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PROFESSOR: In the last lecture,
we discussed discrete
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time processing of continuous
time signals.
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00:01:00,760 --> 00:01:04,069
And, as you know, the basis for
that arises essentially
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00:01:04,069 --> 00:01:06,150
out of a sampling theorem.
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00:01:06,150 --> 00:01:10,240
Now in that context, and also
in its own right, another
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important sampling issue is the
sampling of discrete time
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signals, in other words, the
sampling of a sequence.
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00:01:18,640 --> 00:01:23,000
One common context in which this
arises, for example, is,
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00:01:23,000 --> 00:01:26,690
if we've converted from a
continuous time signal to a
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00:01:26,690 --> 00:01:32,150
sequence, and we then carry out
some additional filtering,
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00:01:32,150 --> 00:01:35,260
then there's the possibility
that we can resample that
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00:01:35,260 --> 00:01:38,460
sequence, and as we'll see as we
go through the discussion,
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00:01:38,460 --> 00:01:42,300
save something in the way
of storage or whatever.
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00:01:42,300 --> 00:01:47,800
So discrete time sampling, as
I indicated, has important
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00:01:47,800 --> 00:01:51,930
application in a context
referred to here, namely
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00:01:51,930 --> 00:01:55,830
resampling after discrete
time filtering.
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00:01:55,830 --> 00:01:59,370
And closely related to that,
as we'll indicate in this
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00:01:59,370 --> 00:02:04,270
lecture, is the concept of using
discrete time sampling
28
00:02:04,270 --> 00:02:08,740
for what's referred to as
sampling rate conversion.
29
00:02:08,740 --> 00:02:15,220
And also closely associated with
both of those ideas is a
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00:02:15,220 --> 00:02:18,120
set of ideas that I'll bring
up in today's lecture,
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00:02:18,120 --> 00:02:22,980
referred to as decimation and
interpolation of discrete time
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00:02:22,980 --> 00:02:25,420
signals or sequences.
33
00:02:25,420 --> 00:02:30,550
Now the basic process for
discrete time sampling is the
34
00:02:30,550 --> 00:02:33,720
same as it is for continuous
time sampling.
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00:02:33,720 --> 00:02:38,880
Namely, we can analyze it and
set it up on the basis of
36
00:02:38,880 --> 00:02:43,330
multiplying or modulating a
discrete time signal by an
37
00:02:43,330 --> 00:02:47,430
impulse train, the impulse train
essentially, or pulse
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00:02:47,430 --> 00:02:51,140
train, pulling out sequence
values at the times that we
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00:02:51,140 --> 00:02:52,950
want to sample.
40
00:02:52,950 --> 00:02:58,400
So the basic block diagram for
the sampling process is to
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00:02:58,400 --> 00:03:01,430
modulate or multiply the
sequence that we want to
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00:03:01,430 --> 00:03:05,520
sample by an impulse train.
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00:03:05,520 --> 00:03:11,380
And here, the impulse train has
impulses spaced by integer
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00:03:11,380 --> 00:03:14,160
multiples of capital N.
This then becomes
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00:03:14,160 --> 00:03:16,170
the sampling period.
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00:03:16,170 --> 00:03:20,010
And the result of that
modulation is then the sample
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00:03:20,010 --> 00:03:23,230
sequence x of p of n.
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00:03:23,230 --> 00:03:28,250
So if we just look at what a
sequence and a sampled version
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00:03:28,250 --> 00:03:33,750
of that sequence might look
like, what we have here is an
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00:03:33,750 --> 00:03:36,220
original sequence x of n.
51
00:03:36,220 --> 00:03:41,790
And then we have the sampling
impulse train, or sampling
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00:03:41,790 --> 00:03:46,120
sequence, and it's the
modulation or product of these
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00:03:46,120 --> 00:03:50,840
two that gives us the sample
sequence x of p of n.
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00:03:50,840 --> 00:03:55,090
And so, as you can see,
multiplying this by this
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00:03:55,090 --> 00:03:59,640
essentially pulls out of the
original sequence sample
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00:03:59,640 --> 00:04:02,740
values at the times that
this pulse train is on.
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00:04:02,740 --> 00:04:06,330
And of course here, I've drawn
this for the case where
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00:04:06,330 --> 00:04:12,570
capital N, the sampling
period, is equal to 3.
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Now the analysis of discrete
time sampling is very similar
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00:04:18,540 --> 00:04:21,579
to the analysis of continuous
time sampling.
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00:04:21,579 --> 00:04:24,100
And let's just quickly
look through the
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steps that are involved.
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We're modulating or
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multiplying in the time domain.
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And what that corresponds to in
the frequency domain is a
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00:04:33,870 --> 00:04:35,330
convolution.
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00:04:35,330 --> 00:04:43,020
And so the spectrum of the
sampled sequence is the
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00:04:43,020 --> 00:04:47,710
periodic convolution of the
spectrum of the sampling
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00:04:47,710 --> 00:04:51,700
sequence and the
spectrum of the
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00:04:51,700 --> 00:04:54,210
sequence that we're sampling.
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00:04:54,210 --> 00:04:59,680
And since the sampling sequence
is an impulse train,
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as we know, the Fourier
transform of an impulse train
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00:05:03,500 --> 00:05:05,560
is itself an impulse train.
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00:05:05,560 --> 00:05:09,500
And so this is the Fourier
transform of
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00:05:09,500 --> 00:05:11,990
the sampling sequence.
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00:05:11,990 --> 00:05:17,340
And now, finally, the Fourier
transform of the resulting
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00:05:17,340 --> 00:05:23,080
sample sequence, being the
convolution of this with the
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00:05:23,080 --> 00:05:26,300
Fourier transform of the
sequence that we're sampling,
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00:05:26,300 --> 00:05:32,090
gives us then a spectrum which
consists of a sum of
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00:05:32,090 --> 00:05:35,370
replicated versions of the
Fourier transform of the
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00:05:35,370 --> 00:05:37,970
sequence that we're sampling.
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00:05:37,970 --> 00:05:40,940
In other words, what we're
doing, very much as we did in
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00:05:40,940 --> 00:05:44,790
continuous time, is, through the
sampling process when we
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00:05:44,790 --> 00:05:49,460
look at it in the frequency
domain, taking the spectrum of
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00:05:49,460 --> 00:05:53,190
the sequence there were sampling
and shifting it and
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00:05:53,190 --> 00:05:54,540
then adding it in--
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00:05:54,540 --> 00:05:57,470
shifting it by integer
multiples of
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00:05:57,470 --> 00:05:58,900
the sampling frequency.
89
00:05:58,900 --> 00:06:02,620
In particular, looking back
at this equation, what we
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00:06:02,620 --> 00:06:09,120
recognize is that this term, k
times 2 pi over capital N, is
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00:06:09,120 --> 00:06:14,460
in fact an integer multiple
of the sampling frequency.
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00:06:14,460 --> 00:06:17,260
And the same thing
is true here.
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00:06:17,260 --> 00:06:23,080
This is k times omega sub s,
where omega sub s, the
94
00:06:23,080 --> 00:06:30,580
sampling frequency, is 2 pi
divided by capital N.
95
00:06:30,580 --> 00:06:34,720
All right, so now let's look at
what this means pictorially
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00:06:34,720 --> 00:06:37,520
or graphically in the
frequency domain.
97
00:06:37,520 --> 00:06:42,110
And as you can imagine, since
the analysis and algebra is
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00:06:42,110 --> 00:06:45,740
similar to what happens in
continuous time, we would
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00:06:45,740 --> 00:06:49,410
expect the pictures to more or
less be identical to what
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00:06:49,410 --> 00:06:51,570
we've seen previously
for continuous time.
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00:06:51,570 --> 00:06:53,660
And indeed that's the case.
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So here we have the spectrum
of the signal
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00:06:58,890 --> 00:07:00,460
that's we're sampling.
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00:07:00,460 --> 00:07:04,160
This is its Fourier transform,
with an assumed highest
105
00:07:04,160 --> 00:07:08,700
frequency omega sub m, highest
frequency over a 2pi range, or
106
00:07:08,700 --> 00:07:10,810
over a range of pi, rather.
107
00:07:10,810 --> 00:07:18,520
And now the spectrum of the
sampling signal is what I show
108
00:07:18,520 --> 00:07:22,680
below, which is an impulse train
with impulses occurring
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00:07:22,680 --> 00:07:27,260
at integer multiples of the
sampling frequency.
