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PROFESSOR: In the last lecture,
I introduced and
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illustrated the kinds of signals
and systems that we'll
12
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be dealing with throughout
this course.
13
00:01:05,010 --> 00:01:08,820
In today's lecture I'd like to
be a little more specific, and
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00:01:08,820 --> 00:01:13,080
in particular, talk about some
of the basic signals, both
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continuous-time and
discrete-time that will form
16
00:01:16,430 --> 00:01:21,020
important building blocks as
the course progresses.
17
00:01:21,020 --> 00:01:25,420
Let's begin with one signal, the
continuous-time sinusoidal
18
00:01:25,420 --> 00:01:27,550
signal, which perhaps
you're already
19
00:01:27,550 --> 00:01:29,650
somewhat familiar with.
20
00:01:29,650 --> 00:01:32,790
Mathematically, the
continuous-time sinusoidal
21
00:01:32,790 --> 00:01:36,500
signal is expressed as
I've indicated here.
22
00:01:36,500 --> 00:01:41,750
There are three parameters,
A, omega_0 and phi.
23
00:01:41,750 --> 00:01:46,760
The parameter A is referred
to as the amplitude, the
24
00:01:46,760 --> 00:01:51,150
parameter omega 0 as the
frequency, and the parameter
25
00:01:51,150 --> 00:01:54,950
phi as the phase.
26
00:01:54,950 --> 00:02:00,010
And graphically, the
continuous-time sinusoidal
27
00:02:00,010 --> 00:02:04,440
signal has the form
shown here.
28
00:02:04,440 --> 00:02:08,759
Now, the sinusoidal signal
has a number of important
29
00:02:08,759 --> 00:02:13,160
properties that we'll find it
convenient to exploit as the
30
00:02:13,160 --> 00:02:18,800
course goes along, one of which
is the fact that the
31
00:02:18,800 --> 00:02:24,250
sinusoidal signal is what is
referred to as periodic.
32
00:02:24,250 --> 00:02:28,920
What I mean by periodic is that
under an appropriate time
33
00:02:28,920 --> 00:02:34,380
shift, which I indicate here as
T_0, the signal replicates
34
00:02:34,380 --> 00:02:36,110
or repeats itself.
35
00:02:36,110 --> 00:02:40,820
Or said another way, if we shift
the time origin by an
36
00:02:40,820 --> 00:02:44,690
appropriate amount T_0, the
smallest value T_0 being
37
00:02:44,690 --> 00:02:48,190
referred to as the period,
then x(t) is
38
00:02:48,190 --> 00:02:51,420
equal to itself, shifted.
39
00:02:51,420 --> 00:02:57,630
And we can demonstrate it
mathematically by simply
40
00:02:57,630 --> 00:03:00,700
substituting into the
mathematical expression for
41
00:03:00,700 --> 00:03:07,630
the sinusoidal signal t
+ T_0, in place of t.
42
00:03:07,630 --> 00:03:11,310
When we carry out the expansion
we then have, for
43
00:03:11,310 --> 00:03:20,040
the argument of the sinusoid,
omega_0 t + omega_0 T_0 + phi.
44
00:03:20,040 --> 00:03:23,350
Now, one of the things that
we know about sinusoidal
45
00:03:23,350 --> 00:03:27,080
functions is that if you change
the argument by any
46
00:03:27,080 --> 00:03:30,630
integer multiple of 2 pi,
then the function
47
00:03:30,630 --> 00:03:32,320
has the same value.
48
00:03:32,320 --> 00:03:36,110
And so we can exploit that
here, in particular with
49
00:03:36,110 --> 00:03:41,400
omega_0 T_0 and integer
multiple of 2 pi.
50
00:03:41,400 --> 00:03:44,700
Then the right-hand side of this
equation is equal to the
51
00:03:44,700 --> 00:03:47,040
left-hand side of
the equation.
52
00:03:47,040 --> 00:03:54,880
So with omega_0 T_0 equal to 2
pi times an integer, or T_0
53
00:03:54,880 --> 00:03:57,420
equal to 2 pi times an
integer divided by
54
00:03:57,420 --> 00:04:00,280
omega_0, the signal repeats.
55
00:04:00,280 --> 00:04:03,700
The period is defined as the
smallest value of T_0.
56
00:04:03,700 --> 00:04:08,750
And so the period is 2 pi
divided by omega_0.
57
00:04:08,750 --> 00:04:11,190
And going back to
our sinusoidal
58
00:04:11,190 --> 00:04:16,160
signal, we can see that--
59
00:04:16,160 --> 00:04:23,100
and I've indicated here, then,
the period as 2 pi / omega_0.
60
00:04:23,100 --> 00:04:26,990
And that's the value under
which the signal repeats.
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00:04:26,990 --> 00:04:30,160
62
00:04:30,160 --> 00:04:35,710
Now in addition, a useful
property of the sinusoidal
63
00:04:35,710 --> 00:04:42,070
signal is the fact that a time
shift of a sinusoid is
64
00:04:42,070 --> 00:04:44,290
equivalent to a phase change.
65
00:04:44,290 --> 00:04:48,670
And we can demonstrate that
again mathematically, in
66
00:04:48,670 --> 00:04:52,610
particular if we put the
sinusoidal signal
67
00:04:52,610 --> 00:04:54,540
under a time shift--
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00:04:54,540 --> 00:04:57,710
I've indicated the time
shift that I'm
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00:04:57,710 --> 00:05:00,680
talking about by t_0--
70
00:05:00,680 --> 00:05:06,570
and expand this out, then we see
that that is equivalent to
71
00:05:06,570 --> 00:05:08,760
a change in phase.
72
00:05:08,760 --> 00:05:12,640
And an important thing to
recognize about this statement
73
00:05:12,640 --> 00:05:18,260
is that not only is a time
shift generating a phase
74
00:05:18,260 --> 00:05:24,890
change, but, in fact, if we
inserted a phase change, there
75
00:05:24,890 --> 00:05:31,570
is always a value of t_0 which
would correspond to an
76
00:05:31,570 --> 00:05:33,420
equivalent time shift.
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00:05:33,420 --> 00:05:39,630
Said another way, if we take
omega_0 t_0 and think of that
78
00:05:39,630 --> 00:05:45,190
as our change in phase, for any
change in phase, we can
79
00:05:45,190 --> 00:05:48,900
solve this equation for a time
shift, or conversely for any
80
00:05:48,900 --> 00:05:51,810
value of time shift,
that represents
81
00:05:51,810 --> 00:05:54,740
an appropriate phase.
82
00:05:54,740 --> 00:05:58,820
So a time shift corresponds to
a phase change, and a phase
83
00:05:58,820 --> 00:06:02,970
change, likewise, corresponds
to time shift.
84
00:06:02,970 --> 00:06:11,160
And so for example, if we look
at the general sinusoidal
85
00:06:11,160 --> 00:06:17,530
signal that we saw previously,
in effect, changing the phase
86
00:06:17,530 --> 00:06:21,090
corresponds to moving
this signal in time
87
00:06:21,090 --> 00:06:22,990
one way or the other.
88
00:06:22,990 --> 00:06:27,110
For example, if we look at the
sinusoidal signal with a phase
89
00:06:27,110 --> 00:06:31,820
equal to 0 that corresponds
to locating the time
90
00:06:31,820 --> 00:06:33,950
origin at this peak.
91
00:06:33,950 --> 00:06:41,820
And I've indicated that on
the following graph.
