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[MUSIC PLAYING]

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ALAN OPPENHEIM:
Well, last time we

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00:00:59,370 --> 00:01:02,640
had discussed, in a very
general way, the field

11
00:01:02,640 --> 00:01:06,180
of digital signal
processing, and hopefully

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00:01:06,180 --> 00:01:08,400
one of the things that
you were convinced of

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00:01:08,400 --> 00:01:12,480
was the importance of
this set of techniques

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00:01:12,480 --> 00:01:14,230
in a general sense.

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00:01:14,230 --> 00:01:17,400
What we'd like to
begin on now is

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00:01:17,400 --> 00:01:21,780
a discussion of some of the
details of digital signal

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00:01:21,780 --> 00:01:23,250
processing.

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00:01:23,250 --> 00:01:26,620
And in particular,
in this lecture,

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00:01:26,620 --> 00:01:31,740
I would like to introduce the
class of discrete time signals,

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00:01:31,740 --> 00:01:36,070
and also the class of
discrete time systems.

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00:01:36,070 --> 00:01:42,330
Well, last time, I
reminded you that I

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00:01:42,330 --> 00:01:49,050
will be assuming a familiarity
with continuous time signals

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00:01:49,050 --> 00:01:50,460
and systems.

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00:01:50,460 --> 00:01:53,580
And what you'll see as we
go through this lecture--

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00:01:53,580 --> 00:01:56,460
and in fact, as we go through
all of these lectures--

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00:01:56,460 --> 00:02:01,710
is a very strong similarity
between the results that you're

27
00:02:01,710 --> 00:02:05,110
familiar with in the
continuous time case,

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00:02:05,110 --> 00:02:09,690
and results that will develop
for the discrete time case.

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00:02:09,690 --> 00:02:14,380
Obviously, of course, there
will also be some differences.

30
00:02:14,380 --> 00:02:16,500
And these differences
are important,

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00:02:16,500 --> 00:02:18,840
and they will be
important to focus on.

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00:02:18,840 --> 00:02:23,340
So there will be a great
deal of similarity, also

33
00:02:23,340 --> 00:02:24,580
a few differences.

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00:02:24,580 --> 00:02:28,440
And these few differences
are very important.

35
00:02:28,440 --> 00:02:31,200
In the discrete
time domain, we're

36
00:02:31,200 --> 00:02:36,150
concerned with processing
signals that are sequences.

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00:02:36,150 --> 00:02:39,870
That is, these signals are
functions of an integer

38
00:02:39,870 --> 00:02:43,650
variable, which we'll call n.

39
00:02:43,650 --> 00:02:47,730
Typically, will depict
sequences graphically,

40
00:02:47,730 --> 00:02:54,570
as I've illustrated here on
this first viewgraph, where I've

41
00:02:54,570 --> 00:02:57,120
denoted a general sequence--

42
00:02:57,120 --> 00:02:59,910
which I call x of n--

43
00:02:59,910 --> 00:03:01,890
a function of the
energy variable

44
00:03:01,890 --> 00:03:08,730
n, and the sequence values,
I've represented by a bar graph

45
00:03:08,730 --> 00:03:11,820
with the height of
each bar corresponding

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00:03:11,820 --> 00:03:14,340
to the sequence value.

47
00:03:14,340 --> 00:03:18,705
So we have, then,
for example, at n

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00:03:18,705 --> 00:03:23,495
equals 0, a line of
height x of 0 at n

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00:03:23,495 --> 00:03:28,380
equals 1 a line of
height x of 1, et cetera.

50
00:03:28,380 --> 00:03:31,740
Of course, as I've drawn
this horizontal axis,

51
00:03:31,740 --> 00:03:35,670
I've drawn it as
a continuous line,

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00:03:35,670 --> 00:03:41,550
but it's important to keep in
mind that the sequences only

53
00:03:41,550 --> 00:03:42,600
are defined--

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00:03:42,600 --> 00:03:44,400
or only make sense--

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00:03:44,400 --> 00:03:47,530
for integer values
of the argument.

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00:03:47,530 --> 00:03:51,940
If n is an integer,
then x of n makes sense.

57
00:03:51,940 --> 00:03:57,010
If n is not an integer,
it's not just that x of n

58
00:03:57,010 --> 00:04:02,200
is 0 or something like that,
it's simply that x of n

59
00:04:02,200 --> 00:04:03,470
is not defined.

60
00:04:03,470 --> 00:04:05,950
In other words, x
of n makes sense,

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00:04:05,950 --> 00:04:09,550
or is defined for n an integer,
and does not make sense,

62
00:04:09,550 --> 00:04:13,940
or is not defined, if
n is not an integer.

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00:04:13,940 --> 00:04:18,279
So this, then, is a
graphical representation

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00:04:18,279 --> 00:04:21,579
of a general sequence.

65
00:04:21,579 --> 00:04:25,430
And just as in the
continuous time case,

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00:04:25,430 --> 00:04:29,680
we have a number of
important basic sequences

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00:04:29,680 --> 00:04:33,930
that we would like to focus on.

68
00:04:33,930 --> 00:04:36,480
The first of these that
I'd like to introduce

69
00:04:36,480 --> 00:04:41,390
is the unit sample,
or impulse sequence,

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00:04:41,390 --> 00:04:45,850
which we'll denote
by delta of n.

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00:04:45,850 --> 00:04:48,540
The unit sample
sequence is a sequence

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00:04:48,540 --> 00:04:54,870
whose value is unity at n
equals 0, and it's equal to 0

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00:04:54,870 --> 00:04:55,800
otherwise.

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00:04:55,800 --> 00:05:00,840
That is delta of n equals 1
at n equals 0, and delta of n

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00:05:00,840 --> 00:05:05,660
is equal to 0 for
n not equal to 0.

76
00:05:05,660 --> 00:05:09,230
The unit sample sequence
plays the same role

77
00:05:09,230 --> 00:05:13,280
in the discrete time case
that the unit impulse time

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00:05:13,280 --> 00:05:16,400
function plays in the
continuous time case.

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00:05:16,400 --> 00:05:20,720
Let me just mention, however,
that the unit sample sequence

80
00:05:20,720 --> 00:05:23,610
is very easily defined.