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00:07:27,260 --> 00:07:31,930
And then finally, the
convolution of these two is
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00:07:31,930 --> 00:07:35,760
simply this one replicated
at the
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00:07:35,760 --> 00:07:37,910
locations of these impulses.
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00:07:37,910 --> 00:07:41,610
And so that's finally
what I show below.
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00:07:41,610 --> 00:07:46,770
And here I made one particular
choice for
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00:07:46,770 --> 00:07:48,460
the sampling period.
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This in particular corresponds
to a sampling period which is
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00:07:54,760 --> 00:08:00,570
capital N equal to 3.
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00:08:00,570 --> 00:08:04,520
And so the sampling frequency,
omega sub s, is
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00:08:04,520 --> 00:08:07,210
2pi divided by 3.
120
00:08:07,210 --> 00:08:09,740
121
00:08:09,740 --> 00:08:16,060
Now when we look at this, what
we recognize is that we have
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00:08:16,060 --> 00:08:20,060
basically the same issue here as
we had in continuous time,
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00:08:20,060 --> 00:08:25,730
in the sense that when these
individual replications of the
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00:08:25,730 --> 00:08:30,600
Fourier transform, when the
sampling frequency is chosen
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00:08:30,600 --> 00:08:34,370
high enough so that they don't
overlap, then we see the
126
00:08:34,370 --> 00:08:37,539
potential for being able to
get one of them back.
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00:08:37,539 --> 00:08:40,659
On the other hand, when they
do overlap then what we'll
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00:08:40,659 --> 00:08:43,309
have is aliasing, in particular,
discrete time
129
00:08:43,309 --> 00:08:46,460
aliasing, much as we had
continuous time aliasing in
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00:08:46,460 --> 00:08:48,030
the continuous time case.
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00:08:48,030 --> 00:08:53,330
Well notice in this picture
that what we have is we've
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00:08:53,330 --> 00:08:59,370
chosen this picture so that
omega sub s minus omega sub m
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00:08:59,370 --> 00:09:04,360
is greater than omega sub m, or
equivalently, so that omega
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00:09:04,360 --> 00:09:11,890
sub s is greater than
2 omega sub m.
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00:09:11,890 --> 00:09:15,240
And so with omega sub s greater
than 2 omega sum m,
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00:09:15,240 --> 00:09:17,600
that corresponds to
this picture.
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00:09:17,600 --> 00:09:23,330
Whereas, if that condition is
violated then, in fact, the
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00:09:23,330 --> 00:09:27,160
picture that we would have is
a picture that looks like.
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00:09:27,160 --> 00:09:32,030
And in this picture, the
individual replications of the
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00:09:32,030 --> 00:09:36,700
Fourier transform of the
original signal overlap.
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00:09:36,700 --> 00:09:41,770
And we can no longer recover the
Fourier transform of the
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00:09:41,770 --> 00:09:43,000
original signal.
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00:09:43,000 --> 00:09:47,390
And this, just as it was in
continuous time, is referred
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00:09:47,390 --> 00:09:50,860
to as aliasing.
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00:09:50,860 --> 00:09:54,870
Now let's look more closely at
the situation in which there
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00:09:54,870 --> 00:09:56,120
is no aliasing.
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00:09:56,120 --> 00:09:58,680
148
00:09:58,680 --> 00:10:07,280
So in that case, what we have is
a Fourier transform for the
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00:10:07,280 --> 00:10:13,980
sampled signal, which is as
I indicated here, and the
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00:10:13,980 --> 00:10:17,380
Fourier transform for the
original signal, as I indicate
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00:10:17,380 --> 00:10:18,710
at the top.
152
00:10:18,710 --> 00:10:22,190
And the question now is
how do we recover this
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00:10:22,190 --> 00:10:23,590
one from this one.
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00:10:23,590 --> 00:10:27,340
Well, the way that we do that,
just as we did in a continuous
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00:10:27,340 --> 00:10:31,010
time case, is by a low
pass filtering.
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00:10:31,010 --> 00:10:34,520
In particular, processing in
the time domain or in the
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00:10:34,520 --> 00:10:40,590
frequency domain, this with an
ideal low pass filter has the
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00:10:40,590 --> 00:10:46,080
effect of extracting that part
of the spectrum that in fact
159
00:10:46,080 --> 00:10:53,400
we identify with the original
signal that we began with.
160
00:10:53,400 --> 00:10:57,470
So what we see, again, is
that the process is
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00:10:57,470 --> 00:10:58,460
very much the same.
162
00:10:58,460 --> 00:11:01,170
As long as there's no aliasing,
we can recover the
163
00:11:01,170 --> 00:11:06,150
original signal by ideal
low pass filtering.
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00:11:06,150 --> 00:11:13,290
So the overall system is, just
to reiterate, a system which
165
00:11:13,290 --> 00:11:20,400
consists of modulating the
original sequence with a pulse
166
00:11:20,400 --> 00:11:22,960
train or impulse train.
167
00:11:22,960 --> 00:11:27,160
And then that is going
to be processed
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00:11:27,160 --> 00:11:30,370
with a low pass filter.
169
00:11:30,370 --> 00:11:34,140
The spectrum of the original
signal x of n
170
00:11:34,140 --> 00:11:36,770
is what I show here.
171
00:11:36,770 --> 00:11:40,020
The spectrum of the sampled
signal, where I'm drawing the
172
00:11:40,020 --> 00:11:43,830
picture on the assumption that
the sampling period is 3, is
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00:11:43,830 --> 00:11:47,470
now what's indicated, where
these are replicated, where
174
00:11:47,470 --> 00:11:50,320
the original spectrum
is replicated.
175
00:11:50,320 --> 00:11:56,740
This is now processed through
a filter which, for exact
176
00:11:56,740 --> 00:12:01,220
reconstruction, is an ideal
low pass filter.
177
00:12:01,220 --> 00:12:04,820
And so we would multiply this
spectrum by this one.
178
00:12:04,820 --> 00:12:08,980
And the result, after doing
that, will generate a
179
00:12:08,980 --> 00:12:11,890
reconstructed spectrum
which, in fact, is
180
00:12:11,890 --> 00:12:14,300
identical to the original.
181
00:12:14,300 --> 00:12:21,450
So the frequency domain
picture is the same.
182
00:12:21,450 --> 00:12:24,740
And what we would expect then
is that the time domain
183
00:12:24,740 --> 00:12:26,110
picture would be the same.
184
00:12:26,110 --> 00:12:29,380
Well, let's in fact look
at the time domain.
185
00:12:29,380 --> 00:12:34,780
And in the time domain, what we
have is an analysis more or
186
00:12:34,780 --> 00:12:38,340
less identical to what we
had in continuous time.
187
00:12:38,340 --> 00:12:41,360
We of course have
the same system.
188
00:12:41,360 --> 00:12:45,260
And in the time domain,
we are multiplying
189
00:12:45,260 --> 00:12:47,550
by an impulse train.
190
00:12:47,550 --> 00:12:54,340
Consequently, the sample
sequence is an impulse train
191
00:12:54,340 --> 00:12:58,820
whose values are samples of x
of at integer multiples of
192
00:12:58,820 --> 00:13:05,310
capital N. For the
reconstruction, this is now
193
00:13:05,310 --> 00:13:10,220
processed through an ideal
low pass filter.
194
00:13:10,220 --> 00:13:11,610
And that implements a
195
00:13:11,610 --> 00:13:13,900
convolution in the time domain.
196
00:13:13,900 --> 00:13:17,700
And so the reconstructed signal
is the convolution of
197
00:13:17,700 --> 00:13:22,720
the sample sequence and the
filter impulse response.
198
00:13:22,720 --> 00:13:26,010
And expressed another way,
namely writing out the
199
00:13:26,010 --> 00:13:29,250
convolution as a sum, we
have this expression.
200
00:13:29,250 --> 00:13:34,010
And so it says then that the
reconstruction is carried out
201
00:13:34,010 --> 00:13:41,430
by replacing the impulses
here, these impulses, by
202
00:13:41,430 --> 00:13:44,460
versions of the filter
impulse response.
203
00:13:44,460 --> 00:13:47,300
204
00:13:47,300 --> 00:13:53,360
Well, if the filter is an ideal
low pass filter, then
205
00:13:53,360 --> 00:13:58,020
that corresponds in the time
domain to sine nx over sine x
206
00:13:58,020 --> 00:13:58,840
kind of function.
