23:11,050
over x function.
338
00:23:11,050 --> 00:23:15,820
Now, the sin nx over sin x
function, which is the
339
00:23:15,820 --> 00:23:19,000
envelope of the Fourier series
coefficients for the
340
00:23:19,000 --> 00:23:24,260
discrete-time periodic square
wave plays the role--
341
00:23:24,260 --> 00:23:26,490
and we'll see it very often
in discrete-time--
342
00:23:26,490 --> 00:23:31,080
that sin x over x does
in continuous-time.
343
00:23:31,080 --> 00:23:33,180
And, in fact, we should
understand right from the
344
00:23:33,180 --> 00:23:37,500
beginning that the sin x over
x envelope couldn't possibly
345
00:23:37,500 --> 00:23:40,820
be the envelope of the
discrete-time Fourier series
346
00:23:40,820 --> 00:23:42,220
coefficients.
347
00:23:42,220 --> 00:23:47,600
And one obvious reason is
that it is not periodic.
348
00:23:47,600 --> 00:23:50,540
What we require, of course, from
the discussion that I've
349
00:23:50,540 --> 00:23:53,530
just gone through is
periodicity of the
350
00:23:53,530 --> 00:23:54,340
coefficients.
351
00:23:54,340 --> 00:23:58,150
And then consequently, also
periodicity of the envelope in
352
00:23:58,150 --> 00:23:59,710
the discrete-time case.
353
00:23:59,710 --> 00:24:03,910
So once again, if we look back
at the algebraic expression
354
00:24:03,910 --> 00:24:08,440
that I have, it's samples of
the sin nx over sine x
355
00:24:08,440 --> 00:24:13,130
function that represent the
Fourier series coefficients of
356
00:24:13,130 --> 00:24:16,160
this periodic square wave.
357
00:24:16,160 --> 00:24:24,650
Now, in the representation in
the continuous-time case, we
358
00:24:24,650 --> 00:24:29,710
essentially had used the concept
of an envelope to
359
00:24:29,710 --> 00:24:33,320
represent the Fourier series
coefficients, and the notion
360
00:24:33,320 --> 00:24:35,990
that the Fourier series
coefficients were samples of
361
00:24:35,990 --> 00:24:36,710
an envelope.
362
00:24:36,710 --> 00:24:41,500
And that is the same notion
that we'll be using in
363
00:24:41,500 --> 00:24:43,110
discrete-time.
364
00:24:43,110 --> 00:24:49,560
So again for this square wave
example, then what we have is
365
00:24:49,560 --> 00:24:54,520
an envelope function, the sin
nx over sin x envelope
366
00:24:54,520 --> 00:24:59,720
function for a particular
value of the period.
367
00:24:59,720 --> 00:25:04,280
Here indicated with a period
of 10 samples.
368
00:25:04,280 --> 00:25:08,530
These samples of this envelope
function would then represent
369
00:25:08,530 --> 00:25:12,340
the Fourier series
coefficients.
370
00:25:12,340 --> 00:25:19,280
If we increased the period, then
we would simply have a
371
00:25:19,280 --> 00:25:23,910
finer spacing on the samples
of the envelope function to
372
00:25:23,910 --> 00:25:26,500
get the Fourier series
coefficients.
373
00:25:26,500 --> 00:25:32,380
And likewise, if we increase the
period still further, what
374
00:25:32,380 --> 00:25:37,260
we would have is an even
finer spacing.
375
00:25:37,260 --> 00:25:40,970
So actually, as the period
increases, and recall we used
376
00:25:40,970 --> 00:25:43,810
this in continuous-time also.
377
00:25:43,810 --> 00:25:47,700
As the period increases, we can
view the Fourier series
378
00:25:47,700 --> 00:25:50,540
coefficients as samples
of an envelope.
379
00:25:50,540 --> 00:25:54,970
And as the period increases,
the sample spacing
380
00:25:54,970 --> 00:25:56,860
gets finer and finer.
381
00:25:56,860 --> 00:26:00,500
And in fact, as the period
goes off essentially to
382
00:26:00,500 --> 00:26:04,380
infinity, the samples of the
envelope, in effect, become
383
00:26:04,380 --> 00:26:06,140
the envelope.
384
00:26:06,140 --> 00:26:10,460
And recall also that this was
essentially the trick that we
385
00:26:10,460 --> 00:26:16,850
used in continuous-time to allow
us to develop or utilize
386
00:26:16,850 --> 00:26:21,590
the Fourier series to provide a
representation of aperiodic
387
00:26:21,590 --> 00:26:24,410
signals as a linear combination
of complex
388
00:26:24,410 --> 00:26:26,490
exponentials.
