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[MUSIC PLAYING]
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PROFESSOR: In the last lecture,
we discussed a number
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of general properties for
systems, which, as you recall,
12
00:01:04,610 --> 00:01:08,710
applied both to continuous-time
and to
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00:01:08,710 --> 00:01:12,060
discrete-time systems.
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00:01:12,060 --> 00:01:15,590
These properties were the
properties associated with a
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system having memory.
16
00:01:17,810 --> 00:01:22,390
The issue of whether a system
is or isn't invertible, we
17
00:01:22,390 --> 00:01:27,180
talked about causality and
stability, and finally we
18
00:01:27,180 --> 00:01:31,170
talked about when linearity
and time invariance.
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00:01:31,170 --> 00:01:34,710
In today's lecture, what
I'd like to do is focus
20
00:01:34,710 --> 00:01:39,210
specifically on linearity and
time invariance, and show how
21
00:01:39,210 --> 00:01:43,420
for systems that have those
properties, we can exploit
22
00:01:43,420 --> 00:01:47,840
them to generate a general
representation.
23
00:01:47,840 --> 00:01:50,610
Let me begin by just
reviewing the two
24
00:01:50,610 --> 00:01:52,700
properties again quickly.
25
00:01:52,700 --> 00:01:57,480
Time invariance, as you recall,
is a property that
26
00:01:57,480 --> 00:01:59,430
applied both to continuous-time
and
27
00:01:59,430 --> 00:02:05,280
discrete-time systems, and in
essence stated that for any
28
00:02:05,280 --> 00:02:10,070
given input and output
relationship if we simply
29
00:02:10,070 --> 00:02:13,470
shift the input,
then the output
30
00:02:13,470 --> 00:02:15,800
shifts by the same amount.
31
00:02:15,800 --> 00:02:20,320
And of course, exactly the same
kind of statement applied
32
00:02:20,320 --> 00:02:22,590
in discrete time.
33
00:02:22,590 --> 00:02:25,510
So time invariance was a
property that said that the
34
00:02:25,510 --> 00:02:28,850
system didn't care about
what the time origin
35
00:02:28,850 --> 00:02:31,540
of the signal is.
36
00:02:31,540 --> 00:02:35,810
Linearity was a property related
to the fact that if we
37
00:02:35,810 --> 00:02:39,390
have a set of outputs
associated with a
38
00:02:39,390 --> 00:02:42,040
given set of inputs--
39
00:02:42,040 --> 00:02:46,910
as I've indicated here with the
inputs as phi_k and the
40
00:02:46,910 --> 00:02:53,160
outputs psi_k, then the property
of linearity states
41
00:02:53,160 --> 00:02:58,250
that if we have an input which
is a linear combination of
42
00:02:58,250 --> 00:03:04,470
those inputs, then the output is
a linear combination of the
43
00:03:04,470 --> 00:03:06,250
associated outputs.
44
00:03:06,250 --> 00:03:11,590
So that linear combination of
inputs generates, for a linear
45
00:03:11,590 --> 00:03:16,600
system, an output which is a
linear combination of the
46
00:03:16,600 --> 00:03:19,600
associated outputs.
47
00:03:19,600 --> 00:03:23,940
Now the question is, how can we
exploit the properties of
48
00:03:23,940 --> 00:03:27,530
linearity and time invariance?
49
00:03:27,530 --> 00:03:31,420
There's a basic strategy which
will flow more or less through
50
00:03:31,420 --> 00:03:33,720
most of this course.
51
00:03:33,720 --> 00:03:39,660
The strategy is to attempt to
decompose a signal, either
52
00:03:39,660 --> 00:03:43,590
continuous-time or
discrete-time, into a set of
53
00:03:43,590 --> 00:03:46,630
basic signals.
54
00:03:46,630 --> 00:03:50,410
I've indicated that here.
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00:03:50,410 --> 00:03:56,000
And the question then is, what
basic signal should we pick?
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00:03:56,000 --> 00:03:59,670
Well, the answer, kind of, is we
should pick a set of basic
57
00:03:59,670 --> 00:04:03,210
signals that provide a certain
degree of analytical
58
00:04:03,210 --> 00:04:04,180
convenience.
59
00:04:04,180 --> 00:04:08,200
So we choose a set of inputs
for the decomposition that
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00:04:08,200 --> 00:04:13,810
provide outputs that we
can easily generate.
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00:04:13,810 --> 00:04:20,279
Now, as we'll see, when we do
this, there are two classes of
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inputs that are particularly
suited to that strategy.
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00:04:24,890 --> 00:04:31,170
One class is the set of delayed
impulses, namely
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00:04:31,170 --> 00:04:35,300
decomposing a signal into a
linear combination of these.
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00:04:35,300 --> 00:04:39,390
And as we'll see, that leads to
a representation for linear
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00:04:39,390 --> 00:04:45,840
time-invariant systems, which is
referred to as convolution.
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00:04:45,840 --> 00:04:50,650
The second is a decomposition
of inputs into complex
68
00:04:50,650 --> 00:04:52,380
exponentials--
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00:04:52,380 --> 00:04:55,450
a linear combination of
complex exponentials--
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00:04:55,450 --> 00:05:00,220
and that leads to a
representation of signals and
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00:05:00,220 --> 00:05:04,045
systems through what we'll refer
to as Fourier analysis.
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00:05:04,045 --> 00:05:06,560
73
00:05:06,560 --> 00:05:10,290
Now, Fourier analysis will
be a topic for a
74
00:05:10,290 --> 00:05:12,030
set of later lectures.
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00:05:12,030 --> 00:05:15,150
What I'd like to begin with is
the representation in terms of
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00:05:15,150 --> 00:05:18,520
impulses and the associated
description of linear
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00:05:18,520 --> 00:05:22,520
time-invariant systems
using convolution.
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00:05:22,520 --> 00:05:27,040
So let's begin with a discussion
of discrete-time
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00:05:27,040 --> 00:05:31,410
signals, and in particular the
issue of how discrete-time
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00:05:31,410 --> 00:05:34,770
signals can be decomposed
as a linear
81
00:05:34,770 --> 00:05:39,740
combination of delayed impulses.
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00:05:39,740 --> 00:05:43,560
Well, in fact, it's relatively
straightforward.
83
00:05:43,560 --> 00:05:49,140
What I've shown here is a
general sequence with values
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00:05:49,140 --> 00:05:52,640
which I've indicated
at the top.
85
00:05:52,640 --> 00:05:57,100
And more or less as we did
when we talked about
86
00:05:57,100 --> 00:06:01,660
representing a unit step in
terms of impulses, we can
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00:06:01,660 --> 00:06:08,170
think of this general sequence
as a sequence of impulses--
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00:06:08,170 --> 00:06:14,060
delayed, namely, occurring at
the appropriate time instant,
89
00:06:14,060 --> 00:06:16,950
and with the appropriate
amplitude.
90
00:06:16,950 --> 00:06:21,950
So we can think of this general
sequence and an
91
00:06:21,950 --> 00:06:28,780
impulse occurring at n = 0 and
with a height of x[0], plus an
92
00:06:28,780 --> 00:06:31,960
impulse of height x[1]
93
00:06:31,960 --> 00:06:35,830
occurring at time n = 1,
and so that's x[1]
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00:06:35,830 --> 00:06:40,850
delta[n-1], an impulse
at -1 with an
95
00:06:40,850 --> 00:06:42,310
amplitude of x[-1], etc.
96
00:06:42,310 --> 00:06:45,030
97
00:06:45,030 --> 00:06:50,500
So if we continued to generate
a set of weighted, delayed
98
00:06:50,500 --> 00:06:55,600
unit samples like that, and if
we added all these together,
99
00:06:55,600 --> 00:07:00,120
then that will generate
the total sequence.
100
00:07:00,120 --> 00:07:04,610
Algebraically, then, what that
corresponds to is representing
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00:07:04,610 --> 00:07:09,500
the sequence as a sum of
individual terms as I've
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00:07:09,500 --> 00:07:16,490
indicated here or in terms of a
general sum, the sum of x[k]
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00:07:16,490 --> 00:07:18,820
delta[n-k].
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00:07:18,820 --> 00:07:19,950
So that's our strategy--
105
00:07:19,950 --> 00:07:25,400
the strategy is to decompose an
arbitrary sequence into a
106
00:07:25,400 --> 00:07:29,540
linear combination of weighted,
delayed impulses.