207
00:13:58,840 --> 00:14:03,570
And that is the interpolation
in between the samples to do
208
00:14:03,570 --> 00:14:05,800
the reconstruction.
209
00:14:05,800 --> 00:14:10,870
Also, as is discussed somewhat
in the text, we can consider
210
00:14:10,870 --> 00:14:14,680
other kinds of interpolation,
for example discrete time zero
211
00:14:14,680 --> 00:14:18,640
order hold or discrete time
first order hold, just as we
212
00:14:18,640 --> 00:14:20,420
had in continuous time.
213
00:14:20,420 --> 00:14:23,530
And the issues and analysis for
the discrete times zero
214
00:14:23,530 --> 00:14:28,170
order hold and first order hold
are very similar to what
215
00:14:28,170 --> 00:14:30,950
they were in continuous time--
the zero order hold just
216
00:14:30,950 --> 00:14:34,340
simply holding the value until
the next sampling instant, and
217
00:14:34,340 --> 00:14:38,160
the first order hold carrying
out linear interpolation in
218
00:14:38,160 --> 00:14:39,410
between the samples.
219
00:14:39,410 --> 00:14:42,400
220
00:14:42,400 --> 00:14:49,610
Now in this sampling process,
if we look again at the wave
221
00:14:49,610 --> 00:14:53,910
forms involved, or sequences
involved, the process
222
00:14:53,910 --> 00:15:01,090
consisted of taking a sequence
and extracting from it
223
00:15:01,090 --> 00:15:03,980
individual values.
224
00:15:03,980 --> 00:15:08,560
And in between those values,
we have sequence
225
00:15:08,560 --> 00:15:10,930
values equal to 0.
226
00:15:10,930 --> 00:15:18,030
So what we're doing in this
case is retaining the same
227
00:15:18,030 --> 00:15:21,950
number of sequence values and
simply setting some number of
228
00:15:21,950 --> 00:15:24,980
them equal to 0.
229
00:15:24,980 --> 00:15:28,380
Well, let's say, for example,
that we want
230
00:15:28,380 --> 00:15:29,930
to carry out sampling.
231
00:15:29,930 --> 00:15:32,200
And what we're talking
about is a sequence.
232
00:15:32,200 --> 00:15:36,170
And let's say this sequence is
stored in a computer memory.
233
00:15:36,170 --> 00:15:40,150
As you can imagine, the notion
of sampling it and actually
234
00:15:40,150 --> 00:15:44,250
replacing some of the values
by zero is somewhat
235
00:15:44,250 --> 00:15:44,790
inefficient.
236
00:15:44,790 --> 00:15:47,550
Namely, it doesn't make sense
to think of storing in the
237
00:15:47,550 --> 00:15:50,900
memory a lot of zeros, when in
fact those are zeros that we
238
00:15:50,900 --> 00:15:52,460
can always put back in.
239
00:15:52,460 --> 00:15:54,670
We know exactly what
the values are.
240
00:15:54,670 --> 00:15:57,920
And if we know what the sampling
rate was in discrete
241
00:15:57,920 --> 00:16:00,760
time, then we would know
when and how to put
242
00:16:00,760 --> 00:16:03,070
the zeros back in.
243
00:16:03,070 --> 00:16:07,700
So actually, in discrete time
sampling, what we've talked
244
00:16:07,700 --> 00:16:10,710
about so far is really
only one part or
245
00:16:10,710 --> 00:16:12,610
one step in the process.
246
00:16:12,610 --> 00:16:16,190
Basically, the other step is to
take those zeros and just
247
00:16:16,190 --> 00:16:18,820
throw them away because we could
put them in any time we
248
00:16:18,820 --> 00:16:22,790
want to and really only retain,
for example in our
249
00:16:22,790 --> 00:16:26,800
computer memory or list of
sequence values or whatever,
250
00:16:26,800 --> 00:16:30,830
only retain the non-zero
values.
251
00:16:30,830 --> 00:16:34,540
So that process and the
resulting sequence that we end
252
00:16:34,540 --> 00:16:39,020
up with is associated with a
concept called decimation.
253
00:16:39,020 --> 00:16:42,050
What I mean by decimation
is very simple.
254
00:16:42,050 --> 00:16:47,200
What we're doing is, instead of
working with this sequence,
255
00:16:47,200 --> 00:16:50,190
we're going to work with
this sequence.
256
00:16:50,190 --> 00:16:55,440
Namely, we'll toss out the
zeros in between here and
257
00:16:55,440 --> 00:17:00,270
collapse the sequence down only
to the sequence values
258
00:17:00,270 --> 00:17:03,860
that are associated with
the original x of n.
259
00:17:03,860 --> 00:17:07,839
Now, in talking about a
decimated sequence, we could
260
00:17:07,839 --> 00:17:12,450
of course do that directly from
this step down to here,
261
00:17:12,450 --> 00:17:15,890
although again in the analysis
it will be somewhat more
262
00:17:15,890 --> 00:17:20,650
convenient to carry that out
by thinking, at least
263
00:17:20,650 --> 00:17:23,839
analytically, in terms
of a 2-step process--
264
00:17:23,839 --> 00:17:27,079
one being a sampling process,
then the other being a
265
00:17:27,079 --> 00:17:28,079
decimation.
266
00:17:28,079 --> 00:17:32,930
But basically, this is a
decimated version of that.
267
00:17:32,930 --> 00:17:38,040
Now for the grammatical purists
out there, the word
268
00:17:38,040 --> 00:17:40,620
decimation of course means
taking every tenth one.
269
00:17:40,620 --> 00:17:43,910
The implication is not that
we're always sampling with a
270
00:17:43,910 --> 00:17:45,220
period of 10.
271
00:17:45,220 --> 00:17:49,650
The idea of decimating is to
pick out every nth sample and
272
00:17:49,650 --> 00:17:53,390
end up with a collapsed
sequence.
273
00:17:53,390 --> 00:17:59,410
Let's now look at a little bit
of the analysis and understand
274
00:17:59,410 --> 00:18:02,870
what the consequence is in
the frequency domain.
275
00:18:02,870 --> 00:18:07,690
In particular what we want to
develop is how the Fourier
276
00:18:07,690 --> 00:18:12,000
transform of the decimated
sequence is related to the
277
00:18:12,000 --> 00:18:15,060
Fourier transform of the
original sequence or the
278
00:18:15,060 --> 00:18:16,540
sample sequence.
279
00:18:16,540 --> 00:18:19,125
So let's look at this in
the frequency domain.
280
00:18:19,125 --> 00:18:21,730
281
00:18:21,730 --> 00:18:28,040
So what we have is a decimated
sequence, which consists of
282
00:18:28,040 --> 00:18:32,250
pulling out every capital
Nth value of x of n.
283
00:18:32,250 --> 00:18:36,080
And of course that's the same as
we can either decimate x of
284
00:18:36,080 --> 00:18:40,710
n or we can decimate
the sample signal.
285
00:18:40,710 --> 00:18:45,960
Now in going through this
analysis, I'll kind of go
286
00:18:45,960 --> 00:18:49,520
through it quickly because again
there's the issue of
287
00:18:49,520 --> 00:18:51,170
some slight mental gymnastics.
288
00:18:51,170 --> 00:18:55,400
And if you're anything like I
am, it's usually best to kind
289
00:18:55,400 --> 00:18:58,390
of try to absorb that by
yourself quietly, rather than
290
00:18:58,390 --> 00:19:00,410
having somebody throw
it at you.
291
00:19:00,410 --> 00:19:03,640
Let me say, though, that the
steps that I'm following here
292
00:19:03,640 --> 00:19:06,090
are slightly different
than the steps that
293
00:19:06,090 --> 00:19:07,120
I use in the text.
294
00:19:07,120 --> 00:19:11,250
It's a slightly different way of
going through the analysis.
295
00:19:11,250 --> 00:19:13,790
I guess you could say for one
thing that if we've gone
296
00:19:13,790 --> 00:19:16,560
through it twice, and it comes
out the same, well of course
297
00:19:16,560 --> 00:19:18,930
it has to be right.
298
00:19:18,930 --> 00:19:24,760
Well anyway, here we have then
the relationship between the
299
00:19:24,760 --> 00:19:29,320
decimated sequence, the original
sequence, and the
300
00:19:29,320 --> 00:19:30,990
sampled sequence.
301
00:19:30,990 --> 00:19:34,260
And we know of course that the
Fourier transform of the
302
00:19:34,260 --> 00:19:38,560
sample sequence is just
simply this summation.
