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[MUSIC PLAYING]
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PROFESSOR: Last time we began
the development of the
10
00:00:58,560 --> 00:01:01,430
discrete-time Fourier
transform.
11
00:01:01,430 --> 00:01:06,350
And just as with the
continuous-time case, we first
12
00:01:06,350 --> 00:01:08,710
treated the notion of
periodic signals.
13
00:01:08,710 --> 00:01:11,060
This led to the Fourier
series.
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00:01:11,060 --> 00:01:15,350
And then we generalized that to
the Fourier transform, and
15
00:01:15,350 --> 00:01:18,080
finally incorporated within the
framework of the Fourier
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00:01:18,080 --> 00:01:23,070
transform both aperiodic
and periodic signals.
17
00:01:23,070 --> 00:01:27,820
In today's lecture, what I'd
like to do is expand on some
18
00:01:27,820 --> 00:01:30,940
of the properties of the
Fourier transform, and
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00:01:30,940 --> 00:01:32,875
indicate how those properties
are used for
20
00:01:32,875 --> 00:01:34,770
a variety of things.
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00:01:34,770 --> 00:01:39,470
Well, let's begin by reviewing
the Fourier transform as we
22
00:01:39,470 --> 00:01:41,310
developed it last time.
23
00:01:41,310 --> 00:01:45,410
It, of course, involves a
synthesis equation and an
24
00:01:45,410 --> 00:01:47,040
analysis equation.
25
00:01:47,040 --> 00:01:51,340
The synthesis equation
expressing x of n, the
26
00:01:51,340 --> 00:01:55,090
sequence, in terms of the
Fourier transform, and the
27
00:01:55,090 --> 00:01:58,880
analysis equation telling us
how to obtain the Fourier
28
00:01:58,880 --> 00:02:02,890
transform from the original
sequence.
29
00:02:02,890 --> 00:02:07,480
And I draw your attention again
to the basic point that
30
00:02:07,480 --> 00:02:11,650
the synthesis equation
essentially corresponds to
31
00:02:11,650 --> 00:02:17,660
decomposing the sequence as a
linear combination of complex
32
00:02:17,660 --> 00:02:21,590
exponentials with amplitudes
that are, in effect,
33
00:02:21,590 --> 00:02:25,820
proportional to the
Fourier transform.
34
00:02:25,820 --> 00:02:30,290
Now, the discrete-time Fourier
transform, just as the
35
00:02:30,290 --> 00:02:34,380
continuous-time Fourier
transform, has a number of
36
00:02:34,380 --> 00:02:37,050
important and useful
properties.
37
00:02:37,050 --> 00:02:41,600
Of course, as I stressed last
time, it's a function of a
38
00:02:41,600 --> 00:02:44,140
continuous variable.
39
00:02:44,140 --> 00:02:48,870
And it's also a complex-valued
function, which means that
40
00:02:48,870 --> 00:02:51,680
when we represent it in
general it requires a
41
00:02:51,680 --> 00:02:56,860
representation in terms of its
real part and imaginary part,
42
00:02:56,860 --> 00:03:00,540
or in terms of magnitude
and angle.
43
00:03:00,540 --> 00:03:06,360
Also, as I indicated last time,
the Fourier transform is
44
00:03:06,360 --> 00:03:12,600
a periodic function of
frequency, and the periodicity
45
00:03:12,600 --> 00:03:16,030
is with a period of 2 pi.
46
00:03:16,030 --> 00:03:20,750
And so it says, in effect, that
the Fourier transform, if
47
00:03:20,750 --> 00:03:24,110
we replace the frequency
variable by an integer
48
00:03:24,110 --> 00:03:28,580
multiple of 2 pi, the
function repeats.
49
00:03:28,580 --> 00:03:34,680
And I stress again that the
underlying basis for this
50
00:03:34,680 --> 00:03:39,080
periodicity property is the
fact that it's the set of
51
00:03:39,080 --> 00:03:41,560
complex exponentials that are
52
00:03:41,560 --> 00:03:44,190
inherently periodic in frequency.
53
00:03:44,190 --> 00:03:47,040
And so, of course, any
representation using them
54
00:03:47,040 --> 00:03:52,660
would, in effect, generate
a periodicity with
55
00:03:52,660 --> 00:03:53,950
this period of 2 pi.
56
00:03:56,780 --> 00:04:01,320
Just as in continuous time,
the Fourier transform has
57
00:04:01,320 --> 00:04:03,560
important symmetry properties.
58
00:04:03,560 --> 00:04:08,060
And in particular, if the
sequence x sub n is
59
00:04:08,060 --> 00:04:12,130
real-valued, then the
Fourier transform
60
00:04:12,130 --> 00:04:14,680
is conjugate symmetric.
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00:04:14,680 --> 00:04:19,420
In other words, if we replace
omega by minus omega, that's
62
00:04:19,420 --> 00:04:22,930
equivalent to applying
complex conjugation
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00:04:22,930 --> 00:04:25,000
to the Fourier transform.
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00:04:25,000 --> 00:04:30,780
And as a consequence of this
conjugate symmetry, this
65
00:04:30,780 --> 00:04:35,880
results in a symmetry in the
real part that is an even
66
00:04:35,880 --> 00:04:41,430
symmetry, or the magnitude has
an even symmetry, whereas the
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00:04:41,430 --> 00:04:47,590
imaginary part or the phase
angle are both odd symmetric.
68
00:04:47,590 --> 00:04:50,850
And these are symmetry
properties, again, that are
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00:04:50,850 --> 00:04:53,480
identical to the symmetry
properties that we saw in
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00:04:53,480 --> 00:04:55,230
continuous time.
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00:04:55,230 --> 00:04:58,820
Well, let's see this in the
context of an example that we
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00:04:58,820 --> 00:05:01,940
worked last time and that we'll
want to draw attention
73
00:05:01,940 --> 00:05:04,960
to in reference to several
issues as this
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00:05:04,960 --> 00:05:06,730
lecture goes along.
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00:05:06,730 --> 00:05:11,890
And that is the Fourier
transform of a real damped
76
00:05:11,890 --> 00:05:13,330
exponential.
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00:05:13,330 --> 00:05:19,330
So the sequence that we are
talking about is a to the n u
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00:05:19,330 --> 00:05:23,590
of n, and let's consider
a to be positive.
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00:05:23,590 --> 00:05:27,060
We saw last time that the
Fourier transform for this
80
00:05:27,060 --> 00:05:31,490
sequence algebraically
is of this form.
81
00:05:31,490 --> 00:05:35,860
And if we look at its magnitude
and angle, the
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00:05:35,860 --> 00:05:40,110
magnitude I show here.
83
00:05:40,110 --> 00:05:43,120
And the magnitude, as
we see, has the
84
00:05:43,120 --> 00:05:45,130
properties that we indicated.
85
00:05:45,130 --> 00:05:48,400
It is an even function
of frequency.
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00:05:48,400 --> 00:05:52,150
Of course, it's a function
of a continuous variable.
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00:05:52,150 --> 00:05:55,400
And it, in addition,
is periodic with a
88
00:05:55,400 --> 00:05:58,300
period of two pi.
89
00:05:58,300 --> 00:06:02,050
On the other hand, if we look
at the phase angle below it,
90
00:06:02,050 --> 00:06:06,310
the phase angle has a symmetry
which is odd symmetric.
91
00:06:06,310 --> 00:06:09,640
And that's indicated clearly
in this picture.
92
00:06:09,640 --> 00:06:12,100
And of course, in addition to
being odd symmetric, it
93
00:06:12,100 --> 00:06:16,700
naturally has to be, again, a
periodic function of frequency
94
00:06:16,700 --> 00:06:20,080
with a period of 2 pi.
95
00:06:20,080 --> 00:06:22,630
OK, so we have some symmetry
properties.
96
00:06:22,630 --> 00:06:25,220
We have this inherent
periodicity in the Fourier
97
00:06:25,220 --> 00:06:31,260
transform, which I'm stressing
very heavily because it forms
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00:06:31,260 --> 00:06:34,060
the basic difference between
continuous time
99
00:06:34,060 --> 00:06:36,170
and discrete time.
