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[MUSIC PLAYING]
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PROFESSOR: In discussing the
sampling theorem, we saw that
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00:00:58,180 --> 00:01:01,650
for a band limited signal,
which is sampled at a
12
00:01:01,650 --> 00:01:05,660
frequency that is at least twice
the highest frequency,
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00:01:05,660 --> 00:01:09,320
we can implement exact
reconstruction of the original
14
00:01:09,320 --> 00:01:16,240
signal by low pass filtering an
impulse train, whose areas
15
00:01:16,240 --> 00:01:19,640
are identical to the
sample values.
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00:01:19,640 --> 00:01:23,200
Well essentially, this low
pass filtering operation
17
00:01:23,200 --> 00:01:26,980
provides for us an interpolation
in between the
18
00:01:26,980 --> 00:01:28,440
sampled values.
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00:01:28,440 --> 00:01:32,160
In other words, the output of a
low pass filter, in fact, is
20
00:01:32,160 --> 00:01:36,820
a continuous curve, which fits
between the sampled values
21
00:01:36,820 --> 00:01:39,980
some continuous function.
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00:01:39,980 --> 00:01:44,170
Now, I'm sure that many of you
are familiar with other kinds
23
00:01:44,170 --> 00:01:47,330
of interpolation that we could
potentially provide in between
24
00:01:47,330 --> 00:01:48,810
sampled values.
25
00:01:48,810 --> 00:01:52,280
And in fact, in today's lecture
what I would like to
26
00:01:52,280 --> 00:01:58,640
do is first of all developed
the interpretation of the
27
00:01:58,640 --> 00:02:03,640
reconstruction as an
interpolation process and then
28
00:02:03,640 --> 00:02:07,370
also see how this exact
interpolation, using a low
29
00:02:07,370 --> 00:02:12,350
pass filter, relates to other
kinds of interpolation, such
30
00:02:12,350 --> 00:02:14,520
as linear interpolation
that you may
31
00:02:14,520 --> 00:02:16,330
already be familiar with.
32
00:02:16,330 --> 00:02:21,220
Well to begin, let's again
review what the overall system
33
00:02:21,220 --> 00:02:25,290
is for exact sampling
and reconstruction.
34
00:02:25,290 --> 00:02:28,670
And so let me remind you that
the overall system for
35
00:02:28,670 --> 00:02:31,960
sampling and desampling, or
reconstruction, is as I
36
00:02:31,960 --> 00:02:34,040
indicate here.
37
00:02:34,040 --> 00:02:37,150
The sampling process consists
of multiplying
38
00:02:37,150 --> 00:02:38,960
by an impulse train.
39
00:02:38,960 --> 00:02:42,200
And then the reconstruction
process corresponds to
40
00:02:42,200 --> 00:02:46,590
processing that impulse train
with a low pass filter.
41
00:02:46,590 --> 00:02:52,420
So if the spectrum of the
original signal is what I
42
00:02:52,420 --> 00:02:57,890
indicate in this diagram, then
after sampling with an impulse
43
00:02:57,890 --> 00:03:02,020
train, that spectrum
is replicated.
44
00:03:02,020 --> 00:03:07,960
And this replicated spectrum
for reconstruction is then
45
00:03:07,960 --> 00:03:10,610
processed through a
low pass filter.
46
00:03:10,610 --> 00:03:14,770
And so, in fact, if this
frequency response is an ideal
47
00:03:14,770 --> 00:03:20,560
low pass filter, as I indicate
on the diagram below, then
48
00:03:20,560 --> 00:03:25,540
multiplying the spectrum of
the sample signal by this
49
00:03:25,540 --> 00:03:31,350
extracts for us just the
portion of the spectrum
50
00:03:31,350 --> 00:03:33,210
centered around the origin.
51
00:03:33,210 --> 00:03:36,920
And what we're left with,
then, is the spectrum,
52
00:03:36,920 --> 00:03:41,490
finally, of the reconstructed
signal, which for the case of
53
00:03:41,490 --> 00:03:46,070
an ideal low pass filter is
exactly equal to the spectrum
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00:03:46,070 --> 00:03:48,160
of the original signal.
55
00:03:48,160 --> 00:03:55,780
Now, that is the frequency
domain picture of the sampling
56
00:03:55,780 --> 00:03:57,220
and reconstruction.
57
00:03:57,220 --> 00:04:00,600
Let's also look at, basically,
the same process.
58
00:04:00,600 --> 00:04:04,920
But let's examine it now
in the time domain.
59
00:04:04,920 --> 00:04:11,270
Well in the time domain, what we
have is our original signal
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00:04:11,270 --> 00:04:14,110
multiplied by an
impulse train.
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00:04:14,110 --> 00:04:19,920
And this then is the sample
signal, or the impulse train
62
00:04:19,920 --> 00:04:23,630
whose areas are equal to
the sample values.
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00:04:23,630 --> 00:04:27,620
And because of the fact that
this is an impulse train, in
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00:04:27,620 --> 00:04:34,160
fact, we can take this term
inside the summation.
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00:04:34,160 --> 00:04:37,130
And of course, what counts
about x of t in this
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00:04:37,130 --> 00:04:41,400
expression is just as values
at the sampling instance,
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00:04:41,400 --> 00:04:47,220
which are displaced in time by
capital T. And so what we can
68
00:04:47,220 --> 00:04:52,350
equivalently write is the
expression for the impulse
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00:04:52,350 --> 00:04:55,880
train samples, or impulse train
of samples, as I've
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00:04:55,880 --> 00:04:57,010
indicated here.
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00:04:57,010 --> 00:05:03,410
Simply an impulse train, whose
areas are the sampled values.
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00:05:03,410 --> 00:05:08,650
Now, in the reconstruction we
process that impulse train
73
00:05:08,650 --> 00:05:10,940
with a low pass filter.
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00:05:10,940 --> 00:05:13,880
That's the basic notion
of the reconstruction.
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00:05:13,880 --> 00:05:18,810
And so in the time domain, the
reconstructed signal is
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00:05:18,810 --> 00:05:23,850
related to the impulse train
of samples through a
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00:05:23,850 --> 00:05:28,280
convolution with the filter
impulse response.
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00:05:28,280 --> 00:05:32,670
And carrying out this
convolution, since this is
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00:05:32,670 --> 00:05:35,880
just a train of pulses, in
effect, what happens in this
80
00:05:35,880 --> 00:05:40,640
convolution is that this
impulse response gets
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00:05:40,640 --> 00:05:45,820
reproduced at each of the
locations of the impulses in x
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00:05:45,820 --> 00:05:48,790
of p of t with the
appropriate area.
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00:05:48,790 --> 00:05:52,580
And finally, then, in the time
domain, the reconstructed
84
00:05:52,580 --> 00:05:59,140
signal is simply a linear
combination of shifted
85
00:05:59,140 --> 00:06:03,280
versions of the impulse response
with amplitudes,
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00:06:03,280 --> 00:06:05,080
which are the sample values.
87
00:06:05,080 --> 00:06:09,410
And so this expression, in
fact then, is our basic
88
00:06:09,410 --> 00:06:14,840
reconstruction expression
in the time domain.
89
00:06:14,840 --> 00:06:20,350
Well in terms of a diagram, we
can think of the original
90
00:06:20,350 --> 00:06:23,150
waveform as I've shown here.
91
00:06:23,150 --> 00:06:30,100
And the red arrows denote the
sampled wave form, or the
92
00:06:30,100 --> 00:06:35,790
train of impulses, whose
amplitudes are the sampled
93
00:06:35,790 --> 00:06:40,310
values of the original
continuous time signal.
94
00:06:40,310 --> 00:06:45,330
And then, I've shown here what
might be a typical impulse
95
00:06:45,330 --> 00:06:49,310
response, particularly typical
in the case where we're
96
00:06:49,310 --> 00:06:53,280
talking about reconstruction
with an ideal low pass filter.
97
00:06:53,280 --> 00:06:58,250
Now, what happens in the
reconstruction is that the
98
00:06:58,250 --> 00:07:04,140
convolution of these impulses
with this impulse response
99
00:07:04,140 --> 00:07:09,240
means that in the
reconstruction, we superimpose
100
00:07:09,240 --> 00:07:10,890
one of these impulse
responses--
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00:07:10,890 --> 00:07:13,810
whatever the filter impulse
response happens to be--
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00:07:13,810 --> 00:07:16,570
at each of these
time instance.
103
00:07:16,570 --> 00:07:20,310
And in doing that, then
those are added up.
104
00:07:20,310 --> 00:07:23,660
And that gives us the total
reconstructed signal.
105
00:07:23,660 --> 00:07:28,730
Of course, for the case in which
the filter is an ideal
106
00:07:28,730 --> 00:07:34,110
low pass filter, then what we
know is that in that case, the
107
00:07:34,110 --> 00:07:38,090
impulse response is of the
form of a sync function.
108
00:07:38,090 --> 00:07:41,880
But generally, we may want to
consider other kinds of
109
00:07:41,880 --> 00:07:43,070
impulse responses.
110
00:07:43,070 --> 00:07:47,420
And so in fact, the
interpolating impulse response
111
00:07:47,420 --> 00:07:50,160
may have and will have, as this
discussion goes along,
112
00:07:50,160 --> 00:07:52,830
some different shapes.