389
00:26:26,490 --> 00:26:35,220
In particular, what we did in
the continuous-time case when
390
00:26:35,220 --> 00:26:41,280
we had an aperiodic signal was
to consider constructing a
391
00:26:41,280 --> 00:26:44,500
periodic signal for which
the aperiodic
392
00:26:44,500 --> 00:26:47,110
signal was one period.
393
00:26:47,110 --> 00:26:52,200
And then we developed the notion
that since the periodic
394
00:26:52,200 --> 00:26:58,310
signal has a Fourier series, and
since as the period of the
395
00:26:58,310 --> 00:27:02,760
periodic signal increases and
goes to infinity, the periodic
396
00:27:02,760 --> 00:27:06,520
signal represents the
aperiodic signal.
397
00:27:06,520 --> 00:27:09,930
Then, essentially, the Fourier
series provides us with a
398
00:27:09,930 --> 00:27:11,740
representation.
399
00:27:11,740 --> 00:27:14,770
Now, we can do exactly
the same thing in the
400
00:27:14,770 --> 00:27:16,420
discrete-time case.
401
00:27:16,420 --> 00:27:19,780
The statement is exactly the
same, except that in the
402
00:27:19,780 --> 00:27:24,530
discrete-time case, instead of t
as the independent variable,
403
00:27:24,530 --> 00:27:29,450
we simply make exactly the same
statement, but with our
404
00:27:29,450 --> 00:27:31,730
discrete-time variable n.
405
00:27:31,730 --> 00:27:37,560
So the basic notion then in
representing a discrete-time
406
00:27:37,560 --> 00:27:46,960
aperiodic signal is to first
construct a periodic signal.
407
00:27:46,960 --> 00:27:50,450
Here we have the aperiodic
signal.
408
00:27:50,450 --> 00:27:54,340
We construct a periodic signal
by simply periodically
409
00:27:54,340 --> 00:27:58,840
replicating the aperiodic
signal.
410
00:27:58,840 --> 00:28:04,890
The periodic signal and the
aperiodic signal are identical
411
00:28:04,890 --> 00:28:07,310
for one period.
412
00:28:07,310 --> 00:28:12,120
And as the period goes off to
infinity, it's the Fourier
413
00:28:12,120 --> 00:28:16,810
series representation of the
periodic signal that provides
414
00:28:16,810 --> 00:28:21,050
a representation of the
aperiodic signal.
415
00:28:21,050 --> 00:28:30,450
Again, to return to the example
that we have been kind
416
00:28:30,450 --> 00:28:31,780
of working through
this lecture.
417
00:28:31,780 --> 00:28:34,730
Namely, the periodic
square wave.
418
00:28:34,730 --> 00:28:38,970
If we have an aperiodic signal,
which is a rectangle,
419
00:28:38,970 --> 00:28:42,330
and we construct a
periodic signal.
420
00:28:42,330 --> 00:28:44,130
And now we consider
letting this
421
00:28:44,130 --> 00:28:46,570
period increase to infinity.
422
00:28:46,570 --> 00:28:51,320
We would first have this set
of samples of the envelope.
423
00:28:51,320 --> 00:28:55,820
As the period increases, we
would decrease the sample
424
00:28:55,820 --> 00:28:58,660
spacing to this set
of samples.
425
00:28:58,660 --> 00:29:03,140
As the period increases further,
it would be this set
426
00:29:03,140 --> 00:29:04,970
of samples.
427
00:29:04,970 --> 00:29:08,850
And as the period goes off to
infinity, it's every point on
428
00:29:08,850 --> 00:29:10,050
the envelope.
429
00:29:10,050 --> 00:29:13,440
In fact, what the representation
of the
430
00:29:13,440 --> 00:29:18,380
aperiodic signal is,
is the envelope.
431
00:29:18,380 --> 00:29:20,670
OK, well, so that's
the basic notion.
432
00:29:20,670 --> 00:29:23,650
It's no different than
what we did in the
433
00:29:23,650 --> 00:29:25,580
continuous-time case.
434
00:29:25,580 --> 00:29:30,770
And mathematically, it develops
in very much the same
435
00:29:30,770 --> 00:29:35,590
way as in the continuous-time
case.
436
00:29:35,590 --> 00:29:42,130
Specifically, here is our
representation through the
437
00:29:42,130 --> 00:29:47,550
Fourier series of the--
438
00:29:47,550 --> 00:29:51,760
here is a representation through
the envelope function.
439
00:29:51,760 --> 00:29:56,500
And this is the Fourier series
synthesis equation where the
440
00:29:56,500 --> 00:30:02,660
equation below tells us how we
get these Fourier coefficients
441
00:30:02,660 --> 00:30:05,970
or the envelope from x of n.