100
00:06:36,170 --> 00:06:39,420
In addition to these properties
of the Fourier
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00:06:39,420 --> 00:06:44,940
transform, there are a number
of other properties that are
102
00:06:44,940 --> 00:06:48,820
particularly useful in the
manipulation of the Fourier
103
00:06:48,820 --> 00:06:53,050
transform, and, in fact, in
using the Fourier transform
104
00:06:53,050 --> 00:06:57,550
to, for example, analyze systems
represented by linear
105
00:06:57,550 --> 00:07:00,660
constant coefficient difference
equations.
106
00:07:00,660 --> 00:07:05,410
There in the text is a longer
list of properties, but let me
107
00:07:05,410 --> 00:07:09,490
just draw your attention
to several of them.
108
00:07:09,490 --> 00:07:13,210
One is the time shifting
property.
109
00:07:13,210 --> 00:07:19,620
And the time shifting property
tells us that if x of omega is
110
00:07:19,620 --> 00:07:24,870
the Fourier transform of x of n,
then the Fourier transform
111
00:07:24,870 --> 00:07:30,090
of x of n shifted in time is
that same Fourier transform
112
00:07:30,090 --> 00:07:34,730
multiplied by this factor,
which is a
113
00:07:34,730 --> 00:07:36,690
linear phase factor.
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00:07:36,690 --> 00:07:41,670
So time shifting introduces
a linear phase term.
115
00:07:41,670 --> 00:07:46,410
And, by the way, recall that in
the continuous-time case we
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00:07:46,410 --> 00:07:49,530
had a similar situation, namely
that a time shift
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00:07:49,530 --> 00:07:53,050
corresponded to a
linear phase.
118
00:07:53,050 --> 00:07:57,260
There also is a dual to the time
shifting property, which
119
00:07:57,260 --> 00:08:01,000
is referred to as the frequency
shifting property,
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00:08:01,000 --> 00:08:06,520
which tells us that if we
multiply a time function by a
121
00:08:06,520 --> 00:08:09,600
complex exponential,
that, in effect,
122
00:08:09,600 --> 00:08:13,030
generates a frequency shift.
123
00:08:13,030 --> 00:08:17,000
And we'll see this frequency
shifting property surface in a
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00:08:17,000 --> 00:08:20,310
slightly different way shortly,
when we talk about
125
00:08:20,310 --> 00:08:24,840
the modulation property in
the discrete-time case.
126
00:08:24,840 --> 00:08:28,080
Another important property that
we'll want to make use of
127
00:08:28,080 --> 00:08:31,390
shortly is linearity, which
follows in a very
128
00:08:31,390 --> 00:08:36,860
straightforward way from the
Fourier transform definition.
129
00:08:36,860 --> 00:08:42,220
And the linearity property says
simply that the Fourier
130
00:08:42,220 --> 00:08:46,510
transform of a sum, or linear
combination, is the same
131
00:08:46,510 --> 00:08:49,020
linear combination of the
Fourier transforms.
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00:08:49,020 --> 00:08:52,750
Again, that's a property that
we saw in continuous time.
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00:08:52,750 --> 00:08:58,570
And, also, among other
properties there is a
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00:08:58,570 --> 00:09:01,790
Parseval's relation for the
discrete-time case that in
135
00:09:01,790 --> 00:09:05,110
effect says something similar
to continuous time,
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00:09:05,110 --> 00:09:10,750
specifically that the energy
in the sequence is
137
00:09:10,750 --> 00:09:15,340
proportional to the energy in
the Fourier transform, the
138
00:09:15,340 --> 00:09:17,370
energy over one period.
139
00:09:17,370 --> 00:09:20,820
Or, said another way, in fact,
or another way that it can be
140
00:09:20,820 --> 00:09:23,760
said, is that the energy
in the time domain is
141
00:09:23,760 --> 00:09:26,070
proportional to the
power in this
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00:09:26,070 --> 00:09:30,060
periodic Fourier transform.
143
00:09:30,060 --> 00:09:33,510
OK, so these are some
of the properties.
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00:09:33,510 --> 00:09:37,750
And, as I indicated, parallel
somewhat properties that we
145
00:09:37,750 --> 00:09:40,140
saw in continuous time.
146
00:09:40,140 --> 00:09:44,730
Two additional properties that
will play important roles in
147
00:09:44,730 --> 00:09:48,530
discrete time just as they did
in continuous time are the
148
00:09:48,530 --> 00:09:53,730
convolution property and the
modulation property.
149
00:09:53,730 --> 00:09:57,760
The convolution property is the
property that tells us how
150
00:09:57,760 --> 00:10:03,100
to relate the Fourier transform
of the convolution
151
00:10:03,100 --> 00:10:06,990
of two sequences to the Fourier
transforms of the
152
00:10:06,990 --> 00:10:08,940
individual sequences.
153
00:10:08,940 --> 00:10:12,240
And, not surprisingly,
what happens--
154
00:10:12,240 --> 00:10:15,530
and this can be demonstrated
algebraically--
155
00:10:15,530 --> 00:10:20,390
not surprisingly, the Fourier
transform of the convolution
156
00:10:20,390 --> 00:10:24,960
is simply the product of
the Fourier transforms.
157
00:10:24,960 --> 00:10:30,460
So, Fourier transform maps
convolution in the time domain
158
00:10:30,460 --> 00:10:33,650
to multiplication in the
frequency domain.
159
00:10:33,650 --> 00:10:37,300
Now convolution, of course,
arises in the context of
160
00:10:37,300 --> 00:10:39,610
linear time-invariant systems.
161
00:10:39,610 --> 00:10:42,120
In particular, if we have
a system with an impulse
162
00:10:42,120 --> 00:10:47,710
response h of n, input x of n,
the output is the convolution.
163
00:10:47,710 --> 00:10:51,860
The convolution property then
tells us that in the frequency
164
00:10:51,860 --> 00:10:57,780
domain, the Fourier transform is
the product of the Fourier
165
00:10:57,780 --> 00:11:01,420
transform of the impulse
response and the Fourier
166
00:11:01,420 --> 00:11:05,120
transform of the input.
167
00:11:05,120 --> 00:11:10,560
Now we also saw and have talked
about a relationship
168
00:11:10,560 --> 00:11:14,950
between the Fourier transform,
the impulse response, and what
169
00:11:14,950 --> 00:11:22,250
we call the frequency response
in the context of the response
170
00:11:22,250 --> 00:11:25,840
of a system to a complex
exponential.
171
00:11:25,840 --> 00:11:29,030
Specifically, complex
exponentials are
172
00:11:29,030 --> 00:11:33,210
eigenfunctions of linear
time-invariant systems.
173
00:11:33,210 --> 00:11:37,810
One of these into the system
gives us, as an output, a
174
00:11:37,810 --> 00:11:42,070
complex exponential with the
same complex frequency
175
00:11:42,070 --> 00:11:45,920
multiplied by what we refer
to as the eigenvalue.
176
00:11:45,920 --> 00:11:52,030
And as you saw in the video
course manual, this
177
00:11:52,030 --> 00:11:56,010
eigenvalue, this constant,
multiplier on the exponential
178
00:11:56,010 --> 00:12:00,530
is, in fact, the Fourier
transform of the impulse
179
00:12:00,530 --> 00:12:05,520
response evaluated at
that frequency.
180
00:12:05,520 --> 00:12:10,700
Now, we saw exactly the same
statement in continuous time.
181
00:12:10,700 --> 00:12:13,280
And, in fact, we used
that statement--
182
00:12:13,280 --> 00:12:17,860
the frequency response
interpretation of the Fourier
183
00:12:17,860 --> 00:12:20,040
transform, the impulse
response--
184
00:12:20,040 --> 00:12:24,710
we use that to motivate an
intuitive interpretation of
185
00:12:24,710 --> 00:12:26,550
the convolution property.
186
00:12:26,550 --> 00:12:30,460
Now, formally the convolution
property can be developed by
187
00:12:30,460 --> 00:12:33,840
taking the convolution sum,
applying the Fourier transform
188
00:12:33,840 --> 00:12:38,240
sum to it, doing the appropriate
substitution of
189
00:12:38,240 --> 00:12:40,810
variables, interchanging order
of summations, et cetera, and
190
00:12:40,810 --> 00:12:44,800
all the algebra works out to
show that it's a product.
191
00:12:44,800 --> 00:12:48,610
But as I stressed when we
discussed this with continuous
192
00:12:48,610 --> 00:12:50,550
time, the interpretation--
193
00:12:50,550 --> 00:12:52,160
the underlying interpretation--
194
00:12:52,160 --> 00:12:55,040
is particularly important
to understand.