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00:07:52,830 --> 00:07:56,750
Now what I'd like to do is
illustrate, or demonstrate,
114
00:07:56,750 --> 00:08:04,240
this process of effectively
doing the interpolation by
115
00:08:04,240 --> 00:08:07,530
replacing each of the impulses
by an appropriate
116
00:08:07,530 --> 00:08:10,250
interpolating impulse response
and adding these up.
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00:08:10,250 --> 00:08:12,690
And I'd like to do this
with a computer
118
00:08:12,690 --> 00:08:14,690
movie that we generated.
119
00:08:14,690 --> 00:08:17,860
And what you'll see in the
computer movie is,
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00:08:17,860 --> 00:08:20,040
essentially, an original
wave form, which is
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00:08:20,040 --> 00:08:21,870
a continuous curve.
122
00:08:21,870 --> 00:08:29,000
And then below that in the movie
is a train of samples.
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00:08:29,000 --> 00:08:33,500
And then below that will be
the reconstructed signal.
124
00:08:33,500 --> 00:08:38,110
And the reconstruction will be
carried out by showing the
125
00:08:38,110 --> 00:08:41,530
location of the impulse response
as it moves along in
126
00:08:41,530 --> 00:08:42,950
the wave form.
127
00:08:42,950 --> 00:08:46,650
And then the reconstructed curve
is simply the summation
128
00:08:46,650 --> 00:08:49,350
of those as that impulse
response moves along.
129
00:08:49,350 --> 00:08:53,930
So what you'll see then is an
impulse response like this--
130
00:08:53,930 --> 00:08:58,340
for the particular case of an
ideal low pass filter for the
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00:08:58,340 --> 00:08:59,590
reconstruction--
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00:08:59,590 --> 00:09:02,140
133
00:09:02,140 --> 00:09:07,480
placed successively at the
locations of these impulses.
134
00:09:07,480 --> 00:09:10,400
And that is the convolution
process.
135
00:09:10,400 --> 00:09:14,490
And below that then will be
the summation of these.
136
00:09:14,490 --> 00:09:17,220
And the summation of those
will then be the
137
00:09:17,220 --> 00:09:18,740
reconstructed signal.
138
00:09:18,740 --> 00:09:23,250
So let's take a look at, first
of all that reconstruction
139
00:09:23,250 --> 00:09:26,150
where the impulse response
corresponds to the impulse
140
00:09:26,150 --> 00:09:29,800
response of an ideal
low pass filter.
141
00:09:29,800 --> 00:09:34,460
Shown here, first, is the
continuous time signal, which
142
00:09:34,460 --> 00:09:39,310
we want to sample and then
reconstruct using band limited
143
00:09:39,310 --> 00:09:42,860
interpolation, or equivalently,
ideal low pass
144
00:09:42,860 --> 00:09:45,120
filtering on the
set of samples.
145
00:09:45,120 --> 00:09:48,230
So the first step then
is to sample this
146
00:09:48,230 --> 00:09:49,940
continuous time signal.
147
00:09:49,940 --> 00:09:53,850
And we see here now the
set of samples.
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00:09:53,850 --> 00:09:56,720
And superimposed on the samples
are the original
149
00:09:56,720 --> 00:10:01,180
continuous time signal to focus
on the fact that those
150
00:10:01,180 --> 00:10:03,420
are samples of the top curve.
151
00:10:03,420 --> 00:10:08,250
Let's now remove the continuous
time envelope of
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00:10:08,250 --> 00:10:09,420
the samples.
153
00:10:09,420 --> 00:10:12,380
And it's this set of samples
that we then want to use for
154
00:10:12,380 --> 00:10:14,490
the reconstruction.
155
00:10:14,490 --> 00:10:17,900
The reconstruction process,
interpreted as interpolation,
156
00:10:17,900 --> 00:10:20,600
consists of replacing
each sample with a
157
00:10:20,600 --> 00:10:22,970
sine x over x function.
158
00:10:22,970 --> 00:10:28,940
And so let's first consider
the sample at t equals 0.
159
00:10:28,940 --> 00:10:32,800
And here is the interpolating
sine x over x function
160
00:10:32,800 --> 00:10:35,530
associated with that sample.
161
00:10:35,530 --> 00:10:41,610
Now, the more general process
then is to place a sine x over
162
00:10:41,610 --> 00:10:45,150
x function at the time location
of each sample and
163
00:10:45,150 --> 00:10:46,400
superimpose those.
164
00:10:46,400 --> 00:10:48,980
165
00:10:48,980 --> 00:10:54,330
Let's begin that process at the
left-hand set of samples.
166
00:10:54,330 --> 00:10:58,800
And in the bottom curve, we'll
build up the reconstruction as
167
00:10:58,800 --> 00:11:02,150
those sine x over x functions
are added together.
168
00:11:02,150 --> 00:11:04,470
So we begin with the
left-hand sample.
169
00:11:04,470 --> 00:11:07,470
And we see there the sine x over
x function on the bottom
170
00:11:07,470 --> 00:11:11,900
curve is the first step
in the reconstruction.
171
00:11:11,900 --> 00:11:14,710
We now have the sine x over x
function associated with the
172
00:11:14,710 --> 00:11:16,070
second sample.
173
00:11:16,070 --> 00:11:18,620
Let's add that in.
174
00:11:18,620 --> 00:11:22,680
Now we move on to the
third sample.
175
00:11:22,680 --> 00:11:27,980
And that sine x over x
function is added in.
176
00:11:27,980 --> 00:11:33,860
Continuing on, the next sample
generates a sine x over x
177
00:11:33,860 --> 00:11:36,670
function, which is superimposed
on the result
178
00:11:36,670 --> 00:11:39,750
that we've accumulated so far.
179
00:11:39,750 --> 00:11:42,720
And now let's just speed
up the process.
180
00:11:42,720 --> 00:11:45,490
We'll move on to the
fifth sample.
181
00:11:45,490 --> 00:11:46,540
Add that in.
182
00:11:46,540 --> 00:11:48,710
The sixth sample, add that in.
183
00:11:48,710 --> 00:11:51,440
And continue on through
the set of samples.
184
00:11:51,440 --> 00:11:54,960
And keep in mind the fact that,
basically, what we're
185
00:11:54,960 --> 00:11:58,860
doing explicitly here is the
convolution of the impulse
186
00:11:58,860 --> 00:12:02,240
train with a sine x
over x function.
187
00:12:02,240 --> 00:12:05,730
And because the set of samples
that we started with were
188
00:12:05,730 --> 00:12:09,610
samples of an exactly band
limited function, what we are
189
00:12:09,610 --> 00:12:14,530
reconstructing exactly is the
original continuous time
190
00:12:14,530 --> 00:12:16,980
signal that we have
on the top trace.
191
00:12:16,980 --> 00:12:24,900
192
00:12:24,900 --> 00:12:31,060
OK, so that then kind of gives
you the picture of doing
193
00:12:31,060 --> 00:12:34,720
interpolation by replacing
the impulses by
194
00:12:34,720 --> 00:12:35,870
a continuous curve.
195
00:12:35,870 --> 00:12:40,010
And that's the way we're fitting
a continuous curve to
196
00:12:40,010 --> 00:12:42,260
the original impulse train.
197
00:12:42,260 --> 00:12:47,870
And let me stress that this
reconstruction process--
198
00:12:47,870 --> 00:12:51,320
by putting the impulses
through a filter--
199
00:12:51,320 --> 00:12:54,690
200
00:12:54,690 --> 00:12:59,320
follows this relationship
whether or not this impulse
201
00:12:59,320 --> 00:13:03,700
response, in fact, corresponds
to an ideal low pass filter.
202
00:13:03,700 --> 00:13:07,210
What this expression always says
is that reconstructing
203
00:13:07,210 --> 00:13:12,440
this way corresponds to
replacing the impulses by a
204
00:13:12,440 --> 00:13:17,940
shifted impulse response with
an amplitude that is an
205
00:13:17,940 --> 00:13:22,220
amplitude corresponding
to the sample value.
206
00:13:22,220 --> 00:13:27,120
Now the kind of reconstruction
that we've just talked about,
207
00:13:27,120 --> 00:13:32,320
and the ideal reconstruction,
is often referred to as band
208
00:13:32,320 --> 00:13:36,420
limited interpolation because
we're interpolating in between
209
00:13:36,420 --> 00:13:39,840
the samples by making the
assumption that the signal is
210
00:13:39,840 --> 00:13:43,840
band limited and using the
impulse response for an ideal
211
00:13:43,840 --> 00:13:48,780
low pass filter, which has a cut
off frequency consistent
212
00:13:48,780 --> 00:13:51,500
with the assumed bandwidth
for the signal.
213
00:13:51,500 --> 00:13:57,590
So if we look here, for example,
at the impulse train,
214
00:13:57,590 --> 00:14:02,170
then in the demonstration that
you just saw, we built up the
215
00:14:02,170 --> 00:14:05,420
reconstructed curve by replacing
each of these
216
00:14:05,420 --> 00:14:07,980
impulses with the
sync function.
217
00:14:07,980 --> 00:14:13,570
And the sum of those built up
the reconstructed curve.