92
00:06:41,820 --> 00:06:46,980
So here we have illustrated a
sinusoid with 0 phase, or a
93
00:06:46,980 --> 00:06:50,650
cosine with 0 phase,
corresponding to taking our
94
00:06:50,650 --> 00:06:52,890
general picture and
shifting it.
95
00:06:52,890 --> 00:06:56,730
Shifting it appropriately
as I've indicated here.
96
00:06:56,730 --> 00:07:00,370
This, of course, still has the
property that it's a periodic
97
00:07:00,370 --> 00:07:05,520
function, since we simply
displaced it in time.
98
00:07:05,520 --> 00:07:09,580
And by looking at the graph,
what we see is that it has
99
00:07:09,580 --> 00:07:12,540
another very important property,
a property referred
100
00:07:12,540 --> 00:07:14,070
to as even.
101
00:07:14,070 --> 00:07:18,180
And that's a property that we'll
find useful, in general,
102
00:07:18,180 --> 00:07:20,970
to refer to in relation
to signals.
103
00:07:20,970 --> 00:07:24,860
A signal is said to be even if,
when we reflect it about
104
00:07:24,860 --> 00:07:27,170
the origin, it looks
exactly the same.
105
00:07:27,170 --> 00:07:30,600
So it's symmetric about
the origin.
106
00:07:30,600 --> 00:07:34,670
And looking at this
sinusoid, that, in
107
00:07:34,670 --> 00:07:37,010
fact, has that property.
108
00:07:37,010 --> 00:07:41,550
And mathematically, the
statement that it's even is
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00:07:41,550 --> 00:07:45,570
equivalent to the statement that
if we replace the time
110
00:07:45,570 --> 00:07:48,630
argument by its negative,
the function
111
00:07:48,630 --> 00:07:50,070
itself doesn't change.
112
00:07:50,070 --> 00:07:53,290
113
00:07:53,290 --> 00:07:56,660
Now this corresponded to a
phase shift of 0 in our
114
00:07:56,660 --> 00:07:58,990
original cosine expression.
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00:07:58,990 --> 00:08:03,350
If instead, we had chosen a
phase shift of, let's say,
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00:08:03,350 --> 00:08:08,275
-pi/2, then instead of a
cosinusoidal signal, what we
117
00:08:08,275 --> 00:08:14,770
would regenerate is a sinusoid
with the appropriate phase.
118
00:08:14,770 --> 00:08:22,260
Or, said another way, if we take
our original cosine and
119
00:08:22,260 --> 00:08:27,120
substitute in for the phase
-pi/2, then of course we have
120
00:08:27,120 --> 00:08:29,540
this mathematical expression.
121
00:08:29,540 --> 00:08:32,900
Using just straightforward
trigonometric identities, we
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00:08:32,900 --> 00:08:37,799
can express that alternately
as sin(omega_0*t).
123
00:08:37,799 --> 00:08:40,990
The frequency and amplitude,
of course, haven't changed.
124
00:08:40,990 --> 00:08:45,750
And that, you can convince
yourself, also is equivalent
125
00:08:45,750 --> 00:08:50,500
to shifting the cosine by an
amount in time that I've
126
00:08:50,500 --> 00:08:54,600
indicated here, namely a
quarter of a period.
127
00:08:54,600 --> 00:09:00,160
So illustrated below is the
graph now, when we have a
128
00:09:00,160 --> 00:09:07,290
phase of -pi/2 in our cosine,
which is a sinusoidal signal.
129
00:09:07,290 --> 00:09:09,270
Of course, it's still
periodic.
130
00:09:09,270 --> 00:09:15,220
It's periodic with a period of
2 pi / omega_0 again, because
131
00:09:15,220 --> 00:09:19,690
all that we've done by
introducing a phase change is
132
00:09:19,690 --> 00:09:20,940
introduced the time shift.
133
00:09:20,940 --> 00:09:23,780
134
00:09:23,780 --> 00:09:27,420
Now, when we look at the
sinusoid in comparison with
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00:09:27,420 --> 00:09:30,730
the cosine, namely with this
particular choice of phase,
136
00:09:30,730 --> 00:09:34,960
this has a different symmetry,
and that symmetry
137
00:09:34,960 --> 00:09:37,020
is referred to odd.
138
00:09:37,020 --> 00:09:41,590
What odd symmetry means,
graphically, is that when we
139
00:09:41,590 --> 00:09:50,230
flip the signal about the time
origin, we also multiply it by
140
00:09:50,230 --> 00:09:51,410
a minus sign.
141
00:09:51,410 --> 00:09:55,110
So that's, in effect,
anti-symmetric.
142
00:09:55,110 --> 00:09:56,880
It's not the mirror image,
but it's the mirror
143
00:09:56,880 --> 00:09:58,390
image flipped over.
144
00:09:58,390 --> 00:10:02,420
And we'll find many occasions,
not only to refer to signals
145
00:10:02,420 --> 00:10:06,610
more general than sinusoidal
signals, as even in some cases
146
00:10:06,610 --> 00:10:08,950
and odd in other cases.
147
00:10:08,950 --> 00:10:14,240
And in general, mathematically,
an odd signal
148
00:10:14,240 --> 00:10:22,190
is one which satisfies the
algebraic expression, x(t).
149
00:10:22,190 --> 00:10:29,660
When you replace t by its
negative, is equal to -x(-t).
150
00:10:29,660 --> 00:10:35,640
So replacing the argument by its
negative corresponds to an
151
00:10:35,640 --> 00:10:36,910
algebraic sign reversal.
152
00:10:36,910 --> 00:10:39,570
153
00:10:39,570 --> 00:10:39,900
OK.
154
00:10:39,900 --> 00:10:43,610
So this is the class of
continuous-time sinusoids.
155
00:10:43,610 --> 00:10:46,470
We'll have a little more
to say about it later.
156
00:10:46,470 --> 00:10:51,720
But I'd now like to turn to
discrete-time sinusoids.
157
00:10:51,720 --> 00:10:54,790
What we'll see is that
discrete-time sinusoids are
158
00:10:54,790 --> 00:10:59,070
very much like continuous-time
ones, but also with some very
159
00:10:59,070 --> 00:11:00,650
important differences.
160
00:11:00,650 --> 00:11:03,920
And we want to focus, not only
on the similarities, but also
161
00:11:03,920 --> 00:11:06,040
on the differences.
162
00:11:06,040 --> 00:11:10,530
Well, let's begin with the
mathematical expression.
163
00:11:10,530 --> 00:11:16,740
A discrete-time sinusoidal
signal, mathematically, is as
164
00:11:16,740 --> 00:11:22,040
I've indicated here, A
cos(omega_0 n + phi).
165
00:11:22,040 --> 00:11:26,050
And just as in the
continuous-time case, the
166
00:11:26,050 --> 00:11:31,360
parameter A is what we'll refer
to as the amplitude,
167
00:11:31,360 --> 00:11:39,820
omega_0 as the frequency,
and phi as the phase.
168
00:11:39,820 --> 00:11:46,420
And I've illustrated here
several discrete-time
169
00:11:46,420 --> 00:11:48,480
sinusoidal signals.
170
00:11:48,480 --> 00:11:51,000
And they kind of look similar.
171
00:11:51,000 --> 00:11:53,720
In fact, if you track what
you might think of as the
172
00:11:53,720 --> 00:11:57,670
envelope, it looks very much
like what a continuous-time
173
00:11:57,670 --> 00:12:00,090
sinusoid might look like.