303
00:19:38,560 --> 00:19:43,320
And now kind of the idea in the
analysis is that we can
304
00:19:43,320 --> 00:19:51,020
collapse this summation by
recognizing that this term is
305
00:19:51,020 --> 00:19:55,420
only non-zero at every
nth value.
306
00:19:55,420 --> 00:19:58,200
And so if we do that,
essentially making a
307
00:19:58,200 --> 00:20:02,260
substitution of variables with
n equal to small m times
308
00:20:02,260 --> 00:20:06,260
capital N, we can turn this
into a summation on m.
309
00:20:06,260 --> 00:20:09,160
And that's what I've
done here.
310
00:20:09,160 --> 00:20:13,170
And we've just simply used the
fact that we can collapse the
311
00:20:13,170 --> 00:20:18,340
sum because of the fact that
all but every nth value is
312
00:20:18,340 --> 00:20:20,080
equal to zero.
313
00:20:20,080 --> 00:20:23,110
So this then is the Fourier
transform all
314
00:20:23,110 --> 00:20:25,590
of the sampled signal.
315
00:20:25,590 --> 00:20:29,920
And now if we look at the
Fourier transform of the
316
00:20:29,920 --> 00:20:34,800
decimated signal, that Fourier
transform, of course, is this
317
00:20:34,800 --> 00:20:38,680
summation on the decimated
sequence.
318
00:20:38,680 --> 00:20:41,480
Well, what we want to look at is
the correspondence between
319
00:20:41,480 --> 00:20:43,790
this equation and the
one above it.
320
00:20:43,790 --> 00:20:47,780
So we want to compare this
equation to this one.
321
00:20:47,780 --> 00:20:51,460
And recognizing that this
decimated sequence is just
322
00:20:51,460 --> 00:20:59,070
simply related to the sample
sequence this way, these two
323
00:20:59,070 --> 00:21:02,960
become equal under a
substitution of variables.
324
00:21:02,960 --> 00:21:09,090
In particular, notice that if
we replace in this equation
325
00:21:09,090 --> 00:21:14,650
omega by omega times capital
N, then these two equations
326
00:21:14,650 --> 00:21:16,740
become equal.
327
00:21:16,740 --> 00:21:20,180
So the consequence of that,
then, what it all boils down
328
00:21:20,180 --> 00:21:27,625
to and says, is that the
relationship between the
329
00:21:27,625 --> 00:21:30,360
Fourier transform of the
decimated sequence and the
330
00:21:30,360 --> 00:21:34,680
Fourier transform of the sampled
sequence is simply a
331
00:21:34,680 --> 00:21:38,540
frequency scaling corresponding
to dividing the
332
00:21:38,540 --> 00:21:41,350
frequency axis by capital N.
333
00:21:41,350 --> 00:21:44,040
So that's essentially
what happens.
334
00:21:44,040 --> 00:21:45,680
That's really all that's
involved in
335
00:21:45,680 --> 00:21:47,420
the decimation process.
336
00:21:47,420 --> 00:21:50,180
And now, again, let's look
at that pictorially
337
00:21:50,180 --> 00:21:52,550
and see what it means.
338
00:21:52,550 --> 00:21:55,780
So what we want to look at, now
that we've looked in the
339
00:21:55,780 --> 00:22:00,540
time domain in this particular
view graph, we now want to
340
00:22:00,540 --> 00:22:04,470
look in the frequency domain.
341
00:22:04,470 --> 00:22:11,620
And in the frequency domain,
we have, again, the Fourier
342
00:22:11,620 --> 00:22:17,470
transform of the original
sequence and we have the
343
00:22:17,470 --> 00:22:23,350
Fourier transform of the
sampled sequence.
344
00:22:23,350 --> 00:22:28,490
And now the Fourier transform
of the decimated sequence is
345
00:22:28,490 --> 00:22:35,030
simply this spectrum with a
linear frequency scaling.
346
00:22:35,030 --> 00:22:38,450
And in particular, it simply
corresponds to multiplying
347
00:22:38,450 --> 00:22:44,150
this frequency axis by capital
N. And notice that this
348
00:22:44,150 --> 00:22:48,840
frequency now, 2 pi over capital
N, that frequency ends
349
00:22:48,840 --> 00:22:55,800
up getting rescaled to
a frequency of 2 pi.
350
00:22:55,800 --> 00:23:00,980
So in fact now, in the
rescaling, it's that this
351
00:23:00,980 --> 00:23:07,200
point in the decimation gets
rescaled to this point.
352
00:23:07,200 --> 00:23:10,480
And correspondingly, of course,
this whole spectrum
353
00:23:10,480 --> 00:23:11,650
broadens out.
354
00:23:11,650 --> 00:23:14,770
Now we can also look at
that in the context of
355
00:23:14,770 --> 00:23:16,010
the original spectrum.
356
00:23:16,010 --> 00:23:19,130
And you can see that the
relationship between the
357
00:23:19,130 --> 00:23:22,020
original spectrum and the
spectrum of the decimated
358
00:23:22,020 --> 00:23:27,730
signal corresponds to simply
linearly scaling this.
359
00:23:27,730 --> 00:23:32,740
But it's important also to
keep in mind that that
360
00:23:32,740 --> 00:23:37,330
analysis, that particular
relationship, assumes that
361
00:23:37,330 --> 00:23:38,850
we've avoided aliasing.
362
00:23:38,850 --> 00:23:41,870
The relationship between the
spectrum of the decimated
363
00:23:41,870 --> 00:23:46,050
signal and the spectrum of the
sample signal is true whether
364
00:23:46,050 --> 00:23:47,700
or not we have aliasing.
365
00:23:47,700 --> 00:23:51,820
But being able to clearly
associate it with just simply
366
00:23:51,820 --> 00:23:56,480
scaling of this spectrum of the
original signal assumes
367
00:23:56,480 --> 00:24:00,380
that the spectrum of the
original signal, the shape of
368
00:24:00,380 --> 00:24:03,950
it, is preserved when we
generate the sample signal.
369
00:24:03,950 --> 00:24:06,940
370
00:24:06,940 --> 00:24:11,870
Well, when might discrete time
sampling, and for that matter,
371
00:24:11,870 --> 00:24:14,080
decimation, be used?
372
00:24:14,080 --> 00:24:18,450
Well, I indicated one context in
which it might be useful at
373
00:24:18,450 --> 00:24:19,710
the beginning of this lecture.
374
00:24:19,710 --> 00:24:22,320
And let me now focus in
on that a little more
375
00:24:22,320 --> 00:24:23,570
specifically.
376
00:24:23,570 --> 00:24:25,730
377
00:24:25,730 --> 00:24:33,450
In particular, suppose that we
have gone through a process in
378
00:24:33,450 --> 00:24:39,730
which the continuous time signal
has been converted to a
379
00:24:39,730 --> 00:24:41,810
discrete time signal.
380
00:24:41,810 --> 00:24:44,200
And we then carry out
some additional
381
00:24:44,200 --> 00:24:45,690
discrete time filtering.
382
00:24:45,690 --> 00:24:49,060
So we have a situation where
we've gone through a
383
00:24:49,060 --> 00:24:52,710
continuous to discrete
time conversion.
384
00:24:52,710 --> 00:24:56,470
And after that conversion,
we carry out some
385
00:24:56,470 --> 00:24:58,035
discrete time filtering.
386
00:24:58,035 --> 00:25:01,380
387
00:25:01,380 --> 00:25:03,940
And in particular, in going
through this part of the
388
00:25:03,940 --> 00:25:09,990
process, we choose the sampling
rate for going from
389
00:25:09,990 --> 00:25:13,710
the continuous time signal to
the sequence so that we don't
390
00:25:13,710 --> 00:25:15,750
violate the sampling theorem.
391
00:25:15,750 --> 00:25:19,500
Well let's suppose, then, that
this is the spectrum of the
392
00:25:19,500 --> 00:25:22,380
continuous time signal.
393
00:25:22,380 --> 00:25:27,170
Below it, we have the spectrum
of the output of the
394
00:25:27,170 --> 00:25:31,200
continuous to discrete
time conversion.
395
00:25:31,200 --> 00:25:36,090
And I've chosen the sampling
frequency to be just high
396
00:25:36,090 --> 00:25:38,560
enough so that I
avoid aliasing.
397
00:25:38,560 --> 00:25:41,350
398
00:25:41,350 --> 00:25:47,220
Well that then is the lowest
sampling frequency I can pick.