442
00:30:05,970 --> 00:30:08,860
Now, x tilde of n is the
periodic signal.
443
00:30:08,860 --> 00:30:11,990
And we know that over one
period, which is the only
444
00:30:11,990 --> 00:30:16,080
interval over which we use it,
in fact, this is identical to
445
00:30:16,080 --> 00:30:17,980
the aperiodic signal.
446
00:30:17,980 --> 00:30:22,610
And so, in fact, we can rewrite
this equation simply
447
00:30:22,610 --> 00:30:27,630
by substituting in instead
of x tilde, the original
448
00:30:27,630 --> 00:30:29,540
aperiodic signal.
449
00:30:29,540 --> 00:30:33,950
And now we can use infinite
limits on this sum.
450
00:30:33,950 --> 00:30:38,180
And what we would want to
examine, mathematically, is
451
00:30:38,180 --> 00:30:46,950
what happens to the top equation
as we let the period
452
00:30:46,950 --> 00:30:49,250
go off to infinity?
453
00:30:49,250 --> 00:30:53,090
And what happens is exactly
identical, mathematically, to
454
00:30:53,090 --> 00:30:54,580
continuous-time.
455
00:30:54,580 --> 00:30:56,970
I won't belabor the details.
456
00:30:56,970 --> 00:31:01,980
Essentially it's this sum that
goes to an integral.
457
00:31:01,980 --> 00:31:05,590
Omega 0, which is the
fundamental frequency, is
458
00:31:05,590 --> 00:31:06,940
going towards 0.
459
00:31:06,940 --> 00:31:10,100
In fact, becomes the
differential in the integral.
460
00:31:10,100 --> 00:31:13,200
And in the second equation,
of course, this then
461
00:31:13,200 --> 00:31:15,210
becomes x of omega.
462
00:31:15,210 --> 00:31:19,980
And as N goes to infinity then,
what the Fourier series
463
00:31:19,980 --> 00:31:25,020
becomes is the Fourier transform
as summarized by the
464
00:31:25,020 --> 00:31:27,610
bottom two equations.
465
00:31:27,610 --> 00:31:33,490
So although there is a little
bit of mathematical trickery.
466
00:31:33,490 --> 00:31:36,900
Or let's not call it trickery,
but subtlety, to be tracked
467
00:31:36,900 --> 00:31:38,330
through in detail.
468
00:31:38,330 --> 00:31:43,300
The important conceptual thing
to think about is this notion
469
00:31:43,300 --> 00:31:46,760
that we take the aperiodic
signal, form a periodic
470
00:31:46,760 --> 00:31:49,230
signal, let the period
go off to infinity.
471
00:31:49,230 --> 00:31:52,920
In which case, the Fourier
series coefficients become
472
00:31:52,920 --> 00:31:54,630
these envelopes functions.
473
00:31:54,630 --> 00:31:56,910
And incidentally,
mathematically, one of the
474
00:31:56,910 --> 00:31:59,890
sums ends up going
to an integral.
475
00:31:59,890 --> 00:32:05,690
So what we have then is the
discrete-time Fourier
476
00:32:05,690 --> 00:32:11,210
transform, which is a
representation of
477
00:32:11,210 --> 00:32:13,080
an aperiodic signal.
478
00:32:13,080 --> 00:32:17,920
And we have the synthesis
equation, which I show as the
479
00:32:17,920 --> 00:32:21,470
top equation on this
transparency.
480
00:32:21,470 --> 00:32:27,940
And this is the integral that
the Fourier series synthesis
481
00:32:27,940 --> 00:32:32,800
equation went to as the period
went off to infinity.
482
00:32:32,800 --> 00:32:36,730
And we have the corresponding
analysis equation, which is
483
00:32:36,730 --> 00:32:41,180
shown below, where this tells
us the Fourier transform.
484
00:32:41,180 --> 00:32:44,890
In effect, the envelope or the
Fourier series coefficients of
485
00:32:44,890 --> 00:32:47,250
that periodic signal.
486
00:32:47,250 --> 00:32:53,550
And here represented in terms
of the aperiodic signal.
487
00:32:53,550 --> 00:32:56,215
So we have the analysis
equation
488
00:32:56,215 --> 00:32:58,710
and synthesis equation.
489
00:32:58,710 --> 00:33:03,160
There are a number of
things to focus on
490
00:33:03,160 --> 00:33:03,980
as you look at this.
491
00:33:03,980 --> 00:33:06,450
And we'll talk about some of its
properties actually in the
492
00:33:06,450 --> 00:33:07,150
next lecture.