174
00:12:00,090 --> 00:12:03,720
But keep in mind that the
independent variable, in this
175
00:12:03,720 --> 00:12:06,890
case, is an integer variable.
176
00:12:06,890 --> 00:12:13,050
And so the sequence only takes
on values at integer values of
177
00:12:13,050 --> 00:12:13,810
the argument.
178
00:12:13,810 --> 00:12:19,540
And we'll see that has a very
important implication, and
179
00:12:19,540 --> 00:12:22,000
we'll see that shortly.
180
00:12:22,000 --> 00:12:24,950
Now, one of the issues that
we addressed in the
181
00:12:24,950 --> 00:12:27,500
continuous-time case
was periodicity.
182
00:12:27,500 --> 00:12:33,150
And I want to return to that
shortly, because that is one
183
00:12:33,150 --> 00:12:35,085
of the areas where there is
an important distinction.
184
00:12:35,085 --> 00:12:37,770
185
00:12:37,770 --> 00:12:42,880
Let's first, though, examine the
statement similar to the
186
00:12:42,880 --> 00:12:46,170
one that we examined for
continuous time, namely the
187
00:12:46,170 --> 00:12:51,730
relationship between a time
shift and a phase change.
188
00:12:51,730 --> 00:12:56,490
Now, in continuous time, of
course, we saw that a time
189
00:12:56,490 --> 00:13:00,360
shift corresponds to a phase
change, and vice versa.
190
00:13:00,360 --> 00:13:04,460
Let's first look at the
relationship between shifting
191
00:13:04,460 --> 00:13:07,470
time and generating
a change in phase.
192
00:13:07,470 --> 00:13:12,400
In particular for discrete time,
if I implement a time
193
00:13:12,400 --> 00:13:16,480
shift that generates
a phase change--
194
00:13:16,480 --> 00:13:21,160
and we can see that easily
by simply inserting a
195
00:13:21,160 --> 00:13:24,300
time shift, n + n_0.
196
00:13:24,300 --> 00:13:29,860
And if we expand out this
argument, we have omega_0 n +
197
00:13:29,860 --> 00:13:31,670
omega_0 n_0.
198
00:13:31,670 --> 00:13:34,300
And so I've done that on the
right-hand side of the
199
00:13:34,300 --> 00:13:36,900
equation here.
200
00:13:36,900 --> 00:13:41,300
And the omega_0 n_0, then,
simply corresponds
201
00:13:41,300 --> 00:13:44,360
to a change in phase.
202
00:13:44,360 --> 00:13:51,390
So clearly, a shift in time
generates a change in phase.
203
00:13:51,390 --> 00:13:57,650
And for example, if we take a
particular sinusoidal signal,
204
00:13:57,650 --> 00:14:01,050
let's say we take the cosine
signal at a particular
205
00:14:01,050 --> 00:14:06,830
frequency, and with a phase
equal to 0, a sequence that we
206
00:14:06,830 --> 00:14:12,370
might generate is one that
I've illustrated here.
207
00:14:12,370 --> 00:14:16,510
So what I'm illustrating
here is the cosine
208
00:14:16,510 --> 00:14:20,230
signal with 0 phase.
209
00:14:20,230 --> 00:14:24,570
And it has a particular behavior
to it, which will
210
00:14:24,570 --> 00:14:27,600
depend somewhat on
the frequency.
211
00:14:27,600 --> 00:14:34,310
If I now take this same sequence
and shift it so that
212
00:14:34,310 --> 00:14:39,230
the time origin is shifted a
quarter of a period away, then
213
00:14:39,230 --> 00:14:41,140
you can convince yourself--
214
00:14:41,140 --> 00:14:43,780
and it's straightforward to
work out-- that that time
215
00:14:43,780 --> 00:14:49,030
shift corresponds to a
phase shift of pi/2.
216
00:14:49,030 --> 00:14:57,580
So in that case, with the cosine
with a phase of -pi/2,
217
00:14:57,580 --> 00:15:05,410
that will correspond to the
expression that I have here.
218
00:15:05,410 --> 00:15:07,900
We could alternately write
that, using again a
219
00:15:07,900 --> 00:15:12,080
trigonometric identity,
as a sine function.
220
00:15:12,080 --> 00:15:17,790
And that, I've stated, is
equivalent to a time shift.
221
00:15:17,790 --> 00:15:23,360
Namely, this shift of pi/2 is
equal to a certain time shift,
222
00:15:23,360 --> 00:15:27,470
and the time shift for this
particular example is a
223
00:15:27,470 --> 00:15:29,560
quarter of a period.
224
00:15:29,560 --> 00:15:35,210
So here, we have the sinusoid.
225
00:15:35,210 --> 00:15:37,700
Previously we had the cosine.
226
00:15:37,700 --> 00:15:40,590
The cosine was exactly the same
sequence, but with the
227
00:15:40,590 --> 00:15:42,520
origin located here.
228
00:15:42,520 --> 00:15:45,310
And in fact, that's exactly the
way we drew this graph.
229
00:15:45,310 --> 00:15:49,120
Namely, we just simply took the
same values and changed
230
00:15:49,120 --> 00:15:51,330
the time origin.
231
00:15:51,330 --> 00:15:56,070
Now, looking at this sequence,
which is the sinusoidal
232
00:15:56,070 --> 00:16:01,120
sequence, the phase of -pi/2,
that has a certain symmetry.
233
00:16:01,120 --> 00:16:05,440
And in fact, what we see is that
it has an odd symmetry,
234
00:16:05,440 --> 00:16:07,790
just as in the continuous-time
case.
235
00:16:07,790 --> 00:16:11,190
Namely, if we take that
sequence, flip it about the
236
00:16:11,190 --> 00:16:15,080
axis, and flip it over in sign,
that we get the same
237
00:16:15,080 --> 00:16:17,380
sequence back again.
238
00:16:17,380 --> 00:16:22,050
Whereas with 0 phase
corresponding to the cosine
239
00:16:22,050 --> 00:16:26,330
that I showed previously, that
has an even symmetry.
240
00:16:26,330 --> 00:16:31,070
Namely, if I flip it about the
time origin and don't do a
241
00:16:31,070 --> 00:16:34,650
sign reversal, then the sequence
is maintained.
242
00:16:34,650 --> 00:16:40,650
So here, we have an odd
symmetry, expressed
243
00:16:40,650 --> 00:16:43,840
mathematically as
I've indicated.
244
00:16:43,840 --> 00:16:46,980
Namely, replacing the
independent variable by its
245
00:16:46,980 --> 00:16:52,370
negative attaches a negative
sign to the whole sequence.
246
00:16:52,370 --> 00:17:00,660
Whereas in the previous case,
what we have is 0 phase and an
247
00:17:00,660 --> 00:17:02,230
even symmetry.
248
00:17:02,230 --> 00:17:06,240
And that's expressed
mathematically as x[n]
249
00:17:06,240 --> 00:17:07,490
= x[-n].
250
00:17:07,490 --> 00:17:11,839
251
00:17:11,839 --> 00:17:14,900
Now, one of the things I've
said so far about
252
00:17:14,900 --> 00:17:19,849
discrete-time sinusoids is that
a time shift corresponds
253
00:17:19,849 --> 00:17:22,140
to a phase change.
254
00:17:22,140 --> 00:17:26,960
And we can then ask whether the
reverse statement is also
255
00:17:26,960 --> 00:17:29,950
true, and we knew that the
reverse statement was true in
256
00:17:29,950 --> 00:17:32,160
continuous time.