218
00:14:13,570 --> 00:14:18,230
Well, there are lots of other
kinds of interpolation that
219
00:14:18,230 --> 00:14:23,180
are perhaps maybe not as exact
but often easier to implement.
220
00:14:23,180 --> 00:14:25,080
And what I'd like to
do is focus our
221
00:14:25,080 --> 00:14:28,300
attention on two of these.
222
00:14:28,300 --> 00:14:31,440
The first that I want to mention
is what's referred to
223
00:14:31,440 --> 00:14:34,760
as the zero order hold, where
in effect, we do the
224
00:14:34,760 --> 00:14:39,100
interpolation in between these
sample values by simply
225
00:14:39,100 --> 00:14:43,610
holding the sample value until
the next sampling instant.
226
00:14:43,610 --> 00:14:48,070
And the reconstruction that we
end up, in that case, will
227
00:14:48,070 --> 00:14:49,260
look something like this.
228
00:14:49,260 --> 00:14:54,590
It's a staircase, or box car,
kind of function where we've
229
00:14:54,590 --> 00:14:57,660
simply held the sample value
until the next sampling
230
00:14:57,660 --> 00:15:01,420
instant and then replaced by
that value, held it until the
231
00:15:01,420 --> 00:15:04,040
next sampling instant,
et cetera.
232
00:15:04,040 --> 00:15:07,750
Now that's one kind
of interpolation.
233
00:15:07,750 --> 00:15:11,030
Another kind of very common
interpolation is what's
234
00:15:11,030 --> 00:15:13,290
referred to as linear
interpolation, where we simply
235
00:15:13,290 --> 00:15:16,910
fit a straight line between
the sampled values.
236
00:15:16,910 --> 00:15:21,240
And in that case, the type of
reconstruction that we would
237
00:15:21,240 --> 00:15:25,060
get would look something like I
indicate here, where we take
238
00:15:25,060 --> 00:15:30,540
a sample value, and the
following sample value, and
239
00:15:30,540 --> 00:15:34,170
simply fit an interpolated curve
between them, which is a
240
00:15:34,170 --> 00:15:36,430
straight line.
241
00:15:36,430 --> 00:15:42,470
Now interestingly, in fact, both
the zero order hold and
242
00:15:42,470 --> 00:15:46,155
the linear interpolation, which
is often referred to as
243
00:15:46,155 --> 00:15:50,600
a first order hold, can also
be either implemented or
244
00:15:50,600 --> 00:15:53,820
interpreted, both implemented
and interpreted, in the
245
00:15:53,820 --> 00:15:56,670
context of the equation that
we just developed.
246
00:15:56,670 --> 00:16:00,450
In particular, the processing
of the impulse train of
247
00:16:00,450 --> 00:16:05,520
samples by a linear time
invariant filter.
248
00:16:05,520 --> 00:16:12,550
Specifically, if we consider
a system where the impulse
249
00:16:12,550 --> 00:16:21,030
response is a rectangular
function, then in fact, if we
250
00:16:21,030 --> 00:16:25,570
processed the train of samples
through a filter with this
251
00:16:25,570 --> 00:16:29,610
impulse response, exactly the
reconstruction that we would
252
00:16:29,610 --> 00:16:32,450
get is what I've shown here.
253
00:16:32,450 --> 00:16:38,810
Alternatively, if we chose an
impulse response which was a
254
00:16:38,810 --> 00:16:44,980
triangular impulse response,
then what in effect happens is
255
00:16:44,980 --> 00:16:49,690
that each of these impulses
activates this triangle.
256
00:16:49,690 --> 00:16:53,210
And when we add up those
triangles at successive
257
00:16:53,210 --> 00:16:57,630
locations, in fact, what we
generate is this linear
258
00:16:57,630 --> 00:16:59,730
interpolation.
259
00:16:59,730 --> 00:17:06,540
So what this says, in fact, is
that either a zero order hold,
260
00:17:06,540 --> 00:17:09,849
which holds the value, or
linear interpolation can
261
00:17:09,849 --> 00:17:15,190
likewise be interpreted as a
process of convulving the
262
00:17:15,190 --> 00:17:18,380
impulse train of samples
with an appropriate
263
00:17:18,380 --> 00:17:21,380
filter impulse response.
264
00:17:21,380 --> 00:17:25,480
Well, what I'd like to do is
demonstrate, as we did with
265
00:17:25,480 --> 00:17:29,860
the band limited interpolation
or the sync interpolation as
266
00:17:29,860 --> 00:17:31,500
it's sometimes called--
interpolating with
267
00:17:31,500 --> 00:17:33,100
a sine x over x--
268
00:17:33,100 --> 00:17:35,260
let me now show the process.
269
00:17:35,260 --> 00:17:41,820
First of all, where we have
a zero order hold as
270
00:17:41,820 --> 00:17:44,280
corresponding to this
impulse response.
271
00:17:44,280 --> 00:17:48,160
In which case, we'll see
basically the same process as
272
00:17:48,160 --> 00:17:52,340
we saw in the computer generated
movie previously.
273
00:17:52,340 --> 00:17:56,080
But now, rather than a sync
function replacing each of
274
00:17:56,080 --> 00:17:59,770
these impulses, we'll have
a rectangular function.
275
00:17:59,770 --> 00:18:04,200
That will generate then our
approximation, which is a zero
276
00:18:04,200 --> 00:18:06,130
order hold.
277
00:18:06,130 --> 00:18:08,900
And following that, we'll do
exactly the same thing with
278
00:18:08,900 --> 00:18:12,820
the same wave form, using
a first order hold or a
279
00:18:12,820 --> 00:18:14,960
triangular impulse response.
280
00:18:14,960 --> 00:18:18,330
In which case, what we'll see
again is that as the triangle
281
00:18:18,330 --> 00:18:23,150
moves along here, and we build
up the running sum or the
282
00:18:23,150 --> 00:18:27,900
convolution, then we'll, in
fact, fit the original curve
283
00:18:27,900 --> 00:18:29,550
with a linear curve.
284
00:18:29,550 --> 00:18:33,980
So now let's again look at that,
remembering that at the
285
00:18:33,980 --> 00:18:36,910
top we'll see the original
continuous curve, exactly the
286
00:18:36,910 --> 00:18:38,820
one that we had before.
287
00:18:38,820 --> 00:18:42,680
Below it, the set of samples
together with the impulse
288
00:18:42,680 --> 00:18:45,070
response moving along.
289
00:18:45,070 --> 00:18:48,650
And then finally below that,
the accumulation of those
290
00:18:48,650 --> 00:18:52,370
impulse responses, or
equivalently the convolution,
291
00:18:52,370 --> 00:18:56,050
or equivalently the
reconstruction.
292
00:18:56,050 --> 00:18:59,600
So we have the same continuous
time signal that we use
293
00:18:59,600 --> 00:19:03,740
previously with band limited
interpolation.
294
00:19:03,740 --> 00:19:08,130
And in this case now, we want to
sample and then interpolate
295
00:19:08,130 --> 00:19:10,810
first with a zero order
hold and then with
296
00:19:10,810 --> 00:19:11,790
a first order hold.
297
00:19:11,790 --> 00:19:15,250
So the first step then
is to sample the
298
00:19:15,250 --> 00:19:17,170
continuous time signal.
299
00:19:17,170 --> 00:19:21,360
And we show here the set
of samples, once again,
300
00:19:21,360 --> 00:19:25,360
superimposed on which we have
the continuous time signal,
301
00:19:25,360 --> 00:19:28,490
which of course is exactly
the same curve as
302
00:19:28,490 --> 00:19:30,650
we have in the top.
303
00:19:30,650 --> 00:19:34,020
Well, let's remove that envelope
so that we focus
304
00:19:34,020 --> 00:19:37,790
attention on the samples that
we're using to interpolate.
305
00:19:37,790 --> 00:19:41,690
And the interpolation process
consists of replacing each
306
00:19:41,690 --> 00:19:47,450
sample by a rectangular signal,
whose amplitude is
307
00:19:47,450 --> 00:19:49,570
equal to the sample size.
308
00:19:49,570 --> 00:19:53,510
So let's put one, first of all,
at t equals 0 associated
309
00:19:53,510 --> 00:19:56,550
with that sample.
310
00:19:56,550 --> 00:20:00,660
And that then would be the
interpolating rectangle
311
00:20:00,660 --> 00:20:04,370
associated with the sample
at t equals 0.
312
00:20:04,370 --> 00:20:07,780
Now to build up the
interpolation, what we'll have
313
00:20:07,780 --> 00:20:09,740
is one of those at each
sample time, and
314
00:20:09,740 --> 00:20:11,890
those are added together.
315
00:20:11,890 --> 00:20:14,930
We'll start that process,
as we did before, at the
316
00:20:14,930 --> 00:20:18,762
left-hand end of the set of
samples and build the
317
00:20:18,762 --> 00:20:20,390
interpolating signal
on the bottom.
318
00:20:20,390 --> 00:20:25,650
So with the left-hand sample,
we have first the rectangle
319
00:20:25,650 --> 00:20:27,640
associated with that.
320
00:20:27,640 --> 00:20:30,510
That's shown now on
the bottom curve.