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00:05:23,610 --> 00:05:26,900
There's no issue here
about the definition.

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00:05:26,900 --> 00:05:30,050
In contrast with the usual
mathematical problems

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00:05:30,050 --> 00:05:32,550
with a continuous time impulse--

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00:05:32,550 --> 00:05:37,310
which is infinitely big at the
origin, and 0 every place else,

85
00:05:37,310 --> 00:05:39,650
but has some area, et cetera.

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00:05:39,650 --> 00:05:43,790
So there's a very precise
definition here of the unit

87
00:05:43,790 --> 00:05:47,570
sample, or unit
impulse sequence,

88
00:05:47,570 --> 00:05:51,380
and it will play a role similar
to the unit impulse time

89
00:05:51,380 --> 00:05:55,430
function in the
continuous time case.

90
00:05:55,430 --> 00:06:01,310
Another important basic sequence
is the unit step sequence,

91
00:06:01,310 --> 00:06:05,180
which I'm denoting by u of n.

92
00:06:05,180 --> 00:06:10,550
And the unit step sequence
is equal to unity for n

93
00:06:10,550 --> 00:06:15,080
greater than or equal
to 0, and is equal to 0

94
00:06:15,080 --> 00:06:18,980
for n less than 0.

95
00:06:18,980 --> 00:06:24,680
So the unit step sequence,
then, is 0 for n less than 0,

96
00:06:24,680 --> 00:06:30,890
it's unity four n greater
than or equal to 0.

97
00:06:30,890 --> 00:06:33,560
This shouldn't be, incidentally,
less than or equal to 0,

98
00:06:33,560 --> 00:06:36,590
this should be n less than 0.

99
00:06:36,590 --> 00:06:44,300
And it is-- again, does not have
the difficulties in definition

100
00:06:44,300 --> 00:06:49,170
that a continuous time
unit step normally has.

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00:06:49,170 --> 00:06:53,420
It's the change in value
from n equals minus 1 to n

102
00:06:53,420 --> 00:06:57,590
equals 0 is precisely
and easily defined.

103
00:06:57,590 --> 00:07:01,520
Here the unit step is 0,
here the unit step is 1.

104
00:07:01,520 --> 00:07:04,850
There's no issue, as there is
in the continuous time case,

105
00:07:04,850 --> 00:07:06,020
about discontinuity.

106
00:07:08,680 --> 00:07:11,100
Just as in the
continuous time case,

107
00:07:11,100 --> 00:07:16,810
there is a simple relationship
between the unit sample,

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00:07:16,810 --> 00:07:20,020
or a unit impulse,
and the unit step.

109
00:07:20,020 --> 00:07:22,420
In a discrete time
case, there is likewise

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00:07:22,420 --> 00:07:26,650
a simple relationship between
the unit sample sequence,

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00:07:26,650 --> 00:07:29,530
and the unit step sequence.

112
00:07:29,530 --> 00:07:33,220
In particular, the
unit sample sequence

113
00:07:33,220 --> 00:07:37,210
can be obtained from
the unit step sequence

114
00:07:37,210 --> 00:07:41,370
by constructing the
first difference.

115
00:07:41,370 --> 00:07:44,410
The unit sample
sequence, delta of n,

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00:07:44,410 --> 00:07:48,310
is equal to u of
n, the unit step,

117
00:07:48,310 --> 00:07:52,690
minus the unit step
delayed by 1 sample.

118
00:07:52,690 --> 00:07:55,030
And we can see
that that obviously

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00:07:55,030 --> 00:07:59,740
is the case, if we look at the
unit step sequence, u of n.

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00:07:59,740 --> 00:08:04,000
Here we have the unit step
sequence delayed by 1--

121
00:08:04,000 --> 00:08:08,132
so this occurs at n equals
0, this is n equals 1.

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00:08:08,132 --> 00:08:11,520
And clearly, the difference
between those two

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00:08:11,520 --> 00:08:15,910
will generate a sample at
n equals 0, 0, of course,

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00:08:15,910 --> 00:08:21,250
for n less than 0, and 0,
likewise, for n greater than 0,

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00:08:21,250 --> 00:08:24,640
because these sequence
values will be canceled out

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00:08:24,640 --> 00:08:26,740
by those sequence values.

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00:08:26,740 --> 00:08:32,230
So the unit sample sequence is
the unit step minus the delayed

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00:08:32,230 --> 00:08:35,110
unit step, or the
unit sample sequence

129
00:08:35,110 --> 00:08:40,150
is equal to the first difference
of the unit step sequence.

130
00:08:40,150 --> 00:08:43,929
Similarly, we can obtain
the unit step sequence

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00:08:43,929 --> 00:08:50,190
from the unit sample sequence
by constructing a running sum.

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00:08:50,190 --> 00:08:53,950
In particular, the
unit step sequence

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00:08:53,950 --> 00:08:59,980
is equal to a running sum
of the unit sample sequence.

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00:08:59,980 --> 00:09:02,650
In other words, we're
constructing here a sum

135
00:09:02,650 --> 00:09:08,750
from k equals minus infinity
to the independent variable n

136
00:09:08,750 --> 00:09:12,280
of the sequence, delta of k.

137
00:09:12,280 --> 00:09:16,060
Here we have the
sequence, delta of k.

138
00:09:16,060 --> 00:09:22,000
If n is less than 0, then
we're adding up sequence values

139
00:09:22,000 --> 00:09:27,160
from minus infinity to
some negative value of k.

140
00:09:27,160 --> 00:09:29,500
And the only values
that we collect when we

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00:09:29,500 --> 00:09:32,230
do that are values equal to 0.

142
00:09:32,230 --> 00:09:38,770
So for n less than 0, we see
that we obtain 0 for this sum.

143
00:09:38,770 --> 00:09:44,140
If n is greater than 0, as I've
indicated here, then the sum,

144
00:09:44,140 --> 00:09:48,490
as we collect sequence values
from minus infinity up to n,

145
00:09:48,490 --> 00:09:53,440
we collect 1 non-0 sequence
value which is equal to 1.