195
00:12:55,040 --> 00:12:58,160
So let me review it again in
the discrete-time case, and
196
00:12:58,160 --> 00:13:01,080
it's exactly the same for
discrete time or for
197
00:13:01,080 --> 00:13:03,330
continuous time.
198
00:13:03,330 --> 00:13:10,100
Specifically, the argument was
that the Fourier transform of
199
00:13:10,100 --> 00:13:14,760
a sequence or signal corresponds
to decomposing it
200
00:13:14,760 --> 00:13:18,330
into a linear combination
of complex exponentials.
201
00:13:18,330 --> 00:13:21,310
What's the amplitude of those
complex exponentials?
202
00:13:21,310 --> 00:13:25,990
It's basically proportional
to the Fourier transform.
203
00:13:25,990 --> 00:13:29,290
If we think of pushing through
the system that linear
204
00:13:29,290 --> 00:13:35,220
combination, then each of those
complex exponentials
205
00:13:35,220 --> 00:13:40,540
gets the amplitude modified, or
multiplied, by the Fourier
206
00:13:40,540 --> 00:13:42,500
transform of--
207
00:13:42,500 --> 00:13:44,110
by the frequency response--
208
00:13:44,110 --> 00:13:46,670
which we saw is the
Fourier transform
209
00:13:46,670 --> 00:13:49,130
of the impulse response.
210
00:13:49,130 --> 00:13:54,170
So the amplitudes of the output
complex exponentials is
211
00:13:54,170 --> 00:13:58,240
then the amplitudes of the input
complex exponentials
212
00:13:58,240 --> 00:14:01,470
multiplied by the frequency
response.
213
00:14:01,470 --> 00:14:06,240
And the Fourier transform of the
output, in effect, is an
214
00:14:06,240 --> 00:14:11,450
expression expressing the
summation, or integration, of
215
00:14:11,450 --> 00:14:14,790
the output as a linear
combination of all of these
216
00:14:14,790 --> 00:14:17,400
exponentials with the
appropriate complex
217
00:14:17,400 --> 00:14:19,580
amplitudes.
218
00:14:19,580 --> 00:14:23,720
So, it's important, in thinking
about the convolution
219
00:14:23,720 --> 00:14:29,880
property, to think about it in
terms of nothing more than the
220
00:14:29,880 --> 00:14:34,740
fact that we've decomposed the
input, and we're now modifying
221
00:14:34,740 --> 00:14:38,410
separately through
multiplication, through
222
00:14:38,410 --> 00:14:43,010
scaling, the amplitudes
of each of the complex
223
00:14:43,010 --> 00:14:44,260
exponential components.
224
00:14:47,230 --> 00:14:50,780
Now what we saw in continuous
time is that this
225
00:14:50,780 --> 00:14:56,390
interpretation and the
convolution property led to an
226
00:14:56,390 --> 00:15:00,640
important concept, namely the
concept of filtering.
227
00:15:00,640 --> 00:15:04,140
Kind of the idea that if we
decompose the input as a
228
00:15:04,140 --> 00:15:08,990
linear combination of complex
exponentials, we can
229
00:15:08,990 --> 00:15:12,320
separately attenuate or
amplify each of those
230
00:15:12,320 --> 00:15:13,910
components.
231
00:15:13,910 --> 00:15:19,190
And, in fact, we could exactly
pass some set of frequencies
232
00:15:19,190 --> 00:15:23,160
and totally eliminate other
set of frequencies.
233
00:15:23,160 --> 00:15:29,430
So, again, just as in continuous
time, we can talk
234
00:15:29,430 --> 00:15:32,130
about an ideal filter.
235
00:15:32,130 --> 00:15:37,820
And what I show here is the
frequency response of an ideal
236
00:15:37,820 --> 00:15:39,990
lowpass filter.
237
00:15:39,990 --> 00:15:45,050
The ideal lowpass filter, of
course, passes exactly, with a
238
00:15:45,050 --> 00:15:51,800
gain of 1, frequencies around
0, and eliminates totally
239
00:15:51,800 --> 00:15:53,050
other frequencies.
240
00:15:55,030 --> 00:15:58,480
However, an important
distinction here between
241
00:15:58,480 --> 00:16:02,530
continuous time and discrete
time is the fact that, whereas
242
00:16:02,530 --> 00:16:06,540
in continuous time when we
talked about an ideal filter,
243
00:16:06,540 --> 00:16:09,110
we passed a band of frequencies
and totally
244
00:16:09,110 --> 00:16:12,070
eliminated everything else
out to infinity.
245
00:16:12,070 --> 00:16:15,070
In the discrete time case, the
246
00:16:15,070 --> 00:16:18,210
frequency response is periodic.
247
00:16:18,210 --> 00:16:22,400
So, obviously, the frequency
response must periodically
248
00:16:22,400 --> 00:16:24,430
repeat for the lowpass filter.
249
00:16:24,430 --> 00:16:26,700
And in fact we see that here.
250
00:16:26,700 --> 00:16:31,570
If we look at the lowpass
filter, then we've eliminated
251
00:16:31,570 --> 00:16:33,240
some frequencies.
252
00:16:33,240 --> 00:16:39,260
But then we pass, of course,
frequencies around 2 pi, and
253
00:16:39,260 --> 00:16:43,160
also frequencies around minus
2 pi, and for that matter
254
00:16:43,160 --> 00:16:46,030
around any multiple of 2 pi.
255
00:16:46,030 --> 00:16:49,770
Although it's important to
recognize that because of the
256
00:16:49,770 --> 00:16:53,310
inherent periodicity of the
complex exponentials, these
257
00:16:53,310 --> 00:16:58,590
frequencies are exactly the
same frequencies as these
258
00:16:58,590 --> 00:16:59,670
frequencies.
259
00:16:59,670 --> 00:17:03,720
So it's lowpass filtering
interpreted in terms of
260
00:17:03,720 --> 00:17:09,180
frequencies over a range
from minus pi to pi.
261
00:17:09,180 --> 00:17:12,670
Well, just as we talk about a
lowpass filter, we can also
262
00:17:12,670 --> 00:17:15,609
talk about a highpass filter.
263
00:17:15,609 --> 00:17:19,500
And a highpass filter, of
course, would pass high
264
00:17:19,500 --> 00:17:21,790
frequencies.
265
00:17:21,790 --> 00:17:24,460
In a continuous-time case,
high frequencies meant
266
00:17:24,460 --> 00:17:28,050
frequencies that go
out to infinity.
267
00:17:28,050 --> 00:17:30,720
In the discrete-time case,
of course, the highest
268
00:17:30,720 --> 00:17:34,800
frequencies we can generate
are frequencies up to pi.
269
00:17:34,800 --> 00:17:39,760
And once our complex
exponentials go past pi, then,
270
00:17:39,760 --> 00:17:42,570
in fact, we start seeing the
lower frequencies again.
271
00:17:42,570 --> 00:17:45,400
Let me indicate what I mean.
272
00:17:45,400 --> 00:17:49,560
If we think in the context of
the lowpass filter, these are
273
00:17:49,560 --> 00:17:51,300
low frequencies.
274
00:17:51,300 --> 00:17:54,030
As we move along the frequency
axis, these become high
275
00:17:54,030 --> 00:17:55,350
frequencies.
276
00:17:55,350 --> 00:17:59,490
And as we move further along the
frequency axis, what we'll
277
00:17:59,490 --> 00:18:03,330
see when we get to, for example,
a frequency of 2 pi
278
00:18:03,330 --> 00:18:09,030
are the same low frequencies
that we see around 0.
279
00:18:09,030 --> 00:18:12,900
In particular then, an ideal
highpass filter in the
280
00:18:12,900 --> 00:18:16,790
discrete-time case would be a
filter that eliminates these
281
00:18:16,790 --> 00:18:20,660
frequencies and passes
frequencies around pi.
282
00:18:23,620 --> 00:18:28,860
OK, so we've seen the
convolution property and its
283
00:18:28,860 --> 00:18:31,760
interpretation in terms
of filtering.