257
00:17:32,160 --> 00:17:38,920
Specifically, is it true that
a phase change always
258
00:17:38,920 --> 00:17:40,580
corresponds to a time shift?
259
00:17:40,580 --> 00:17:43,100
Now, we know that that is
true, namely, that this
260
00:17:43,100 --> 00:17:47,160
statement works both ways
in continuous time.
261
00:17:47,160 --> 00:17:49,300
Does it in discrete time?
262
00:17:49,300 --> 00:17:54,040
Well, the answer, somewhat
interestingly or surprisingly
263
00:17:54,040 --> 00:17:58,480
until you sit down and think
about it, is no.
264
00:17:58,480 --> 00:18:02,460
It is not necessarily true in
discrete time that any phase
265
00:18:02,460 --> 00:18:06,830
change can be interpreted
as a simple time
266
00:18:06,830 --> 00:18:08,430
shift of the sequence.
267
00:18:08,430 --> 00:18:12,210
And let me just indicate
what the problem is.
268
00:18:12,210 --> 00:18:17,880
If we look at the relationship
between the left side and the
269
00:18:17,880 --> 00:18:21,660
right side of this equation,
expanding this out as we did
270
00:18:21,660 --> 00:18:27,950
previously, we have that omega_0
n + omega_0 n_0 must
271
00:18:27,950 --> 00:18:31,370
correspond to omega_0 n + phi.
272
00:18:31,370 --> 00:18:35,200
And so omega_0 n_0
must correspond
273
00:18:35,200 --> 00:18:37,630
to the phase change.
274
00:18:37,630 --> 00:18:43,200
Now, what you can see pretty
clearly is that depending on
275
00:18:43,200 --> 00:18:48,730
the relationship between phi
and omega_0, n_0 may or may
276
00:18:48,730 --> 00:18:51,950
not come out to be an integer.
277
00:18:51,950 --> 00:18:56,400
Now, in continuous time, the
amount of time shift did not
278
00:18:56,400 --> 00:18:58,350
have to be an integer amount.
279
00:18:58,350 --> 00:19:01,520
In discrete time, when we talk
about a time shift, the amount
280
00:19:01,520 --> 00:19:03,760
of time shift-- obviously,
because of the nature of
281
00:19:03,760 --> 00:19:05,270
discrete time signals--
282
00:19:05,270 --> 00:19:06,640
must be an integer.
283
00:19:06,640 --> 00:19:12,570
So the phase changes related
to time shifts must satisfy
284
00:19:12,570 --> 00:19:14,580
this particular relationship.
285
00:19:14,580 --> 00:19:19,380
Namely, that omega_0 n_0, where
n_0 is an integer, is
286
00:19:19,380 --> 00:19:21,145
equal to the change in phase.
287
00:19:21,145 --> 00:19:24,940
288
00:19:24,940 --> 00:19:25,480
OK.
289
00:19:25,480 --> 00:19:29,350
Now, that's one distinction
between continuous time and
290
00:19:29,350 --> 00:19:30,290
discrete time.
291
00:19:30,290 --> 00:19:34,530
Let's now focus on another
one, namely the issue of
292
00:19:34,530 --> 00:19:36,190
periodicity.
293
00:19:36,190 --> 00:19:39,930
And what we'll see is that
again, whereas in continuous
294
00:19:39,930 --> 00:19:44,930
time, all continuous-time
sinusoids are periodic, in the
295
00:19:44,930 --> 00:19:48,220
discrete-time case that is
not necessarily true.
296
00:19:48,220 --> 00:19:51,480
297
00:19:51,480 --> 00:19:56,960
To explore that a little more
carefully, let's look at the
298
00:19:56,960 --> 00:20:00,750
expression, again, for a general
sinusoidal signal with
299
00:20:00,750 --> 00:20:05,520
an arbitrary amplitude,
frequency, and phase.
300
00:20:05,520 --> 00:20:10,590
And for this to be periodic,
what we require is that there
301
00:20:10,590 --> 00:20:15,850
be some value, N, under which,
when we shift the sequence by
302
00:20:15,850 --> 00:20:19,150
that amount, we get the same
sequence back again.
303
00:20:19,150 --> 00:20:22,000
And the smallest-value
N is what we've
304
00:20:22,000 --> 00:20:24,050
defined as the period.
305
00:20:24,050 --> 00:20:27,870
Now, when we try that on a
sinusoid, we of course
306
00:20:27,870 --> 00:20:35,460
substitute in for n, n + N.
And when we expand out the
307
00:20:35,460 --> 00:20:41,910
argument here, we'll get the
argument that I have on the
308
00:20:41,910 --> 00:20:44,210
right-hand side.
309
00:20:44,210 --> 00:20:48,960
And in order for this to repeat,
in other words, in
310
00:20:48,960 --> 00:20:55,630
order for us to discard this
term, omega_0 N, where N is
311
00:20:55,630 --> 00:21:00,100
the period, must be an integer
multiple of 2 pi.
312
00:21:00,100 --> 00:21:06,720
And in that case, it's periodic
as long as omega_0 N,
313
00:21:06,720 --> 00:21:11,550
N being the period, is 2
pi times an integer.
314
00:21:11,550 --> 00:21:15,550
Just simply dividing this out,
we have N, the period, is 2 pi
315
00:21:15,550 --> 00:21:20,900
m / omega_0.
316
00:21:20,900 --> 00:21:23,880
Well, you could say, OK
what's the big deal?
317
00:21:23,880 --> 00:21:26,690
Whatever N happens to come out
to be when we do that little
318
00:21:26,690 --> 00:21:29,010
bit of algebra, that's
the period.
319
00:21:29,010 --> 00:21:35,910
But in fact, N, or 2 pi m /
omega_0, may not ever come out
320
00:21:35,910 --> 00:21:37,270
to be an integer.
321
00:21:37,270 --> 00:21:38,970
Or it may not come out
to be the one that
322
00:21:38,970 --> 00:21:39,850
you thought it might.
323
00:21:39,850 --> 00:21:43,340
For example, let's look at some
324
00:21:43,340 --> 00:21:45,895
particular sinusoidal signals.
325
00:21:45,895 --> 00:21:49,050
326
00:21:49,050 --> 00:21:49,430
Let's see.
327
00:21:49,430 --> 00:21:53,160
We have the first one
here, which is a
328
00:21:53,160 --> 00:21:55,880
sinusoid, as I've shown.
329
00:21:55,880 --> 00:21:59,110
And it has a frequency, what
I've referred to as the
330
00:21:59,110 --> 00:22:05,130
frequency, omega_0
= 2 pi / 12.
331
00:22:05,130 --> 00:22:11,720
And what we'd like to look at
is 2 pi / omega_0, then find
332
00:22:11,720 --> 00:22:14,070
an integer to multiply
that by in order
333
00:22:14,070 --> 00:22:15,740
to get another integer.
334
00:22:15,740 --> 00:22:17,110
Let's just try that here.
335
00:22:17,110 --> 00:22:24,760
If we look at 2 pi / omega_0, 2
pi / omega_0, for this case,
336
00:22:24,760 --> 00:22:27,130
is equal to 12.
337
00:22:27,130 --> 00:22:28,530
Well, that's fine.
338
00:22:28,530 --> 00:22:29,890
12 is an integer.
339
00:22:29,890 --> 00:22:35,030
So what that says is that this
sinusoidal signal is periodic.
340
00:22:35,030 --> 00:22:39,990
And in fact, it's periodic
with a period of 12.