493
00:33:07,150 --> 00:33:11,120
But some of the points that I'd
like you to think about
494
00:33:11,120 --> 00:33:16,700
and focus on is the fact that
now there is somewhat of an
495
00:33:16,700 --> 00:33:20,420
imbalance or lack of duality
between the time domain and
496
00:33:20,420 --> 00:33:21,890
frequency domain.
497
00:33:21,890 --> 00:33:24,530
x of n, which is our
aperiodic signal,
498
00:33:24,530 --> 00:33:27,350
is of course, discrete.
499
00:33:27,350 --> 00:33:32,100
It's Fourier transform, x of
omega, is a function of a
500
00:33:32,100 --> 00:33:33,320
continuous variable.
501
00:33:33,320 --> 00:33:36,870
Omega is a continuous
variable.
502
00:33:36,870 --> 00:33:41,420
That is essentially what
represents the envelope.
503
00:33:41,420 --> 00:33:46,060
Also, in the time domain
x of n is aperiodic.
504
00:33:46,060 --> 00:33:49,270
It's not a periodic function.
505
00:33:49,270 --> 00:33:52,440
However, in the frequency
domain, remember that the
506
00:33:52,440 --> 00:33:55,960
Fourier series coefficients
were always periodic.
507
00:33:55,960 --> 00:34:00,090
Well, this envelope function
then is also periodic with a
508
00:34:00,090 --> 00:34:03,820
period in omega of 2 pi.
509
00:34:03,820 --> 00:34:07,820
Once again, the reason for the
periodicity, it all stems back
510
00:34:07,820 --> 00:34:12,449
to the fact that when we talk
about complex exponentials--
511
00:34:12,449 --> 00:34:15,199
and recall back to the
early lectures.
512
00:34:15,199 --> 00:34:19,300
In discrete-time, as the
frequency variable covers a
513
00:34:19,300 --> 00:34:25,980
range of 2 pi, when you proceed
past that range, you
514
00:34:25,980 --> 00:34:28,610
simply see the same complex
exponentials
515
00:34:28,610 --> 00:34:30,350
over and over again.
516
00:34:30,350 --> 00:34:33,300
And so obviously, anything that
we do with them would
517
00:34:33,300 --> 00:34:37,480
have to be periodic in that
frequency variable.
518
00:34:37,480 --> 00:34:42,480
All right, notationally, we'll,
again, represent the
519
00:34:42,480 --> 00:34:44,520
discrete-time Fourier
transform pair
520
00:34:44,520 --> 00:34:46,179
as I indicated here.
521
00:34:46,179 --> 00:34:49,980
And since it's a complex
function of frequency may, on
522
00:34:49,980 --> 00:34:54,530
occasion, want to either
represent it in rectangular
523
00:34:54,530 --> 00:34:59,180
form as I indicate in this
equation, or in polar form as
524
00:34:59,180 --> 00:35:02,040
I indicate in this equation.
525
00:35:02,040 --> 00:35:04,930
Let's look at an example.
526
00:35:04,930 --> 00:35:10,590
And, of course, one example that
we can look at is the one
527
00:35:10,590 --> 00:35:14,480
that has kind of been tracking
us through this lecture, which
528
00:35:14,480 --> 00:35:18,260
is the example of a rectangle.
529
00:35:18,260 --> 00:35:24,130
Now, the rectangle, if we refer
back to our argument of
530
00:35:24,130 --> 00:35:26,720
how we get a Fourier
representation for an
531
00:35:26,720 --> 00:35:30,550
aperiodic signal, we would form
a periodic signal where
532
00:35:30,550 --> 00:35:31,620
this is repeated.
533
00:35:31,620 --> 00:35:34,110
And that's our square
wave example.
534
00:35:34,110 --> 00:35:36,740
As the period goes to infinity,
the Fourier
535
00:35:36,740 --> 00:35:40,370
transform of this is represented
by the envelope of
536
00:35:40,370 --> 00:35:44,140
those Fourier series
coefficients, and that was our
537
00:35:44,140 --> 00:35:48,210
sin nx over sin x function,
which in this particular case,
538
00:35:48,210 --> 00:35:54,320
for these particular numbers, is
sin 5 omega over 2 divided
539
00:35:54,320 --> 00:35:57,890
by sin omega over 2.
540
00:35:57,890 --> 00:36:02,490
And notice, of course,
as we would expect--
541
00:36:02,490 --> 00:36:07,800
notice that this is a periodic
function of the frequency
542
00:36:07,800 --> 00:36:13,830
variable omega repeating, of
course, with a period of 2 pi.