146
00:09:53,440 --> 00:09:56,110
Consequently for
n greater than 0,

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00:09:56,110 --> 00:10:00,790
we obtain, in this running
sum, u of n equal to 1.

148
00:10:00,790 --> 00:10:03,130
This plays a role,
this running sum

149
00:10:03,130 --> 00:10:07,690
plays a role, similar to the
integral in the continuous time

150
00:10:07,690 --> 00:10:08,720
case.

151
00:10:08,720 --> 00:10:11,020
And so now we have
a relationship

152
00:10:11,020 --> 00:10:12,970
between the unit
sample sequence,

153
00:10:12,970 --> 00:10:16,250
and the unit step sequence.

154
00:10:16,250 --> 00:10:18,490
There are other
basic sequences that

155
00:10:18,490 --> 00:10:21,430
will play an important
role, just as they

156
00:10:21,430 --> 00:10:25,660
do in the continuous time case.

157
00:10:25,660 --> 00:10:32,110
In particular, we have the
real exponential sequence.

158
00:10:32,110 --> 00:10:38,560
The sequence x of n is
equal to alpha to the n.

159
00:10:38,560 --> 00:10:42,040
I've depicted the
exponential sequence here

160
00:10:42,040 --> 00:10:48,040
for the case in which alpha is
positive but less than unity,

161
00:10:48,040 --> 00:10:53,710
so that as n increases,
the exponential decreases.

162
00:10:53,710 --> 00:10:57,910
So alpha to the n, if
alpha is between 0 and 1,

163
00:10:57,910 --> 00:11:00,280
decreases with increasing n.

164
00:11:00,280 --> 00:11:03,910
If alpha is greater than 1,
then the exponential sequence

165
00:11:03,910 --> 00:11:07,660
grows, of course, exponentially.

166
00:11:07,660 --> 00:11:12,640
We have, also, the
sinusoidal sequence.

167
00:11:12,640 --> 00:11:15,220
The general form of
the sinusoidal sequence

168
00:11:15,220 --> 00:11:21,730
is x of n is equal to a
cosine omega 0 n plus phi--

169
00:11:21,730 --> 00:11:27,100
phi a phase angle, omega 0 we'll
refer to as the frequency, a,

170
00:11:27,100 --> 00:11:29,500
of course, is the amplitude.

171
00:11:29,500 --> 00:11:33,640
I've illustrated a
sinusoidal sequence here

172
00:11:33,640 --> 00:11:35,890
for a particular
choice of omega 0

173
00:11:35,890 --> 00:11:39,310
and phi, namely omega
0 equal to pi over 4,

174
00:11:39,310 --> 00:11:42,580
and phi equal minus pi over 8.

175
00:11:42,580 --> 00:11:49,990
And in looking at this, we see
that we get a periodic sequence

176
00:11:49,990 --> 00:11:52,420
that is, for this
choice of parameters,

177
00:11:52,420 --> 00:11:55,390
this sequence is periodic.

178
00:11:55,390 --> 00:11:57,250
However, an
important distinction

179
00:11:57,250 --> 00:12:01,330
between the continuous time
case and the discrete time case,

180
00:12:01,330 --> 00:12:06,310
is that a discrete time
sinusoidal signal is not

181
00:12:06,310 --> 00:12:09,070
necessarily periodic.

182
00:12:09,070 --> 00:12:11,260
Particular, I've
illustrated here

183
00:12:11,260 --> 00:12:15,130
another sinusoidal sequence
with a different choice

184
00:12:15,130 --> 00:12:17,560
for omega 0 and phi.

185
00:12:17,560 --> 00:12:22,840
In this case, omega 0
equal to 3 pi over 7.

186
00:12:22,840 --> 00:12:25,120
With that choice
of omega 0, these

187
00:12:25,120 --> 00:12:27,760
are the sequence
values we obtain.

188
00:12:27,760 --> 00:12:31,990
And it should be obvious
by examining this sequence

189
00:12:31,990 --> 00:12:35,170
that this sequence is
no longer periodic,

190
00:12:35,170 --> 00:12:37,930
whereas this sequence was.

191
00:12:37,930 --> 00:12:42,190
So the sinusoidal sequence
may or may not be periodic.

192
00:12:42,190 --> 00:12:46,780
Of course, let me remind you
that if n was allowed to vary

193
00:12:46,780 --> 00:12:49,780
continuously-- if it was
not an integer variable--

194
00:12:49,780 --> 00:12:52,290
x of n would be periodic.

195
00:12:52,290 --> 00:12:56,800
The periodicity is lost
because n is now constrained

196
00:12:56,800 --> 00:12:59,710
to be an integer variable.

197
00:12:59,710 --> 00:13:02,440
And that is one of the
important distinctions

198
00:13:02,440 --> 00:13:06,310
between continuous time signals
and systems, and discrete time

199
00:13:06,310 --> 00:13:08,500
signals and systems.

200
00:13:08,500 --> 00:13:13,120
Another point which
I'll suggest now,

201
00:13:13,120 --> 00:13:17,020
although it's a point
that I'll want to refer to

202
00:13:17,020 --> 00:13:21,400
in more detail in some
of the future lectures,

203
00:13:21,400 --> 00:13:31,480
is that the sinusoidal sequence
is only distinguishable

204
00:13:31,480 --> 00:13:37,240
as omega 0 runs from
over the range 0 to 2 pi

205
00:13:37,240 --> 00:13:40,060
or minus pi to pi.

206
00:13:40,060 --> 00:13:45,340
If omega 0-- if we were to
replace omega 0 by omega 0

207
00:13:45,340 --> 00:13:50,830
plus 2 pi, then we would have
in this argument omega 0 n plus

208
00:13:50,830 --> 00:13:52,180
2 pi n.

209
00:13:52,180 --> 00:13:54,370
The 2 pi n, of course,
would have no effect,

210
00:13:54,370 --> 00:13:56,480
and we'd end up with
the same sequence.

211
00:13:56,480 --> 00:14:01,590
So in fact as, we varied omega
0 between minus pi and plus pi,

212
00:14:01,590 --> 00:14:06,820
you will have seen all of
the sinusoidal sequences

213
00:14:06,820 --> 00:14:08,830
with this amplitude
and this phase

214
00:14:08,830 --> 00:14:12,820
that we can possibly generate.