107
00:07:29,540 --> 00:07:34,150
And here again is the
representation, which we just
108
00:07:34,150 --> 00:07:37,160
finished generating.
109
00:07:37,160 --> 00:07:41,370
Now, why is this representation
useful?
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00:07:41,370 --> 00:07:45,380
It's useful because we now have
a decomposition of the
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00:07:45,380 --> 00:07:49,790
sequence as a linear combination
of basic
112
00:07:49,790 --> 00:07:53,080
sequences, namely the
delayed impulses.
113
00:07:53,080 --> 00:07:58,920
And if we are talking about a
linear system, the response to
114
00:07:58,920 --> 00:08:01,390
that linear combination
is a linear
115
00:08:01,390 --> 00:08:04,270
combination of the responses.
116
00:08:04,270 --> 00:08:10,830
So if we denote the response
to a delayed impulse as
117
00:08:10,830 --> 00:08:19,020
h_k[n], then the response to
this general input is what
118
00:08:19,020 --> 00:08:24,720
I've indicated here, where y[n],
of course, is the output
119
00:08:24,720 --> 00:08:27,900
due to the general input x[n].
120
00:08:27,900 --> 00:08:28,720
h_k[n]
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00:08:28,720 --> 00:08:34,169
in is the output due to the
delayed impulse, and these are
122
00:08:34,169 --> 00:08:37,299
simply the coefficients
in the weighting.
123
00:08:37,299 --> 00:08:40,169
124
00:08:40,169 --> 00:08:45,340
So for a linear system, we
have this representation.
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00:08:45,340 --> 00:08:50,830
And if now, in addition, the
system is time-invariant, we
126
00:08:50,830 --> 00:08:54,810
can, in fact, relate the
outputs due to these
127
00:08:54,810 --> 00:08:57,240
individual delayed impulses.
128
00:08:57,240 --> 00:09:01,220
Specifically, if the system is
time-invariant, then the
129
00:09:01,220 --> 00:09:08,330
response to an impulse at time
k is exactly the same as the
130
00:09:08,330 --> 00:09:13,470
response to an impulse at time
0, shifted over to time k.
131
00:09:13,470 --> 00:09:16,890
Said another way, h_k[n]
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00:09:16,890 --> 00:09:24,450
is simply h_0[n-k], where h_0 is
the response of the system
133
00:09:24,450 --> 00:09:27,140
to an impulse at n = 0.
134
00:09:27,140 --> 00:09:32,060
And it's generally useful to,
rather than carrying around
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00:09:32,060 --> 00:09:36,800
h_0[n], just simply
define h_0[n]
136
00:09:36,800 --> 00:09:42,700
as h[n], which is the unit
sample or unit impulse
137
00:09:42,700 --> 00:09:45,070
response of the system.
138
00:09:45,070 --> 00:09:49,440
And so the consequence, then, is
for a linear time-invariant
139
00:09:49,440 --> 00:09:54,520
system, the output can
be expressed as
140
00:09:54,520 --> 00:09:58,510
this sum where h[n-k]
141
00:09:58,510 --> 00:10:03,600
is the response to an impulse
occurring at time n = k.
142
00:10:03,600 --> 00:10:10,780
And this is referred to as
the convolution sum.
143
00:10:10,780 --> 00:10:12,450
Now, we can--
144
00:10:12,450 --> 00:10:17,570
just to emphasize how we've gone
about this, let me show
145
00:10:17,570 --> 00:10:18,820
it from another perspective.
146
00:10:18,820 --> 00:10:22,040
147
00:10:22,040 --> 00:10:28,080
We of course have taken the
sequence x[n], we have
148
00:10:28,080 --> 00:10:33,180
decomposed it as a linear
combination of these weighted,
149
00:10:33,180 --> 00:10:36,010
delayed impulses.
150
00:10:36,010 --> 00:10:39,400
When these are added together,
those correspond to the
151
00:10:39,400 --> 00:10:42,810
original sequence x[n].
152
00:10:42,810 --> 00:10:48,200
If this impulse, for example,
generates a
153
00:10:48,200 --> 00:10:51,990
response which is x[0]
154
00:10:51,990 --> 00:10:53,290
h[n], where h[n]
155
00:10:53,290 --> 00:10:58,750
is the response to a unit
impulse at n = 0, and the
156
00:10:58,750 --> 00:11:03,150
second one generates a delayed
weighted response, and the
157
00:11:03,150 --> 00:11:07,770
third one similarly, and we
generate these individual
158
00:11:07,770 --> 00:11:12,050
responses, these are all added
together, and it's that linear
159
00:11:12,050 --> 00:11:16,180
combination that forms
the final output.
160
00:11:16,180 --> 00:11:18,450
So that's really kind of the way
we're thinking about it.
161
00:11:18,450 --> 00:11:22,870
We have a general sequence,
we're thinking of each
162
00:11:22,870 --> 00:11:26,160
individual sample individually,
each one of
163
00:11:26,160 --> 00:11:29,890
those pops the system, and
because of linearity, the
164
00:11:29,890 --> 00:11:34,930
response is the sum of those
individual responses.
165
00:11:34,930 --> 00:11:38,720
That's what happens in discrete
time, and pretty much
166
00:11:38,720 --> 00:11:43,050
the same strategy works
in continuous time.
167
00:11:43,050 --> 00:11:51,380
In particular, we can begin in
continuous time with the
168
00:11:51,380 --> 00:11:57,600
notion of decomposing a
continuous-time signal into a
169
00:11:57,600 --> 00:12:02,060
succession of arbitrarily
narrow rectangles.
170
00:12:02,060 --> 00:12:06,670
And as the width of the
rectangles goes to 0, the
171
00:12:06,670 --> 00:12:09,080
approximation gets better.
172
00:12:09,080 --> 00:12:11,410
Essentially what's going to
happen is that each of those
173
00:12:11,410 --> 00:12:15,510
individual rectangles, as they
get narrower and narrower,
174
00:12:15,510 --> 00:12:18,770
correspond more and more
to an impulse.
175
00:12:18,770 --> 00:12:21,970
Let me show you what I mean.
176
00:12:21,970 --> 00:12:29,540
Here we have a continuous-time
signal, and I've approximated
177
00:12:29,540 --> 00:12:33,530
it by a staircase.
178
00:12:33,530 --> 00:12:39,970
So in essence I can think of
this as individual rectangles
179
00:12:39,970 --> 00:12:42,710
of heights associated with the
height of the continuous
180
00:12:42,710 --> 00:12:46,220
curve, and so I've indicated
that down below.
181
00:12:46,220 --> 00:12:50,120
Here, for example, is the
impulse corresponding to the
182
00:12:50,120 --> 00:12:56,110
rectangle between t = -2
Delta and t = -Delta.
183
00:12:56,110 --> 00:13:01,830
Here's the one from -Delta to 0,
and as we continue on down,
184
00:13:01,830 --> 00:13:06,410
we get impulses, or rather
rectangles, from successive
185
00:13:06,410 --> 00:13:08,810
parts of the wave form.
186
00:13:08,810 --> 00:13:14,270
Now let's look specifically at
the rectangle, for example,
187
00:13:14,270 --> 00:13:19,410
starting at 0 and ending at
Delta, and the amplitude of it
188
00:13:19,410 --> 00:13:20,660
is x(Delta).
189
00:13:20,660 --> 00:13:24,730
190
00:13:24,730 --> 00:13:26,760
So what we have--
191
00:13:26,760 --> 00:13:32,040
actually, this should be x(0),
and so let me just
192
00:13:32,040 --> 00:13:33,480
correct that here.
193
00:13:33,480 --> 00:13:35,280
That's x(0).
194
00:13:35,280 --> 00:13:40,660
And so we have a rectangle
height x(0), and recall the
195
00:13:40,660 --> 00:13:46,740
function that I defined last
time as delta_Delta(t), which
196
00:13:46,740 --> 00:13:50,010
had height 1 / delta
and width delta.
197
00:13:50,010 --> 00:13:55,060
So multiplying finally by this
last little Delta, then, this
198
00:13:55,060 --> 00:13:58,580
is a representation
for the rectangle
199
00:13:58,580 --> 00:14:00,110
that I've shown there.
200
00:14:00,110 --> 00:14:03,010
Now there's a little bit of
algebra there to kind of track
201
00:14:03,010 --> 00:14:07,510
through, but what we're really
doing is just simply
202
00:14:07,510 --> 00:14:10,490
representing this in terms
of rectangles.