543
00:36:13,830 --> 00:36:17,680
Whereas, in the time domain,
the function was not a
544
00:36:17,680 --> 00:36:19,020
periodic function,
it's aperiodic.
545
00:36:22,750 --> 00:36:26,170
Now, let's look at
another example.
546
00:36:26,170 --> 00:36:31,880
Let's look at an example which
is another signal that has
547
00:36:31,880 --> 00:36:34,780
kind of popped its head up
from time to time as the
548
00:36:34,780 --> 00:36:36,720
lectures have gone along.
549
00:36:36,720 --> 00:36:40,110
A signal which is another
aperiodic signal, which is a
550
00:36:40,110 --> 00:36:47,010
decaying exponential of this
form with the factor a chosen
551
00:36:47,010 --> 00:36:49,890
between 0 and 1.
552
00:36:49,890 --> 00:36:54,670
And you can work out the algebra
at your leisure.
553
00:36:54,670 --> 00:36:58,220
Basically, if we substitute
into the Fourier transform
554
00:36:58,220 --> 00:37:02,830
analysis equation, it's this
sum that we evaluate.
555
00:37:02,830 --> 00:37:07,665
Because we have a unit step here
which shuts this off for
556
00:37:07,665 --> 00:37:11,250
n less than 0, we can change
the limits on the sum.
557
00:37:11,250 --> 00:37:16,790
This then corresponds to the sum
over an infinite number of
558
00:37:16,790 --> 00:37:19,160
terms of a geometric series.
559
00:37:19,160 --> 00:37:23,690
And that, as we've seen before,
is 1 divided by 1
560
00:37:23,690 --> 00:37:27,340
minus a e to the
minus j omega.
561
00:37:27,340 --> 00:37:32,610
So let's look at what
that looks like.
562
00:37:32,610 --> 00:37:40,100
Here then we have, again, the
expression in the time domain
563
00:37:40,100 --> 00:37:43,330
and the expression in the
frequency domain.
564
00:37:43,330 --> 00:37:47,580
And let's, in particular, focus
on what the magnitude of
565
00:37:47,580 --> 00:37:49,960
the Fourier transform
looks like.
566
00:37:49,960 --> 00:37:54,790
It's as we show here.
567
00:37:54,790 --> 00:37:59,380
And for the particular values
of a that I pick, namely
568
00:37:59,380 --> 00:38:03,460
between 0 and 1, it's
larger at the origin
569
00:38:03,460 --> 00:38:05,340
than it is at pi.
570
00:38:05,340 --> 00:38:09,090
And then, of course,
it is periodic.
571
00:38:09,090 --> 00:38:12,080
And the periodicity is inherent
in the Fourier
572
00:38:12,080 --> 00:38:16,150
transform in discrete-time, so
we really might only need to
573
00:38:16,150 --> 00:38:21,160
look at this either from minus
pi to pi, or from 0 to 2 pi.
574
00:38:21,160 --> 00:38:24,710
The periodicity, of course,
would imply what the rest of
575
00:38:24,710 --> 00:38:27,900
this is for other
values of omega.
576
00:38:27,900 --> 00:38:31,290
Let me also draw your attention
while we're on it to
577
00:38:31,290 --> 00:38:33,900
the fact that--
578
00:38:33,900 --> 00:38:38,960
observe that if a were, in
fact, negative, then this
579
00:38:38,960 --> 00:38:42,700
value would be less
than this value.
580
00:38:42,700 --> 00:38:47,090
And in fact, for a negative, the
magnitude of the frequency
581
00:38:47,090 --> 00:38:51,630
response would look like this
except shifted by an amount in
582
00:38:51,630 --> 00:38:54,380
omega equal to pi.
583
00:38:54,380 --> 00:38:59,050
And this example will come up
and play an important role in
584
00:38:59,050 --> 00:39:03,650
our discussion next time, so
try to keep it in mind.
585
00:39:03,650 --> 00:39:06,710
And in fact, work it out
more carefully between
586
00:39:06,710 --> 00:39:08,220
now and next time.
587
00:39:08,220 --> 00:39:12,800
And also, if you have a chance,
focus on this issue of
588
00:39:12,800 --> 00:39:18,500
how it looks with a positive as
compared with a negative.
589
00:39:18,500 --> 00:39:23,860
Now, we developed the Fourier
transform by beginning with
590
00:39:23,860 --> 00:39:25,370
the Fourier series.
591
00:39:25,370 --> 00:39:29,350
We did that in continuous-time
also.
592
00:39:29,350 --> 00:39:32,920
What I'd like to do now, just as
we did in continuous-time,
593
00:39:32,920 --> 00:39:38,800
is now absorb the Fourier series
within the broader
594
00:39:38,800 --> 00:39:41,540
framework of the Fourier
transform.