284
00:18:31,760 --> 00:18:36,040
More broadly, the convolution
property in combination with a
285
00:18:36,040 --> 00:18:38,470
number of the other properties
that I introduced, in
286
00:18:38,470 --> 00:18:42,040
particular the time shifting and
linearity property, allows
287
00:18:42,040 --> 00:18:49,290
us to generate or analyze
systems that are described by
288
00:18:49,290 --> 00:18:52,100
linear constant coefficient
difference equations.
289
00:18:52,100 --> 00:18:56,330
And this, again, parallels very
strongly the discussion
290
00:18:56,330 --> 00:19:00,650
we carried out in the
continuous-time case.
291
00:19:00,650 --> 00:19:08,510
In particular, let's think of a
discrete-time system that is
292
00:19:08,510 --> 00:19:09,900
described by a linear constant
293
00:19:09,900 --> 00:19:11,910
coefficient difference equation.
294
00:19:11,910 --> 00:19:15,470
And we'll restrict the initial
conditions on the equation
295
00:19:15,470 --> 00:19:19,340
such that it corresponds to a
linear time-invariant system.
296
00:19:19,340 --> 00:19:23,360
And recall that, in fact, in
our discussion of linear
297
00:19:23,360 --> 00:19:26,810
constant coefficient difference
equations, it is
298
00:19:26,810 --> 00:19:31,230
the condition of initial rest
that-- on the equation--
299
00:19:31,230 --> 00:19:35,240
that guarantees for us that
the system will be causal,
300
00:19:35,240 --> 00:19:38,720
linear, and time invariant.
301
00:19:38,720 --> 00:19:41,790
OK, now let's consider a
first-order difference
302
00:19:41,790 --> 00:19:43,340
equation, a system
described by a
303
00:19:43,340 --> 00:19:44,740
first-order difference equation.
304
00:19:44,740 --> 00:19:46,260
And we've talked about
the solution of
305
00:19:46,260 --> 00:19:48,160
this equation before.
306
00:19:48,160 --> 00:19:51,140
Essentially, we run the
solution recursively.
307
00:19:51,140 --> 00:19:55,370
Let's now consider generating
the solution by taking
308
00:19:55,370 --> 00:19:59,270
advantage of the properties
of the Fourier transform.
309
00:19:59,270 --> 00:20:02,360
Well, just as we did in
continuous time, we can
310
00:20:02,360 --> 00:20:05,870
consider applying the Fourier
transform to both sides of
311
00:20:05,870 --> 00:20:07,210
this equation.
312
00:20:07,210 --> 00:20:10,520
And the Fourier transform
of y of n, of
313
00:20:10,520 --> 00:20:12,680
course, is Y of omega.
314
00:20:12,680 --> 00:20:16,090
And then using the shifting
property, the time shifting
315
00:20:16,090 --> 00:20:21,270
property, the Fourier transform
of y of n minus 1 is
316
00:20:21,270 --> 00:20:25,520
Y of omega multiplied by
e to the minus j omega.
317
00:20:25,520 --> 00:20:30,180
And so we have this, using a
linearity property we can
318
00:20:30,180 --> 00:20:32,840
carry down the scale factor, and
add these two together as
319
00:20:32,840 --> 00:20:34,290
they're added here.
320
00:20:34,290 --> 00:20:39,020
And the Fourier transform
of x of n is X of omega.
321
00:20:39,020 --> 00:20:43,150
Well, we can solve this equation
for the Fourier
322
00:20:43,150 --> 00:20:47,630
transform of the output in terms
of the Fourier transform
323
00:20:47,630 --> 00:20:51,650
of the input and an appropriate
324
00:20:51,650 --> 00:20:53,220
complex scale factor.
325
00:20:53,220 --> 00:20:58,170
And simply solving this
for Y of omega yields
326
00:20:58,170 --> 00:21:00,800
what we have here.
327
00:21:00,800 --> 00:21:04,740
Now what we've used in going
from this point to this point
328
00:21:04,740 --> 00:21:11,750
is both the shifting property
and we've also used the
329
00:21:11,750 --> 00:21:13,000
linearity property.
330
00:21:15,860 --> 00:21:19,940
At this point, we can recognize
that here the
331
00:21:19,940 --> 00:21:22,810
Fourier transform of the output
is the product of the
332
00:21:22,810 --> 00:21:28,860
Fourier transform of the input
and some complex function.
333
00:21:28,860 --> 00:21:33,300
And from the convolution
property, then, that complex
334
00:21:33,300 --> 00:21:38,480
function must in fact correspond
to the frequency
335
00:21:38,480 --> 00:21:43,440
response, or equivalently, the
Fourier transform of the
336
00:21:43,440 --> 00:21:45,620
impulse response.
337
00:21:45,620 --> 00:21:50,460
So if we want to determine the
Fourier transform of the--
338
00:21:50,460 --> 00:21:55,170
or the impulse response of the
system, let's say for example,
339
00:21:55,170 --> 00:21:59,980
then it becomes a matter of
having identified the Fourier
340
00:21:59,980 --> 00:22:02,080
transform of the impulse
response, which is the
341
00:22:02,080 --> 00:22:03,500
frequency response.
342
00:22:03,500 --> 00:22:05,800
We now want to inverse
transform to
343
00:22:05,800 --> 00:22:08,600
get the impulse response.
344
00:22:08,600 --> 00:22:10,730
Well, how do we inverse
transform?
345
00:22:10,730 --> 00:22:15,420
Of course, we could do it by
attempting to go through the
346
00:22:15,420 --> 00:22:18,880
synthesis equation for the
Fourier transform.
347
00:22:18,880 --> 00:22:22,520
Or we can do as we did in the
continuous-time case which is
348
00:22:22,520 --> 00:22:25,362
to take advantage
of what we know.
349
00:22:25,362 --> 00:22:30,450
And in particular, we know that
from an example that we
350
00:22:30,450 --> 00:22:36,910
worked before, this is in fact
the Fourier transform of a
351
00:22:36,910 --> 00:22:43,960
sequence which is a to
the n times u of n.
352
00:22:43,960 --> 00:22:46,860
And so, in essence,
by inspection--
353
00:22:46,860 --> 00:22:51,770
very similar to what has gone
on in continuous time--
354
00:22:51,770 --> 00:22:55,900
essentially by inspection, we
can then solve for the impulse
355
00:22:55,900 --> 00:22:58,400
response to the system.
356
00:22:58,400 --> 00:23:02,940
OK, so that procedure follows
very much the kind of
357
00:23:02,940 --> 00:23:06,630
procedure that we've carried
out in continuous time.
358
00:23:06,630 --> 00:23:08,630
And this, of course, is
discussed in more
359
00:23:08,630 --> 00:23:11,480
detail in the text.
360
00:23:11,480 --> 00:23:16,380
Well, let's look at
that example then.
361
00:23:16,380 --> 00:23:21,530
Here we have the impulse
response for that, associated
362
00:23:21,530 --> 00:23:22,960
with the system described
by that
363
00:23:22,960 --> 00:23:26,340
particular difference equation.
364
00:23:26,340 --> 00:23:29,630
And to the right of
that, we have the
365
00:23:29,630 --> 00:23:32,140
associated frequency response.
366
00:23:32,140 --> 00:23:34,150
And one of the things
that we notice--
367
00:23:34,150 --> 00:23:39,080
and this is drawn for a positive
between 0 and 1--
368
00:23:39,080 --> 00:23:44,650
what we notice, in fact, is that
it is an approximation to
369
00:23:44,650 --> 00:23:48,670
a lowpass filter, because it
tends to attenuate the high
370
00:23:48,670 --> 00:23:52,380
frequencies and retain and,
in fact, amplify the low
371
00:23:52,380 --> 00:23:54,520
frequencies.
372
00:23:54,520 --> 00:23:59,840
Now if instead, actually, the
impulse response was such that
373
00:23:59,840 --> 00:24:04,540
we picked a to be negative
between minus 1 and 0, then
374
00:24:04,540 --> 00:24:08,110
the impulse response in the time
domain looks like this.
375
00:24:08,110 --> 00:24:10,380
And the corresponding frequency
376
00:24:10,380 --> 00:24:12,980
response looks like this.
377
00:24:12,980 --> 00:24:15,920
And that becomes an
approximation
378
00:24:15,920 --> 00:24:19,790
to a highpass filter.