203
00:14:10,490 --> 00:14:15,390
What I want to do is describe
each rectangle as in terms of
204
00:14:15,390 --> 00:14:19,360
that function delta_Delta(t),
which in the limit, then,
205
00:14:19,360 --> 00:14:21,330
becomes an impulse.
206
00:14:21,330 --> 00:14:23,290
So let's track that through
a little further.
207
00:14:23,290 --> 00:14:25,800
208
00:14:25,800 --> 00:14:29,490
When we have that linear
combination, then, we're
209
00:14:29,490 --> 00:14:34,300
saying that x(t) can be
represented by a sum as I
210
00:14:34,300 --> 00:14:42,180
indicate here, which I can then
write more generally in
211
00:14:42,180 --> 00:14:46,650
this form, just indicating that
this is an infinite sum.
212
00:14:46,650 --> 00:14:51,110
We now want to take the limit
as Delta goes to 0, and as
213
00:14:51,110 --> 00:14:56,110
Delta goes to 0, notice
that this term
214
00:14:56,110 --> 00:14:59,370
becomes arbitrarily narrow.
215
00:14:59,370 --> 00:15:02,820
This goes to our impulse
function, and this, of course,
216
00:15:02,820 --> 00:15:05,150
goes to x of tau.
217
00:15:05,150 --> 00:15:10,840
And in fact, in the limit, a
sum of this form is exactly
218
00:15:10,840 --> 00:15:13,270
the way an integral
is defined.
219
00:15:13,270 --> 00:15:17,040
So we have an expression
for y(t) in terms
220
00:15:17,040 --> 00:15:18,800
of an impulse function.
221
00:15:18,800 --> 00:15:22,460
There, I have to admit, is a
little bit of detail to kind
222
00:15:22,460 --> 00:15:26,700
of focus on at your leisure, but
this is the general flow
223
00:15:26,700 --> 00:15:28,300
of the strategy.
224
00:15:28,300 --> 00:15:32,500
So what we have now is an
integral that tells us that
225
00:15:32,500 --> 00:15:38,640
tells us how x(t) can be
described as a sum or linear
226
00:15:38,640 --> 00:15:43,120
combination involving
impulses.
227
00:15:43,120 --> 00:15:46,430
This bottom equation, by the
way, is often referred to as
228
00:15:46,430 --> 00:15:48,170
the sifting integral.
229
00:15:48,170 --> 00:15:52,130
In essence, what it says is that
if I take a time function
230
00:15:52,130 --> 00:15:56,720
x(t) and put it through that
integral, the impulse as it
231
00:15:56,720 --> 00:16:02,070
zips by generates x(t)
all over again.
232
00:16:02,070 --> 00:16:06,000
Now, at first glance, what it
could look like is that we've
233
00:16:06,000 --> 00:16:10,610
taken a time function x(t) and
proceeded to represent it in a
234
00:16:10,610 --> 00:16:14,780
very complicated way, in terms
of itself, and one could ask,
235
00:16:14,780 --> 00:16:16,720
why bother doing that?
236
00:16:16,720 --> 00:16:22,940
And the reason, going back to
what our strategy was, is that
237
00:16:22,940 --> 00:16:29,240
what we want to do is exploit
the property of linearity.
238
00:16:29,240 --> 00:16:34,630
So by describing a time
function as a linear
239
00:16:34,630 --> 00:16:39,370
combination of weighted, delayed
impulses, as in effect
240
00:16:39,370 --> 00:16:44,350
we've done through this
summation that corresponds to
241
00:16:44,350 --> 00:16:49,580
a decomposition in terms of
impulses, we can now exploit
242
00:16:49,580 --> 00:16:54,350
linearity, specifically
recognizing that the output of
243
00:16:54,350 --> 00:17:00,440
a linear system is the sum
of the responses to these
244
00:17:00,440 --> 00:17:02,120
individual inputs.
245
00:17:02,120 --> 00:17:09,750
So with h_kDelta(t)
corresponding to the response
246
00:17:09,750 --> 00:17:15,770
to delta_Delta(t-kDelta) and
the rest of this stuff, the
247
00:17:15,770 --> 00:17:18,700
x(kDelta) and this
little Delta are
248
00:17:18,700 --> 00:17:21,140
basically scale factors--
249
00:17:21,140 --> 00:17:24,839
for a linear system, then, if
the input is expressed in this
250
00:17:24,839 --> 00:17:29,610
form, the output is expressed in
this form, and again taking
251
00:17:29,610 --> 00:17:33,460
the limit as Delta goes to
0, by definition, this
252
00:17:33,460 --> 00:17:35,710
corresponds to an integral.
253
00:17:35,710 --> 00:17:40,380
It's the integral that I
indicate here with h_tau(t)
254
00:17:40,380 --> 00:17:47,500
corresponding to the impulse
response due to an impulse
255
00:17:47,500 --> 00:17:50,360
occurring at time tau.
256
00:17:50,360 --> 00:17:54,580
Now, again, we can do
the same thing.
257
00:17:54,580 --> 00:17:58,760
In particular, if the system
is time-invariant, then the
258
00:17:58,760 --> 00:18:02,920
response to each of these
delayed impulses is simply a
259
00:18:02,920 --> 00:18:07,870
delayed version of the impulse
response, and so we can relate
260
00:18:07,870 --> 00:18:10,160
these individual terms.
261
00:18:10,160 --> 00:18:13,500
And in particular, then, the
response to an impulse
262
00:18:13,500 --> 00:18:18,850
occurring at time t = tau is
simply the response to an
263
00:18:18,850 --> 00:18:23,020
impulse occurring at time 0
shifted over to the time
264
00:18:23,020 --> 00:18:24,840
origin tau.
265
00:18:24,840 --> 00:18:30,020
Again, as we did before, we'll
drop this subscript h_0, so
266
00:18:30,020 --> 00:18:37,090
h_0(t) we'll simply
define as h(t).
267
00:18:37,090 --> 00:18:42,110
What we're left with when we do
that is the description of
268
00:18:42,110 --> 00:18:46,690
a linear time-invariant system
through this integral, which
269
00:18:46,690 --> 00:18:53,240
tells us how the output is
related to the input and to
270
00:18:53,240 --> 00:18:54,990
the impulse response.
271
00:18:54,990 --> 00:18:58,910
Again, let's just quickly look
at this from another
272
00:18:58,910 --> 00:19:01,240
perspective as we did
in discrete time.
273
00:19:01,240 --> 00:19:03,840
274
00:19:03,840 --> 00:19:07,740
Recall that what we've done
is to take the continuous
275
00:19:07,740 --> 00:19:14,510
function, decompose it in terms
of rectangles, and then
276
00:19:14,510 --> 00:19:19,970
each of these rectangles
generates its individual
277
00:19:19,970 --> 00:19:23,600
response, and then
these individual
278
00:19:23,600 --> 00:19:25,225
responses are added together.
279
00:19:25,225 --> 00:19:27,930
280
00:19:27,930 --> 00:19:31,480
And as we go through that
process, of course, there's a
281
00:19:31,480 --> 00:19:37,260
process whereby we let the
approximation go to the
282
00:19:37,260 --> 00:19:40,920
representation of
a smooth curve.
283
00:19:40,920 --> 00:19:45,890
Now, again, I stress if there is
certainly a fair amount to
284
00:19:45,890 --> 00:19:50,050
kind of examine carefully there,
but it's important to
285
00:19:50,050 --> 00:19:53,980
also reflect on what we've done,
which is really pretty
286
00:19:53,980 --> 00:19:55,240
significant.
287
00:19:55,240 --> 00:19:58,790
What we've managed to accomplish
is to exploit the
288
00:19:58,790 --> 00:20:03,720
properties of linearity and time
invariance, so that the
289
00:20:03,720 --> 00:20:09,840
system could be represented in
terms only of its response to
290
00:20:09,840 --> 00:20:12,380
an impulse at time 0.
291
00:20:12,380 --> 00:20:14,410
So for a linear time-invariant
system--
292
00:20:14,410 --> 00:20:16,410
quite amazingly, actually--
293
00:20:16,410 --> 00:20:20,670
if you know its response to an
impulse at t = 0 or n = 0,
294
00:20:20,670 --> 00:20:24,490
depending on discrete or
continuous time, then in fact,
295
00:20:24,490 --> 00:20:27,380
through the convolution sum
in discrete time or the
296
00:20:27,380 --> 00:20:30,260
convolution integral in
continuous time, you can
297
00:20:30,260 --> 00:20:33,110
generate the response to
an arbitrary input.