595
00:39:41,540 --> 00:39:44,360
And there are two relationships
between the
596
00:39:44,360 --> 00:39:48,000
Fourier series and the Fourier
transform, which are identical
597
00:39:48,000 --> 00:39:54,250
to relationships that we had in
the continuous-time case.
598
00:39:54,250 --> 00:40:00,980
Let me remind you that in
continuous-time we had the
599
00:40:00,980 --> 00:40:09,290
statement that if we have a
periodic signal, that in fact
600
00:40:09,290 --> 00:40:12,040
the Fourier series coefficients
of that periodic
601
00:40:12,040 --> 00:40:19,340
signal is proportional to
samples of the Fourier
602
00:40:19,340 --> 00:40:23,140
transform of one period.
603
00:40:23,140 --> 00:40:28,510
Well, in fact, let me remind you
flows easily from all the
604
00:40:28,510 --> 00:40:33,210
things that we built up so far,
because of the fact that
605
00:40:33,210 --> 00:40:36,670
the Fourier transform
essentially, by definition, of
606
00:40:36,670 --> 00:40:44,600
the way we developed it, is
what we get as the Fourier
607
00:40:44,600 --> 00:40:48,780
series coefficients, as we focus
on one period, and then
608
00:40:48,780 --> 00:40:51,050
let the period go
off to infinity.
609
00:40:51,050 --> 00:40:55,510
Well, looking at one period, the
Fourier transform of that
610
00:40:55,510 --> 00:40:59,490
then is the envelope of the
Fourier series coefficients.
611
00:40:59,490 --> 00:41:05,610
And so in continuous-time, we
have this relationship.
612
00:41:05,610 --> 00:41:10,500
And in discrete-time, we
have precisely the same
613
00:41:10,500 --> 00:41:16,240
relationship, except that here
we're talking about an integer
614
00:41:16,240 --> 00:41:20,090
variable as opposed to the
continuous variable, and a
615
00:41:20,090 --> 00:41:28,440
period of capital N as opposed
to a period of t0.
616
00:41:28,440 --> 00:41:38,720
OK, so once again, if we return
to our example, or if
617
00:41:38,720 --> 00:41:40,790
we return to a periodic
signal.
618
00:41:40,790 --> 00:41:46,240
If we have a periodic signal
and we consider the Fourier
619
00:41:46,240 --> 00:41:52,100
transform of one period, the
Fourier series coefficients of
620
00:41:52,100 --> 00:41:56,880
this periodic signal are,
in fact, samples--
621
00:41:56,880 --> 00:42:01,800
as stated mathematically in the
bottom equation, samples
622
00:42:01,800 --> 00:42:07,340
of the Fourier transform
of one period.
623
00:42:07,340 --> 00:42:11,180
So x of omega is the Fourier
transform of one period.
624
00:42:11,180 --> 00:42:13,760
a sub k's are the Fourier series
coefficients of the
625
00:42:13,760 --> 00:42:15,350
periodic signal.
626
00:42:15,350 --> 00:42:19,190
And this relationship simply
says they're related except
627
00:42:19,190 --> 00:42:23,030
for scale factor through
samples along
628
00:42:23,030 --> 00:42:25,820
the frequency axis.
629
00:42:25,820 --> 00:42:33,370
And, of course, we saw this
in the context of
630
00:42:33,370 --> 00:42:36,350
our square wave example.
631
00:42:36,350 --> 00:42:42,080
In the square wave example, we
have a periodic signal, which
632
00:42:42,080 --> 00:42:45,690
is a periodic square wave.
633
00:42:45,690 --> 00:42:51,800
And the Fourier transform
of one period, in fact,
634
00:42:51,800 --> 00:42:54,790
represents the envelope.
635
00:42:54,790 --> 00:42:57,180
And here we have the
envelope function.
636
00:42:57,180 --> 00:43:02,520
Represents the envelope of the
Fourier series coefficients.
637
00:43:02,520 --> 00:43:05,710
And the Fourier series
coefficients are samples.
638
00:43:08,390 --> 00:43:14,460
So what we have then is a
relationship back to the
639
00:43:14,460 --> 00:43:18,120
Fourier series coefficients
from the Fourier transform
640
00:43:18,120 --> 00:43:21,750
that tells us that for a
periodic signal now, the
641
00:43:21,750 --> 00:43:23,420
periodic signal--
642
00:43:23,420 --> 00:43:27,170
the Fourier series coefficients
are related, are
643
00:43:27,170 --> 00:43:31,920
samples of the Fourier transform
of one period.