399
00:25:47,220 --> 00:25:51,860
But now, if we go through some
additional low pass filtering,
400
00:25:51,860 --> 00:25:53,580
then let's see what happens.
401
00:25:53,580 --> 00:25:59,790
If I now low pass filter the
sequence x of n, then in
402
00:25:59,790 --> 00:26:03,340
effect, I'm multiplying
the sequence
403
00:26:03,340 --> 00:26:06,490
spectrum by this filter.
404
00:26:06,490 --> 00:26:10,470
And so the result of that,
the product of the filter
405
00:26:10,470 --> 00:26:14,380
frequency response and the
Fourier transform of x of n
406
00:26:14,380 --> 00:26:20,130
would have a shape somewhat
like I indicate below.
407
00:26:20,130 --> 00:26:25,490
Now notice that in this
spectrum, although in the
408
00:26:25,490 --> 00:26:31,490
input to the filter this entire
band was filled up, in
409
00:26:31,490 --> 00:26:38,380
the output of the filter, there
is a band that in fact
410
00:26:38,380 --> 00:26:40,190
has zero energy in it.
411
00:26:40,190 --> 00:26:45,310
So what I can consider doing is
taking the output sequence
412
00:26:45,310 --> 00:26:49,590
from the filter and in fact
resampling it, in other words
413
00:26:49,590 --> 00:26:53,520
sampling it, which would be more
or less associated with a
414
00:26:53,520 --> 00:26:56,230
different sampling rate
for the continuous
415
00:26:56,230 --> 00:26:58,560
time signals involved.
416
00:26:58,560 --> 00:27:02,830
So I could now go through a
process which is commonly
417
00:27:02,830 --> 00:27:05,900
referred to as down sampling
that is lowering
418
00:27:05,900 --> 00:27:07,270
the sampling rate.
419
00:27:07,270 --> 00:27:10,570
When we do that, of course,
what's going to happen is that
420
00:27:10,570 --> 00:27:13,690
in fact this spectral
energy will now fill
421
00:27:13,690 --> 00:27:16,110
out more of the band.
422
00:27:16,110 --> 00:27:21,950
And for example, if this was a
third, then in fact if I down
423
00:27:21,950 --> 00:27:25,470
sampled by a factor of three,
then I would fill up the
424
00:27:25,470 --> 00:27:27,400
entire band with this energy.
425
00:27:27,400 --> 00:27:29,870
But since I've done some
additional low pass filtering,
426
00:27:29,870 --> 00:27:34,260
as I indicate here, there's
no problem with aliasing.
427
00:27:34,260 --> 00:27:38,230
If I had, let's say, down
sampled by a factor of three
428
00:27:38,230 --> 00:27:40,880
and I'm now taking that signal
and converting it back to a
429
00:27:40,880 --> 00:27:45,430
continuous time signal, then
of course the way I can do
430
00:27:45,430 --> 00:27:51,170
that is by simply running my
output clock for the discrete
431
00:27:51,170 --> 00:27:52,760
to continuous time converter.
432
00:27:52,760 --> 00:27:56,600
I can run my output clock
at a third the rate
433
00:27:56,600 --> 00:27:58,380
of the input clock.
434
00:27:58,380 --> 00:28:00,250
And that, in effect,
takes care of the
435
00:28:00,250 --> 00:28:01,500
bookkeeping for me.
436
00:28:01,500 --> 00:28:03,930
437
00:28:03,930 --> 00:28:11,000
So here we have now the notion
of sampling a sequence, and
438
00:28:11,000 --> 00:28:14,250
very closely tied in with that,
the notion of decimating
439
00:28:14,250 --> 00:28:19,000
a sequence, and related to both
of those, the notion of
440
00:28:19,000 --> 00:28:22,630
down sampling, that is changing
the sampling rates so
441
00:28:22,630 --> 00:28:25,880
that, if we were trying this
in with continuous time
442
00:28:25,880 --> 00:28:30,170
signals, we've essentially
changed our clock rate.
443
00:28:30,170 --> 00:28:35,030
And we might also want to, and
it's important to, consider
444
00:28:35,030 --> 00:28:36,800
the opposite of that.
445
00:28:36,800 --> 00:28:41,140
So now a question is what's the
opposite of decimation.
446
00:28:41,140 --> 00:28:45,080
Suppose that we had a sequence
and we decimate it.
447
00:28:45,080 --> 00:28:49,470
Thinking about it as a 2-step
process, that would correspond
448
00:28:49,470 --> 00:28:52,170
to first multiplying by an
impulse train, where there are
449
00:28:52,170 --> 00:28:56,920
bunch of zeros in there, and
then choosing, throwing away
450
00:28:56,920 --> 00:29:00,880
the zeros and keeping only the
values that are non-zero,
451
00:29:00,880 --> 00:29:03,370
because the zeros we can
always recreate.
452
00:29:03,370 --> 00:29:06,570
Well, in fact, the inverse
process is very specifically a
453
00:29:06,570 --> 00:29:10,955
process of recreating the
zeros and then doing the
454
00:29:10,955 --> 00:29:12,860
desampling.
455
00:29:12,860 --> 00:29:22,610
So in the opposite operation,
what we would do is undo the
456
00:29:22,610 --> 00:29:24,890
decimation step.
457
00:29:24,890 --> 00:29:28,330
And that would consist of
converting the decimated
458
00:29:28,330 --> 00:29:35,790
sequence back to an impulse
train and then processing that
459
00:29:35,790 --> 00:29:42,470
impulse train by an ideal low
pass filter to do the
460
00:29:42,470 --> 00:29:46,500
interpolation or reconstruction,
filling in the
461
00:29:46,500 --> 00:29:51,610
values which, in this impulse
train, are equal to zero.
462
00:29:51,610 --> 00:29:53,640
So we now have the two steps.
463
00:29:53,640 --> 00:29:56,490
We take the decimated sequence
and we expand it
464
00:29:56,490 --> 00:29:58,950
out, putting in zeros.
465
00:29:58,950 --> 00:30:03,380
And then we desample that by
processing it through a low
466
00:30:03,380 --> 00:30:05,150
pass filter.
467
00:30:05,150 --> 00:30:12,680
So just kind of looking at
sequences again, what we have
468
00:30:12,680 --> 00:30:18,360
is an original sequence,
the sequence x of n.
469
00:30:18,360 --> 00:30:23,510
And then the sample sequence
is simply a sequence which
470
00:30:23,510 --> 00:30:26,100
alternates, in this particular
case, those
471
00:30:26,100 --> 00:30:28,300
sequence values was zero.
472
00:30:28,300 --> 00:30:35,610
Here what we're assuming is that
the sampling period is 2.
473
00:30:35,610 --> 00:30:40,160
And so every other value
here is equal to zero.
474
00:30:40,160 --> 00:30:45,350
The decimated sequence then is
this sequence, collapsed as I
475
00:30:45,350 --> 00:30:48,400
show in the sequence above.
476
00:30:48,400 --> 00:30:52,960
And so it's, in effect, time
compressing the sample
477
00:30:52,960 --> 00:30:57,160
sequence or the original
sequence so that we throw out
478
00:30:57,160 --> 00:30:59,890
the sequence values which
were equal to zero
479
00:30:59,890 --> 00:31:02,500
in the sample sequence.
480
00:31:02,500 --> 00:31:06,270
Now in recovering the original
sequence from the decimated
481
00:31:06,270 --> 00:31:09,400
sequence, we can think
of a 2-step process.
482
00:31:09,400 --> 00:31:15,640
Namely, we spread this out
alternating with zeros, and
483
00:31:15,640 --> 00:31:18,050
again, keeping in mind that
this is drawn for the case
484
00:31:18,050 --> 00:31:19,950
where capital N is 2.
485
00:31:19,950 --> 00:31:23,630
And then finally, we interpolate
between the
486
00:31:23,630 --> 00:31:28,700
non-zero values here by going
through a low pass filter to
487
00:31:28,700 --> 00:31:32,240
reconstruct the original
sequence.
488
00:31:32,240 --> 00:31:36,650
And that's what we show finally
on the bottom curve.
489
00:31:36,650 --> 00:31:39,730
So that's what we would see
in the time domain.
490
00:31:39,730 --> 00:31:43,950
Let's look at what we would see
in the frequency domain.