298
00:20:33,110 --> 00:20:36,130
299
00:20:36,130 --> 00:20:41,560
Let me just introduce a small
amount of notation.
300
00:20:41,560 --> 00:20:46,660
Again, reminding you of the
convolution sum in the
301
00:20:46,660 --> 00:20:51,080
discrete-time case, which looks
as I've indicated here--
302
00:20:51,080 --> 00:20:52,510
the sum of x[k]
303
00:20:52,510 --> 00:20:54,980
h[n-k]
304
00:20:54,980 --> 00:20:59,990
will have the requirement of
making such frequent reference
305
00:20:59,990 --> 00:21:04,190
to convolution that it's
convenient to notationally
306
00:21:04,190 --> 00:21:07,380
represent it as I have here
with an asterisk.
307
00:21:07,380 --> 00:21:09,350
So x[n]
308
00:21:09,350 --> 00:21:12,200
* h[n]
309
00:21:12,200 --> 00:21:15,060
means or denotes the
convolution of x[n]
310
00:21:15,060 --> 00:21:17,060
with h[n].
311
00:21:17,060 --> 00:21:21,590
And correspondingly in the
continuous-time case, we have
312
00:21:21,590 --> 00:21:26,230
the convolution integral, which
here is the sifting
313
00:21:26,230 --> 00:21:26,670
integral
314
00:21:26,670 --> 00:21:30,660
as we talked about, representing
x(t) in terms of
315
00:21:30,660 --> 00:21:34,800
itself as a linear combination
of delayed impulses.
316
00:21:34,800 --> 00:21:38,580
Here we have the convolution
integral, and again we'll use
317
00:21:38,580 --> 00:21:43,040
the asterisk to denote
convolution.
318
00:21:43,040 --> 00:21:46,950
Now, there's a lot about
convolution that we'll want to
319
00:21:46,950 --> 00:21:47,750
talk about.
320
00:21:47,750 --> 00:21:50,240
There are properties of
convolution which tell us
321
00:21:50,240 --> 00:21:53,540
about properties of linear
time-invariant systems.
322
00:21:53,540 --> 00:21:57,920
Also, it's important to focus
on the mechanics of
323
00:21:57,920 --> 00:21:59,370
implementing a convolution--
324
00:21:59,370 --> 00:22:02,690
in other words, understanding
and generating some fluency
325
00:22:02,690 --> 00:22:08,860
and insight into what these
particular sum and integral
326
00:22:08,860 --> 00:22:11,080
expressions mean.
327
00:22:11,080 --> 00:22:15,410
So let's first look at
discrete-time convolution and
328
00:22:15,410 --> 00:22:20,680
examine more specifically what
in essence the sum tells us to
329
00:22:20,680 --> 00:22:22,935
do in terms of manipulating
the sequences.
330
00:22:22,935 --> 00:22:25,960
331
00:22:25,960 --> 00:22:35,040
So returning to the expression
for the convolution sum, as I
332
00:22:35,040 --> 00:22:39,960
show here, the sum of x[k]
333
00:22:39,960 --> 00:22:42,430
h[n-k]--
334
00:22:42,430 --> 00:22:47,080
let's focus in on an example
where we choose x[n]
335
00:22:47,080 --> 00:22:49,950
as a unit step and h[n]
336
00:22:49,950 --> 00:22:53,900
as a real exponential times
the a unit step.
337
00:22:53,900 --> 00:22:56,090
So the sequence x[n]
338
00:22:56,090 --> 00:23:00,950
is as I indicate here,
and the sequence h[n]
339
00:23:00,950 --> 00:23:05,760
is an exponential for positive
time and 0 for negative time.
340
00:23:05,760 --> 00:23:07,105
So we have x[n]
341
00:23:07,105 --> 00:23:12,840
and h[n], but now let's look
back at the equation and let
342
00:23:12,840 --> 00:23:16,960
me stress that what we
want is not x[n]
343
00:23:16,960 --> 00:23:18,340
and h[n]--
344
00:23:18,340 --> 00:23:21,440
we want x[k], because
we're going to sum
345
00:23:21,440 --> 00:23:24,660
over k, and not h[k]
346
00:23:24,660 --> 00:23:25,910
but h[n-k].
347
00:23:25,910 --> 00:23:27,970
348
00:23:27,970 --> 00:23:33,970
So we have then from x[n],
it's straightforward to
349
00:23:33,970 --> 00:23:35,220
generate x[k].
350
00:23:35,220 --> 00:23:39,580
It's simply changing the
index of summation.
351
00:23:39,580 --> 00:23:42,310
And what's h[n-k]?
352
00:23:42,310 --> 00:23:44,740
Well what's h[-k]?
353
00:23:44,740 --> 00:23:46,100
h[-k]
354
00:23:46,100 --> 00:23:47,640
is h[k]
355
00:23:47,640 --> 00:23:49,120
flipped over.
356
00:23:49,120 --> 00:23:53,550
So if this is what h[k]
357
00:23:53,550 --> 00:24:00,140
looks like, then this
is what h[n-k]
358
00:24:00,140 --> 00:24:01,690
looks like.
359
00:24:01,690 --> 00:24:05,920
In essence, what the operation
of convolution or the
360
00:24:05,920 --> 00:24:11,390
mechanics of convolution tells
us to do is to take the
361
00:24:11,390 --> 00:24:16,710
sequence h[n-k],
which is h[-k]
362
00:24:16,710 --> 00:24:21,870
positioned with its origin at
k = n, and multiply this
363
00:24:21,870 --> 00:24:27,700
sequence by this sequence and
sum the product from k =
364
00:24:27,700 --> 00:24:29,780
-infinity to +infinity.
365
00:24:29,780 --> 00:24:38,080
So if we were to compute, for
example, the output at n = 0--
366
00:24:38,080 --> 00:24:43,000
as I positioned this sequence
here, this is at n = 0--
367
00:24:43,000 --> 00:24:46,820
I would take this and multiply
it by this and sum from
368
00:24:46,820 --> 00:24:49,020
-infinity to +infinity.
369
00:24:49,020 --> 00:24:52,900
Or, for n = 1, I would position
it here, for n = 2, I
370
00:24:52,900 --> 00:24:55,470
would position it here.
371
00:24:55,470 --> 00:24:58,780
Well, you can kind of see
what the idea is.
372
00:24:58,780 --> 00:25:02,770
Let's look at this a little more
dynamically and see, in
373
00:25:02,770 --> 00:25:07,180
fact, how one sequence slides
past the other, and how the
374
00:25:07,180 --> 00:25:08,590
output y[n]
375
00:25:08,590 --> 00:25:11,910
builds up to the
correct answer.
376
00:25:11,910 --> 00:25:14,950
377
00:25:14,950 --> 00:25:18,100
So the input that we're
considering is a step input,
378
00:25:18,100 --> 00:25:22,680
which I show here, and the
impulse response that we will
379
00:25:22,680 --> 00:25:26,830
convolve this with is a
decaying exponential.
380
00:25:26,830 --> 00:25:31,340
Now, to form the convolution, we
want the product of x[k]--
381
00:25:31,340 --> 00:25:35,990
not with h[k], but with h[n-k],
corresponding to
382
00:25:35,990 --> 00:25:37,330
taking h[k]
383
00:25:37,330 --> 00:25:41,020
and reflecting it about the
origin and then shifting it
384
00:25:41,020 --> 00:25:42,130
appropriately.
385
00:25:42,130 --> 00:25:44,320
So here we see h[n-k]
386
00:25:44,320 --> 00:25:52,050
for n = 0, namely h[-k], and now
h[1-k], which we'll show
387
00:25:52,050 --> 00:25:57,670
next, is this shifted to
the right by one point.
388
00:25:57,670 --> 00:26:00,720
Here we have h[1-k]--
389
00:26:00,720 --> 00:26:07,420
shifting to the right by one
more point is h[2-k], and
390
00:26:07,420 --> 00:26:10,370
shifting again to the right
we'll have h[3-k].
391
00:26:10,370 --> 00:26:12,880
392
00:26:12,880 --> 00:26:20,480
Now let's shift back to the left
until n is negative, and
393
00:26:20,480 --> 00:26:22,210
then we'll begin the
convolution.