644
00:43:31,920 --> 00:43:36,830
Now, finally, to kind of bring
things back in a circle and
645
00:43:36,830 --> 00:43:40,100
exactly identical to what we
did in the continuous-time
646
00:43:40,100 --> 00:43:46,400
case, we can finally absorb
the Fourier series in
647
00:43:46,400 --> 00:43:47,310
discrete-time.
648
00:43:47,310 --> 00:43:50,480
We can absorb it into
the framework
649
00:43:50,480 --> 00:43:53,300
of the Fourier transform.
650
00:43:53,300 --> 00:44:00,800
Now, remember or recall how we
did that when we tried to do a
651
00:44:00,800 --> 00:44:04,970
similar sort of thing
in continuous-time.
652
00:44:04,970 --> 00:44:11,430
In continuous-time, what we
essentially did is to develop
653
00:44:11,430 --> 00:44:15,040
that, more or less,
by definition.
654
00:44:15,040 --> 00:44:18,150
We have a periodic signal.
655
00:44:18,150 --> 00:44:20,800
The periodic signal is
represented through a Fourier
656
00:44:20,800 --> 00:44:24,360
series and Fourier series
coefficients.
657
00:44:24,360 --> 00:44:27,570
Essentially what I pointed
out at that time
658
00:44:27,570 --> 00:44:30,180
was that if we define--
659
00:44:30,180 --> 00:44:33,780
take it as a definition, the
Fourier transform of the
660
00:44:33,780 --> 00:44:39,220
periodic signal as an impulse
train where the amplitudes of
661
00:44:39,220 --> 00:44:42,640
the impulses are proportional
to the Fourier series
662
00:44:42,640 --> 00:44:44,680
coefficients.
663
00:44:44,680 --> 00:44:50,430
If we take that impulse train
representation and simply plug
664
00:44:50,430 --> 00:44:58,460
it into the Fourier transform
synthesis equation, what we
665
00:44:58,460 --> 00:45:04,860
end up with is the Fourier
series synthesis equation.
666
00:45:04,860 --> 00:45:17,320
So in continuous-time, we had
used this definition of the
667
00:45:17,320 --> 00:45:20,060
continuous-time Fourier
transform
668
00:45:20,060 --> 00:45:22,610
of a periodic signal.
669
00:45:22,610 --> 00:45:27,670
And again, in discrete-time,
it's simply a matter of using
670
00:45:27,670 --> 00:45:30,110
exactly the same expression.
671
00:45:30,110 --> 00:45:35,400
And using, instead, the
appropriate variables related
672
00:45:35,400 --> 00:45:38,430
to discrete-time rather than
the variables related to
673
00:45:38,430 --> 00:45:39,810
continuous-time.
674
00:45:39,810 --> 00:45:43,120
So in discrete-time, if we have
a periodic signal, the
675
00:45:43,120 --> 00:45:47,150
Fourier transform of that
periodic signal is defined as
676
00:45:47,150 --> 00:45:53,290
an impulse train where the
amplitudes of the impulses are
677
00:45:53,290 --> 00:45:57,500
proportional to the Fourier
series coefficients.
678
00:45:57,500 --> 00:46:00,660
If this expression is
substituted into the synthesis
679
00:46:00,660 --> 00:46:05,340
equation for the Fourier
transform, that will simply
680
00:46:05,340 --> 00:46:08,440
then reduce to the synthesis
equation
681
00:46:08,440 --> 00:46:11,240
for the Fourier series.
682
00:46:11,240 --> 00:46:17,310
So once more returning to our
example, which is the square
683
00:46:17,310 --> 00:46:19,170
wave example that we've
carried through these
684
00:46:19,170 --> 00:46:23,780
lectures, or through this
lecture, we can see that
685
00:46:23,780 --> 00:46:25,870
really what we're talking
about really is
686
00:46:25,870 --> 00:46:27,120
a notational change.
687
00:46:30,740 --> 00:46:35,290
Here is the periodic signal and
below it are the Fourier
688
00:46:35,290 --> 00:46:39,330
series coefficients, where
I've removed the envelope
689
00:46:39,330 --> 00:46:43,420
function and just indicate the
amplitudes of the coefficients
690
00:46:43,420 --> 00:46:46,610
indexed, of course, on the
coefficient number.
691
00:46:46,610 --> 00:46:49,550
And so this represents
a bar graph.
692
00:46:49,550 --> 00:46:52,760
And if instead of talking
about the Fourier series
693
00:46:52,760 --> 00:46:59,040
coefficients, what I want to
talk about is the Fourier
694
00:46:59,040 --> 00:47:03,300
transform, the Fourier
transform, in essence,
695
00:47:03,300 --> 00:47:08,450
corresponds to simply redrawing
that using impulses
696
00:47:08,450 --> 00:47:14,420
and using an axis that is
essentially indexed on the
697
00:47:14,420 --> 00:47:18,910
fundamental frequency omega 0,
rather than on the Fourier
698
00:47:18,910 --> 00:47:20,433
series coefficient number k.