321
00:20:30,510 --> 00:20:36,740
We now have an interpolating
rectangle with a second sample
322
00:20:36,740 --> 00:20:40,200
that gets added into
the bottom curve.
323
00:20:40,200 --> 00:20:44,210
Similarly, an interpolating
rectangle with the zero order
324
00:20:44,210 --> 00:20:46,430
hold with the third sample.
325
00:20:46,430 --> 00:20:49,440
We add that into the
bottom curve.
326
00:20:49,440 --> 00:20:52,570
And as we proceed, we're
building a staircase
327
00:20:52,570 --> 00:20:53,920
approximation.
328
00:20:53,920 --> 00:20:58,630
On to the next sample, that gets
added in as we see there.
329
00:20:58,630 --> 00:21:00,720
And now let's speed
up the process.
330
00:21:00,720 --> 00:21:04,920
And we'll see the staircase
approximation building up.
331
00:21:04,920 --> 00:21:09,390
And notice in this case, as in
the previous case, that what
332
00:21:09,390 --> 00:21:13,780
we're basically watching
dynamically is the convolution
333
00:21:13,780 --> 00:21:17,370
of the impulse train of samples
with the impulse
334
00:21:17,370 --> 00:21:20,420
response of the interpolating
filter, which in this
335
00:21:20,420 --> 00:21:24,800
particular case is just
a rectangular pulse.
336
00:21:24,800 --> 00:21:27,140
And so this staircase
approximation that we're
337
00:21:27,140 --> 00:21:32,040
generating is the zero order
hold interpolation between the
338
00:21:32,040 --> 00:21:37,070
samples of the band limited
signal, which is at the top.
339
00:21:37,070 --> 00:21:41,890
340
00:21:41,890 --> 00:21:44,800
Now let's do the same thing
with a first order hold.
341
00:21:44,800 --> 00:21:49,610
So in this case, we want to
interpolate using a triangular
342
00:21:49,610 --> 00:21:53,760
impulse response rather then
the sine x over x, or
343
00:21:53,760 --> 00:21:56,940
rectangular impulse responses
that we showed previously.
344
00:21:56,940 --> 00:22:02,670
So first, let's say with the
sample at t equals 0, we would
345
00:22:02,670 --> 00:22:08,520
replace that with a triangular
interpolating function.
346
00:22:08,520 --> 00:22:12,050
And more generally, each impulse
or sample is replaced
347
00:22:12,050 --> 00:22:14,950
with a triangular interpolating
function of a
348
00:22:14,950 --> 00:22:17,550
height equal to the
sample type.
349
00:22:17,550 --> 00:22:20,050
And these are superimposed
to generate the linear
350
00:22:20,050 --> 00:22:21,680
interpolation.
351
00:22:21,680 --> 00:22:26,010
We'll begin this process with
the leftmost sample.
352
00:22:26,010 --> 00:22:29,950
And we'll build the
superposition below in the
353
00:22:29,950 --> 00:22:31,240
bottom curve.
354
00:22:31,240 --> 00:22:34,810
So here is the interpolating
triangle for
355
00:22:34,810 --> 00:22:36,970
the leftmost sample.
356
00:22:36,970 --> 00:22:40,750
And now it's reproduced below.
357
00:22:40,750 --> 00:22:43,760
With the second sample, we
have an interpolating
358
00:22:43,760 --> 00:22:47,490
triangle, which is added
into the bottom curve.
359
00:22:47,490 --> 00:22:50,670
And now on to the
third sample.
360
00:22:50,670 --> 00:22:55,800
And again, that interpolating
triangle will be added on to
361
00:22:55,800 --> 00:22:59,550
the curve that we've
developed so far.
362
00:22:59,550 --> 00:23:03,300
And now onto the next sample.
363
00:23:03,300 --> 00:23:05,630
We add that in.
364
00:23:05,630 --> 00:23:07,480
Then we'll speed
up the process.
365
00:23:07,480 --> 00:23:13,700
And as we proceed through, we
are building, basically, a
366
00:23:13,700 --> 00:23:18,670
linear interpolation in between
the sample points,
367
00:23:18,670 --> 00:23:20,550
essentially corresponding to--
368
00:23:20,550 --> 00:23:22,290
if one wants to think
of it this way--
369
00:23:22,290 --> 00:23:23,730
connecting the dots.
370
00:23:23,730 --> 00:23:26,550
And what you're watching, once
again, is essentially the
371
00:23:26,550 --> 00:23:31,220
convolution process convulving
the impulse train with the
372
00:23:31,220 --> 00:23:34,640
impulse response of the
interpolating filter.
373
00:23:34,640 --> 00:23:39,010
And what we're generating,
then, is a linear
374
00:23:39,010 --> 00:23:44,850
approximation to the band
limited continuous time curve
375
00:23:44,850 --> 00:23:46,100
at the top.
376
00:23:46,100 --> 00:23:48,230
377
00:23:48,230 --> 00:23:54,500
OK, so what we have then is
several other kinds of
378
00:23:54,500 --> 00:23:59,940
interpolation, which fit within
the same context as
379
00:23:59,940 --> 00:24:02,210
exact band limited
interpolation.
380
00:24:02,210 --> 00:24:06,020
One being interpolation in the
time domain with an impulse
381
00:24:06,020 --> 00:24:08,130
response, which is
a rectangle.
382
00:24:08,130 --> 00:24:12,420
The second being interpolation
in the time domain with an
383
00:24:12,420 --> 00:24:15,520
impulse response, which
is a triangle.
384
00:24:15,520 --> 00:24:19,380
And in fact, it's interesting
to also look at the
385
00:24:19,380 --> 00:24:22,710
relationship between that and
band limited interpolation.
386
00:24:22,710 --> 00:24:27,190
Look at it, specifically,
in the frequency domain.
387
00:24:27,190 --> 00:24:32,170
Well, in the frequency domain,
what we know, of course, is
388
00:24:32,170 --> 00:24:36,810
that for exact interpolation,
what we want as our
389
00:24:36,810 --> 00:24:40,780
interpolating filter is an
ideal low pass filter.
390
00:24:40,780 --> 00:24:43,850
Now keep in mind, by the way,
that an ideal low pass filter
391
00:24:43,850 --> 00:24:46,350
is an abstraction, as I've
stressed several
392
00:24:46,350 --> 00:24:47,850
times in the past.
393
00:24:47,850 --> 00:24:52,700
An ideal low pass filter is a
non-causal filter and, in
394
00:24:52,700 --> 00:24:55,730
fact, infinite extent, which is
one of the reasons why in
395
00:24:55,730 --> 00:24:58,710
any case we would use some
approximation to it.
396
00:24:58,710 --> 00:25:05,700
But here, what we have is the
exact interpolating filter.
397
00:25:05,700 --> 00:25:11,030
And that corresponds to an
ideal low pass filter.
398
00:25:11,030 --> 00:25:17,260
If, instead, we carried out the
interpolating using the
399
00:25:17,260 --> 00:25:21,280
zero order hold, the zero order
hold has a rectangular
400
00:25:21,280 --> 00:25:22,740
impulse response.
401
00:25:22,740 --> 00:25:25,600
And that means in the frequency
domain, its
402
00:25:25,600 --> 00:25:30,410
frequency response is of the
form of a sync function, or
403
00:25:30,410 --> 00:25:31,890
sine x over x.
404
00:25:31,890 --> 00:25:35,910
And so this, in fact, when we're
doing the reconstruction
405
00:25:35,910 --> 00:25:40,230
with a zero order hold, is the
associated frequency response.
406
00:25:40,230 --> 00:25:43,040
Now notice that it does some
407
00:25:43,040 --> 00:25:44,930
approximate low pass filtering.
408
00:25:44,930 --> 00:25:51,430
But of course, it permits
significant energy outside the
409
00:25:51,430 --> 00:25:53,670
past band of the filter.
410
00:25:53,670 --> 00:25:56,680
Well, instead of the zero order
hold, if we used the
411
00:25:56,680 --> 00:25:59,610
first order hold corresponding
to the triangular impulse
412
00:25:59,610 --> 00:26:03,630
response, in that case then in
the frequency domain, the
413
00:26:03,630 --> 00:26:07,560
associated frequency response
would be the Fourier transform
414
00:26:07,560 --> 00:26:09,090
of the triangle.
415
00:26:09,090 --> 00:26:13,520
And the Fourier transform of a
triangle is a sine squared x
416
00:26:13,520 --> 00:26:16,310
over x squared kind
of function.
417
00:26:16,310 --> 00:26:20,770
And so in that case, what we
would have for the frequency
418
00:26:20,770 --> 00:26:24,070
response, associated with the
first order hold, is a
419
00:26:24,070 --> 00:26:27,750
frequency response
as I show here.
420
00:26:27,750 --> 00:26:33,330
And the fact that there's
somewhat more attenuation
421
00:26:33,330 --> 00:26:37,620
outside the past band of the
ideal filter is what suggests,
422
00:26:37,620 --> 00:26:42,270
in fact, that the first order
hold, or linear interpolation,
423
00:26:42,270 --> 00:26:46,280
gives us a somewhat smoother
approximation to the original
424
00:26:46,280 --> 00:26:49,620
signal than the zero
order hold does.
425
00:26:49,620 --> 00:26:53,680
And so, in fact, just to compare
these two, we can see
426
00:26:53,680 --> 00:26:57,990
that here is the ideal filter.