394
00:26:22,210 --> 00:26:32,890
So here's n = 0, n = -1,
n = -2, and n = -3.
395
00:26:32,890 --> 00:26:36,530
Now, to form the convolution,
we want the product of x[k]
396
00:26:36,530 --> 00:26:38,470
with h[n-k]
397
00:26:38,470 --> 00:26:43,090
summed from -infinity
to +infinity.
398
00:26:43,090 --> 00:26:47,360
For n negative, that product is
0, and therefore the result
399
00:26:47,360 --> 00:26:49,440
of the convolution is 0.
400
00:26:49,440 --> 00:26:52,130
As we shift to the right,
we'll build up the
401
00:26:52,130 --> 00:26:55,510
convolution, and the result of
the convolution will be shown
402
00:26:55,510 --> 00:26:57,330
on the bottom trace.
403
00:26:57,330 --> 00:27:01,780
So we begin the process
with n negative, and
404
00:27:01,780 --> 00:27:03,730
here we have n = -1.
405
00:27:03,730 --> 00:27:09,210
At n = 0, we get our first
non-zero contribution.
406
00:27:09,210 --> 00:27:13,310
Now as we shift further to the
right corresponding to
407
00:27:13,310 --> 00:27:18,820
increasing n, we will accumulate
more and more terms
408
00:27:18,820 --> 00:27:22,960
in the sum, and the convolution
will build up.
409
00:27:22,960 --> 00:27:25,670
In particular for this example,
the result of the
410
00:27:25,670 --> 00:27:30,040
convolution increases
monotonically, asymptotically
411
00:27:30,040 --> 00:27:34,790
approaching a constant, and
that constant, in fact, is
412
00:27:34,790 --> 00:27:38,630
just simply the accumulation
of the values under the
413
00:27:38,630 --> 00:27:39,880
exponential.
414
00:27:39,880 --> 00:27:48,780
415
00:27:48,780 --> 00:27:51,800
Now let's carry out the
convolution this time with an
416
00:27:51,800 --> 00:27:55,870
input which is a rectangular
pulse instead of a step input.
417
00:27:55,870 --> 00:27:58,510
Again, the same impulse
response, namely a decaying
418
00:27:58,510 --> 00:28:03,370
exponential, and so we want
to begin with h[n-k]
419
00:28:03,370 --> 00:28:06,750
and again with n negative
shown here.
420
00:28:06,750 --> 00:28:10,710
Again, with n negative, there
are no non-zero terms in the
421
00:28:10,710 --> 00:28:14,510
product, and so the convolution
for n negative
422
00:28:14,510 --> 00:28:17,390
will be 0 as it was in
the previous case.
423
00:28:17,390 --> 00:28:21,230
Again, on the bottom trace we'll
show the result of the
424
00:28:21,230 --> 00:28:25,470
convolution as the impulse
response slides along.
425
00:28:25,470 --> 00:28:30,040
At n = 0, we get our first
non-zero term.
426
00:28:30,040 --> 00:28:35,860
As n increases past 0, we will
begin to generate an output,
427
00:28:35,860 --> 00:28:39,350
basically the same as the output
that we generated with
428
00:28:39,350 --> 00:28:45,100
a step input, until the impulse
response reaches a
429
00:28:45,100 --> 00:28:50,280
point where as we slide further,
we slide outside the
430
00:28:50,280 --> 00:28:53,490
interval where the rectangle
is non-zero.
431
00:28:53,490 --> 00:28:57,690
So when we slide one point
further from what's shown
432
00:28:57,690 --> 00:29:03,370
here, the output will now decay,
corresponding to the
433
00:29:03,370 --> 00:29:09,100
fact that the impulse response
is sliding outside the
434
00:29:09,100 --> 00:29:12,790
interval in which the
input is non-zero.
435
00:29:12,790 --> 00:29:15,480
So, on the bottom trace we
now see the result of the
436
00:29:15,480 --> 00:29:16,730
convolution.
437
00:29:16,730 --> 00:29:20,470
438
00:29:20,470 --> 00:29:21,010
OK.
439
00:29:21,010 --> 00:29:27,450
So what you've seen, then, is
an example of discrete-time
440
00:29:27,450 --> 00:29:29,010
convolution.
441
00:29:29,010 --> 00:29:33,470
Let's now look at an example
of continuous-time
442
00:29:33,470 --> 00:29:35,340
convolution.
443
00:29:35,340 --> 00:29:38,140
As you might expect,
continuous-time convolution
444
00:29:38,140 --> 00:29:42,030
operates in exactly
the same way.
445
00:29:42,030 --> 00:29:44,530
Continuous-time convolution--
446
00:29:44,530 --> 00:29:51,750
we have the expression again
y(t) is an integral with now
447
00:29:51,750 --> 00:29:55,270
x(tau) and h(t-tau).
448
00:29:55,270 --> 00:30:01,450
It has exactly the same kind of
form as we had previously
449
00:30:01,450 --> 00:30:05,190
for discrete-time
convolution--
450
00:30:05,190 --> 00:30:08,190
and in fact, the mechanics
of the continuous-time
451
00:30:08,190 --> 00:30:10,930
convolution are identical.
452
00:30:10,930 --> 00:30:16,440
So here is our example with x(t)
equal to a unit step and
453
00:30:16,440 --> 00:30:22,280
h(t) now a real exponential
times a unit step.
454
00:30:22,280 --> 00:30:28,330
I show here x(t), which is
the unit step function.
455
00:30:28,330 --> 00:30:34,590
Here we have h(t), which is an
exponential for positive time
456
00:30:34,590 --> 00:30:38,670
and 0 for negative time.
457
00:30:38,670 --> 00:30:43,960
Again, looking back at the
expression for convolution,
458
00:30:43,960 --> 00:30:48,770
it's not x(t) that we want,
it x(tau) that we want.
459
00:30:48,770 --> 00:30:54,480
And it's not h(t) or h(tau) that
we want, it's h(t-tau).
460
00:30:54,480 --> 00:30:57,310
461
00:30:57,310 --> 00:31:01,880
We plan to multiply these
together and integrate over
462
00:31:01,880 --> 00:31:06,270
the variable tau, and that gives
us the output at any
463
00:31:06,270 --> 00:31:07,700
given time.
464
00:31:07,700 --> 00:31:12,260
If we want it at another time,
we change the value of t as an
465
00:31:12,260 --> 00:31:14,580
argument inside this integral.
466
00:31:14,580 --> 00:31:21,340
So here we have x(t), and here
we have h(t), which isn't
467
00:31:21,340 --> 00:31:23,210
quite what we wanted.
468
00:31:23,210 --> 00:31:26,430
Here we have x(tau), and that's
fine-- it's just x(t)
469
00:31:26,430 --> 00:31:29,410
with t relabeled as tau.
470
00:31:29,410 --> 00:31:33,800
Now, what is h(t-tau)?
471
00:31:33,800 --> 00:31:39,590
Well, here's h(tau), and
if we simply turn that
472
00:31:39,590 --> 00:31:43,820
over, here is h(t-tau).
473
00:31:43,820 --> 00:31:52,470
And h(t-tau) is positioned,
then, at tau equal to t.
474
00:31:52,470 --> 00:31:56,320
As we change the value of t that
change the position of
475
00:31:56,320 --> 00:32:03,020
this signal, now we multiply
these two together and
476
00:32:03,020 --> 00:32:08,970
integrate from -infinity to
+infinity with h(t-tau)
477
00:32:08,970 --> 00:32:11,370
positioned at the appropriate
value of t.
478
00:32:11,370 --> 00:32:14,600
479
00:32:14,600 --> 00:32:18,700
Again, it's best really to see
this example and get the
480
00:32:18,700 --> 00:32:23,080
notion of the signal being
flipped and the two signals
481
00:32:23,080 --> 00:32:28,020
sliding past each other,
multiplying and integrating by
482
00:32:28,020 --> 00:32:31,260
looking at it dynamically
and observing how the
483
00:32:31,260 --> 00:32:33,970
answer builds up.
484
00:32:33,970 --> 00:32:38,300
Again, the input that we
consider is a step input.
485
00:32:38,300 --> 00:32:41,800
And again, we use an impulse
response which is a decaying
486
00:32:41,800 --> 00:32:44,250
exponential.
487
00:32:44,250 --> 00:32:48,840
To form the convolution, we want
the product of x(tau)--
488
00:32:48,840 --> 00:32:52,040
not with h(tau), but
with h(t-tau).