491
00:31:43,950 --> 00:31:48,860
In the frequency domain, we
have to begin with the
492
00:31:48,860 --> 00:31:52,020
sequence on the bottom, or the
spectrum on the bottom, which
493
00:31:52,020 --> 00:31:56,400
would correspond to the
original spectrum.
494
00:31:56,400 --> 00:31:59,710
Then, through the sampling
process, that is periodically
495
00:31:59,710 --> 00:32:00,910
replicated.
496
00:32:00,910 --> 00:32:04,990
Again, this is drawn on the
assumption that the sampling
497
00:32:04,990 --> 00:32:09,870
frequency is pi or the sampling
period is equal to 2.
498
00:32:09,870 --> 00:32:12,480
And so this is now replicated.
499
00:32:12,480 --> 00:32:17,310
And then, in going from this
to the spectrum of the
500
00:32:17,310 --> 00:32:23,990
decimated sequence, we would
rescale the frequency axis so
501
00:32:23,990 --> 00:32:29,590
that the frequency pi now gets
rescaled in the spectrum for
502
00:32:29,590 --> 00:32:34,090
the decimated sequence to a
frequency which is 2 pi.
503
00:32:34,090 --> 00:32:36,900
And so this now is
the spectrum of
504
00:32:36,900 --> 00:32:38,970
the decimated sequence.
505
00:32:38,970 --> 00:32:44,990
If we now want to reconvert to
the original sequence we would
506
00:32:44,990 --> 00:32:49,210
first intersperse in the
time domain with zeros,
507
00:32:49,210 --> 00:32:54,910
corresponding to compressing
in the frequency domain.
508
00:32:54,910 --> 00:32:58,610
This would then be low
pass filtered.
509
00:32:58,610 --> 00:33:02,680
And the low pass filtering would
consist of throwing away
510
00:33:02,680 --> 00:33:06,600
this replication, accounting
for a factor which is the
511
00:33:06,600 --> 00:33:10,630
factor capital N, and extracting
the portion of the
512
00:33:10,630 --> 00:33:16,680
spectrum which is associated
with the spectrum of the
513
00:33:16,680 --> 00:33:20,110
original signal which
we began with.
514
00:33:20,110 --> 00:33:24,420
So once again, we have
decimation and interpolation.
515
00:33:24,420 --> 00:33:30,150
And the decimation can be
thought of as a time
516
00:33:30,150 --> 00:33:34,250
compression that corresponds to
a frequency expansion then.
517
00:33:34,250 --> 00:33:37,890
And the interpolation process
is then just the reverse.
518
00:33:37,890 --> 00:33:41,080
519
00:33:41,080 --> 00:33:45,130
Now there are lots of situations
in which decimation
520
00:33:45,130 --> 00:33:48,980
and interpolation and discrete
time sampling are useful.
521
00:33:48,980 --> 00:33:52,390
And one context that I just
want to quickly draw your
522
00:33:52,390 --> 00:33:56,480
attention to is the use of
decimation and interpolation
523
00:33:56,480 --> 00:34:00,030
in what is commonly referred
to as sampling rate
524
00:34:00,030 --> 00:34:01,480
conversion.
525
00:34:01,480 --> 00:34:05,710
What the basic issue and
sampling rate conversion is is
526
00:34:05,710 --> 00:34:10,650
that, in some situations, and
a very common one is digital
527
00:34:10,650 --> 00:34:15,330
audio, a continuous time
signal is sampled.
528
00:34:15,330 --> 00:34:18,850
And those sampled values
are stored or whatever.
529
00:34:18,850 --> 00:34:22,699
And kind of the notion is that,
perhaps when that is
530
00:34:22,699 --> 00:34:26,530
played back, it's played back
through a different system.
531
00:34:26,530 --> 00:34:31,350
And the different system has a
different assumed sampling
532
00:34:31,350 --> 00:34:33,480
frequency or sampling period.
533
00:34:33,480 --> 00:34:36,630
So that's kind of the
issue and the idea.
534
00:34:36,630 --> 00:34:41,090
We have, let's say, a continuous
time signal which
535
00:34:41,090 --> 00:34:45,070
we've converted to a sequence
through a sampling process
536
00:34:45,070 --> 00:34:48,830
using an assumed sampling
period of T1.
537
00:34:48,830 --> 00:34:54,420
And these sequence values may
then, for example, be put into
538
00:34:54,420 --> 00:34:55,710
digital storage.
539
00:34:55,710 --> 00:34:59,360
In the case of a digital audio
system, it may, for example,
540
00:34:59,360 --> 00:35:01,720
go onto a digital record.
541
00:35:01,720 --> 00:35:06,660
And it might be the output of
this that we want to recreate.
542
00:35:06,660 --> 00:35:11,350
Or we might in fact follow
that with some additional
543
00:35:11,350 --> 00:35:14,120
processing, whatever that
additional processing is.
544
00:35:14,120 --> 00:35:17,400
And I'll kind of put a question
mark in there because
545
00:35:17,400 --> 00:35:21,030
we don't know exactly
what that might be.
546
00:35:21,030 --> 00:35:25,900
And then, in any case, the
result of that is going to be
547
00:35:25,900 --> 00:35:30,540
converted back to a continuous
time signal.
548
00:35:30,540 --> 00:35:36,470
But it might be converted
through a system that has a
549
00:35:36,470 --> 00:35:40,660
different assumed
sampling period.
550
00:35:40,660 --> 00:35:44,950
And so a very common issue, and
it comes up as I indicated
551
00:35:44,950 --> 00:35:50,190
particularly in digital audio,
a very common issue is to be
552
00:35:50,190 --> 00:35:55,490
able to convert from one assumed
sampling period, T1,
553
00:35:55,490 --> 00:35:58,660
our sampling frequency,
to another
554
00:35:58,660 --> 00:36:00,780
assumed sampling period.
555
00:36:00,780 --> 00:36:02,120
Now how do we do that?
556
00:36:02,120 --> 00:36:07,460
Well in fact, we do that by
using the ideas of decimation
557
00:36:07,460 --> 00:36:09,140
and interpolation.
558
00:36:09,140 --> 00:36:14,110
In particular, if we had, for
example, a situation where we
559
00:36:14,110 --> 00:36:19,180
wanted to convert from a
sampling period, T1, to a
560
00:36:19,180 --> 00:36:24,210
sampling period which was twice
as long as that, then
561
00:36:24,210 --> 00:36:29,340
essentially, we're going to take
the sequence and process
562
00:36:29,340 --> 00:36:32,710
it in a way that would, in
effect, correspond to assuming
563
00:36:32,710 --> 00:36:36,190
that we had sampled at half
the original frequency.
564
00:36:36,190 --> 00:36:37,440
Well how do we do that?
565
00:36:37,440 --> 00:36:40,210
The way we do it is we take the
sequence we have and we
566
00:36:40,210 --> 00:36:43,460
just throw away every
other value.
567
00:36:43,460 --> 00:36:47,370
So in that case, we would then,
for this sampling rate
568
00:36:47,370 --> 00:36:51,560
conversion, down sample
and decimate.
569
00:36:51,560 --> 00:36:54,770
Or actually, we might not go
through this step formally.
570
00:36:54,770 --> 00:36:57,520
We might just simply decimate.
571
00:36:57,520 --> 00:37:02,590
Now we might have an alternative
situation where in
572
00:37:02,590 --> 00:37:05,680
fact the new sampling period, or
the sampling period of the
573
00:37:05,680 --> 00:37:09,230
output, is half the sampling
period of the input,
574
00:37:09,230 --> 00:37:13,670
corresponding to an assumed
sampling frequency, which is
575
00:37:13,670 --> 00:37:15,550
twice as high.
576
00:37:15,550 --> 00:37:20,010
And in that case, then., we
would go through a process of
577
00:37:20,010 --> 00:37:21,000
interpolation.
578
00:37:21,000 --> 00:37:25,730
And in particular, we would up
sample and interpolate by a
579
00:37:25,730 --> 00:37:27,780
factor of 2 to one.
580
00:37:27,780 --> 00:37:31,090
So in one case, we're
simply throwing
581
00:37:31,090 --> 00:37:32,260
away every other value.
582
00:37:32,260 --> 00:37:34,080
In the other case, what we're
going to do is take our
583
00:37:34,080 --> 00:37:37,630
sequence, put in zeros, put it
through a low pass filter to
584
00:37:37,630 --> 00:37:39,470
interpolate.