427
00:26:57,990 --> 00:27:03,300
Here is the zero order hold,
corresponding to generating a
428
00:27:03,300 --> 00:27:05,790
box car kind of reconstruction.
429
00:27:05,790 --> 00:27:09,710
And here is the first order
hold, corresponding to a
430
00:27:09,710 --> 00:27:12,500
linear interpolation.
431
00:27:12,500 --> 00:27:17,460
Now in fact, in many sampling
systems, in any sampling
432
00:27:17,460 --> 00:27:22,200
system really, we need to use
some approximation to the low
433
00:27:22,200 --> 00:27:23,520
pass filter.
434
00:27:23,520 --> 00:27:27,340
And very often, in fact, what
is done in many sampling
435
00:27:27,340 --> 00:27:32,130
systems, is to first use just
the zero order hold, and then
436
00:27:32,130 --> 00:27:34,300
follow the zero order
hold with some
437
00:27:34,300 --> 00:27:37,830
additional low pass filtering.
438
00:27:37,830 --> 00:27:43,240
Well, to illustrate some of
these ideas and the notion of
439
00:27:43,240 --> 00:27:46,380
doing a reconstruction with a
zero order hold or first order
440
00:27:46,380 --> 00:27:50,700
hold and then in fact adding
to that some additional low
441
00:27:50,700 --> 00:27:56,220
pass filtering, what I'd like
to do is demonstrate, or
442
00:27:56,220 --> 00:27:59,820
illustrate, sampling and
interpolation in the context
443
00:27:59,820 --> 00:28:01,230
of some images.
444
00:28:01,230 --> 00:28:05,380
An image, of course, is a
two-dimensional signal.
445
00:28:05,380 --> 00:28:07,490
The independent variables
are spatial
446
00:28:07,490 --> 00:28:09,650
variables not time variables.
447
00:28:09,650 --> 00:28:13,050
And of course, we can sample
in both of the spatial
448
00:28:13,050 --> 00:28:16,020
dimensions, both in x and y.
449
00:28:16,020 --> 00:28:22,000
And what I've chosen as a
possibly appropriate choice
450
00:28:22,000 --> 00:28:26,140
for an image is, again,
our friend and
451
00:28:26,140 --> 00:28:29,350
colleague J.B.J. Fourier.
452
00:28:29,350 --> 00:28:33,800
So let's begin with the original
image, which we then
453
00:28:33,800 --> 00:28:36,350
want to sample and
reconstruct.
454
00:28:36,350 --> 00:28:40,590
And the sampling is done by
effectively multiplying by a
455
00:28:40,590 --> 00:28:43,510
pulse both horizontally
and vertically.
456
00:28:43,510 --> 00:28:47,840
The sample picture is then
the next one that I show.
457
00:28:47,840 --> 00:28:51,930
And as you can see, this
corresponds, in effect, to
458
00:28:51,930 --> 00:28:54,280
extracting small brightness
elements out of
459
00:28:54,280 --> 00:28:55,080
the original image.
460
00:28:55,080 --> 00:28:58,200
In fact, let's look in
a little closer.
461
00:28:58,200 --> 00:29:01,500
And what you can see,
essentially, is that what we
462
00:29:01,500 --> 00:29:05,220
have, of course, are not
impulses spatially but small
463
00:29:05,220 --> 00:29:10,490
spatial pillars that implement
the sampling for us.
464
00:29:10,490 --> 00:29:15,450
OK, now going back to the
original sample picture, we
465
00:29:15,450 --> 00:29:18,920
know that a picture can be
reconstructed by low pass
466
00:29:18,920 --> 00:29:20,200
filtering from the samples.
467
00:29:20,200 --> 00:29:23,850
And in fact, we can do that
optically in this case by
468
00:29:23,850 --> 00:29:26,480
simply defocusing the camera.
469
00:29:26,480 --> 00:29:30,270
And when we do that, what
happens is that we smear out
470
00:29:30,270 --> 00:29:34,430
the picture, or effectively
convulve the impulses with the
471
00:29:34,430 --> 00:29:36,620
point spread function of
the optical system.
472
00:29:36,620 --> 00:29:40,340
And this then is not too
bad a reconstruction.
473
00:29:40,340 --> 00:29:44,420
So that's an approximate
reconstruction.
474
00:29:44,420 --> 00:29:47,840
And focusing back now
what we have again
475
00:29:47,840 --> 00:29:49,830
is the sample picture.
476
00:29:49,830 --> 00:29:53,990
477
00:29:53,990 --> 00:29:58,590
Now these images are, in fact,
taken off a computer display.
478
00:29:58,590 --> 00:30:02,620
And a common procedure in
computer generated or
479
00:30:02,620 --> 00:30:08,090
displayed images is in fact the
use of a zero order hold.
480
00:30:08,090 --> 00:30:11,640
And if the sampling rate is
high enough, then that
481
00:30:11,640 --> 00:30:13,490
actually works reasonably
well.
482
00:30:13,490 --> 00:30:16,650
So now let's look at the result
of applying a zero
483
00:30:16,650 --> 00:30:21,350
order hold to the sample image
that I just showed.
484
00:30:21,350 --> 00:30:24,580
485
00:30:24,580 --> 00:30:27,760
The zero order hold corresponds
to replacing the
486
00:30:27,760 --> 00:30:30,910
impulses by rectangles.
487
00:30:30,910 --> 00:30:35,180
And you can see that what that
generates is a mosaic effect,
488
00:30:35,180 --> 00:30:37,490
as you would expect.
489
00:30:37,490 --> 00:30:41,860
And in fact, let's go in a
little closer and emphasize
490
00:30:41,860 --> 00:30:42,840
the mosaic effect.
491
00:30:42,840 --> 00:30:45,390
You can see that, essentially,
where there were impulses
492
00:30:45,390 --> 00:30:49,200
previously, there are now
rectangles with those
493
00:30:49,200 --> 00:30:50,870
brightness values.
494
00:30:50,870 --> 00:30:55,300
A very common procedure with
computer generated images is
495
00:30:55,300 --> 00:30:59,110
to first do a zero order hold,
as we've done here, and then
496
00:30:59,110 --> 00:31:02,460
follow that with some additional
low pass filtering.
497
00:31:02,460 --> 00:31:05,900
And fact, we can do that low
pass filtering now again by
498
00:31:05,900 --> 00:31:07,930
defocusing the camera.
499
00:31:07,930 --> 00:31:12,890
And you can begin to see that
with the zero order hold plus
500
00:31:12,890 --> 00:31:14,140
the low pass filtering, the
501
00:31:14,140 --> 00:31:17,230
reconstruction is not that bad.
502
00:31:17,230 --> 00:31:21,450
Well, let's go back to
the full image with
503
00:31:21,450 --> 00:31:23,930
the zero order hold.
504
00:31:23,930 --> 00:31:29,200
And again, now the effect of
low pass filtering will be
505
00:31:29,200 --> 00:31:30,100
somewhat better.
506
00:31:30,100 --> 00:31:33,310
And let's defocus again here.
507
00:31:33,310 --> 00:31:37,210
And you can begin to see that
this is a reasonable
508
00:31:37,210 --> 00:31:39,370
reconstruction.
509
00:31:39,370 --> 00:31:44,050
With the mosaic, in fact, with
this back in focus, you can
510
00:31:44,050 --> 00:31:47,440
apply your own low pass
filtering to it either by
511
00:31:47,440 --> 00:31:51,040
squinting, or if you have the
right or wrong kind of
512
00:31:51,040 --> 00:31:55,800
eyeglasses, either taking them
off or putting them on.
513
00:31:55,800 --> 00:31:59,780
Now, in addition to the zero
order hold, we can, of course,
514
00:31:59,780 --> 00:32:01,260
apply a first order hold.
515
00:32:01,260 --> 00:32:05,550
And that would correspond to
replacing the impulses,
516
00:32:05,550 --> 00:32:09,310
instead of with rectangles as
we have here, replacing them
517
00:32:09,310 --> 00:32:10,880
with triangles.
518
00:32:10,880 --> 00:32:14,930
And so now let's take a look at
the result of a first order
519
00:32:14,930 --> 00:32:18,430
hold applied to the
original samples.
520
00:32:18,430 --> 00:32:22,410
And you can see now that the
reconstruction is somewhat
521
00:32:22,410 --> 00:32:24,960
smoother because of the fact
that we're using an impulse
522
00:32:24,960 --> 00:32:28,750
response that's somewhat
smoother or a corresponding
523
00:32:28,750 --> 00:32:31,310
frequency response that
has a sharper cut off.
524
00:32:31,310 --> 00:32:34,580
I emphasize again that this is
a somewhat low pass filtered
525
00:32:34,580 --> 00:32:38,220
version of the original because
we have under sampled
526
00:32:38,220 --> 00:32:41,870
somewhat spatially to bring out
the point that I want to
527
00:32:41,870 --> 00:32:43,120
illustrate.
528
00:32:43,120 --> 00:32:46,840
529
00:32:46,840 --> 00:32:52,120
OK, to emphasize these effects
even more, what I'd like to do
530
00:32:52,120 --> 00:32:55,750
is go through, basically,
the same sequence again.