489
00:32:52,040 --> 00:32:56,890
So we want h(t) time-reversed,
and then shifted appropriately
490
00:32:56,890 --> 00:32:59,190
depending on the value of t.
491
00:32:59,190 --> 00:33:04,480
Let's first just look at
h(t-tau) for t positive
492
00:33:04,480 --> 00:33:09,510
corresponding to shifting
h(-tau) out to the right, and
493
00:33:09,510 --> 00:33:12,155
here we have t increasing.
494
00:33:12,155 --> 00:33:15,810
495
00:33:15,810 --> 00:33:20,310
Here is t decreasing, and
we'll want to begin the
496
00:33:20,310 --> 00:33:25,430
convolution with t negative,
corresponding to shifting
497
00:33:25,430 --> 00:33:29,030
h(-tau) to the left.
498
00:33:29,030 --> 00:33:30,660
Now to form the convolution,
we want the
499
00:33:30,660 --> 00:33:32,290
product of these two.
500
00:33:32,290 --> 00:33:38,820
For t negative, there are no
non-zero contributions to the
501
00:33:38,820 --> 00:33:42,580
integral, and so the convolution
will be 0 for t
502
00:33:42,580 --> 00:33:44,090
less than 0.
503
00:33:44,090 --> 00:33:48,090
On the bottom trace, we show the
result of the convolution,
504
00:33:48,090 --> 00:33:55,030
here for t negative, and for t
less than 0, we will continue
505
00:33:55,030 --> 00:33:58,490
to have 0 in the convolution.
506
00:33:58,490 --> 00:34:04,910
Now as t increases past 0, we
begin to get some non-zero
507
00:34:04,910 --> 00:34:09,040
contribution in the product,
indicated by the fact that the
508
00:34:09,040 --> 00:34:09,989
convolution--
509
00:34:09,989 --> 00:34:11,270
the result of the
convolution--
510
00:34:11,270 --> 00:34:14,370
starts to build up.
511
00:34:14,370 --> 00:34:19,780
As t increases further, we
will get more and more
512
00:34:19,780 --> 00:34:23,540
non-zero contribution
in the integrand.
513
00:34:23,540 --> 00:34:28,270
So, the result of the
convolution will be a
514
00:34:28,270 --> 00:34:31,469
monotonically increasing
function for this particular
515
00:34:31,469 --> 00:34:36,100
example, which asymptotically
approaches a constant.
516
00:34:36,100 --> 00:34:40,320
That constant will be
proportional to the area under
517
00:34:40,320 --> 00:34:43,360
the impulse response, because
of the fact that we're
518
00:34:43,360 --> 00:34:45,090
convolving with a step input.
519
00:34:45,090 --> 00:34:59,270
520
00:34:59,270 --> 00:35:02,160
Now let's carry out the
convolution with an input
521
00:35:02,160 --> 00:35:05,110
which is a rectangular pulse--
522
00:35:05,110 --> 00:35:08,420
again, an impulse response
which is an exponential.
523
00:35:08,420 --> 00:35:13,710
So to form the convolution, we
want x(tau) with h(t-tau)--
524
00:35:13,710 --> 00:35:18,010
h(t-tau) shown here
for t negative.
525
00:35:18,010 --> 00:35:21,880
To form the convolution, we
take the integral of the
526
00:35:21,880 --> 00:35:25,920
product of these two, which
again will be 0
527
00:35:25,920 --> 00:35:28,360
for t less than 0.
528
00:35:28,360 --> 00:35:31,360
The bottom trace shows the
result of the convolution
529
00:35:31,360 --> 00:35:36,630
here, shown as 0, and it will
continue to be 0 until t
530
00:35:36,630 --> 00:35:40,990
becomes positive, at which
point we build up some
531
00:35:40,990 --> 00:35:47,040
non-zero term in
the integrand.
532
00:35:47,040 --> 00:35:53,540
Now as we slide further, until
the impulse response shifts
533
00:35:53,540 --> 00:35:58,870
outside the interval in which
the pulse is non-zero, the
534
00:35:58,870 --> 00:36:00,670
output will build up.
535
00:36:00,670 --> 00:36:05,530
But here we've now begun to
leave that interval, and so
536
00:36:05,530 --> 00:36:09,700
the output will start to
decay exponentially.
537
00:36:09,700 --> 00:36:13,680
As the impulse response slides
further and further
538
00:36:13,680 --> 00:36:20,410
corresponding to increasing t,
then the output will decay
539
00:36:20,410 --> 00:36:24,980
exponentially, representing the
fact that there is less
540
00:36:24,980 --> 00:36:31,200
and less area in the product
of x(tau) and h(t-tau).
541
00:36:31,200 --> 00:36:34,580
542
00:36:34,580 --> 00:36:37,680
Asymptotically, this output
will then approach 0.
543
00:36:37,680 --> 00:36:40,830
544
00:36:40,830 --> 00:36:41,140
OK.
545
00:36:41,140 --> 00:36:45,360
So you've seen convolution,
you've seen the derivation of
546
00:36:45,360 --> 00:36:49,120
convolution, and kind of the
graphical representation of
547
00:36:49,120 --> 00:36:50,820
convolution.
548
00:36:50,820 --> 00:36:56,880
Finally, let's work again with
these two examples, and let's
549
00:36:56,880 --> 00:37:00,870
go through those two examples
analytically so that we
550
00:37:00,870 --> 00:37:05,980
finally see how, analytically,
the result develops for those
551
00:37:05,980 --> 00:37:07,230
same examples.
552
00:37:07,230 --> 00:37:10,450
553
00:37:10,450 --> 00:37:13,400
Well, we have first the
discrete-time case, and let's
554
00:37:13,400 --> 00:37:16,960
take our discrete-time
example.
555
00:37:16,960 --> 00:37:23,490
In general, the convolution sum
is as I've indicated here.
556
00:37:23,490 --> 00:37:27,860
This is just our expression
from before, which is the
557
00:37:27,860 --> 00:37:31,140
convolution sum.
558
00:37:31,140 --> 00:37:33,080
If we take our two examples--
559
00:37:33,080 --> 00:37:39,760
the example of an input which is
a unit step, and an impulse
560
00:37:39,760 --> 00:37:44,220
response, which is a real
exponential multiplied by a
561
00:37:44,220 --> 00:37:46,110
unit step--
562
00:37:46,110 --> 00:37:50,990
we have then replacing x[k]
563
00:37:50,990 --> 00:37:56,230
by what we know the input
to be, and h[n-k]
564
00:37:56,230 --> 00:38:00,700
by what we know the impulse
response to be, the output is
565
00:38:00,700 --> 00:38:02,290
the expression that
we have here.
566
00:38:02,290 --> 00:38:05,020
567
00:38:05,020 --> 00:38:07,240
Now, in this expression--
568
00:38:07,240 --> 00:38:11,430
and you'll see this very
generally and with some more
569
00:38:11,430 --> 00:38:17,210
complicated examples when
you look at the text--
570
00:38:17,210 --> 00:38:22,720
as you go to evaluate these
expressions, generally what
571
00:38:22,720 --> 00:38:26,050
happens is that the signals
have different analytical
572
00:38:26,050 --> 00:38:28,380
forms in different regions.
573
00:38:28,380 --> 00:38:31,100
That's, in fact, what
we have here.
574
00:38:31,100 --> 00:38:36,670
In particular, let's look at the
sum, and what we observe
575
00:38:36,670 --> 00:38:42,660
first of all is that the limits
on this sum are going
576
00:38:42,660 --> 00:38:47,200
to be modified, depending
on where this unit
577
00:38:47,200 --> 00:38:50,390
step is 0 and non-zero.
578
00:38:50,390 --> 00:38:54,820
In particular, if we first
consider what will turn out to
579
00:38:54,820 --> 00:38:55,980
be the simple case--
580
00:38:55,980 --> 00:38:59,030
namely, n less than 0--
581
00:38:59,030 --> 00:39:08,790
for n less than 0, this
unit step is 0 for k
582
00:39:08,790 --> 00:39:10,640
greater than n.
583
00:39:10,640 --> 00:39:17,460
With n less than 0, that means
that this unit step never is
584
00:39:17,460 --> 00:39:21,480
non-zero for k positive.
585
00:39:21,480 --> 00:39:27,360
On the other hand, this unit
step is never non-zero or
586
00:39:27,360 --> 00:39:31,350
always 0 for k negative.