215
00:14:12,820 --> 00:14:20,020
One of the important of
this set of basic signals

216
00:14:20,020 --> 00:14:24,850
is that they can be used to
represent a more general class

217
00:14:24,850 --> 00:14:28,880
of signals, just as in
the continuous time case,

218
00:14:28,880 --> 00:14:33,040
we use impulses, or we
use complex exponentials,

219
00:14:33,040 --> 00:14:37,570
or we use sinusoids to
represent very general signals,

220
00:14:37,570 --> 00:14:41,500
we can develop similar
representations here.

221
00:14:41,500 --> 00:14:47,050
Let me illustrate this
with an example where

222
00:14:47,050 --> 00:14:55,440
I consider a general
sequence here, x of n.

223
00:14:55,440 --> 00:15:01,180
The sequence values are x of
0, x of 1, x of 2, et cetera.

224
00:15:01,180 --> 00:15:05,860
And I mean to suggest here
a very general sequence.

225
00:15:05,860 --> 00:15:10,360
Now, I can decompose
this sequence

226
00:15:10,360 --> 00:15:16,590
into a linear combination
of weighted, delayed unit

227
00:15:16,590 --> 00:15:20,920
samples, unit sample
sequences, by simply

228
00:15:20,920 --> 00:15:24,910
extracting individual
sequence values out of x of n.

229
00:15:24,910 --> 00:15:29,420
For example, let's
consider the sequence

230
00:15:29,420 --> 00:15:34,340
x of 0 times delta of n,
as I've illustrated here.

231
00:15:34,340 --> 00:15:38,890
X of 0 times delta of n is
equal to the original sequence

232
00:15:38,890 --> 00:15:43,600
x of n at n equals 0, and
it's equal to 0 otherwise.

233
00:15:46,290 --> 00:15:50,860
A second sequence, x of 1
times delta of n minus 1,

234
00:15:50,860 --> 00:15:55,510
is a unit sample
sequence, delayed by 1

235
00:15:55,510 --> 00:15:58,600
and having an amplitude
equal to x of 1.

236
00:15:58,600 --> 00:16:03,100
So this sequence is equal
to this one at n equals 1,

237
00:16:03,100 --> 00:16:05,320
and it's equal to 0, otherwise.

238
00:16:05,320 --> 00:16:07,960
The sum of these
two, of course, is

239
00:16:07,960 --> 00:16:14,020
equal to this x of n at these
two values, and equal to 0,

240
00:16:14,020 --> 00:16:15,100
otherwise.

241
00:16:15,100 --> 00:16:19,660
Similarly, we can consider
x of minus 1 times

242
00:16:19,660 --> 00:16:21,890
delta of n plus 1.

243
00:16:21,890 --> 00:16:24,820
That's a unit sample
at n equals minus 1,

244
00:16:24,820 --> 00:16:28,070
with an amplitude
of x of minus 1,

245
00:16:28,070 --> 00:16:31,060
which picks up this
sequence value.

246
00:16:31,060 --> 00:16:37,600
x of minus 2 times delta of n
plus 2 is a unit sample at n

247
00:16:37,600 --> 00:16:42,650
equals minus 2 multiplied by
an amplitude x of minus 2,

248
00:16:42,650 --> 00:16:47,450
which picks up this
sequence value, et cetera.

249
00:16:47,450 --> 00:16:52,870
So I think you can see, then,
that as we add these up,

250
00:16:52,870 --> 00:16:57,710
what we'll generate is this
arbitrary sequence, x of n.

251
00:16:57,710 --> 00:17:03,490
In other words, we can construct
x of n, an arbitrary sequence,

252
00:17:03,490 --> 00:17:09,060
as a linear combination of
weighted delayed unit samples--

253
00:17:09,060 --> 00:17:15,700
x of 0 times delta of n plus x
of 1 times delta of n minus 1

254
00:17:15,700 --> 00:17:19,060
plus x of minus 1 delta
n plus 1, et cetera.

255
00:17:19,060 --> 00:17:22,710
Or, more generally,
the sum from k

256
00:17:22,710 --> 00:17:27,640
equals minus infinity to
plus infinity of x of k times

257
00:17:27,640 --> 00:17:30,010
delta of n minus k.

258
00:17:30,010 --> 00:17:36,490
This, then, corresponds to
a general representation

259
00:17:36,490 --> 00:17:41,320
of an arbitrary sequence in
terms of weighted delayed unit

260
00:17:41,320 --> 00:17:42,740
samples.

261
00:17:42,740 --> 00:17:46,600
And it's a representation that
we'll want to refer back to

262
00:17:46,600 --> 00:17:47,980
in a few minutes.

263
00:17:47,980 --> 00:17:51,400
It's not, as I've indicated,
the only representation

264
00:17:51,400 --> 00:17:53,490
of arbitrary sequences--

265
00:17:53,490 --> 00:17:56,230
we'll also be developing
representations

266
00:17:56,230 --> 00:17:58,450
in terms of complex
exponentials,

267
00:17:58,450 --> 00:18:03,700
or real exponentials, or in
terms of sines and cosines.

268
00:18:03,700 --> 00:18:12,760
OK, that is an introduction
to the basic class of signals

269
00:18:12,760 --> 00:18:14,800
and the notion of sequences.

270
00:18:14,800 --> 00:18:19,540
What I'd like to do now is focus
on the class of discrete time

271
00:18:19,540 --> 00:18:23,800
systems, and then refer back
to some of these results

272
00:18:23,800 --> 00:18:25,840
to develop a general
representation

273
00:18:25,840 --> 00:18:28,555
for a special class of
discrete time systems.

274
00:18:35,160 --> 00:18:41,560
First of all, let me begin
with a general system, which

275
00:18:41,560 --> 00:18:47,540
is a discrete time system
that has an input--

276
00:18:47,540 --> 00:18:50,200
which is a sequence x of n.

277
00:18:50,200 --> 00:18:53,680
It has an output, which
is a sequence y of n.