699
00:47:23,884 --> 00:47:31,540
OK, so to summarize,
what we've done is
700
00:47:31,540 --> 00:47:34,190
to pretty much parallel--
701
00:47:34,190 --> 00:47:38,290
somewhat more quickly, the kind
of development that we
702
00:47:38,290 --> 00:47:41,970
went through for continuous-time
representation
703
00:47:41,970 --> 00:47:45,070
through complex exponentials,
paralleled that for the
704
00:47:45,070 --> 00:47:47,390
discrete-time case.
705
00:47:47,390 --> 00:47:51,340
And pretty much the conceptual
underpinnings of the
706
00:47:51,340 --> 00:47:55,230
development are identical
in discrete-time and in
707
00:47:55,230 --> 00:47:56,480
continuous-time.
708
00:47:58,750 --> 00:48:04,510
We saw that there are some
major differences, or
709
00:48:04,510 --> 00:48:06,700
important differences between
continuous-time and
710
00:48:06,700 --> 00:48:08,560
discrete-time.
711
00:48:08,560 --> 00:48:11,240
And the difference, essentially
712
00:48:11,240 --> 00:48:14,080
relates to two aspects.
713
00:48:14,080 --> 00:48:18,790
One aspect is the fact that in
discrete-time, we have a
714
00:48:18,790 --> 00:48:22,690
discrete representation in the
time domain, whereas the
715
00:48:22,690 --> 00:48:25,020
independent variable in the
frequency domain is a
716
00:48:25,020 --> 00:48:26,750
continuous variable.
717
00:48:26,750 --> 00:48:30,860
Whereas in continuous-time for
the Fourier transform, we had
718
00:48:30,860 --> 00:48:35,120
a duality between the time
domain and frequency domain.
719
00:48:35,120 --> 00:48:38,170
The other very important
difference tied back to the
720
00:48:38,170 --> 00:48:41,310
difference between complex
exponentials, continuous-time
721
00:48:41,310 --> 00:48:43,050
and discrete-time.
722
00:48:43,050 --> 00:48:45,980
In continuous-time, complex
exponentials, as you vary the
723
00:48:45,980 --> 00:48:52,940
frequency, generate distinct
time functions.
724
00:48:52,940 --> 00:48:56,500
In discrete-time, as you vary
the frequency, once you've
725
00:48:56,500 --> 00:49:01,120
covered a frequency interval of
2 pi, then you've seen all
726
00:49:01,120 --> 00:49:02,240
the ones there are to see.
727
00:49:02,240 --> 00:49:03,680
There are no more.
728
00:49:03,680 --> 00:49:09,610
And this, in effect, imposes a
periodicity on the Fourier
729
00:49:09,610 --> 00:49:13,390
domain representation of
discrete-time signals.
730
00:49:13,390 --> 00:49:15,710
And some of those differences
and, of course, lots of the
731
00:49:15,710 --> 00:49:20,380
similarities will surface,
both as we use this
732
00:49:20,380 --> 00:49:26,120
representation and as we develop
further properties.
733
00:49:26,120 --> 00:49:30,290
In the next lecture, what
we'll do is to focus in,
734
00:49:30,290 --> 00:49:33,540
again, on the Fourier transform,
the discrete-time
735
00:49:33,540 --> 00:49:39,050
Fourier transform, develop
or illuminate some of the
736
00:49:39,050 --> 00:49:42,770
properties of the Fourier
transform, and then see how
737
00:49:42,770 --> 00:49:46,850
these properties can be used
for a number of things.
738
00:49:46,850 --> 00:49:49,360
For example, how the properties
as they were in
739
00:49:49,360 --> 00:49:55,280
continuous-time can be used to
efficiently generate the
740
00:49:55,280 --> 00:49:58,430
solution and analyze linear
constant coefficient
741
00:49:58,430 --> 00:49:59,950
difference equations.
742
00:49:59,950 --> 00:50:03,680
And then beyond that, the
concepts of filtering and
743
00:50:03,680 --> 00:50:05,090
modulation.
744
00:50:05,090 --> 00:50:09,220
And both the properties and
interpretation, which will
745
00:50:09,220 --> 00:50:12,750
very strongly parallel the kinds
of developments along
746
00:50:12,750 --> 00:50:14,650
those lines that we did
in the last lecture.
747
00:50:14,650 --> 00:50:15,900
Thank you.