379
00:24:19,790 --> 00:24:24,330
So, in fact, a first-order
difference equation, as we
380
00:24:24,330 --> 00:24:28,250
see, has a frequency response,
depending on the value of a,
381
00:24:28,250 --> 00:24:31,930
that either looks approximately
like a lowpass
382
00:24:31,930 --> 00:24:34,540
filter for a positive or
a highpass filter for a
383
00:24:34,540 --> 00:24:38,360
negative, very much like the
first-order differential
384
00:24:38,360 --> 00:24:41,560
equation looked like a
lowpass filter in the
385
00:24:41,560 --> 00:24:43,710
continuous-time case.
386
00:24:43,710 --> 00:24:48,950
And, in fact, what I'd like to
illustrate is the filtering
387
00:24:48,950 --> 00:24:50,340
characteristics--
388
00:24:50,340 --> 00:24:52,680
or an example of filtering--
389
00:24:52,680 --> 00:24:55,600
using a first-order difference
equation.
390
00:24:55,600 --> 00:25:00,400
And the example that I'll
illustrate is a filtering of a
391
00:25:00,400 --> 00:25:04,190
sequence that in fact is
filtered very often for very
392
00:25:04,190 --> 00:25:07,880
practical reasons, namely a
sequence which represents the
393
00:25:07,880 --> 00:25:12,030
Dow Jones Industrial Average
over a fairly long period.
394
00:25:12,030 --> 00:25:17,040
And we'll process the Dow Jones
Industrial Average first
395
00:25:17,040 --> 00:25:19,610
through a first-order difference
equation, where, if
396
00:25:19,610 --> 00:25:23,460
we begin with a equals 0,
then, referring to the
397
00:25:23,460 --> 00:25:27,760
frequency response that we have
here, a equals 0 would
398
00:25:27,760 --> 00:25:30,960
simply be passing
all frequencies.
399
00:25:30,960 --> 00:25:37,140
As a is positive we start to
retain mostly low frequencies,
400
00:25:37,140 --> 00:25:44,370
and the larger a gets, but still
less than 1, the more it
401
00:25:44,370 --> 00:25:46,770
attenuates high frequencies
at the expense of low
402
00:25:46,770 --> 00:25:48,270
frequencies.
403
00:25:48,270 --> 00:25:53,490
So let's watch the filtering,
first with a positive and
404
00:25:53,490 --> 00:25:56,500
we'll see it behave as a lowpass
filter, and then with
405
00:25:56,500 --> 00:26:02,270
a negative and we'll see the
difference equation behaving
406
00:26:02,270 --> 00:26:05,020
as a highpass filter.
407
00:26:05,020 --> 00:26:09,520
What we see here is the Dow
Jones Industrial Average over
408
00:26:09,520 --> 00:26:15,100
roughly a five-year period
from 1927 to 1932.
409
00:26:15,100 --> 00:26:19,470
And, in fact, that big dip in
the middle is the famous stock
410
00:26:19,470 --> 00:26:22,010
market crash of 1929.
411
00:26:22,010 --> 00:26:25,190
And we can see that following
that, in fact, the market
412
00:26:25,190 --> 00:26:29,030
continued a very long
downward trend.
413
00:26:29,030 --> 00:26:34,600
And what we now want to do
is process this through a
414
00:26:34,600 --> 00:26:36,720
difference equation.
415
00:26:36,720 --> 00:26:40,950
Above the Dow Jones average we
show the impulse response of
416
00:26:40,950 --> 00:26:42,020
the difference equation.
417
00:26:42,020 --> 00:26:46,080
Here we've chosen the parameter
a equal to 0.
418
00:26:46,080 --> 00:26:50,570
And the impulse response will
be displayed on an expanded
419
00:26:50,570 --> 00:26:57,700
scale in relation to the scale
of the input and, for that
420
00:26:57,700 --> 00:27:00,300
matter, the scale
of the output.
421
00:27:00,300 --> 00:27:03,900
Now with the impulse response
shown here which is just an
422
00:27:03,900 --> 00:27:09,160
impulse, in fact, the output
shown on the bottom trace is
423
00:27:09,160 --> 00:27:11,880
exactly identical
to the input.
424
00:27:11,880 --> 00:27:16,730
And what we'll want to do now
is increase, first, the
425
00:27:16,730 --> 00:27:22,000
parameter a, and the impulse
response will begin to look
426
00:27:22,000 --> 00:27:27,140
like an exponential with a
duration that's longer and
427
00:27:27,140 --> 00:27:30,670
longer as a moves from 0 to 1.
428
00:27:33,180 --> 00:27:35,850
Correspondingly we'll get more
and more lowpass filtering as
429
00:27:35,850 --> 00:27:39,300
the coefficient a increases
from 0 toward 1.
430
00:27:39,300 --> 00:27:43,340
So now we are increasing
the parameter a.
431
00:27:43,340 --> 00:27:47,300
We see that the bottom trace
in relation to the middle
432
00:27:47,300 --> 00:27:52,480
trace in fact is looking more
and more smoothed or
433
00:27:52,480 --> 00:27:53,700
lowpass-filtered.
434
00:27:53,700 --> 00:27:57,440
And here now we have a fair
amount of smoothing, to the
435
00:27:57,440 --> 00:28:02,480
point where the stock market
crash of 1929 is totally lost.
436
00:28:02,480 --> 00:28:05,470
And in fact I'm sure there are
many people who wish that
437
00:28:05,470 --> 00:28:09,090
through filtering we could, in
fact, have avoided the stock
438
00:28:09,090 --> 00:28:11,940
market crash altogether.
439
00:28:11,940 --> 00:28:19,490
Now, let's decrease a from
1 back towards 0.
440
00:28:19,490 --> 00:28:22,410
And as we do that,
we will be taking
441
00:28:22,410 --> 00:28:25,930
out the lowpass filtering.
442
00:28:25,930 --> 00:28:29,840
And when a finally reaches 0,
the impulse response of the
443
00:28:29,840 --> 00:28:34,180
filter will again be an impulse,
and so the output
444
00:28:34,180 --> 00:28:38,650
will be once again identical
to the input.
445
00:28:38,650 --> 00:28:42,000
And that's where we are now.
446
00:28:42,000 --> 00:28:46,930
All right now we want to
continue to decrease a so that
447
00:28:46,930 --> 00:28:51,160
it becomes negative, moving
from 0 toward minus 1.
448
00:28:51,160 --> 00:28:55,890
And what we will see in that
case is more and more highpass
449
00:28:55,890 --> 00:29:00,550
filtering on the output in
relation to the input.
450
00:29:00,550 --> 00:29:04,500
And this will be particularly
evident in, again, the region
451
00:29:04,500 --> 00:29:07,090
of high frequencies represented
by sharp
452
00:29:07,090 --> 00:29:10,160
transitions which, of course,
the market crash
453
00:29:10,160 --> 00:29:12,870
of 1929 would represent.
454
00:29:12,870 --> 00:29:18,740
So here, now, a is decreasing
toward minus 1.
455
00:29:18,740 --> 00:29:22,140
We see that the high
frequencies, or rapid
456
00:29:22,140 --> 00:29:29,230
variations are emphasized., And
finally, let's move from
457
00:29:29,230 --> 00:29:35,620
minus 1 back towards 0, taking
out the highpass filtering and
458
00:29:35,620 --> 00:29:40,390
ending up with a equal to 0,
corresponding to an impulse
459
00:29:40,390 --> 00:29:42,290
response which is an
impulse, in other
460
00:29:42,290 --> 00:29:44,010
words, an identity system.
461
00:29:44,010 --> 00:29:47,050
And let me stress once again
that the time scale on which
462
00:29:47,050 --> 00:29:51,030
we displayed the impulse
response is an expanded time
463
00:29:51,030 --> 00:29:55,320
scale in relation to the time
scale on which we displayed
464
00:29:55,320 --> 00:29:56,820
the input and the output.
465
00:29:59,650 --> 00:30:04,090
OK, so we see that, in fact,
a first-order difference
466
00:30:04,090 --> 00:30:07,370
equation is a filter.
467
00:30:07,370 --> 00:30:10,270
And, in fact, it's a very
important class of filters,
468
00:30:10,270 --> 00:30:13,320
and it's used very often to
do approximate lowpass and
469
00:30:13,320 --> 00:30:14,570
highpass filtering.
470
00:30:17,200 --> 00:30:24,310
Now, in addition to the
convolution property, another
471
00:30:24,310 --> 00:30:27,420
important property that we had
in continuous time, and that
472
00:30:27,420 --> 00:30:31,500
we have in discrete time, is
the modulation property.