278
00:18:53,680 --> 00:18:56,150
And it has a system
transformation,

279
00:18:56,150 --> 00:19:00,760
which I've denoted by T.
Of course, there isn't much

280
00:19:00,760 --> 00:19:03,850
that you can do with
a general system--

281
00:19:03,850 --> 00:19:07,960
the difficulty with trying
to describe a general system

282
00:19:07,960 --> 00:19:10,450
in general is that
there are no properties

283
00:19:10,450 --> 00:19:12,920
that you can take advantage of.

284
00:19:12,920 --> 00:19:16,240
So always, in
characterizing systems,

285
00:19:16,240 --> 00:19:19,630
it's useful to specialize
the class of systems--

286
00:19:19,630 --> 00:19:22,060
that is, impose properties
on the system which

287
00:19:22,060 --> 00:19:25,330
you can exploit to
represent the system,

288
00:19:25,330 --> 00:19:28,930
or to implement the
system, et cetera.

289
00:19:28,930 --> 00:19:31,900
The special class that
we'll want to consider

290
00:19:31,900 --> 00:19:36,250
is the class of systems which
are, first of all, linear.

291
00:19:36,250 --> 00:19:40,210
And second of all,
shift invariant.

292
00:19:40,210 --> 00:19:44,650
And this class corresponds,
in the continuous time case,

293
00:19:44,650 --> 00:19:47,140
to the class of systems
that we normally

294
00:19:47,140 --> 00:19:50,470
refer to as linear
and time invariant.

295
00:19:50,470 --> 00:19:53,700
We'll tend to refer to these
as linear and shift invariant,

296
00:19:53,700 --> 00:20:00,700
and for a shorthand notation,
just express this as well LSI.

297
00:20:00,700 --> 00:20:03,610
When I refer to an LSI
system, what I mean

298
00:20:03,610 --> 00:20:07,180
is a system that is linear
and shift invariant.

299
00:20:07,180 --> 00:20:09,740
Well, let's define these terms.

300
00:20:09,740 --> 00:20:11,530
First of all, linearity.

301
00:20:14,170 --> 00:20:18,690
Linearity states
that if I excite

302
00:20:18,690 --> 00:20:24,750
the system with a sequence x 1
of n, and I get at the output

303
00:20:24,750 --> 00:20:30,120
a sequence y 1 of n, and
if I excite with x 2 of n,

304
00:20:30,120 --> 00:20:37,470
and get at the output y 2 of n,
then the condition of linearity

305
00:20:37,470 --> 00:20:43,530
is that the linear combination,
a x 1 of n plus b x 2 of n,

306
00:20:43,530 --> 00:20:48,570
produces at the output a
y1 of n plus b y 2 of n.

307
00:20:48,570 --> 00:20:53,630
That is, the response of the
system to a sum of inputs,

308
00:20:53,630 --> 00:20:55,860
or a linear
combination of inputs,

309
00:20:55,860 --> 00:21:00,130
is a linear combination of
the corresponding outputs.

310
00:21:00,130 --> 00:21:04,230
Now, by repeated application
of just this statement,

311
00:21:04,230 --> 00:21:07,120
we can make a more
general statement,

312
00:21:07,120 --> 00:21:14,910
which is that the sum of a sub
k times x sub k of n produces,

313
00:21:14,910 --> 00:21:24,090
as an output, the sum of a
sub k times y sub k of n.

314
00:21:24,090 --> 00:21:28,050
Linearity is stated here,
of course, for two inputs,

315
00:21:28,050 --> 00:21:32,730
but we can obviously extend
that to an arbitrary number

316
00:21:32,730 --> 00:21:36,360
of inputs so that the statement
of linearity, as we'll

317
00:21:36,360 --> 00:21:39,300
refer to it, will
generally be that

318
00:21:39,300 --> 00:21:43,260
a general linear combination of
inputs produces, at the output,

319
00:21:43,260 --> 00:21:47,820
the same linear combination
of the corresponding outputs.

320
00:21:47,820 --> 00:21:51,630
So that's the
condition of linearity.

321
00:21:51,630 --> 00:21:53,460
The second condition
that we want

322
00:21:53,460 --> 00:21:56,040
to impose on our
class of systems

323
00:21:56,040 --> 00:21:58,880
is the condition of
shift invariance.

324
00:21:58,880 --> 00:22:05,630
What shift invariance says,
simply, is that if x of n

325
00:22:05,630 --> 00:22:10,680
produces y of n,
then x of n minus n 0

326
00:22:10,680 --> 00:22:13,710
produces y of n minus n 0.

327
00:22:13,710 --> 00:22:17,160
It's basically a statement that
the system doesn't particularly

328
00:22:17,160 --> 00:22:20,460
care what you call n equals 0.

329
00:22:20,460 --> 00:22:24,120
In other words, if we
shift the input in n,

330
00:22:24,120 --> 00:22:25,830
we shift the output in n.

331
00:22:25,830 --> 00:22:28,740
It's exactly like the
condition of time invariance

332
00:22:28,740 --> 00:22:30,000
in the continuous time case.

333
00:22:32,570 --> 00:22:37,920
For example, if we excited
the system with a unit sample,

334
00:22:37,920 --> 00:22:40,985
to get the unit
sample response, h

335
00:22:40,985 --> 00:22:46,490
of n, then the response of the
system, if the system is shift

336
00:22:46,490 --> 00:22:49,340
invariant, to a
delayed unit sample

337
00:22:49,340 --> 00:22:54,150
is the delayed version of
the unit sample response.

338
00:22:54,150 --> 00:22:56,390
So this is shift invariance.

339
00:22:56,390 --> 00:22:58,460
We had the condition
of linearity.

340
00:22:58,460 --> 00:23:00,810
These are two
independent conditions

341
00:23:00,810 --> 00:23:03,020
which we'll now
want to put together

342
00:23:03,020 --> 00:23:07,310
to develop a general
representation for linear shift

343
00:23:07,310 --> 00:23:10,200
invariant systems.