587
00:39:31,350 --> 00:39:34,480
Let me just stress that by
looking at the particular
588
00:39:34,480 --> 00:39:40,840
graphs, here is the
unit step u[k].
589
00:39:40,840 --> 00:39:48,420
Here is the unit step u[n-k],
and for n less than 0, so that
590
00:39:48,420 --> 00:39:52,550
this point comes before this
point, the product of these
591
00:39:52,550 --> 00:39:54,960
two is equal to 0.
592
00:39:54,960 --> 00:39:58,730
That means there is no overlap
between these two terms, and
593
00:39:58,730 --> 00:40:03,380
so it says that y[n],
the output, is 0
594
00:40:03,380 --> 00:40:05,940
for n less than 0.
595
00:40:05,940 --> 00:40:08,980
Well, that was an easy one.
596
00:40:08,980 --> 00:40:13,320
For n greater than 0, it's not
quite as straightforward as
597
00:40:13,320 --> 00:40:15,520
coming out with the answer 0.
598
00:40:15,520 --> 00:40:19,270
So now let's look at what
happens when the two unit
599
00:40:19,270 --> 00:40:25,430
steps overlap, and this would
correspond to what I've
600
00:40:25,430 --> 00:40:30,470
labeled here as interval 2,
namely for n greater than 0.
601
00:40:30,470 --> 00:40:36,840
If we just look back at the
summation that we had, the
602
00:40:36,840 --> 00:40:44,430
summation now corresponds to
this unit step and this unit
603
00:40:44,430 --> 00:40:48,260
step, having some overlap.
604
00:40:48,260 --> 00:40:53,960
So for interval 2, corresponding
to n greater
605
00:40:53,960 --> 00:41:00,150
than 0, we have u[k],
the unit step.
606
00:41:00,150 --> 00:41:05,980
We have u[n-k], which is a unit
step going backward in
607
00:41:05,980 --> 00:41:13,870
time, but which extends for
positive values of n.
608
00:41:13,870 --> 00:41:17,970
If we think about multiplying
these two together, we will
609
00:41:17,970 --> 00:41:24,320
get in the product unity
for what values of k?
610
00:41:24,320 --> 00:41:27,900
Well, for k starting at 0
corresponding to one of the
611
00:41:27,900 --> 00:41:31,320
unit steps and ending at
n corresponding to
612
00:41:31,320 --> 00:41:33,270
the other unit step.
613
00:41:33,270 --> 00:41:39,242
So we have an overlap between
these for k equal to 0, et
614
00:41:39,242 --> 00:41:41,630
cetera, up through
the value n.
615
00:41:41,630 --> 00:41:44,630
616
00:41:44,630 --> 00:41:49,990
Now, that means that in terms
of the original sum, we can
617
00:41:49,990 --> 00:41:53,170
get rid of the unit steps
involved by simply changing
618
00:41:53,170 --> 00:41:56,210
the limits on the sum.
619
00:41:56,210 --> 00:42:02,790
The limits now are from 0 to
n, of the term alpha^(n-k).
620
00:42:02,790 --> 00:42:05,680
621
00:42:05,680 --> 00:42:07,250
We had before a u[k]
622
00:42:07,250 --> 00:42:11,360
and u[n-k], and that disappeared
because we dealt
623
00:42:11,360 --> 00:42:14,740
with that simply by modifying
the limits.
624
00:42:14,740 --> 00:42:19,590
We now pull out the term
alpha^n, because the summation
625
00:42:19,590 --> 00:42:24,350
is on k, not on n, so
we can simply pull
626
00:42:24,350 --> 00:42:25,750
that term of the sum.
627
00:42:25,750 --> 00:42:29,190
628
00:42:29,190 --> 00:42:33,810
We now have alpha^(-k), which
we can rewrite as
629
00:42:33,810 --> 00:42:36,480
(alpha^(-1))^k.
630
00:42:36,480 --> 00:42:40,100
The upshot of all of
this is that y[n]
631
00:42:40,100 --> 00:42:45,600
now we can reexpress as alpha^n,
the sum from 0 to n
632
00:42:45,600 --> 00:42:46,850
of (alpha^(-1)^k.
633
00:42:46,850 --> 00:42:49,460
634
00:42:49,460 --> 00:42:54,850
The question is, how do
we evaluate that?
635
00:42:54,850 --> 00:42:58,960
It essentially corresponds to a
finite number of terms in a
636
00:42:58,960 --> 00:43:01,060
geometric series.
637
00:43:01,060 --> 00:43:05,140
That, by the way, is a summation
that will recur over
638
00:43:05,140 --> 00:43:08,900
and over and over and over
again, and it's one that you
639
00:43:08,900 --> 00:43:11,870
should write down, write on your
back pocket, write on the
640
00:43:11,870 --> 00:43:15,650
palm of your hand, or whatever
it takes to remember it.
641
00:43:15,650 --> 00:43:20,700
What you'll see is it that
will recur more or less
642
00:43:20,700 --> 00:43:24,970
throughout the course, and so
it's one worth remembering.
643
00:43:24,970 --> 00:43:31,520
In particular, what the sum of a
geometric series is, is what
644
00:43:31,520 --> 00:43:32,950
I've indicated here.
645
00:43:32,950 --> 00:43:35,930
We have the sum from
0 to r, of beta^k.
646
00:43:35,930 --> 00:43:38,830
647
00:43:38,830 --> 00:43:42,430
It's 1 - beta^(r+1)--
648
00:43:42,430 --> 00:43:48,460
this is one more than the upper
limit on the summation--
649
00:43:48,460 --> 00:43:51,640
and in the denominator
is 1 - beta.
650
00:43:51,640 --> 00:43:55,100
So, this equation
is important.
651
00:43:55,100 --> 00:43:57,720
There's no point in attempting
to derive it.
652
00:43:57,720 --> 00:44:03,460
However you get to it, it's
important to retain it.
653
00:44:03,460 --> 00:44:08,595
We can now use that summation
in the expression
654
00:44:08,595 --> 00:44:11,170
that we just developed.
655
00:44:11,170 --> 00:44:16,930
So let's proceed to evaluate
that sum in closed form.
656
00:44:16,930 --> 00:44:21,520
We now go back to the expression
that we just
657
00:44:21,520 --> 00:44:22,910
worked out-- y[n]
658
00:44:22,910 --> 00:44:27,150
is alpha^n, the sum from
0 to n, (alpha^(-1))^k.
659
00:44:27,150 --> 00:44:31,950
660
00:44:31,950 --> 00:44:35,280
This plays the role of beta
in the term that I just--
661
00:44:35,280 --> 00:44:39,120
in the expression then
I just presented.
662
00:44:39,120 --> 00:44:45,170
So, using that result, we can
rewrite this summation as I
663
00:44:45,170 --> 00:44:46,420
indicate here.
664
00:44:46,420 --> 00:44:49,000
665
00:44:49,000 --> 00:44:52,550
The final result that we end up
with after a certain amount
666
00:44:52,550 --> 00:44:55,180
of algebra is y[n]
667
00:44:55,180 --> 00:45:02,230
equal to (1 - alpha^(n+1))
/ (1 - alpha).
668
00:45:02,230 --> 00:45:09,560
Let me just kind of indicate
with a few dots here that
669
00:45:09,560 --> 00:45:13,970
there is a certain amount of
algebra required in going from
670
00:45:13,970 --> 00:45:18,470
this step to this step, and I'd
like to leave you with the
671
00:45:18,470 --> 00:45:20,740
fun and opportunity of doing
that at your leisure.
672
00:45:20,740 --> 00:45:23,860
673
00:45:23,860 --> 00:45:28,770
The expression we have
now for y[n], is y[n]
674
00:45:28,770 --> 00:45:33,560
= (1 - alpha^(n+1))
/ (1 - alpha).
675
00:45:33,560 --> 00:45:35,900
That's for n greater than 0.
676
00:45:35,900 --> 00:45:37,980
We had found out previously
there it was 0
677
00:45:37,980 --> 00:45:40,070
for n less than 0.
678
00:45:40,070 --> 00:45:45,000
Finally, if we were to plot
this, what we would get is the
679
00:45:45,000 --> 00:45:48,310
graph that I indicate here.