341
00:22:39,990 --> 00:22:41,820
Let's look at the next one.
342
00:22:41,820 --> 00:22:46,810
The next one, we would have
2 pi / omega_0 again.
343
00:22:46,810 --> 00:22:48,060
And that's equal to 31/4.
344
00:22:48,060 --> 00:22:50,560
345
00:22:50,560 --> 00:22:57,080
So what that says is that
the period is 31/4.
346
00:22:57,080 --> 00:22:58,440
But wait a minute.
347
00:22:58,440 --> 00:23:00,730
31/4 isn't an integer.
348
00:23:00,730 --> 00:23:02,820
We have to multiply
that by an integer
349
00:23:02,820 --> 00:23:04,730
to get another integer.
350
00:23:04,730 --> 00:23:09,680
Well, we'd multiply that by 4,
so (2 pi / omega_0) times 4 is
351
00:23:09,680 --> 00:23:11,790
31, 31 is an integer.
352
00:23:11,790 --> 00:23:16,290
And so what that says is this is
periodic, not with a period
353
00:23:16,290 --> 00:23:22,130
of 2 pi / omega_0, but with a
period of (2 pi / omega_0)
354
00:23:22,130 --> 00:23:29,950
times 4, namely with
a period of 31.
355
00:23:29,950 --> 00:23:34,310
Finally, let's take the example
where omega_0 is equal
356
00:23:34,310 --> 00:23:37,030
to 1/6, as I've shown here.
357
00:23:37,030 --> 00:23:40,330
That actually looks, if you
track it with your eye, like
358
00:23:40,330 --> 00:23:41,970
it's periodic.
359
00:23:41,970 --> 00:23:47,660
2 pi / omega_0, in that case,
is equal to 12 pi.
360
00:23:47,660 --> 00:23:53,070
Well, what integer can I
multiply 12 pi by and get
361
00:23:53,070 --> 00:23:54,400
another integer?
362
00:23:54,400 --> 00:23:58,350
The answer is none, because pi
is an irrational number.
363
00:23:58,350 --> 00:24:03,860
So in fact, what that says is
that if you look at this
364
00:24:03,860 --> 00:24:09,380
sinusoidal signal, it's not
periodic at all, even though
365
00:24:09,380 --> 00:24:13,120
you might fool yourself into
thinking it is simply because
366
00:24:13,120 --> 00:24:15,830
the envelope looks periodic.
367
00:24:15,830 --> 00:24:19,510
Namely, the continuous-time
equivalent of this is
368
00:24:19,510 --> 00:24:22,580
periodic, the discrete-time
sequence is not.
369
00:24:22,580 --> 00:24:26,750
370
00:24:26,750 --> 00:24:27,180
OK.
371
00:24:27,180 --> 00:24:31,860
Well, we've seen, then, some
important distinctions between
372
00:24:31,860 --> 00:24:34,780
continuous-time sinusoidal
signals and discrete-time
373
00:24:34,780 --> 00:24:36,730
sinusoidal signals.
374
00:24:36,730 --> 00:24:41,900
The first one is the fact that
in the continuous-time case, a
375
00:24:41,900 --> 00:24:45,830
time shift and phase change
are always equivalent.
376
00:24:45,830 --> 00:24:49,210
Whereas in the discrete-time
case, in effect, it works one
377
00:24:49,210 --> 00:24:52,400
way but not the other way.
378
00:24:52,400 --> 00:24:58,060
We've also seen that for a
continuous-time signal, the
379
00:24:58,060 --> 00:25:01,990
continuous-time signal is always
periodic, whereas the
380
00:25:01,990 --> 00:25:04,850
discrete-time signal
is not necessarily.
381
00:25:04,850 --> 00:25:08,640
In particular, for the
continuous-time case, if we
382
00:25:08,640 --> 00:25:12,320
have a general expression for
the sinusoidal signal that
383
00:25:12,320 --> 00:25:15,650
I've indicated here,
that's periodic for
384
00:25:15,650 --> 00:25:18,440
any choice of omega_0.
385
00:25:18,440 --> 00:25:30,880
Whereas in the discrete-time
case, it's periodic only if 2
386
00:25:30,880 --> 00:25:34,740
pi / omega_0 can be multiplied
by an integer
387
00:25:34,740 --> 00:25:37,020
to get another integer.
388
00:25:37,020 --> 00:25:41,470
Now, another important and,
as it turns out, useful
389
00:25:41,470 --> 00:25:44,940
distinction between the
continuous-time and
390
00:25:44,940 --> 00:25:49,370
discrete-time case is the fact
that in the discrete-time
391
00:25:49,370 --> 00:25:56,550
case, as we vary what I've
called the frequency omega_0,
392
00:25:56,550 --> 00:26:02,390
we only see distinct
signals as omega_0
393
00:26:02,390 --> 00:26:04,900
varies over a 2 pi interval.
394
00:26:04,900 --> 00:26:09,940
And if we let omega_0 vary
outside the range of, let's
395
00:26:09,940 --> 00:26:15,130
say, -pi to pi, or 0 to 2 pi,
we'll see the same sequences
396
00:26:15,130 --> 00:26:18,870
all over again, even though at
first glance, the mathematical
397
00:26:18,870 --> 00:26:20,870
expression might
look different.
398
00:26:20,870 --> 00:26:26,980
So in the discrete-time case,
this class of signals is
399
00:26:26,980 --> 00:26:33,340
identical for values of omega_0
separated by 2 pi,
400
00:26:33,340 --> 00:26:37,890
whereas in the continuous-time
case, that is not true.
401
00:26:37,890 --> 00:26:41,020
In particular, if I consider
these sinusoidal
402
00:26:41,020 --> 00:26:48,260
continuous-time signals, as I
vary omega_0, what will happen
403
00:26:48,260 --> 00:26:53,200
is that I will always see
different sinusoidal signals.
404
00:26:53,200 --> 00:26:55,210
Namely, these won't be equal.
405
00:26:55,210 --> 00:27:00,970
And in effect, we can justify
that statement algebraically.
406
00:27:00,970 --> 00:27:04,410
And I won't take the time
to do it carefully.
407
00:27:04,410 --> 00:27:09,670
But let's look, first of all,
at the discrete-time case.
408
00:27:09,670 --> 00:27:15,130
And the statement that I'm
making is that if I have two
409
00:27:15,130 --> 00:27:18,700
discrete-time sinusoidal signals
at two different
410
00:27:18,700 --> 00:27:24,980
frequencies, and if these
frequencies are separated by
411
00:27:24,980 --> 00:27:30,170
an integer multiple of 2 pi--
namely if omega_2 is equal to
412
00:27:30,170 --> 00:27:35,650
omega_1 + 2 pi times
an integer m--
413
00:27:35,650 --> 00:27:40,070
when I substitute this into this
expression, because of
414
00:27:40,070 --> 00:27:46,330
the fact that n is also an
integer, I'll have m * n as an
415
00:27:46,330 --> 00:27:48,270
integer multiple of 2 pi.
416
00:27:48,270 --> 00:27:50,520
And that term, of course, will
disappear because of the
417
00:27:50,520 --> 00:27:53,370
periodicity of the sinusoid,
and these two
418
00:27:53,370 --> 00:27:56,350
sequences will be equal.
419
00:27:56,350 --> 00:28:03,450
On the other hand in the
continuous-time case, since t
420
00:28:03,450 --> 00:28:06,560
is not restricted to be an
integer variable, for
421
00:28:06,560 --> 00:28:11,230
different values of omega_1 and
omega_2, these sinusoidal
422
00:28:11,230 --> 00:28:13,770
signals will always
be different.