344
00:23:10,200 --> 00:23:14,990
Well, let me remind you
from the last viewgraph

345
00:23:14,990 --> 00:23:19,820
that we had looked at, that we
had a general representation

346
00:23:19,820 --> 00:23:25,040
for a sequence in terms of
weighted delayed units samples.

347
00:23:25,040 --> 00:23:30,180
That is, previously we had
developed a representation,

348
00:23:30,180 --> 00:23:35,510
x of n, of this form, which
expresses an arbitrary

349
00:23:35,510 --> 00:23:40,370
sequence in terms of weighted
delayed unit samples.

350
00:23:40,370 --> 00:23:43,370
Well, this, then,
says that x of n

351
00:23:43,370 --> 00:23:48,260
is expressed as a linear
combination of basic inputs,

352
00:23:48,260 --> 00:23:53,210
and if we restrict
ourselves to linear systems,

353
00:23:53,210 --> 00:23:59,870
then the output must correspond
to the same linear combination

354
00:23:59,870 --> 00:24:02,000
of the corresponding outputs.

355
00:24:02,000 --> 00:24:07,910
Well, delta of n minus k into
a linear shift invariant system

356
00:24:07,910 --> 00:24:13,160
will produce, at the
output, by virtue

357
00:24:13,160 --> 00:24:18,380
of the property of shift
invariance, the sequence

358
00:24:18,380 --> 00:24:21,500
h of n minus k.

359
00:24:21,500 --> 00:24:25,840
And the linear combination
of these delayed unit

360
00:24:25,840 --> 00:24:30,380
samples will produce at the
output, by virtue of linearity,

361
00:24:30,380 --> 00:24:38,000
the same linear combination
of the responses

362
00:24:38,000 --> 00:24:40,130
to delta of n minus k.

363
00:24:40,130 --> 00:24:45,200
So if we consider a linear shift
invariant system, and because

364
00:24:45,200 --> 00:24:48,110
of this representation,
we can say

365
00:24:48,110 --> 00:24:52,190
that the response due
to an arbitrary input

366
00:24:52,190 --> 00:24:58,550
is equal to a linear combination
of delayed unit sample

367
00:24:58,550 --> 00:24:59,900
responses.

368
00:24:59,900 --> 00:25:02,067
Where, in general,
this is a sum from k

369
00:25:02,067 --> 00:25:05,990
equals minus infinity
to plus infinity.

370
00:25:05,990 --> 00:25:08,450
Well, this is the key result--

371
00:25:08,450 --> 00:25:11,480
this is a statement
that says that if we

372
00:25:11,480 --> 00:25:14,750
talk about linear shift
invariant systems,

373
00:25:14,750 --> 00:25:17,480
then all that we need
to know about the system

374
00:25:17,480 --> 00:25:21,980
to characterize it is its
response to a unit sample.

375
00:25:21,980 --> 00:25:24,410
If we have its response
to a unit sample,

376
00:25:24,410 --> 00:25:27,620
we can construct y
of n in general--

377
00:25:27,620 --> 00:25:30,980
that is, we can construct
the output for an arbitrary

378
00:25:30,980 --> 00:25:32,000
input, x of k.

379
00:25:34,670 --> 00:25:40,150
This is generally referred
to as the convolution sum.

380
00:25:40,150 --> 00:25:44,750
In analogy with the convolution
integral in the continuous time

381
00:25:44,750 --> 00:25:49,880
case, this is one way of
writing the convolution sum.

382
00:25:49,880 --> 00:25:53,450
There is an
alternative, which is

383
00:25:53,450 --> 00:25:57,350
interesting in that it suggests
some important properties

384
00:25:57,350 --> 00:25:59,810
of linear shift
invariant systems,

385
00:25:59,810 --> 00:26:01,790
and it's obtained
simply by applying

386
00:26:01,790 --> 00:26:05,120
a substitution of variables
to this expression.

387
00:26:05,120 --> 00:26:10,970
In particular, suppose
that we replace n minus k

388
00:26:10,970 --> 00:26:16,880
by the variable r,
or equivalently, k

389
00:26:16,880 --> 00:26:20,510
is equal to n minus r.

390
00:26:20,510 --> 00:26:27,050
Then what we obtain is y of
n is now equal to a sum--

391
00:26:27,050 --> 00:26:29,480
and this is now a sum on r.

392
00:26:29,480 --> 00:26:31,400
For k, we have n minus r.

393
00:26:31,400 --> 00:26:36,416
So this is x of
n minus r times h

394
00:26:36,416 --> 00:26:41,030
of n minus k, which
is equal to r.

395
00:26:41,030 --> 00:26:44,060
So this h of r.

396
00:26:44,060 --> 00:26:46,790
Well, it's a simple
step to take,

397
00:26:46,790 --> 00:26:52,730
but in fact, what it says
is that the system doesn't

398
00:26:52,730 --> 00:26:57,560
particularly care what you
call the input to the system,

399
00:26:57,560 --> 00:27:01,780
and what you call the unit
sample response of the system.

400
00:27:01,780 --> 00:27:07,760
Said another way, it says that
convolution is commutative--

401
00:27:07,760 --> 00:27:12,380
that is, if we represent
the convolution of x of n

402
00:27:12,380 --> 00:27:16,220
with h of n with
an asterisk, then,

403
00:27:16,220 --> 00:27:19,730
because of the fact that
we were able to interchange

404
00:27:19,730 --> 00:27:23,120
the roles of x of n and
h of n, that implies,

405
00:27:23,120 --> 00:27:27,020
essentially, that x of
n convolved with h o n

406
00:27:27,020 --> 00:27:31,940
is the same thing as h of
n convolved with x of n.

407
00:27:31,940 --> 00:27:35,450
That implies, as I
indicated, that if we

408
00:27:35,450 --> 00:27:39,890
had a system with an impulse
response or unit sample

409
00:27:39,890 --> 00:27:46,670
response, h of n, and input
x of n, and output y of n,

410
00:27:46,670 --> 00:27:50,840
that if I call this the input,
and I call this the unit sample

411
00:27:50,840 --> 00:27:51,950
response--

412
00:27:51,950 --> 00:27:53,690
as I've done here--

413
00:27:53,690 --> 00:27:55,220
we obtain the same output.