585
00:37:39,470 --> 00:37:43,300
Now life would be simple if
everything happened in simple
586
00:37:43,300 --> 00:37:45,070
integer amounts like that.
587
00:37:45,070 --> 00:37:49,690
A more common situation is that
we may have an assumed
588
00:37:49,690 --> 00:37:55,250
output sampling period
which is 3/2 of the
589
00:37:55,250 --> 00:37:57,250
input sampling period.
590
00:37:57,250 --> 00:38:00,530
And now the question is what are
we going to do to convert
591
00:38:00,530 --> 00:38:04,520
from this sampling period
to this sampling period.
592
00:38:04,520 --> 00:38:09,850
Well, in fact, the answer to
that is to use a combination
593
00:38:09,850 --> 00:38:13,090
of down sampling and up
sampling, or up sampling and
594
00:38:13,090 --> 00:38:17,470
down sampling, equivalently
interpolation and decimation.
595
00:38:17,470 --> 00:38:21,900
And for this particular case,
in fact, what we would do is
596
00:38:21,900 --> 00:38:28,630
to first take the data, up
sample by a factor of 2, and
597
00:38:28,630 --> 00:38:33,370
then down sample the result
of that by a factor of 3.
598
00:38:33,370 --> 00:38:37,290
And what that would give us is
a sampling rate conversion,
599
00:38:37,290 --> 00:38:42,750
overall, of 3/2, or a sampling
period conversion of 3/2.
600
00:38:42,750 --> 00:38:45,730
And more generally, what you
could think of is how you
601
00:38:45,730 --> 00:38:51,240
might do this if, in general,
the relationship between the
602
00:38:51,240 --> 00:38:54,090
input and output sampling
periods was some rational
603
00:38:54,090 --> 00:38:56,190
number p/q.
604
00:38:56,190 --> 00:39:00,350
And so in fact, in many systems,
in hardware systems
605
00:39:00,350 --> 00:39:04,370
related to digital audio, very
often the sampling rate
606
00:39:04,370 --> 00:39:07,160
conversion, most typically the
sampling rate conversion, is
607
00:39:07,160 --> 00:39:12,570
done through a process of up
sampling or interpolating and
608
00:39:12,570 --> 00:39:15,280
then down sampling by
some other amount.
609
00:39:15,280 --> 00:39:18,470
610
00:39:18,470 --> 00:39:23,970
Now what we've seen, what we
talked about in a set of
611
00:39:23,970 --> 00:39:29,830
lectures, is the concepts
of sampling a signal.
612
00:39:29,830 --> 00:39:32,640
And what we've seen is that the
signal can be represented
613
00:39:32,640 --> 00:39:35,140
by samples under certain
conditions.
614
00:39:35,140 --> 00:39:38,240
And the sampling that we've been
talking about is sampling
615
00:39:38,240 --> 00:39:39,020
in the time domain.
616
00:39:39,020 --> 00:39:41,920
And we've done that for
continuous time and we've done
617
00:39:41,920 --> 00:39:45,150
it for discrete time.
618
00:39:45,150 --> 00:39:49,450
Now we know that there is some
type of duality both
619
00:39:49,450 --> 00:39:52,600
continuous time and discrete
time, some type of duality,
620
00:39:52,600 --> 00:39:55,240
between the time domain
and frequency domain.
621
00:39:55,240 --> 00:40:01,220
And so, as you can imagine, we
can also talk about sampling
622
00:40:01,220 --> 00:40:07,720
in the frequency domain and
expect that, more or less, the
623
00:40:07,720 --> 00:40:11,640
kinds of properties and analysis
will be similar to
624
00:40:11,640 --> 00:40:15,400
those related to sampling
in the time domain.
625
00:40:15,400 --> 00:40:21,020
Well I want to talk just briefly
about that and leave
626
00:40:21,020 --> 00:40:24,810
the more detailed discussion
to the text and
627
00:40:24,810 --> 00:40:26,550
video course manual.
628
00:40:26,550 --> 00:40:29,980
But let me indicate, for
example, one context in which
629
00:40:29,980 --> 00:40:32,370
frequency domain sampling
is important.
630
00:40:32,370 --> 00:40:38,220
Suppose that you have a signal
and what you'd like to measure
631
00:40:38,220 --> 00:40:41,770
is its Fourier transform,
its spectrum.
632
00:40:41,770 --> 00:40:46,130
Well of course, if you want to
measure it or calculate it,
633
00:40:46,130 --> 00:40:49,150
you can never do that exactly
at every single frequency.
634
00:40:49,150 --> 00:40:50,630
There are too many frequencies,
namely, an
635
00:40:50,630 --> 00:40:52,280
infinite number of them.
636
00:40:52,280 --> 00:40:55,670
And so, in fact, all that you
can really calculate or
637
00:40:55,670 --> 00:41:00,920
measure is the Fourier transform
at a set of sample
638
00:41:00,920 --> 00:41:02,170
frequencies.
639
00:41:02,170 --> 00:41:06,570
So essentially, if you are going
to look at a spectrum,
640
00:41:06,570 --> 00:41:09,950
continuous time or discrete
time, you can only really look
641
00:41:09,950 --> 00:41:11,060
at samples.
642
00:41:11,060 --> 00:41:15,210
And a reasonable question to
ask, then, is when does a set
643
00:41:15,210 --> 00:41:19,500
of samples in fact tell you
everything that there is to
644
00:41:19,500 --> 00:41:23,450
know about the Fourier
transform.
645
00:41:23,450 --> 00:41:27,100
That, and the answer to that, is
very closely related to the
646
00:41:27,100 --> 00:41:31,140
concept of frequency
domain sampling.
647
00:41:31,140 --> 00:41:33,770
Well, frequency domain sampling,
just to kind of
648
00:41:33,770 --> 00:41:37,900
introduce the topic, corresponds
and can be
649
00:41:37,900 --> 00:41:44,300
analyzed in terms doing
modulation in the frequency
650
00:41:44,300 --> 00:41:48,280
domain, very much like the
modulation that we carried out
651
00:41:48,280 --> 00:41:51,160
in the time domain for
time domain sampling.
652
00:41:51,160 --> 00:41:57,440
And so we would multiply the
Fourier transform of the
653
00:41:57,440 --> 00:42:02,200
signal whose spectrum is
to be sampled by an
654
00:42:02,200 --> 00:42:05,060
impulse train in frequency.
655
00:42:05,060 --> 00:42:10,520
And so shown below is what
might be a representative
656
00:42:10,520 --> 00:42:13,660
spectrum for the input signal.
657
00:42:13,660 --> 00:42:19,550
And the spectrum, then for the
signal associated with the
658
00:42:19,550 --> 00:42:23,760
frequency domain sampling,
consists of multiplying the
659
00:42:23,760 --> 00:42:27,670
frequency domain by this
impulse train.
660
00:42:27,670 --> 00:42:32,950
Or correspondingly, the Fourier
transform of the
661
00:42:32,950 --> 00:42:39,430
resulting signal is an impulse
train in frequency with an
662
00:42:39,430 --> 00:42:42,670
envelope which is the
original spectrum
663
00:42:42,670 --> 00:42:45,420
that we were sampling.
664
00:42:45,420 --> 00:42:48,360
Well, this of course is
what we would do in
665
00:42:48,360 --> 00:42:49,620
the frequency domain.
666
00:42:49,620 --> 00:42:52,560
It's modulation by
an impulse train.
667
00:42:52,560 --> 00:42:55,690
What does this mean in
the time domain?
668
00:42:55,690 --> 00:42:57,070
Well, let's see.
669
00:42:57,070 --> 00:42:59,690
Multiplication in the time
domain is convolution in the
670
00:42:59,690 --> 00:43:01,220
frequency domain.
671
00:43:01,220 --> 00:43:04,210
Convolution in the frequency
domain is multiplication--
672
00:43:04,210 --> 00:43:04,880
I'm sorry.
673
00:43:04,880 --> 00:43:07,700
Multiplication in the frequency
domain, then, is
674
00:43:07,700 --> 00:43:09,660
convolution in the
time domain.
675
00:43:09,660 --> 00:43:12,490
And in fact, the process
in the time domain is a
676
00:43:12,490 --> 00:43:14,120
convolution process.
677
00:43:14,120 --> 00:43:22,020
Namely, the time domain signal
is replicated at integer
678
00:43:22,020 --> 00:43:26,840
amounts of a particular time
associated with the spacing in
679
00:43:26,840 --> 00:43:29,050
frequency under which
we're doing the
680
00:43:29,050 --> 00:43:31,340
frequency domain sampling.