423
00:28:13,770 --> 00:28:17,280
424
00:28:17,280 --> 00:28:17,650
OK.
425
00:28:17,650 --> 00:28:21,540
Now, many of the issues that
I've raised so far, in
426
00:28:21,540 --> 00:28:24,550
relation to sinusoidal signals,
are elaborated on in
427
00:28:24,550 --> 00:28:26,240
more detail in the text.
428
00:28:26,240 --> 00:28:30,890
And of course, you'll have an
opportunity to exercise some
429
00:28:30,890 --> 00:28:36,150
of this as you work through
the video course manual.
430
00:28:36,150 --> 00:28:40,550
Let me stress that sinusoidal
signals will play an extremely
431
00:28:40,550 --> 00:28:44,260
important role for us as
building blocks for general
432
00:28:44,260 --> 00:28:47,770
signals and descriptions of
systems, and leads to the
433
00:28:47,770 --> 00:28:51,870
whole concept Fourier analysis,
which is very
434
00:28:51,870 --> 00:28:53,755
heavily exploited throughout
the course.
435
00:28:53,755 --> 00:28:56,760
436
00:28:56,760 --> 00:29:02,870
What I'd now like to turn to is
another class of important
437
00:29:02,870 --> 00:29:03,780
building blocks.
438
00:29:03,780 --> 00:29:06,820
And in fact, we'll see that
under certain conditions,
439
00:29:06,820 --> 00:29:10,970
these relate strongly to
sinusoidal signals, namely the
440
00:29:10,970 --> 00:29:14,580
class of real and complex
exponentials.
441
00:29:14,580 --> 00:29:19,810
Let me begin, first of all, with
the real exponential, and
442
00:29:19,810 --> 00:29:23,420
in particular, in the
continuous-time case.
443
00:29:23,420 --> 00:29:26,980
A real continuous-time
exponential is mathematically
444
00:29:26,980 --> 00:29:36,000
expressed, as I indicate here,
x(t) = C e ^ (a t), where for
445
00:29:36,000 --> 00:29:40,320
the real exponential, C and
a are real numbers.
446
00:29:40,320 --> 00:29:43,900
And that's what we mean by
the real exponential.
447
00:29:43,900 --> 00:29:47,010
Shortly, we'll also consider
complex exponentials, where
448
00:29:47,010 --> 00:29:50,240
these numbers can then
become complex.
449
00:29:50,240 --> 00:29:54,120
So this is an exponential
function.
450
00:29:54,120 --> 00:29:59,220
And for example, if the
parameter a is positive, that
451
00:29:59,220 --> 00:30:03,320
means that we have a growing
exponential function.
452
00:30:03,320 --> 00:30:08,410
If the parameter a is negative,
then that means that
453
00:30:08,410 --> 00:30:13,760
we have a decaying exponential
function.
454
00:30:13,760 --> 00:30:17,750
Now, somewhat as an aside, it's
kind of interesting to
455
00:30:17,750 --> 00:30:23,160
note that for exponentials, a
time shift corresponds to a
456
00:30:23,160 --> 00:30:26,950
scale change, which is somewhat
different than what
457
00:30:26,950 --> 00:30:29,700
happens with sinusoids.
458
00:30:29,700 --> 00:30:33,380
In the sinusoidal case, we saw
that a time shift corresponded
459
00:30:33,380 --> 00:30:34,940
to a phase change.
460
00:30:34,940 --> 00:30:37,850
With the real exponential, a
time shift, as it turns out,
461
00:30:37,850 --> 00:30:41,070
corresponds to simply
changing the scale.
462
00:30:41,070 --> 00:30:45,730
There's nothing particularly
crucial or
463
00:30:45,730 --> 00:30:47,300
exciting about that.
464
00:30:47,300 --> 00:30:51,050
And in fact, perhaps stressing
it is a little misleading.
465
00:30:51,050 --> 00:30:53,620
For general functions, of
course, about all that you can
466
00:30:53,620 --> 00:30:58,430
say about what happens when you
implement a time shift is
467
00:30:58,430 --> 00:31:02,720
that it implements
a time shift.
468
00:31:02,720 --> 00:31:03,060
OK.
469
00:31:03,060 --> 00:31:05,890
So here's the real
exponential.
470
00:31:05,890 --> 00:31:08,600
Just C e ^ (a t).
471
00:31:08,600 --> 00:31:12,160
Let's look at the real
exponential, now, in the
472
00:31:12,160 --> 00:31:14,480
discrete-time case.
473
00:31:14,480 --> 00:31:21,550
And in the discrete-time case,
we have several alternate ways
474
00:31:21,550 --> 00:31:23,910
of expressing it.
475
00:31:23,910 --> 00:31:29,020
We can express the real
exponential in the form C e ^
476
00:31:29,020 --> 00:31:35,120
(beta n), or as we'll find more
convenient, in part for a
477
00:31:35,120 --> 00:31:38,810
reason at I'll indicate shortly,
we can rewrite this
478
00:31:38,810 --> 00:31:44,820
as C alpha ^ n, where of course,
alpha = e ^ beta.
479
00:31:44,820 --> 00:31:47,610
More typically in the
discrete-time case, we'll
480
00:31:47,610 --> 00:31:53,720
express the exponential
as C alpha ^ n.
481
00:31:53,720 --> 00:31:57,690
So for example, this becomes,
essentially, a geometric
482
00:31:57,690 --> 00:32:01,920
series or progression
as n continues for
483
00:32:01,920 --> 00:32:03,740
certain values of alpha.
484
00:32:03,740 --> 00:32:11,070
Here for example, we have for
alpha greater than 0, first of
485
00:32:11,070 --> 00:32:15,010
all on the top, the case where
the magnitude of alpha is
486
00:32:15,010 --> 00:32:21,430
greater than 1, so that the
sequence is exponentially or
487
00:32:21,430 --> 00:32:23,420
geometrically growing.
488
00:32:23,420 --> 00:32:27,770
On the bottom, again with alpha
positive, but now with
489
00:32:27,770 --> 00:32:32,517
its magnitude less than 1, we
have a geometric progression
490
00:32:32,517 --> 00:32:38,360
that is exponentially or
geometrically decaying.
491
00:32:38,360 --> 00:32:38,650
OK.
492
00:32:38,650 --> 00:32:42,350
So this, in both of these
cases, is with alpha
493
00:32:42,350 --> 00:32:45,070
greater than 0.
494
00:32:45,070 --> 00:32:48,480
Now the function that we're
talking about is alpha ^ n.
495
00:32:48,480 --> 00:32:51,310
And of course, what you can
see is that if alpha is
496
00:32:51,310 --> 00:32:57,910
negative instead of positive,
then when n is even, that
497
00:32:57,910 --> 00:32:59,620
minus sign is going
to disappear.
498
00:32:59,620 --> 00:33:02,320
When n is odd, there will
be a minus sign.
499
00:33:02,320 --> 00:33:06,610
And so for alpha negative, the
sequence is going to alternate
500
00:33:06,610 --> 00:33:08,500
positive and negative values.
501
00:33:08,500 --> 00:33:15,870
So for example, here we have
alpha negative, with its
502
00:33:15,870 --> 00:33:17,230
magnitude less than 1.
503
00:33:17,230 --> 00:33:21,130
And you can see that, again,
its envelope decays
504
00:33:21,130 --> 00:33:25,320
geometrically, and the values
alternate in sign.