414
00:27:55,220 --> 00:27:59,960
That is, h of n into a system
with unit sample response, x

415
00:27:59,960 --> 00:28:05,600
of n, gives us at the
output, y of n, also.

416
00:28:05,600 --> 00:28:10,190
An implication of that is that
linear shift invariant systems

417
00:28:10,190 --> 00:28:15,050
in cascade don't
particularly care which order

418
00:28:15,050 --> 00:28:17,300
the systems are cascaded in.

419
00:28:17,300 --> 00:28:21,470
That is, if I have
a system h 1 of n--

420
00:28:21,470 --> 00:28:24,360
that is with unit sample
response, h 1 of n--

421
00:28:24,360 --> 00:28:30,710
and cascade with a system with
unit sample response h 2 of n,

422
00:28:30,710 --> 00:28:34,550
and input x of n,
the unit sample

423
00:28:34,550 --> 00:28:37,480
response of this
overall system is h 1

424
00:28:37,480 --> 00:28:41,090
of n convolved with h 2 of n.

425
00:28:41,090 --> 00:28:43,250
But since convolution
is commutative,

426
00:28:43,250 --> 00:28:46,700
we could just as easily
convolve h 2 with h 1,

427
00:28:46,700 --> 00:28:52,340
and that corresponds to a
cascade of the system h 2 of n

428
00:28:52,340 --> 00:28:54,380
with the system h 1 of n.

429
00:28:54,380 --> 00:28:58,400
So simply because of the
fact that convolution

430
00:28:58,400 --> 00:29:06,010
is commutative, that
implies that, for example,

431
00:29:06,010 --> 00:29:09,860
that linear shift invariant
systems in cascade

432
00:29:09,860 --> 00:29:13,720
don't particularly care in
which order their cascaded.

433
00:29:13,720 --> 00:29:16,420
There are lots of other
properties of convolution

434
00:29:16,420 --> 00:29:20,200
and simple properties of
linear shift invariant systems

435
00:29:20,200 --> 00:29:24,910
that are a direct
consequence of convolution,

436
00:29:24,910 --> 00:29:28,300
and some of the properties that
we've mentioned of convolution.

437
00:29:28,300 --> 00:29:32,470
And a number of these, we'll see
in the lectures as we go along,

438
00:29:32,470 --> 00:29:35,290
and also a number of
them you'll have a chance

439
00:29:35,290 --> 00:29:39,780
to wrestle with in
the study guide.

440
00:29:39,780 --> 00:29:45,790
OK, so this is a development
of the convolution sum.

441
00:29:45,790 --> 00:29:50,500
One of the important aspects
of the convolution sum

442
00:29:50,500 --> 00:29:54,970
is the steps involved in
actually computing it--

443
00:29:54,970 --> 00:29:59,650
that is, the manipulations
involved in forming this sum.

444
00:29:59,650 --> 00:30:03,410
And as the last point that I'd
like to make in this lecture,

445
00:30:03,410 --> 00:30:10,000
I'd like to illustrate, first
with a viewgraph, and then

446
00:30:10,000 --> 00:30:14,740
with a movie, the computation,
or the implementation,

447
00:30:14,740 --> 00:30:16,600
of the convolution sum.

448
00:30:16,600 --> 00:30:19,600
So let's return
to the viewgraph.

449
00:30:30,850 --> 00:30:36,550
And I've indicated here,
again, the convolution sum as--

450
00:30:36,550 --> 00:30:41,980
we've derived it-- y of n is the
sum of x of k h of n minus k.

451
00:30:41,980 --> 00:30:44,890
Now, in implementing
the convolution sum,

452
00:30:44,890 --> 00:30:50,590
let me stress that
it is for n, which

453
00:30:50,590 --> 00:30:52,540
is the independent
variable for which

454
00:30:52,540 --> 00:30:54,370
we are computing the output--

455
00:30:54,370 --> 00:30:58,090
that is, k is just
simply a dummy variable

456
00:30:58,090 --> 00:31:00,670
inside this summation.

457
00:31:00,670 --> 00:31:06,160
Well, I've illustrated here, for
a specific example, a sequence,

458
00:31:06,160 --> 00:31:12,070
x of k, which is a
constant from 0 to I

459
00:31:12,070 --> 00:31:18,610
guess n equals 10, equal to
unity for n equals 0 to 10,

460
00:31:18,610 --> 00:31:20,660
and 0 otherwise.

461
00:31:20,660 --> 00:31:24,310
And I've indicated the sequence
a little differently here than

462
00:31:24,310 --> 00:31:28,120
I had previously, since I
want to distinguish between

463
00:31:28,120 --> 00:31:30,610
the x's, h's, and the y's.

464
00:31:30,610 --> 00:31:35,020
The sequence x of k, I'm
denoting with little x's--

465
00:31:35,020 --> 00:31:38,350
the sequence h, I'll denote
note with little h's, and later

466
00:31:38,350 --> 00:31:42,190
on, the sequence y, we'll
denote with little y's.

467
00:31:42,190 --> 00:31:48,790
All right, so here we have the
sequence a, that's this one.

468
00:31:48,790 --> 00:31:53,530
And plotted as a function of k.

469
00:31:53,530 --> 00:31:57,130
Here, I have the
sequence, h of k,

470
00:31:57,130 --> 00:31:59,710
which I've chosen to
be an exponential for k

471
00:31:59,710 --> 00:32:03,460
greater than 0, and
0 for k less than 0.

472
00:32:03,460 --> 00:32:07,360
But it's not h of k that
we want in this sum.

473
00:32:07,360 --> 00:32:09,840
It's h of n minus k.

474
00:32:09,840 --> 00:32:10,930
And what's n?

475
00:32:10,930 --> 00:32:14,470
Well, n is whatever value
of the output sequence

476
00:32:14,470 --> 00:32:16,340
we're trying to compute.

477
00:32:16,340 --> 00:32:20,000
So if n was equal
to 0, for example,

478
00:32:20,000 --> 00:32:23,680
we would want the
sequence, h of 0 minus k.

479
00:32:23,680 --> 00:32:25,880
And that's this sequence.