473
00:30:31,500 --> 00:30:35,080
The modulation property tells
us what happens in the
474
00:30:35,080 --> 00:30:38,730
frequency domain when
you multiply
475
00:30:38,730 --> 00:30:41,700
signals in the time domain.
476
00:30:41,700 --> 00:30:45,100
In continuous time, the
modulation property
477
00:30:45,100 --> 00:30:48,760
corresponded to the statement
that if we multiply the time
478
00:30:48,760 --> 00:30:54,400
domain, we convolve the Fourier
transforms in the
479
00:30:54,400 --> 00:30:56,470
frequency domain.
480
00:30:56,470 --> 00:31:02,730
And in discrete time we have
very much the same kind of
481
00:31:02,730 --> 00:31:05,060
relationship.
482
00:31:05,060 --> 00:31:11,090
The only real distinction
between these is that in the
483
00:31:11,090 --> 00:31:15,160
discrete-time case, in carrying
out the convolution,
484
00:31:15,160 --> 00:31:18,870
it's an integration only
over a 2 pi interval.
485
00:31:18,870 --> 00:31:25,120
And what that corresponds to
is what's referred to as a
486
00:31:25,120 --> 00:31:31,550
periodic convolution, as opposed
to the continuous-time
487
00:31:31,550 --> 00:31:37,800
case where what we have is
a convolution that is an
488
00:31:37,800 --> 00:31:41,050
aperiodic convolution.
489
00:31:41,050 --> 00:31:44,930
So, again, we have a convolution
property in
490
00:31:44,930 --> 00:31:48,680
discrete time that is very
much like the convolution
491
00:31:48,680 --> 00:31:51,060
property in continuous time.
492
00:31:51,060 --> 00:31:54,570
The only real difference is
that here we're convolving
493
00:31:54,570 --> 00:31:56,300
periodic functions.
494
00:31:56,300 --> 00:32:01,050
And so it's a periodic
convolution which involves an
495
00:32:01,050 --> 00:32:04,510
integration only over a 2 pi
interval, rather than an
496
00:32:04,510 --> 00:32:08,570
integration from minus infinity
to plus infinity.
497
00:32:08,570 --> 00:32:14,650
Well, let's take a look at an
example of the modulation
498
00:32:14,650 --> 00:32:20,070
property, which will then lead
to one particular application,
499
00:32:20,070 --> 00:32:24,110
and a very useful application,
of the modulation property in
500
00:32:24,110 --> 00:32:26,180
discrete time.
501
00:32:26,180 --> 00:32:31,580
The example that I want to pick
is an example in which we
502
00:32:31,580 --> 00:32:35,590
consider modulating
a signal with--
503
00:32:35,590 --> 00:32:40,150
a signal with another signal,
x of n, or x1 of n as I
504
00:32:40,150 --> 00:32:44,170
indicated here, which
is minus 1 to the n.
505
00:32:44,170 --> 00:32:48,050
Essentially what that says is
that any signal which I
506
00:32:48,050 --> 00:32:51,640
modulate with this in effect
corresponds to taking the
507
00:32:51,640 --> 00:32:56,110
original signal and then going
through that signal
508
00:32:56,110 --> 00:33:00,690
alternating the algebraic
signs.
509
00:33:00,690 --> 00:33:03,200
Now we--
510
00:33:03,200 --> 00:33:06,430
in applying the modulation
property, of course, what we
511
00:33:06,430 --> 00:33:09,390
need to do is develop
the Fourier
512
00:33:09,390 --> 00:33:12,410
transform of this signal.
513
00:33:12,410 --> 00:33:14,820
This signal which I rewrite--
514
00:33:14,820 --> 00:33:17,450
I can write either as minus 1
to the n or rewrite as e to
515
00:33:17,450 --> 00:33:22,520
the j pi n since e to the j
pi is equal to minus 1--
516
00:33:22,520 --> 00:33:25,210
is a periodic signal.
517
00:33:25,210 --> 00:33:28,700
And it's the periodic signal
that I show here.
518
00:33:28,700 --> 00:33:32,860
And recall that to get the
Fourier transform of a
519
00:33:32,860 --> 00:33:39,020
periodic signal, one way to do
it is to generate the Fourier
520
00:33:39,020 --> 00:33:42,360
series coefficients for the
periodic signal, and then
521
00:33:42,360 --> 00:33:47,450
identify the Fourier transform
as an impulse train where the
522
00:33:47,450 --> 00:33:49,900
heights of the impulses in
the impulse train are
523
00:33:49,900 --> 00:33:52,510
proportional, with a
proportionality factor of 2
524
00:33:52,510 --> 00:33:58,200
pi, proportional to the Fourier
series coefficients.
525
00:33:58,200 --> 00:34:01,380
So let's first work out what the
Fourier series is and for
526
00:34:01,380 --> 00:34:04,580
this example, in fact,
it's fairly easy.
527
00:34:04,580 --> 00:34:08,886
Here is the general
synthesis equation
528
00:34:08,886 --> 00:34:12,429
for the Fourier series.
529
00:34:12,429 --> 00:34:18,969
And if we take our particular
example where, if we look back
530
00:34:18,969 --> 00:34:25,130
at the curve above, what we
recognize is that the period
531
00:34:25,130 --> 00:34:29,139
is equal to 2, namely it
repeats after 2 points.
532
00:34:29,139 --> 00:34:34,389
Then capital N is equal to 2,
and so we can just write this
533
00:34:34,389 --> 00:34:36,389
out with the two terms.
534
00:34:36,389 --> 00:34:40,900
And the two terms involved are
x1 of n is a0, the 0-th
535
00:34:40,900 --> 00:34:46,520
coefficient, that's with k
equals 0, and a1, and this is
536
00:34:46,520 --> 00:34:49,719
with k equals 1, and
we substituted in
537
00:34:49,719 --> 00:34:52,520
capital N equal to 2.
538
00:34:52,520 --> 00:34:55,290
All right, well, we can do a
little bit of algebra here,
539
00:34:55,290 --> 00:34:58,800
obviously cross off
the factors of 2.
540
00:34:58,800 --> 00:35:04,370
And what we recognize, if we
compare this expression with
541
00:35:04,370 --> 00:35:08,840
the original signal which is e
to the j pi n, then we can
542
00:35:08,840 --> 00:35:13,780
simply identify the fact that
a0, the 0-th coefficient is 0,
543
00:35:13,780 --> 00:35:15,510
that's the DC term.
544
00:35:15,510 --> 00:35:19,760
And the coefficient
a1 is equal to 1.
545
00:35:19,760 --> 00:35:25,160
So we've done it simply by
essentially inspecting the
546
00:35:25,160 --> 00:35:30,950
Fourier series synthesis
equation.
547
00:35:30,950 --> 00:35:35,380
OK, now, if we want to get the
Fourier transform for this, we
548
00:35:35,380 --> 00:35:40,470
take those coefficients and
essentially generate an
549
00:35:40,470 --> 00:35:46,830
impulse train where we choose
as values for the impulses 2
550
00:35:46,830 --> 00:35:50,680
pi times the Fourier series
coefficients.
551
00:35:50,680 --> 00:35:54,010
So, the Fourier series
coefficients are a0 is equal
552
00:35:54,010 --> 00:35:56,630
to 0 and a1 is equal to 1.
553
00:35:56,630 --> 00:36:03,370
So, notice that in the plot that
I've shown here of the
554
00:36:03,370 --> 00:36:09,460
Fourier transform of x1 of n, we
have the 0-th coefficient,
555
00:36:09,460 --> 00:36:14,320
which happens to be 0, and so
I have it indicated, an
556
00:36:14,320 --> 00:36:16,440
impulse there.
557
00:36:16,440 --> 00:36:23,360
We have the coefficient a1, and
the coefficient a1 occurs
558
00:36:23,360 --> 00:36:28,540
at a frequency which is omega
0, and omega 0 in fact is
559
00:36:28,540 --> 00:36:33,100
equal to pi because the signal
is e to the j pi n.
560
00:36:33,100 --> 00:36:37,730
Well, what's this impulse
over here?
561
00:36:37,730 --> 00:36:41,150
Well, that impulse is a--
562
00:36:41,150 --> 00:36:42,940
corresponds to the
Fourier series
563
00:36:42,940 --> 00:36:45,560
coefficient a sub minus 1.