531
00:32:55,750 --> 00:32:58,990
But in this case, what we'll
do is double the sample
532
00:32:58,990 --> 00:33:02,830
spacing both horizontal
and vertically.
533
00:33:02,830 --> 00:33:05,550
This of course, means that
we'll be even more highly
534
00:33:05,550 --> 00:33:09,970
under sampled than in the ones
I previously showed.
535
00:33:09,970 --> 00:33:12,970
And so the result of the
reconstructions with some low
536
00:33:12,970 --> 00:33:18,910
pass filtering will be a much
more low pass filtered image.
537
00:33:18,910 --> 00:33:21,590
So we now have the
sampled picture.
538
00:33:21,590 --> 00:33:24,900
But I've now under sampled
considerably more.
539
00:33:24,900 --> 00:33:28,460
And you can see the effect
of the sampling.
540
00:33:28,460 --> 00:33:33,510
And if we now apply a zero order
hold to this picture, we
541
00:33:33,510 --> 00:33:34,910
will again get a mosaic.
542
00:33:34,910 --> 00:33:37,200
And let's look at that.
543
00:33:37,200 --> 00:33:40,300
And that mosaic, of
course, looks even
544
00:33:40,300 --> 00:33:42,290
blockier than the original.
545
00:33:42,290 --> 00:33:46,110
And again, it emphasizes the
fact that the zero order hold
546
00:33:46,110 --> 00:33:50,140
simply corresponds to filling
in squares, or replacing the
547
00:33:50,140 --> 00:33:53,120
impulses, by squares, with the
548
00:33:53,120 --> 00:33:55,990
corresponding brightness values.
549
00:33:55,990 --> 00:34:00,540
Finally, if we, instead of a
zero order hold, use a first
550
00:34:00,540 --> 00:34:04,910
order hold, corresponding to two
dimensional triangles in
551
00:34:04,910 --> 00:34:06,750
place of these original
blocks.
552
00:34:06,750 --> 00:34:09,920
What we get is the next image.
553
00:34:09,920 --> 00:34:13,840
And that, again, is a smoother
reconstruction consistent with
554
00:34:13,840 --> 00:34:16,040
the fact that the triangles
are smoother than the
555
00:34:16,040 --> 00:34:17,270
rectangles.
556
00:34:17,270 --> 00:34:20,530
Again, I emphasize that this
looks so highly low pass
557
00:34:20,530 --> 00:34:24,280
filtered because of the fact
that we've under sampled so
558
00:34:24,280 --> 00:34:27,720
severely to essentially
emphasize the effect.
559
00:34:27,720 --> 00:34:31,110
560
00:34:31,110 --> 00:34:35,469
As I mentioned, the images that
we just looked at were
561
00:34:35,469 --> 00:34:38,250
taken from a computer, although
of course the
562
00:34:38,250 --> 00:34:41,659
original images were continuous
time images or more
563
00:34:41,659 --> 00:34:43,880
specifically, continuous
space.
564
00:34:43,880 --> 00:34:45,280
That is the independent
variable
565
00:34:45,280 --> 00:34:47,830
is a spatial variable.
566
00:34:47,830 --> 00:34:52,900
Now, computer processing of
signals, pictures, speech, or
567
00:34:52,900 --> 00:34:57,040
whatever the signals are is
very important and useful
568
00:34:57,040 --> 00:35:00,070
because it offers a lot
of flexibility.
569
00:35:00,070 --> 00:35:02,220
And in fact, the kinds of things
that I showed with
570
00:35:02,220 --> 00:35:07,120
these pictures would have been
very hard to do without, in
571
00:35:07,120 --> 00:35:10,620
fact, doing computer
processing.
572
00:35:10,620 --> 00:35:15,250
Well, in computer processing
of any kind of signal,
573
00:35:15,250 --> 00:35:20,800
basically what's required is
that we do the processing in
574
00:35:20,800 --> 00:35:24,280
the context of discrete time
signals and discrete time
575
00:35:24,280 --> 00:35:27,860
processing because of the
fact that a computer
576
00:35:27,860 --> 00:35:29,310
is run off a clock.
577
00:35:29,310 --> 00:35:35,630
And essentially, things happen
in the computer as a sequence
578
00:35:35,630 --> 00:35:39,400
of numbers and as a sequence
of events.
579
00:35:39,400 --> 00:35:43,290
Well, it turns out that the
sampling theorem, in fact, as
580
00:35:43,290 --> 00:35:47,280
I've indicated previously,
provides us with a very nice
581
00:35:47,280 --> 00:35:53,170
mechanism for converting our
continuous time signals into
582
00:35:53,170 --> 00:35:54,830
discrete time signals.
583
00:35:54,830 --> 00:35:58,810
For example, for computer
processing or, in fact, if
584
00:35:58,810 --> 00:36:02,230
it's not a computer for some
other kind of discrete time or
585
00:36:02,230 --> 00:36:04,930
perhaps digital processing.
586
00:36:04,930 --> 00:36:11,700
Well, the basic idea, as I've
indicated previously, is to
587
00:36:11,700 --> 00:36:15,520
carry out discrete time
processing of continuous time
588
00:36:15,520 --> 00:36:22,010
signals by first converting the
continuous time signal to
589
00:36:22,010 --> 00:36:27,890
a discrete time signal, carry
out the appropriate discrete
590
00:36:27,890 --> 00:36:33,310
time processing of the discrete
time signal, and then
591
00:36:33,310 --> 00:36:37,210
after we're done with that
processing, converting from
592
00:36:37,210 --> 00:36:41,840
the discrete time sequence
back to a continuous time
593
00:36:41,840 --> 00:36:45,400
signal, corresponding to the
output that we have here.
594
00:36:45,400 --> 00:36:48,010
595
00:36:48,010 --> 00:36:51,500
Well in the remainder of this
lecture, what I'd like to
596
00:36:51,500 --> 00:36:55,830
analyze is the first step in
that process, namely the
597
00:36:55,830 --> 00:36:59,710
conversion from a continuous
time signal to a discrete time
598
00:36:59,710 --> 00:37:05,630
signal and understand how the
two relate both in the time
599
00:37:05,630 --> 00:37:07,900
domain and in the frequency
domain.
600
00:37:07,900 --> 00:37:11,410
And in the next lecture, we'll
be analyzing and demonstrating
601
00:37:11,410 --> 00:37:15,750
the overall system, including
some intermediate processing.
602
00:37:15,750 --> 00:37:20,560
So the first step in the process
is the conversion from
603
00:37:20,560 --> 00:37:24,670
a continuous time signal to
a discrete time signal.
604
00:37:24,670 --> 00:37:28,160
And that can be thought of as
a process that involves two
605
00:37:28,160 --> 00:37:31,490
steps, although in practical
terms it may not be
606
00:37:31,490 --> 00:37:34,680
implemented specifically
as these two steps.
607
00:37:34,680 --> 00:37:39,500
The two steps are to first
convert from the continuous
608
00:37:39,500 --> 00:37:45,010
time, or continuous time
continuous signal, to an
609
00:37:45,010 --> 00:37:52,150
impulse train through a sampling
process and then to
610
00:37:52,150 --> 00:37:56,950
convert that impulse train to
a discrete time sequence.
611
00:37:56,950 --> 00:38:02,640
And the discrete time sequence
x of n is simply then a
612
00:38:02,640 --> 00:38:08,320
sequence of values which are the
samples of the continuous
613
00:38:08,320 --> 00:38:09,450
time signal.
614
00:38:09,450 --> 00:38:12,550
And as we'll see as we walk
through this, basically the
615
00:38:12,550 --> 00:38:16,960
step of going from the impulse
train to the sequence
616
00:38:16,960 --> 00:38:22,320
corresponds principally to a
relabeling step where we pick
617
00:38:22,320 --> 00:38:29,250
off the impulse values and use
those as the sequence values
618
00:38:29,250 --> 00:38:31,670
for the discrete time signal.
619
00:38:31,670 --> 00:38:35,880
So what I'd like to do as a
first step in understanding
620
00:38:35,880 --> 00:38:40,980
this process is to analyze
it in particular with our
621
00:38:40,980 --> 00:38:44,160
attention focused on trying
to understand what the
622
00:38:44,160 --> 00:38:48,040
relationship is in the frequency
domain between the
623
00:38:48,040 --> 00:38:52,370
discrete time Fourier transform
of the sequence,
624
00:38:52,370 --> 00:38:56,380
discrete time signal, and the
continuous time Fourier
625
00:38:56,380 --> 00:39:00,630
transform of the original
unsampled, and then the
626
00:39:00,630 --> 00:39:02,170
sampled signal.
627
00:39:02,170 --> 00:39:04,430
So let's go through that.
628
00:39:04,430 --> 00:39:10,760
And in particular, what we have
is a process where the
629
00:39:10,760 --> 00:39:15,020
continuous time signal is,
of course, modulated or
630
00:39:15,020 --> 00:39:17,740
multiplied by an
impulse train.
631
00:39:17,740 --> 00:39:20,060
And that gives us,
then, another
632
00:39:20,060 --> 00:39:21,290
continuous time signal.
633
00:39:21,290 --> 00:39:23,470
We're still in the continuous
time domain.