480
00:32:25,880 --> 00:32:32,410
It's h of k, flipped around in
k, because of this minus sign,

481
00:32:32,410 --> 00:32:34,300
and not shifted one
way or the other

482
00:32:34,300 --> 00:32:36,890
simply because we have a 0 here.

483
00:32:36,890 --> 00:32:40,870
So here's h of k, but
that's not what you want.

484
00:32:40,870 --> 00:32:44,260
It's h of 0 minus k.

485
00:32:44,260 --> 00:32:50,140
And now to compute y of 0, we
would multiply this sequence

486
00:32:50,140 --> 00:32:54,400
by this one, and compute
the sum from minus infinity

487
00:32:54,400 --> 00:32:55,720
to plus infinity.

488
00:32:55,720 --> 00:32:58,810
That would give us
the value, y of 0.

489
00:32:58,810 --> 00:33:01,330
For a different
value of n, we would

490
00:33:01,330 --> 00:33:05,290
have to look at h of n minus
k, for whichever value of n

491
00:33:05,290 --> 00:33:06,960
we were computing this for.

492
00:33:06,960 --> 00:33:10,660
For example, if n
is equal to minus 4,

493
00:33:10,660 --> 00:33:14,830
the sequence that we want
is h of minus 4 minus k,

494
00:33:14,830 --> 00:33:19,930
and that is this sequence
shifted to the left by 4.

495
00:33:19,930 --> 00:33:24,190
So to compute y of
minus 4, we want

496
00:33:24,190 --> 00:33:28,060
this sequence
multiplied by this one,

497
00:33:28,060 --> 00:33:31,570
and the sum computed from minus
infinity to plus infinity.

498
00:33:31,570 --> 00:33:34,960
And you can see, obviously,
for that particular case--

499
00:33:34,960 --> 00:33:37,750
that is n equal to minus 4--

500
00:33:37,750 --> 00:33:40,300
the product of
this and this is 0,

501
00:33:40,300 --> 00:33:41,730
and the sum will be equal to 0.

502
00:33:41,730 --> 00:33:49,180
Well, let's look at this example
a little more dynamically,

503
00:33:49,180 --> 00:33:54,370
with a movie that was prepared
at Bell Telephone Laboratories

504
00:33:54,370 --> 00:33:55,610
by Dr. Ronald Shaefer.

505
00:34:06,710 --> 00:34:08,630
OK, the convolution
sum, then, we

506
00:34:08,630 --> 00:34:12,469
have is the sum of x
of k, h of n minus k.

507
00:34:12,469 --> 00:34:15,110
And so what we would
like to illustrate

508
00:34:15,110 --> 00:34:19,370
is the operation of
evaluating this sum.

509
00:34:19,370 --> 00:34:21,770
On the top, we have x of k.

510
00:34:21,770 --> 00:34:26,300
On the bottom we have h of k--
he of k being an exponential.

511
00:34:26,300 --> 00:34:31,909
And now, we see h of minus k,
namely h of k flipped over.

512
00:34:31,909 --> 00:34:35,489
As we shift h of k to the
left, corresponding to n

513
00:34:35,489 --> 00:34:42,969
equals minus 1, then back
to 0, and now to the right.

514
00:34:42,969 --> 00:34:45,929
So we have h of 1 minus k.

515
00:34:45,929 --> 00:34:51,570
And then to 4 y of n, we want
the product of h of minus k

516
00:34:51,570 --> 00:34:55,110
shifted with x of
k, that product sum

517
00:34:55,110 --> 00:34:58,360
from minus infinity
to plus infinity.

518
00:34:58,360 --> 00:35:02,880
So here, we see x of k
times h of 1 minus k, 2

519
00:35:02,880 --> 00:35:06,300
minus k, 3 minus k, et cetera.

520
00:35:06,300 --> 00:35:08,880
Those multiplied and then
summed from minus infinity

521
00:35:08,880 --> 00:35:12,980
to plus infinity, we
see during this portion

522
00:35:12,980 --> 00:35:14,400
that more and more
values of h are

523
00:35:14,400 --> 00:35:17,730
engaged with non-0 values
of x, and so y of n

524
00:35:17,730 --> 00:35:21,720
grows until we reach the
point where values fall off

525
00:35:21,720 --> 00:35:28,320
the end of the non-0 values
of x, where y of n decays.

526
00:35:28,320 --> 00:35:31,410
So this, then, is
an illustration

527
00:35:31,410 --> 00:35:36,660
of the linear convolution
of x of k with h of k.

528
00:35:40,830 --> 00:35:43,850
OK, this completes
our introduction

529
00:35:43,850 --> 00:35:47,150
to discrete time
signals and systems.

530
00:35:47,150 --> 00:35:50,480
There were a number
of important points

531
00:35:50,480 --> 00:35:53,010
that we've made
during this lecture.

532
00:35:53,010 --> 00:35:56,990
But the key result,
and the result

533
00:35:56,990 --> 00:36:00,290
that it's important to
develop some experience with,

534
00:36:00,290 --> 00:36:03,140
is the convolution sum.

535
00:36:03,140 --> 00:36:07,490
Next time, we'll introduce
some additional considerations,

536
00:36:07,490 --> 00:36:11,510
namely the considerations
of stability and causality

537
00:36:11,510 --> 00:36:13,760
for discrete time systems.

538
00:36:13,760 --> 00:36:17,960
And we'll also discuss, briefly,
the class of linear shift

539
00:36:17,960 --> 00:36:21,410
invariant systems
that are represented

540
00:36:21,410 --> 00:36:25,320
by linear constant coefficient
difference equations.

541
00:36:25,320 --> 00:36:29,220
Finally, we'll try to
tie some of this together

542
00:36:29,220 --> 00:36:33,770
and in particular, present what
will be called the frequency

543
00:36:33,770 --> 00:36:35,660
response of the systems.

544
00:36:35,660 --> 00:36:38,360
And this will eventually
lead into a discussion

545
00:36:38,360 --> 00:36:40,146
of Fourier transforms.

546
00:36:40,146 --> 00:36:41,138
Thank you.

547
00:36:42,626 --> 00:36:45,364
[MUSIC PLAYING]