564
00:36:45,560 --> 00:36:49,300
And, of course, if we drew
this out over a longer
565
00:36:49,300 --> 00:36:52,850
frequency axis, we would see
lots of other impulses because
566
00:36:52,850 --> 00:36:56,900
of the fact that the Fourier
transform periodically repeats
567
00:36:56,900 --> 00:36:59,660
or, equivalently, the Fourier
series coefficients
568
00:36:59,660 --> 00:37:01,790
periodically repeat.
569
00:37:01,790 --> 00:37:08,580
So this is the coefficient a0,
This is the coefficient a1
570
00:37:08,580 --> 00:37:12,180
with a factor of 2 pi,
this is 2 pi times a0
571
00:37:12,180 --> 00:37:15,680
and 2 pi times a1.
572
00:37:15,680 --> 00:37:20,260
And then this is simply an
indication that it's
573
00:37:20,260 --> 00:37:21,510
periodically repeated.
574
00:37:24,220 --> 00:37:24,670
All right.
575
00:37:24,670 --> 00:37:28,830
Now, let's consider what happens
if we take a signal
576
00:37:28,830 --> 00:37:34,470
and multiply it, modulate
it, by minus 1 to the n.
577
00:37:34,470 --> 00:37:37,540
Well in the frequency domain
that corresponds to a
578
00:37:37,540 --> 00:37:39,800
convolution.
579
00:37:39,800 --> 00:37:43,820
Let's consider a signal x2
of n which has a Fourier
580
00:37:43,820 --> 00:37:47,300
transform as I've
indicated here.
581
00:37:47,300 --> 00:37:52,460
Then the Fourier transform of
the product of x1 of n and x2
582
00:37:52,460 --> 00:37:57,270
of n is the convolution
of these two spectra.
583
00:37:57,270 --> 00:38:01,850
And recall that if you could
convolve something with an
584
00:38:01,850 --> 00:38:05,800
impulse train, as this is, that
simply corresponds to
585
00:38:05,800 --> 00:38:10,920
taking the something and placing
it at the positions of
586
00:38:10,920 --> 00:38:12,830
each of the impulses.
587
00:38:12,830 --> 00:38:17,850
So, in fact, the result of the
convolution of this with this
588
00:38:17,850 --> 00:38:23,320
would then be the spectrum that
I indicate here, namely
589
00:38:23,320 --> 00:38:28,920
this spectrum shifted up to pi
and of course to minus pi.
590
00:38:28,920 --> 00:38:34,550
And then of course to not only
pi but 3 pi and 5 pi, et
591
00:38:34,550 --> 00:38:36,180
cetera, et cetera.
592
00:38:36,180 --> 00:38:42,020
And so this spectrum, finally,
corresponds to the Fourier
593
00:38:42,020 --> 00:38:48,010
transform of minus 1 to the n
times x2 of n where x2 of n is
594
00:38:48,010 --> 00:38:53,780
the sequence whose spectrum
was X2 of omega.
595
00:38:53,780 --> 00:38:57,750
OK, now, this is in fact an
important, useful, and
596
00:38:57,750 --> 00:38:58,570
interesting point.
597
00:38:58,570 --> 00:39:03,210
What it says is if I have a
signal with a certain spectrum
598
00:39:03,210 --> 00:39:05,230
and if I modulate--
599
00:39:05,230 --> 00:39:06,180
multiply--
600
00:39:06,180 --> 00:39:10,130
that signal by minus 1 to the
n, meaning that I alternate
601
00:39:10,130 --> 00:39:14,470
the signs, then it takes
the low frequencies--
602
00:39:14,470 --> 00:39:17,160
in effect, it shifts
the spectrum by pi.
603
00:39:17,160 --> 00:39:19,120
So it takes the low frequencies
and moves them up
604
00:39:19,120 --> 00:39:22,470
to high frequencies, and will
incidentally take the high
605
00:39:22,470 --> 00:39:26,440
frequencies and move them
to low frequencies.
606
00:39:26,440 --> 00:39:30,400
So in fact we, in essence,
saw this when we took--
607
00:39:30,400 --> 00:39:33,920
or when I talked about the
example of a sequence which
608
00:39:33,920 --> 00:39:36,120
was a to the n times u of n.
609
00:39:36,120 --> 00:39:36,730
Notice--
610
00:39:36,730 --> 00:39:40,670
let me draw your attention to
the fact that when a is
611
00:39:40,670 --> 00:39:46,460
positive, we have this sequence
and its Fourier
612
00:39:46,460 --> 00:39:52,220
transform is as I show
on the right.
613
00:39:52,220 --> 00:40:00,820
For a negative, the sequence is
identical to a positive but
614
00:40:00,820 --> 00:40:03,030
with alternating sines.
615
00:40:03,030 --> 00:40:06,730
And the Fourier transform of
that you can now see, and
616
00:40:06,730 --> 00:40:12,900
verify also algebraically if
you'd like, is identical to
617
00:40:12,900 --> 00:40:17,640
this spectrum, simply
shifted by pi.
618
00:40:17,640 --> 00:40:21,280
So it says in fact that
multiplying that impulse
619
00:40:21,280 --> 00:40:25,530
response, or if we think of a
positive and a negative, that
620
00:40:25,530 --> 00:40:29,260
is algebraically similar to
multiplying the impulse
621
00:40:29,260 --> 00:40:31,650
response by minus 1 to the n.
622
00:40:31,650 --> 00:40:35,150
And in the frequency domain,
the effect of that,
623
00:40:35,150 --> 00:40:38,250
essentially, is shifting
the spectrum by pi.
624
00:40:38,250 --> 00:40:40,440
And we can interpret that
in the context of
625
00:40:40,440 --> 00:40:43,060
the modulation property.
626
00:40:43,060 --> 00:40:50,260
Now it's interesting that what
that says is that if we have a
627
00:40:50,260 --> 00:40:58,070
system which corresponds to a
lowpass filter, as I indicate
628
00:40:58,070 --> 00:41:03,320
here, with an impulse
response h of n.
629
00:41:03,320 --> 00:41:06,530
And it can be any approximation
to a lowpass
630
00:41:06,530 --> 00:41:09,860
filter and even an ideal
lowpass filter.
631
00:41:09,860 --> 00:41:14,900
If we want to convert that to
a highpass filter, we can do
632
00:41:14,900 --> 00:41:19,980
that by generating a new system
whose impulse response
633
00:41:19,980 --> 00:41:24,010
is minus 1 to the n times the
impulse response of the
634
00:41:24,010 --> 00:41:25,490
lowpass filter.
635
00:41:25,490 --> 00:41:31,920
And this modulation by minus
1 to the n will take the
636
00:41:31,920 --> 00:41:36,980
frequency response of this
system and shift it by pi so
637
00:41:36,980 --> 00:41:40,500
that what's going on here at low
frequencies will now go on
638
00:41:40,500 --> 00:41:41,750
here at high frequencies.
639
00:41:44,600 --> 00:41:53,010
This also says, incidentally,
that if we look at an ideal
640
00:41:53,010 --> 00:41:58,610
lowpass filter and an ideal
highpass filter, and we choose
641
00:41:58,610 --> 00:42:02,270
the cutoff frequencies for
comparison, or the bandwidth
642
00:42:02,270 --> 00:42:04,650
of the filter to be equal.
643
00:42:04,650 --> 00:42:10,190
Since this ideal highpass filter
is this ideal lowpass
644
00:42:10,190 --> 00:42:16,580
filter with the frequency
response shifted by pi, the
645
00:42:16,580 --> 00:42:20,640
modulation property tells us
that in the time domain, what
646
00:42:20,640 --> 00:42:25,520
that corresponds to is an
impulse response multiplied by
647
00:42:25,520 --> 00:42:26,950
minus 1 to the n.
648
00:42:26,950 --> 00:42:32,500
So it says that the impulse
response of the highpass
649
00:42:32,500 --> 00:42:36,230
filter, or equivalently the
inverse Fourier transform of
650
00:42:36,230 --> 00:42:40,720
the highpass filter frequency
response, is minus 1 to the n
651
00:42:40,720 --> 00:42:43,780
times the impulse response
for the lowpass filter.
652
00:42:43,780 --> 00:42:47,860
That all follows from the
modulation property.