634
00:39:23,470 --> 00:39:26,200
It gives us another continuous
time signal, which is an
635
00:39:26,200 --> 00:39:28,120
impulse train.
636
00:39:28,120 --> 00:39:31,250
And in fact, we've gone
through this analysis
637
00:39:31,250 --> 00:39:32,540
previously.
638
00:39:32,540 --> 00:39:37,830
And what we have is this
multiplication or taking this
639
00:39:37,830 --> 00:39:42,630
term inside the summation and
recognizing that the impulse
640
00:39:42,630 --> 00:39:48,660
train is simply an impulse
train with areas of the
641
00:39:48,660 --> 00:39:51,680
impulses, which are
the samples of the
642
00:39:51,680 --> 00:39:53,510
continuous time function.
643
00:39:53,510 --> 00:39:56,190
We can then carry out
the analysis in
644
00:39:56,190 --> 00:39:59,540
the frequency domain.
645
00:39:59,540 --> 00:40:03,080
Now in the time domain, we have
a multiplication process.
646
00:40:03,080 --> 00:40:06,680
So in the frequency domain, we
have a convolution of the
647
00:40:06,680 --> 00:40:12,010
Fourier transform of the
continuous time signal, the
648
00:40:12,010 --> 00:40:14,820
original signal, and the Fourier
transform of the
649
00:40:14,820 --> 00:40:18,590
impulse train, which is itself
an impulse train.
650
00:40:18,590 --> 00:40:22,950
So in the frequency domain then,
the Fourier transform of
651
00:40:22,950 --> 00:40:26,590
the sampled signal, which is
an impulse train, is the
652
00:40:26,590 --> 00:40:30,780
convolution of the Fourier
transform of the sampling
653
00:40:30,780 --> 00:40:34,310
function P of t and the Fourier
transform of the
654
00:40:34,310 --> 00:40:36,440
sampled signal.
655
00:40:36,440 --> 00:40:41,710
Since the sampling signal is a
periodic impulse train, its
656
00:40:41,710 --> 00:40:44,750
Fourier transform is
an impulse train.
657
00:40:44,750 --> 00:40:49,700
And consequently, carrying out
this convolution in effect
658
00:40:49,700 --> 00:40:53,300
says that this Fourier
transform simply gets
659
00:40:53,300 --> 00:40:57,600
replicated at each of the
locations of these impulses.
660
00:40:57,600 --> 00:41:02,390
And finally, what we end up
with then is a Fourier
661
00:41:02,390 --> 00:41:08,730
transform after the sampling
process, which is the original
662
00:41:08,730 --> 00:41:13,650
Fourier transform of the
continuous signal but added to
663
00:41:13,650 --> 00:41:17,750
itself shifted by integer
multiples of
664
00:41:17,750 --> 00:41:19,350
the sampling frequency.
665
00:41:19,350 --> 00:41:23,040
And so this is the basic
equation then that tells us in
666
00:41:23,040 --> 00:41:28,280
the frequency domain what
happens through the first part
667
00:41:28,280 --> 00:41:30,510
of this two step process.
668
00:41:30,510 --> 00:41:33,030
Now I emphasize that it's
a two step process.
669
00:41:33,030 --> 00:41:38,500
The first process is sampling,
where we're still essentially
670
00:41:38,500 --> 00:41:41,240
in the continuous time world.
671
00:41:41,240 --> 00:41:46,050
The next step is essentially a
relabeling process, where we
672
00:41:46,050 --> 00:41:50,650
convert that impulse train
simply to a sequence.
673
00:41:50,650 --> 00:41:53,770
So let's look at
the next step.
674
00:41:53,770 --> 00:41:57,780
The next step is to take the
impulse train and convert it
675
00:41:57,780 --> 00:42:01,850
through a process
to a sequence.
676
00:42:01,850 --> 00:42:06,880
And the sequence values are
simply then samples of the
677
00:42:06,880 --> 00:42:09,550
original continuous signal.
678
00:42:09,550 --> 00:42:13,840
And so now we can
analyze this.
679
00:42:13,840 --> 00:42:18,860
And what we want to relate is
the discrete time Fourier
680
00:42:18,860 --> 00:42:22,930
transform of this and the
continuous time Fourier
681
00:42:22,930 --> 00:42:25,920
transform of this, or in fact,
the continuous time Fourier
682
00:42:25,920 --> 00:42:30,660
transform of x of C of T.
683
00:42:30,660 --> 00:42:34,930
OK, we have the impulse train.
684
00:42:34,930 --> 00:42:40,570
And it's Fourier transform we
can get by simply evaluating
685
00:42:40,570 --> 00:42:42,000
the Fourier transform.
686
00:42:42,000 --> 00:42:45,430
And since the Fourier
transform of this--
687
00:42:45,430 --> 00:42:48,170
since this corresponds
to an impulse train--
688
00:42:48,170 --> 00:42:51,140
the Fourier transform, by the
time we change some sums and
689
00:42:51,140 --> 00:42:56,340
integrals, will then have this
impulse replaced by the
690
00:42:56,340 --> 00:42:59,420
Fourier transform of the shifted
impulse, which is this
691
00:42:59,420 --> 00:43:01,570
exponential factor.
692
00:43:01,570 --> 00:43:04,950
So this expression is the
Fourier transform of the
693
00:43:04,950 --> 00:43:08,850
impulse train, the continuous
time Fourier transform.
694
00:43:08,850 --> 00:43:13,750
And alternatively, we can look
at the Fourier transform of
695
00:43:13,750 --> 00:43:15,080
the sequence.
696
00:43:15,080 --> 00:43:17,540
And this, of course,
is a discrete
697
00:43:17,540 --> 00:43:21,110
time Fourier transform.
698
00:43:21,110 --> 00:43:25,310
So we have the continuous time
Fourier transform of the
699
00:43:25,310 --> 00:43:29,570
impulse train, we have the
discrete Fourier transform of
700
00:43:29,570 --> 00:43:30,360
the sequence.
701
00:43:30,360 --> 00:43:33,970
And now we want to look at
how those two relate.
702
00:43:33,970 --> 00:43:37,750
Well, it pretty much falls out
of just comparing these two
703
00:43:37,750 --> 00:43:39,020
summations.
704
00:43:39,020 --> 00:43:44,870
In particular, this term and
this term are identical.
705
00:43:44,870 --> 00:43:52,100
That's just a relabeling of what
the sequence values are.
706
00:43:52,100 --> 00:43:55,540
And notice that when we compare
these exponential
707
00:43:55,540 --> 00:44:01,350
factors, they're identical as
long as we associate capital
708
00:44:01,350 --> 00:44:05,650
omega with little omega times
capital T. In other words, if
709
00:44:05,650 --> 00:44:09,370
we were to replace here capital
omega by little omega
710
00:44:09,370 --> 00:44:15,060
times capital T, and replace x
of n by x of c of nt, then
711
00:44:15,060 --> 00:44:20,220
this expression would be
identical to this expression.
712
00:44:20,220 --> 00:44:26,010
So in fact, these two are equal
with a relabeling, or
713
00:44:26,010 --> 00:44:29,970
with a transformation, between
small omega and capital omega.
714
00:44:29,970 --> 00:44:34,350
And so in fact, the relationship
that we have is
715
00:44:34,350 --> 00:44:39,000
that the discrete time Fourier
transform of the sequence of
716
00:44:39,000 --> 00:44:43,990
samples is equal to the
continuous time Fourier
717
00:44:43,990 --> 00:44:50,440
transform of the impulse train
of samples where we associate
718
00:44:50,440 --> 00:44:54,460
the continuous time frequency
variable and the discrete time
719
00:44:54,460 --> 00:44:58,200
frequency variable through
a frequency scaling as I
720
00:44:58,200 --> 00:44:59,520
indicate here.
721
00:44:59,520 --> 00:45:04,130
Or said another way, the
discrete time spectrum is the
722
00:45:04,130 --> 00:45:10,090
continuous time spectrum of the
samples with small omega
723
00:45:10,090 --> 00:45:12,440
replaced by capital
omega divided by
724
00:45:12,440 --> 00:45:15,480
capital T. All right.
725
00:45:15,480 --> 00:45:19,660
So we have then this
two step process.
726
00:45:19,660 --> 00:45:22,960
The first step is taking the
continuous time signal,
727
00:45:22,960 --> 00:45:26,130
sampling it with an
impulse train.
728
00:45:26,130 --> 00:45:31,480
In the frequency domain, that
corresponds to replicating the
729
00:45:31,480 --> 00:45:34,200
Fourier transform
of the original
730
00:45:34,200 --> 00:45:36,540
continuous time signal.
731
00:45:36,540 --> 00:45:41,780
The second step is relabeling
that, in effect turning it
732
00:45:41,780 --> 00:45:43,170
into a sequence.
733
00:45:43,170 --> 00:45:46,580
And what that does in the
frequency domain is provide us
734
00:45:46,580 --> 00:45:50,140
with a rescaling of the
frequency axis, or as we'll
735
00:45:50,140 --> 00:45:55,040
see a frequency normalization,
which is associated with the
736
00:45:55,040 --> 00:45:56,130
corresponding time
737
00:45:56,130 --> 00:45:58,740
normalization in the time domain.