681
00:43:31,340 --> 00:43:39,590
So in fact, if we look at this
in the time domain, the
682
00:43:39,590 --> 00:43:46,170
resulting picture corresponds
to an original signal whose
683
00:43:46,170 --> 00:43:50,150
spectrum or Fourier transform
we've sampled.
684
00:43:50,150 --> 00:43:53,980
And a consequence of the
sampling is that the
685
00:43:53,980 --> 00:43:58,220
associated time domain signal
is just like the original
686
00:43:58,220 --> 00:44:02,490
signal, but periodically
replicated, in time now, not
687
00:44:02,490 --> 00:44:07,020
frequency, but in time, at
integer multiples of 2 pi
688
00:44:07,020 --> 00:44:12,720
divided by the spectral sampling
interval omega 0.
689
00:44:12,720 --> 00:44:18,050
And so this then is the time
function associated with the
690
00:44:18,050 --> 00:44:20,630
sample frequency function.
691
00:44:20,630 --> 00:44:24,640
Now, that's not surprising
because what we've done is
692
00:44:24,640 --> 00:44:27,750
generated an impulse
train and frequency
693
00:44:27,750 --> 00:44:29,660
with a certain envelope.
694
00:44:29,660 --> 00:44:32,460
We know that an impulse train
in frequency is the Fourier
695
00:44:32,460 --> 00:44:36,340
transform of a periodic
time function.
696
00:44:36,340 --> 00:44:39,180
And so in fact, we have a
periodic time function.
697
00:44:39,180 --> 00:44:43,240
We also know that the envelope
of those impulses--
698
00:44:43,240 --> 00:44:46,500
we know this from way back when
we talked about Fourier
699
00:44:46,500 --> 00:44:47,670
transforms--
700
00:44:47,670 --> 00:44:49,560
the envelope, in fact,
is the Fourier
701
00:44:49,560 --> 00:44:51,060
transform of one period.
702
00:44:51,060 --> 00:44:54,400
And so all of this, of course,
fits together as it should in
703
00:44:54,400 --> 00:44:56,620
a consistent way.
704
00:44:56,620 --> 00:45:02,340
Now given that we have this
periodic time function whose
705
00:45:02,340 --> 00:45:05,460
Fourier transform is the samples
in the frequency
706
00:45:05,460 --> 00:45:10,250
domain, how do we get back the
original time function?
707
00:45:10,250 --> 00:45:17,070
Well, with time domain sampling,
what we did was to
708
00:45:17,070 --> 00:45:21,370
multiply in the frequency domain
by a gate, or window,
709
00:45:21,370 --> 00:45:24,060
to extract that part
of the spectrum.
710
00:45:24,060 --> 00:45:29,080
What we do here is exactly the
same thing, namely multiply in
711
00:45:29,080 --> 00:45:34,650
the time domain by a time window
which extracts just one
712
00:45:34,650 --> 00:45:38,280
period of this periodic signal,
which would then give
713
00:45:38,280 --> 00:45:42,930
us back the original signal
that we started with.
714
00:45:42,930 --> 00:45:48,550
Now also let's keep in mind,
going back to this time
715
00:45:48,550 --> 00:45:52,400
function and the relationship
between them, then again,
716
00:45:52,400 --> 00:45:56,090
there is the potential, if this
time function is too long
717
00:45:56,090 --> 00:46:00,780
in relation to 2 pi divided by
omega 0, there's the potential
718
00:46:00,780 --> 00:46:02,420
for these to overlap.
719
00:46:02,420 --> 00:46:06,880
And so what this means is that,
in fact, what we can end
720
00:46:06,880 --> 00:46:11,320
up with, if the sample spacing
and the frequency is not small
721
00:46:11,320 --> 00:46:15,100
enough, what we can end up
with is an overlap in the
722
00:46:15,100 --> 00:46:17,530
replication in the
time domain.
723
00:46:17,530 --> 00:46:21,220
And what that corresponds to
and what it's called is, in
724
00:46:21,220 --> 00:46:23,200
fact, time aliasing.
725
00:46:23,200 --> 00:46:27,590
So we can have time aliasing
with frequency domain sampling
726
00:46:27,590 --> 00:46:30,440
just as we can have
frequency aliasing
727
00:46:30,440 --> 00:46:33,440
with time domain sampling.
728
00:46:33,440 --> 00:46:37,730
Finally, let me just indicate
very quickly that, although
729
00:46:37,730 --> 00:46:41,780
we're not going through this in
any detail, the same basic
730
00:46:41,780 --> 00:46:44,820
idea applies in discrete time.
731
00:46:44,820 --> 00:46:50,280
Namely, if we have a discrete
time signal and if the
732
00:46:50,280 --> 00:46:55,870
discrete time signal is a finite
length, if we sample
733
00:46:55,870 --> 00:47:01,260
its Fourier transform, the time
function associated with
734
00:47:01,260 --> 00:47:05,980
those samples is a periodic
replication.
735
00:47:05,980 --> 00:47:11,270
And we can now extract, from
this periodic signal, the
736
00:47:11,270 --> 00:47:16,080
original signal by multiplying
by an appropriate time window,
737
00:47:16,080 --> 00:47:19,660
the product of that giving
us the reconstructed time
738
00:47:19,660 --> 00:47:21,630
function as I indicate below.
739
00:47:21,630 --> 00:47:24,730
740
00:47:24,730 --> 00:47:29,100
So we've now seen a little bit
of the notion of frequency
741
00:47:29,100 --> 00:47:31,690
domain sampling, as well as
time domain sampling.
742
00:47:31,690 --> 00:47:34,440
And let me stress that, although
I haven't gone into
743
00:47:34,440 --> 00:47:38,800
this in a lot of detail,
it's important.
744
00:47:38,800 --> 00:47:40,510
It's used very often.
745
00:47:40,510 --> 00:47:43,020
It's naturally important
to understand it.
746
00:47:43,020 --> 00:47:46,550
But, in fact, there is so much
duality between the time
747
00:47:46,550 --> 00:47:49,460
domain and frequency domain,
that a thorough understanding
748
00:47:49,460 --> 00:47:53,270
of time domain sampling just
naturally leads to a thorough
749
00:47:53,270 --> 00:47:55,520
understanding of frequency
domain sampling.
750
00:47:55,520 --> 00:47:58,810
751
00:47:58,810 --> 00:48:01,680
Now we've talked a lot
about sampling.
752
00:48:01,680 --> 00:48:06,840
And this now concludes our
discussion of sampling.
753
00:48:06,840 --> 00:48:10,170
I've stressed many times in the
lectures associated with
754
00:48:10,170 --> 00:48:16,020
this that sampling is a very
important topic in the context
755
00:48:16,020 --> 00:48:19,440
of our whole discussion, in part
because it forms such an
756
00:48:19,440 --> 00:48:22,520
important bridge between
continuous time and discrete
757
00:48:22,520 --> 00:48:23,890
time ideas.
758
00:48:23,890 --> 00:48:27,150
And your picture now should kind
of be a global one that
759
00:48:27,150 --> 00:48:31,060
sees how continuous time and
discrete time fit together,
760
00:48:31,060 --> 00:48:33,395
not just analytically,
but also practically.
761
00:48:33,395 --> 00:48:36,510
762
00:48:36,510 --> 00:48:42,560
Beginning in the next lecture,
what I will introduce is the
763
00:48:42,560 --> 00:48:46,740
Laplace transform and, beyond
that, the Z transform.
764
00:48:46,740 --> 00:48:51,460
And what those will correspond
to are generalizations of the
765
00:48:51,460 --> 00:48:52,930
Fourier transform.
766
00:48:52,930 --> 00:48:55,730
So we now want to turn our
attention back to some
767
00:48:55,730 --> 00:48:59,270
analytical tools, in particular
developing some
768
00:48:59,270 --> 00:49:03,830
generalizations of the Fourier
transform in both continuous
769
00:49:03,830 --> 00:49:05,780
time and discrete time.
770
00:49:05,780 --> 00:49:11,140
And what we'll see is that those
generalizations provide
771
00:49:11,140 --> 00:49:16,510
us with considerably enhanced
flexibility in dealing with
772
00:49:16,510 --> 00:49:21,210
and analyzing both signals and
linear time invariant systems.
773
00:49:21,210 --> 00:49:22,460
Thank you.
774
00:49:22,460 --> 00:49:23,541