680
00:45:48,310 --> 00:45:53,350
The first non-zero value occurs
at n = 0, and it has a
681
00:45:53,350 --> 00:45:59,750
height of 1, and then the next
non-zero value at 1, and this
682
00:45:59,750 --> 00:46:05,490
has a height of 1 + alpha, and
this is 1 + alpha + alpha^2.
683
00:46:05,490 --> 00:46:08,760
684
00:46:08,760 --> 00:46:13,960
The sequence continues on like
that and asymptotically
685
00:46:13,960 --> 00:46:17,980
approaches, as n goes to
infinity, asymptotically
686
00:46:17,980 --> 00:46:22,630
approaches 1 / (1 - alpha),
which is consistent with the
687
00:46:22,630 --> 00:46:27,780
algebraic expression that we
have, that we developed, and
688
00:46:27,780 --> 00:46:32,370
obviously of course is also
consistent with the movie.
689
00:46:32,370 --> 00:46:35,760
That's our discrete-time
example, which we kind of went
690
00:46:35,760 --> 00:46:39,720
through graphically with the
transparencies, and we went
691
00:46:39,720 --> 00:46:41,870
through graphically with the
movie, and now we've gone
692
00:46:41,870 --> 00:46:43,500
through analytically.
693
00:46:43,500 --> 00:46:48,690
Now let's look analytically at
the continuous-time example,
694
00:46:48,690 --> 00:46:52,090
which pretty much flows
in the same way as
695
00:46:52,090 --> 00:46:55,010
we've just gone through.
696
00:46:55,010 --> 00:46:59,820
Again, we have the convolution
integral, which is the
697
00:46:59,820 --> 00:47:02,275
integral indicated at the top.
698
00:47:02,275 --> 00:47:05,070
699
00:47:05,070 --> 00:47:11,250
Our example, you recall, was
with x(t) as a unit step, and
700
00:47:11,250 --> 00:47:15,850
h(t) as an exponential
times a unit step.
701
00:47:15,850 --> 00:47:20,830
So when we substitute those in,
this then corresponds to
702
00:47:20,830 --> 00:47:24,730
x(t) and this corresponds
to h(t-tau).
703
00:47:24,730 --> 00:47:27,600
704
00:47:27,600 --> 00:47:33,230
Again, we have the same issue
more or less, which is that
705
00:47:33,230 --> 00:47:36,690
inside that integral, there
are two steps, one of them
706
00:47:36,690 --> 00:47:39,480
going forward in time and one
of them going backward in
707
00:47:39,480 --> 00:47:44,340
time, and we need to examine
when they overlap and when
708
00:47:44,340 --> 00:47:44,810
they don't.
709
00:47:44,810 --> 00:47:48,310
When they don't overlap, the
product, of course, is 0, and
710
00:47:48,310 --> 00:47:50,920
there's no point doing any
integration because the
711
00:47:50,920 --> 00:47:52,720
integrand is 0.
712
00:47:52,720 --> 00:47:58,160
So if we track it through, we
have again Interval 1, which
713
00:47:58,160 --> 00:48:00,300
is t less than 0.
714
00:48:00,300 --> 00:48:06,940
For t less than 0, this unit
step, which only begins at tau
715
00:48:06,940 --> 00:48:13,990
= 0, and this unit step which is
0, by the time tau gets up
716
00:48:13,990 --> 00:48:16,170
to t and beyond.
717
00:48:16,170 --> 00:48:22,860
For t less than 0, there's no
overlap between the unit step
718
00:48:22,860 --> 00:48:25,040
going forward in time
and the unit step
719
00:48:25,040 --> 00:48:27,210
going backward in time.
720
00:48:27,210 --> 00:48:31,940
Consequently, the integrand is
equal to 0, and consequently,
721
00:48:31,940 --> 00:48:33,450
the output is equal to 0.
722
00:48:33,450 --> 00:48:40,190
723
00:48:40,190 --> 00:48:45,520
We can likewise look at the
interval where the two unit
724
00:48:45,520 --> 00:48:48,070
steps do overlap.
725
00:48:48,070 --> 00:48:51,380
In that case what happens again
is that the overlap, in
726
00:48:51,380 --> 00:48:55,940
essence of the unit step, tells
us, gives us a range on
727
00:48:55,940 --> 00:48:59,320
the integration--
728
00:48:59,320 --> 00:49:03,600
in particular, the two steps
overlap when t is greater than
729
00:49:03,600 --> 00:49:10,840
0 from tau = 0 to tau = t.
730
00:49:10,840 --> 00:49:15,140
For Interval 2, for
t greater than 0--
731
00:49:15,140 --> 00:49:19,370
again, of course, we have
this expression.
732
00:49:19,370 --> 00:49:25,200
This product of this unit step
and this unit step is equal to
733
00:49:25,200 --> 00:49:34,310
1 in this range, and so that
allows us, then, to change the
734
00:49:34,310 --> 00:49:36,800
limits on the integral--
735
00:49:36,800 --> 00:49:40,290
instead of from -infinity to
+infinity, we know that the
736
00:49:40,290 --> 00:49:45,590
integrand is non-zero only
over this range.
737
00:49:45,590 --> 00:49:48,990
Looking at this integral, we
can now pull out the term
738
00:49:48,990 --> 00:49:55,240
which corresponds to e^-at, just
as we pulled out a term
739
00:49:55,240 --> 00:49:56,725
in the discrete-time case.
740
00:49:56,725 --> 00:49:59,310
741
00:49:59,310 --> 00:50:05,190
We notice in the integral that
we have e^(-a*-tau), so that
742
00:50:05,190 --> 00:50:09,740
gives us the integral from
0 to t of e^(a*tau).
743
00:50:09,740 --> 00:50:13,600
If we perform that integration,
we end up with
744
00:50:13,600 --> 00:50:15,220
this expression.
745
00:50:15,220 --> 00:50:22,500
Finally, multiplying this by
e^-at, we have for y(t), for t
746
00:50:22,500 --> 00:50:28,270
greater than 0, the algebraic
expression that I've indicated
747
00:50:28,270 --> 00:50:29,520
on the bottom.
748
00:50:29,520 --> 00:50:33,540
749
00:50:33,540 --> 00:50:36,500
So we had worked out t
less than 0, and we
750
00:50:36,500 --> 00:50:38,170
come out with 0.
751
00:50:38,170 --> 00:50:43,630
We work out t greater than 0,
and we come out with this
752
00:50:43,630 --> 00:50:44,880
algebraic expression.
753
00:50:44,880 --> 00:50:47,250
754
00:50:47,250 --> 00:50:50,560
If we plot this algebraic
expression as a function of
755
00:50:50,560 --> 00:50:54,690
time, we find that what it
corresponds to is an
756
00:50:54,690 --> 00:51:00,210
exponential behavior starting
at zero and exponentially
757
00:51:00,210 --> 00:51:04,800
heading asymptotically toward
the value 1 / a.
758
00:51:04,800 --> 00:51:09,980
759
00:51:09,980 --> 00:51:12,430
We've gone through these
examples several ways, and one
760
00:51:12,430 --> 00:51:14,470
is analytically.
761
00:51:14,470 --> 00:51:19,780
In order to develop a feel and
fluency for convolution, it's
762
00:51:19,780 --> 00:51:24,000
absolutely essential to work
through a variety of examples,
763
00:51:24,000 --> 00:51:27,410
both understanding them
graphically and understanding
764
00:51:27,410 --> 00:51:31,790
them as we did here
analytically.
765
00:51:31,790 --> 00:51:34,860
You'll have an opportunity to
do that through the problems
766
00:51:34,860 --> 00:51:38,950
that I've suggested in the
video course manual.
767
00:51:38,950 --> 00:51:43,980
In the next lecture, what we'll
turn to are some general
768
00:51:43,980 --> 00:51:48,210
properties of convolution, and
show how this rather amazing
769
00:51:48,210 --> 00:51:52,310
representation of linear
time-invariant systems in fact
770
00:51:52,310 --> 00:51:55,580
leads to a variety of
properties of linear
771
00:51:55,580 --> 00:51:57,620
time-invariant systems.
772
00:51:57,620 --> 00:52:02,690
We'll find that convolution is
fairly rich in its properties,
773
00:52:02,690 --> 00:52:09,180
and what this leads to are some
very nice and desirable
774
00:52:09,180 --> 00:52:12,680
and exploitable properties of
linear time-invariant systems.
775
00:52:12,680 --> 00:52:13,930
Thank you.
776
00:52:13,930 --> 00:52:16,704