738
00:45:58,740 --> 00:46:01,810
Well, let's look at those
statements a little more
739
00:46:01,810 --> 00:46:04,350
specifically.
740
00:46:04,350 --> 00:46:06,900
What I show here
is the original
741
00:46:06,900 --> 00:46:09,750
continuous time signal.
742
00:46:09,750 --> 00:46:16,020
And then below it is
the sampled signal.
743
00:46:16,020 --> 00:46:20,770
And these two are signals in
the continuous time domain.
744
00:46:20,770 --> 00:46:23,310
Now, what is the conversion
from this
745
00:46:23,310 --> 00:46:25,660
impulse train to a sequence?
746
00:46:25,660 --> 00:46:31,040
Well, it's simply taking these
impulse areas, or these sample
747
00:46:31,040 --> 00:46:39,070
values, and relabeling them, in
effect as I show below, as
748
00:46:39,070 --> 00:46:41,850
sequence values.
749
00:46:41,850 --> 00:46:48,610
And essentially, I'm now
replacing the impulse by the
750
00:46:48,610 --> 00:46:50,700
designation of a
sequence value.
751
00:46:50,700 --> 00:46:51,830
That's one step.
752
00:46:51,830 --> 00:46:56,930
But the other important step to
focus on is that whereas in
753
00:46:56,930 --> 00:47:00,370
the impulse train, these
impulses are spaced by integer
754
00:47:00,370 --> 00:47:05,000
multiples of the sampling
period capital T. In the
755
00:47:05,000 --> 00:47:08,390
sequence, of course, because
of the way that we label
756
00:47:08,390 --> 00:47:13,210
sequences, these are always
spaced by simply integer
757
00:47:13,210 --> 00:47:14,750
multiples of one.
758
00:47:14,750 --> 00:47:18,160
So in effect, you could say that
the step in going from
759
00:47:18,160 --> 00:47:23,180
here to here corresponds to
normalizing out in the time
760
00:47:23,180 --> 00:47:27,590
domain the sampling
period capital T.
761
00:47:27,590 --> 00:47:31,570
To stress that another way, if
the sampling period were
762
00:47:31,570 --> 00:47:35,810
doubled so that in this picture,
the spacing stretched
763
00:47:35,810 --> 00:47:38,280
out by a factor of two.
764
00:47:38,280 --> 00:47:45,320
Nevertheless, for the discrete
time signal, the spacing would
765
00:47:45,320 --> 00:47:46,740
remain as one.
766
00:47:46,740 --> 00:47:52,470
And essentially, it's the
envelope of those sequence
767
00:47:52,470 --> 00:47:56,460
values that would then get
compressed in time.
768
00:47:56,460 --> 00:48:00,370
So you can think of the step
in going from the impulse
769
00:48:00,370 --> 00:48:04,430
train to the samples as,
essentially, a time
770
00:48:04,430 --> 00:48:05,740
normalization.
771
00:48:05,740 --> 00:48:08,470
Now let's look at this in
the frequency domain.
772
00:48:08,470 --> 00:48:11,320
In the frequency domain, what
we have is the Fourier
773
00:48:11,320 --> 00:48:16,110
transform of our original
continuous signal.
774
00:48:16,110 --> 00:48:21,220
After sampling with an impulse
train, this spectrum retains
775
00:48:21,220 --> 00:48:26,280
its shape but is replicated at
integer multiples of the
776
00:48:26,280 --> 00:48:31,350
sampling frequency 2 pi over
capital T, as I indicate here.
777
00:48:31,350 --> 00:48:38,000
Now, we know that a discrete
time spectrum must be periodic
778
00:48:38,000 --> 00:48:40,580
in frequency with a
period of 2 pi.
779
00:48:40,580 --> 00:48:42,560
Here, we have the periodicity.
780
00:48:42,560 --> 00:48:45,590
But it's not periodic with
a period of 2 pi.
781
00:48:45,590 --> 00:48:47,750
It's periodic with a period,
which is equal to
782
00:48:47,750 --> 00:48:50,680
the sampling frequency.
783
00:48:50,680 --> 00:48:55,930
However, in converting from the
samples to the sequence
784
00:48:55,930 --> 00:48:58,610
values, we go through
another step.
785
00:48:58,610 --> 00:48:59,490
What's the other step?
786
00:48:59,490 --> 00:49:02,940
The other step is a time
normalization, where we take
787
00:49:02,940 --> 00:49:05,890
the impulses, which are spaced
by the sampling period.
788
00:49:05,890 --> 00:49:08,810
And we rescale that, essentially
in the time
789
00:49:08,810 --> 00:49:11,400
domain, to a spacing
which is unity.
790
00:49:11,400 --> 00:49:19,990
So we're dividing out in the
time domain by a factor, which
791
00:49:19,990 --> 00:49:23,050
is equal to the sampling
period.
792
00:49:23,050 --> 00:49:28,740
Well, dividing out in the time
domain by capital T would
793
00:49:28,740 --> 00:49:33,510
correspond to multiplying in
the frequency domain the
794
00:49:33,510 --> 00:49:38,060
frequency axis by capital T.
And indeed, what happens is
795
00:49:38,060 --> 00:49:42,270
that in going from the impulse
train to the sequence values,
796
00:49:42,270 --> 00:49:49,250
we now rescale this axis so
that, in fact, the axis gets
797
00:49:49,250 --> 00:49:53,550
stretched by capital T. And
the frequency, which
798
00:49:53,550 --> 00:49:57,380
corresponded to 2 pi over
capital T, now gets
799
00:49:57,380 --> 00:50:00,980
renormalized to 2 pi.
800
00:50:00,980 --> 00:50:04,740
So just looking at this again,
and perhaps with the overall
801
00:50:04,740 --> 00:50:08,940
picture, in the time domain,
we've gone from a continuous
802
00:50:08,940 --> 00:50:13,380
curve to samples, relabeled
those, and in effect
803
00:50:13,380 --> 00:50:15,780
implemented a time
normalization.
804
00:50:15,780 --> 00:50:20,180
Corresponding in the frequency
domain, we have replicated the
805
00:50:20,180 --> 00:50:24,690
spectrum through the initial
sampling process and then
806
00:50:24,690 --> 00:50:29,020
rescaled the frequency axis
so that, in fact, now this
807
00:50:29,020 --> 00:50:33,480
periodicity corresponds to a
periodicity here, which is 2
808
00:50:33,480 --> 00:50:37,870
pi, and here, which is the
sampling frequency.
809
00:50:37,870 --> 00:50:41,470
So very often, in fact--
and we'll be
810
00:50:41,470 --> 00:50:44,000
doing this next time--
811
00:50:44,000 --> 00:50:49,630
when you think of continuous
time signals, which have been
812
00:50:49,630 --> 00:50:52,280
converted to discrete time
signals, when you look at the
813
00:50:52,280 --> 00:50:56,520
discrete time frequency axis,
the frequency 2 pi is
814
00:50:56,520 --> 00:51:03,290
associated with the sampling
frequency as it was applied to
815
00:51:03,290 --> 00:51:06,530
the original continuous
time signal.
816
00:51:06,530 --> 00:51:13,010
Now as I indicated, what we'll
want to go on to from here is
817
00:51:13,010 --> 00:51:15,880
an understanding of what
happens when we take a
818
00:51:15,880 --> 00:51:19,180
continuous time signal, convert
it to a discrete time
819
00:51:19,180 --> 00:51:22,440
signal as I've just gone
through, do some discrete time
820
00:51:22,440 --> 00:51:25,890
processing with a linear time
invariant system, and then
821
00:51:25,890 --> 00:51:29,520
carry that back into the
continuous time world.
822
00:51:29,520 --> 00:51:34,190
That is a procedure that we'll
go through, and analyze, and
823
00:51:34,190 --> 00:51:38,660
in fact, illustrate in some
detail next time.
824
00:51:38,660 --> 00:51:42,970
In preparation for that, what I
would be eager to encourage
825
00:51:42,970 --> 00:51:46,650
you to do using the study guide
and in reviewing this
826
00:51:46,650 --> 00:51:53,340
lecture, is to begin the next
lecture with a careful and
827
00:51:53,340 --> 00:51:56,220
thorough understanding
of the arguments that
828
00:51:56,220 --> 00:51:57,500
I've just gone through.
829
00:51:57,500 --> 00:52:02,210
In particular, understanding the
process that's involved in
830
00:52:02,210 --> 00:52:06,320
going from a continuous time
signal through sampling to a
831
00:52:06,320 --> 00:52:08,390
discrete time signal.
832
00:52:08,390 --> 00:52:11,860
And what that means in the
frequency domain in terms of
833
00:52:11,860 --> 00:52:15,710
taking the original spectrum,
replicating it because of the
834
00:52:15,710 --> 00:52:20,590
sampling process, and then
rescaling that so that the
835
00:52:20,590 --> 00:52:24,710
periodicity gets rescaled so
that it's periodic with a
836
00:52:24,710 --> 00:52:26,010
period of 2 pi.
837
00:52:26,010 --> 00:52:31,120
So we'll continue with that next
time, focusing now on the
838
00:52:31,120 --> 00:52:33,610
subsequent steps in
the processing.
839
00:52:33,610 --> 00:52:34,860
Thank you.
840
00:52:34,860 --> 00:52:36,139