505
00:33:25,320 --> 00:33:29,310
And here we have the magnitude
of alpha greater than 1, with
506
00:33:29,310 --> 00:33:30,600
alpha negative.
507
00:33:30,600 --> 00:33:33,360
Again, they alternate in sign,
and of course it's growing
508
00:33:33,360 --> 00:33:35,740
geometrically.
509
00:33:35,740 --> 00:33:42,910
Now, if you think about alpha
positive and go back to the
510
00:33:42,910 --> 00:33:48,640
expression that I have at the
top, namely C alpha ^ n.
511
00:33:48,640 --> 00:33:54,670
With alpha positive, you can
see a straightforward
512
00:33:54,670 --> 00:33:58,440
relationship between
alpha and beta.
513
00:33:58,440 --> 00:34:01,830
Namely, beta is the natural
logarithm of alpha.
514
00:34:01,830 --> 00:34:04,520
515
00:34:04,520 --> 00:34:06,830
Something to think
about is what
516
00:34:06,830 --> 00:34:09,070
happens if alpha is negative?
517
00:34:09,070 --> 00:34:13,230
Which is, of course, a very
important and useful class of
518
00:34:13,230 --> 00:34:16,190
real discrete-time exponentials
also.
519
00:34:16,190 --> 00:34:19,560
Well, it turns out that with
alpha negative, if you try to
520
00:34:19,560 --> 00:34:25,020
express it as C e ^ (beta n),
then beta comes out to be an
521
00:34:25,020 --> 00:34:26,659
imaginary number.
522
00:34:26,659 --> 00:34:30,889
And that is one, but not the
only reason why, in the
523
00:34:30,889 --> 00:34:36,389
discrete-time case, it's often
most convenient to phrase real
524
00:34:36,389 --> 00:34:42,420
exponentials in the form alpha ^
n, rather than e ^ (beta n).
525
00:34:42,420 --> 00:34:47,650
In other words, to express them
in this form rather than
526
00:34:47,650 --> 00:34:48,900
in this form.
527
00:34:48,900 --> 00:34:54,520
528
00:34:54,520 --> 00:34:56,699
Those are real exponentials,
continuous-time and
529
00:34:56,699 --> 00:34:57,960
discrete-time.
530
00:34:57,960 --> 00:35:01,710
Now let's look at the
continuous-time complex
531
00:35:01,710 --> 00:35:03,130
exponential.
532
00:35:03,130 --> 00:35:09,910
And what I mean by a complex
exponential, again, is an
533
00:35:09,910 --> 00:35:15,750
exponential of the
form C e ^ (a t).
534
00:35:15,750 --> 00:35:21,050
But in this case, we allow the
parameters C and a to be
535
00:35:21,050 --> 00:35:22,970
complex numbers.
536
00:35:22,970 --> 00:35:25,850
And let's just track this
through algebraically.
537
00:35:25,850 --> 00:35:28,820
If C and a are complex numbers,
let's write C in
538
00:35:28,820 --> 00:35:33,350
polar form, so it has a
magnitude and an angle.
539
00:35:33,350 --> 00:35:36,800
Let's write a in rectangular
form, so it has a real part
540
00:35:36,800 --> 00:35:38,950
and an imaginary part.
541
00:35:38,950 --> 00:35:43,060
And when we substitute these
two in here, combine some
542
00:35:43,060 --> 00:35:44,680
things together--
543
00:35:44,680 --> 00:35:46,100
well actually, I haven't
combined yet.
544
00:35:46,100 --> 00:35:52,160
I have this for the amplitude
factor, and this for the
545
00:35:52,160 --> 00:35:53,960
exponential factor.
546
00:35:53,960 --> 00:35:58,597
I can now pull out of this the
term corresponding to e ^ (r
547
00:35:58,597 --> 00:36:02,320
t), and combine the imaginary
parts together.
548
00:36:02,320 --> 00:36:05,760
And I come down to the
expression that I have here.
549
00:36:05,760 --> 00:36:12,800
So following this further, an
exponential of this form, e ^
550
00:36:12,800 --> 00:36:19,640
(j omega) or e ^ (j phi), using
Euler's relation, can be
551
00:36:19,640 --> 00:36:28,150
expressed as the sum of a cosine
plus j times a sine.
552
00:36:28,150 --> 00:36:31,560
And so that corresponds
to this factor.
553
00:36:31,560 --> 00:36:35,290
And then there is this
time-varying amplitude factor
554
00:36:35,290 --> 00:36:36,420
on top of it.
555
00:36:36,420 --> 00:36:39,390
Finally putting those together,
we end up with the
556
00:36:39,390 --> 00:36:42,580
expression that I show
on the bottom.
557
00:36:42,580 --> 00:36:49,010
And what this corresponds to are
two sinusoidal signals, 90
558
00:36:49,010 --> 00:36:51,733
degrees out of phase, as
indicated by the fact that
559
00:36:51,733 --> 00:36:54,220
there's a cosine and a sine.
560
00:36:54,220 --> 00:36:57,560
So there's a real part and
an imaginary part, with
561
00:36:57,560 --> 00:37:03,210
sinusoidal components 90 degrees
out of phase, and a
562
00:37:03,210 --> 00:37:07,140
time-varying amplitude factor,
which is a real exponential.
563
00:37:07,140 --> 00:37:11,670
So it's a sinusoid multiplied by
a real exponential in both
564
00:37:11,670 --> 00:37:14,870
the real part and the
imaginary part.
565
00:37:14,870 --> 00:37:19,070
And let's just see what one of
those terms might look like.
566
00:37:19,070 --> 00:37:29,290
What I've indicated at the top
is a sinusoidal signal with a
567
00:37:29,290 --> 00:37:32,330
time-varying exponential
envelope, or an envelope which
568
00:37:32,330 --> 00:37:37,780
is a real exponential, and in
particular which is growing,
569
00:37:37,780 --> 00:37:40,960
namely with r greater than 0.
570
00:37:40,960 --> 00:37:44,430
And on the bottom, I've
indicated the same thing with
571
00:37:44,430 --> 00:37:46,240
r less than 0.
572
00:37:46,240 --> 00:37:51,300
And this kind of sinusoidal
signal, by the way, is
573
00:37:51,300 --> 00:37:54,730
typically referred to as
a damped sinusoid.
574
00:37:54,730 --> 00:37:58,110
So with r negative, what we have
in the real and imaginary
575
00:37:58,110 --> 00:38:01,130
parts are damped sinusoids.
576
00:38:01,130 --> 00:38:07,270
And the sinusoidal components of
that are 90 degrees out of
577
00:38:07,270 --> 00:38:12,260
phase, in the real part and
in the imaginary part.
578
00:38:12,260 --> 00:38:12,560
OK.
579
00:38:12,560 --> 00:38:17,040
Now, in the discrete-time case,
we have more or less the
580
00:38:17,040 --> 00:38:19,350
same kind of outcome.
581
00:38:19,350 --> 00:38:26,220
In particular we'll make
reference to our complex
582
00:38:26,220 --> 00:38:28,860
exponentials in the
discrete-time case.
583
00:38:28,860 --> 00:38:33,200
The expression for the complex
exponential looks very much
584
00:38:33,200 --> 00:38:36,430
like the expression for the real
exponential, except that
585
00:38:36,430 --> 00:38:40,460
now we have complex factors.
586
00:38:40,460 --> 00:38:46,040
So C and alpha are
complex numbers.
587
00:38:46,040 --> 00:38:53,050
And again, if we track through
the algebra, and get to a
588
00:38:53,050 --> 00:38:57,740
point where we have a real
exponential multiplied by a
589
00:38:57,740 --> 00:39:03,930
factor which is a purely
imaginary exponential, apply
590
00:39:03,930 --> 00:39:10,430
Euler's relationship to this, we
then finally come down to a
591
00:39:10,430 --> 00:39:16,980
sequence, which has a real
exponential amplitude
592
00:39:16,980 --> 00:39:21,480
multiplying one sinusoid
in the real part.
593
00:39:21,480 --> 00:39:25,120
And in the imaginary part,
exactly the same kind of
594
00:39:25,120 --> 00:39:30,560
exponential multiplying a
sinusoid that's 90 degrees out
595
00:39:30,560 --> 00:39:33,150
of phase from that.
596
00:39:33,150 --> 00:39:37,750
And so if we look at what one
of these factors might look
597
00:39:37,750 --> 00:39:42,510
like, it's what we would expect
given the analogy with
598
00:39:42,510 --> 00:39:44,580
the continuous-time case.
599
00:39:44,580 --> 00:39:53,140
Namely, it's a sinusoidal
sequence with a real
600
00:39:53,140 --> 00:39:55,060
exponential envelope.
601
00:39:55,060 --> 00:39:58,500
In the case where alpha
is positive, then
602
00:39:58,500 --> 00:40:00,250
it's a growing envelope.
603
00:40:00,250 --> 00:40:03,050
In the case where alpha
is negative--
604
00:40:03,050 --> 00:40:03,990
I'm sorry--
605
00:40:03,990 --> 00:40:08,200
where the magnitude of alpha
is greater than 1, it's a
606
00:40:08,200 --> 00:40:10,070
growing exponential envelope.
607
00:40:10,070 --> 00:40:14,700
Where the magnitude of alpha
is less than 1, it's a
608
00:40:14,700 --> 00:40:17,570
decaying exponential envelope.
609
00:40:17,570 --> 00:40:20,370
And so I've illustrated
that here.
610
00:40:20,370 --> 00:40:24,050
Here we have the magnitude
of alpha greater than 1.
611
00:40:24,050 --> 00:40:27,510
And here we have the magnitude
of alpha less than 1.
612
00:40:27,510 --> 00:40:30,960
In both cases, sinusoidal
sequences underneath the
613
00:40:30,960 --> 00:40:35,920
envelope, and then an envelope
that is dictated by what the
614
00:40:35,920 --> 00:40:39,710
magnitude of alpha is.
615
00:40:39,710 --> 00:40:40,200
OK.
616
00:40:40,200 --> 00:40:44,900
Now, in the discrete-time case,
then, we have results
617
00:40:44,900 --> 00:40:47,080
similar to the continuous-time
case.
618
00:40:47,080 --> 00:40:53,170
Namely, components in a real and
imaginary part that have a
619
00:40:53,170 --> 00:40:57,370
real exponential factor
times a sinusoid.
620
00:40:57,370 --> 00:41:07,240
Of course, if the magnitude of
alpha is equal to 1, then this
621
00:41:07,240 --> 00:41:10,010
factor disappears,
or is equal to 1.
622
00:41:10,010 --> 00:41:12,350
And this factor is equal to 1.
623
00:41:12,350 --> 00:41:15,250
And so we have sinusoids
in both the real
624
00:41:15,250 --> 00:41:17,620
and imaginary parts.
625
00:41:17,620 --> 00:41:22,660
Now, one can ask whether,
in general, the complex
626
00:41:22,660 --> 00:41:26,580
exponential with the magnitude
of alpha equal to 1 is
627
00:41:26,580 --> 00:41:28,640
periodic or not periodic.
628
00:41:28,640 --> 00:41:34,450
And the clue to that can be
inferred by examining this
629
00:41:34,450 --> 00:41:36,680
expression.
630
00:41:36,680 --> 00:41:39,600
In particular, in the
discrete-time case with the
631
00:41:39,600 --> 00:41:44,600
magnitude of alpha equal to 1,
we have pure sinusoids in the
632
00:41:44,600 --> 00:41:48,250
real part and the
imaginary part.
633
00:41:48,250 --> 00:41:51,240
And in fact, in a
continuous-time case with r
634
00:41:51,240 --> 00:41:56,910
equal to 0, we have sinusoids
in the real part and the
635
00:41:56,910 --> 00:41:59,570
imaginary part.
636
00:41:59,570 --> 00:42:02,520
In a continuous-time case when
we have a pure complex
637
00:42:02,520 --> 00:42:07,580
exponential, so that the terms
aren't exponentially growing
638
00:42:07,580 --> 00:42:09,410
or decaying, those
639
00:42:09,410 --> 00:42:13,420
exponentials are always periodic.
640
00:42:13,420 --> 00:42:17,630
Because, of course, the real
and imaginary sinusoidal
641
00:42:17,630 --> 00:42:19,640
components are periodic.
642
00:42:19,640 --> 00:42:23,100
In the discrete-time case, we
know that the sinusoids may or
643
00:42:23,100 --> 00:42:28,150
may not be periodic, depending
on the value of omega_0.
644
00:42:28,150 --> 00:42:31,390
And so in fact, in the
discrete-time case, the
645
00:42:31,390 --> 00:42:38,630
exponential e ^ (j omega_0 n),
that I've indicated here, may
646
00:42:38,630 --> 00:42:42,780
or may not be periodic depending
on what the value of
647
00:42:42,780 --> 00:42:44,030
omega_0 is.
648
00:42:44,030 --> 00:42:47,140
649
00:42:47,140 --> 00:42:47,660
OK.
650
00:42:47,660 --> 00:42:53,410
Now, to summarize, in this
lecture I've introduced and
651
00:42:53,410 --> 00:42:58,000
discussed a number of important
basic signals.
652
00:42:58,000 --> 00:43:04,710
In particular, sinusoids and
real and complex exponentials.
653
00:43:04,710 --> 00:43:07,660
One of the important outcomes of
the discussion, emphasized
654
00:43:07,660 --> 00:43:11,190
further in the text, is that
there are some very important
655
00:43:11,190 --> 00:43:13,060
similarities between them.
656
00:43:13,060 --> 00:43:16,700
But there are also some very
important differences.
657
00:43:16,700 --> 00:43:20,770
And these differences will
surface when we exploit
658
00:43:20,770 --> 00:43:26,220
sinusoids and complex
exponentials as basic building
659
00:43:26,220 --> 00:43:29,350
blocks for more general
continuous-time and
660
00:43:29,350 --> 00:43:32,160
discrete-time signals.
661
00:43:32,160 --> 00:43:35,900
In the next lecture, what I'll
discuss are some other very
662
00:43:35,900 --> 00:43:40,070
important building blocks,
namely, what are referred to
663
00:43:40,070 --> 00:43:43,230
as step signals and
impulse signals.
664
00:43:43,230 --> 00:43:47,970
And those, together with the
sinusoidal signals and
665
00:43:47,970 --> 00:43:52,160
exponentials as we've talked
about today, will really form
666
00:43:52,160 --> 00:43:57,300
the cornerstone for,
essentially, all of the signal
667
00:43:57,300 --> 00:44:00,270
and system analysis that we'll
be dealing with for the
668
00:44:00,270 --> 00:44:01,780
remainder of course.
669
00:44:01,780 --> 00:44:03,030
Thank you.
670
00:44:03,030 --> 00:44:07,381