653
00:42:47,860 --> 00:42:51,710
Now there's another way, an
interesting and useful way,
654
00:42:51,710 --> 00:42:57,200
that modulation can be used to
implement or convert from
655
00:42:57,200 --> 00:43:00,710
lowpass filtering to
highpass filtering.
656
00:43:00,710 --> 00:43:04,300
The modulation property tells us
about multiplying the time
657
00:43:04,300 --> 00:43:07,170
domain is shifting in the
frequency domain.
658
00:43:07,170 --> 00:43:09,810
And in the example that we
happened to pick said if you
659
00:43:09,810 --> 00:43:14,120
multiply or modulate by minus
1 to the n, that takes low
660
00:43:14,120 --> 00:43:17,580
frequencies and shifts them
to high frequencies.
661
00:43:17,580 --> 00:43:23,280
What that tells us, as a
practical and useful notion,
662
00:43:23,280 --> 00:43:24,520
is the following.
663
00:43:24,520 --> 00:43:28,540
Suppose we have a system that
we know is a lowpass filter,
664
00:43:28,540 --> 00:43:31,410
and it's a good lowpass
filter.
665
00:43:31,410 --> 00:43:34,800
How might we use it as
a highpass filter?
666
00:43:34,800 --> 00:43:38,080
Well, one way to do it, instead
of shifting its
667
00:43:38,080 --> 00:43:43,810
frequency response, is to take
the original signal, shift its
668
00:43:43,810 --> 00:43:46,080
low frequencies to high
frequencies and its high
669
00:43:46,080 --> 00:43:49,500
frequencies to low frequencies
by multiplying the input
670
00:43:49,500 --> 00:43:54,160
signal, the original signal, by
minus 1 to the n, process
671
00:43:54,160 --> 00:43:57,070
that with a lowpass filter where
now what's sitting at
672
00:43:57,070 --> 00:44:00,430
the low frequencies were
the high frequencies.
673
00:44:00,430 --> 00:44:05,320
And then unscramble it all at
the output so that we put the
674
00:44:05,320 --> 00:44:07,760
frequencies back where
they belong.
675
00:44:07,760 --> 00:44:10,060
And I summarize that here.
676
00:44:10,060 --> 00:44:13,860
Let's suppose, for example, that
this system was a lowpass
677
00:44:13,860 --> 00:44:17,840
filter, and so it
lowpass-filters
678
00:44:17,840 --> 00:44:20,120
whatever comes into it.
679
00:44:20,120 --> 00:44:24,560
Down below, I indicate taking
the input and first
680
00:44:24,560 --> 00:44:28,830
interchanging the high and low
frequencies through modulation
681
00:44:28,830 --> 00:44:32,460
with minus 1 to the n.
682
00:44:32,460 --> 00:44:35,220
Doing the lowpass filtering,
which--
683
00:44:35,220 --> 00:44:37,660
and what's sitting at the low
frequencies here were the high
684
00:44:37,660 --> 00:44:39,990
frequencies of this signal.
685
00:44:39,990 --> 00:44:42,830
And then after the lowpass
filtering, moving the
686
00:44:42,830 --> 00:44:46,860
frequencies back where they
belong by again modulating
687
00:44:46,860 --> 00:44:48,560
with minus 1 to the n.
688
00:44:48,560 --> 00:44:56,150
And that, in fact, turns out to
be a very useful notion for
689
00:44:56,150 --> 00:45:01,240
applying a fixed lowpass
filter to do highpass
690
00:45:01,240 --> 00:45:02,550
filtering and vice versa.
691
00:45:06,740 --> 00:45:15,470
OK, now, what we've seen and
what we've talked about are
692
00:45:15,470 --> 00:45:20,740
the Fourier representation for
discrete-time signals, and
693
00:45:20,740 --> 00:45:24,080
prior to that, continuous-time
signals.
694
00:45:24,080 --> 00:45:27,250
And we've seen some very
important similarities and
695
00:45:27,250 --> 00:45:28,470
differences.
696
00:45:28,470 --> 00:45:34,660
And what I'd like to do is
conclude this lecture by
697
00:45:34,660 --> 00:45:38,910
summarizing those various
relationships kind of all in
698
00:45:38,910 --> 00:45:43,550
one package, and in fact drawing
your attention to both
699
00:45:43,550 --> 00:45:46,250
the similarities and differences
and comparisons
700
00:45:46,250 --> 00:45:48,930
between them.
701
00:45:48,930 --> 00:45:54,710
Well, let's begin this summary
by first looking at the
702
00:45:54,710 --> 00:45:57,240
continuous-time Fourier
series.
703
00:45:57,240 --> 00:46:02,280
In the continuous-time Fourier
series, we have a periodic
704
00:46:02,280 --> 00:46:06,940
time function expanded as
a linear combination of
705
00:46:06,940 --> 00:46:10,000
harmonically-related complex
exponentials.
706
00:46:10,000 --> 00:46:12,420
And there are an infinite
number of these that are
707
00:46:12,420 --> 00:46:15,710
required to do the
decomposition.
708
00:46:15,710 --> 00:46:20,230
And we saw an analysis equation
which tells us how to
709
00:46:20,230 --> 00:46:24,380
get these Fourier series
coefficients through an
710
00:46:24,380 --> 00:46:28,330
integration on the original
time function.
711
00:46:28,330 --> 00:46:32,260
And notice in this that what
we have is a continuous
712
00:46:32,260 --> 00:46:34,250
periodic time function.
713
00:46:34,250 --> 00:46:38,990
What we end up with in the
frequency domain is a sequence
714
00:46:38,990 --> 00:46:42,670
of Fourier series coefficients
which in fact is an infinite
715
00:46:42,670 --> 00:46:46,700
sequence, namely, requires all
values of k in general.
716
00:46:49,380 --> 00:46:53,650
We had then generalized that to
the continuous-time Fourier
717
00:46:53,650 --> 00:46:57,750
transform, and, in effect, in
doing that what happened is
718
00:46:57,750 --> 00:47:07,580
that the synthesis equation in
the Fourier series became an
719
00:47:07,580 --> 00:47:11,830
integral relationship in
the Fourier transform.
720
00:47:11,830 --> 00:47:17,360
And we now have a
continuous-time function which
721
00:47:17,360 --> 00:47:21,290
is no longer periodic, this was
for the aperiodic case,
722
00:47:21,290 --> 00:47:25,080
represented as a linear
combination of infinitesimally
723
00:47:25,080 --> 00:47:29,870
close-in-frequency complex
exponentials with complex
724
00:47:29,870 --> 00:47:35,850
amplitudes given by X of omega
d omega divided by 2 pi.
725
00:47:35,850 --> 00:47:39,210
And we had of course the
corresponding analysis
726
00:47:39,210 --> 00:47:43,360
equation that told us how
to get X of omega.
727
00:47:43,360 --> 00:47:48,060
Here we have a continuous-time
function which is aperiodic,
728
00:47:48,060 --> 00:47:53,485
and a continuous function of
frequency which is aperiodic.
729
00:47:56,990 --> 00:48:01,780
The conceptual strategy in the
discrete-time case was very
730
00:48:01,780 --> 00:48:08,640
similar, with some differences
resulting in the relationships
731
00:48:08,640 --> 00:48:11,920
because of some inherent
differences between continuous
732
00:48:11,920 --> 00:48:15,110
time and discrete time.
733
00:48:15,110 --> 00:48:19,560
We began with the discrete-time
Fourier series,
734
00:48:19,560 --> 00:48:24,360
corresponding to representing a
periodic sequence through a
735
00:48:24,360 --> 00:48:29,780
set of complex exponentials,
where now we only required a
736
00:48:29,780 --> 00:48:33,990
finite number of these because
of the fact that, in fact,
737
00:48:33,990 --> 00:48:36,830
there are only a finite number
of harmonically-related
738
00:48:36,830 --> 00:48:38,580
complex exponentials.
739
00:48:38,580 --> 00:48:41,710
That's an inherent property
of discrete-time complex
740
00:48:41,710 --> 00:48:43,070
exponentials.
741
00:48:43,070 --> 00:48:49,640
And so we have a discrete,
periodic time function.
742
00:48:49,640 --> 00:48:53,270
And we ended up with a set of
Fourier series coefficients,
743
00:48:53,270 --> 00:48:56,670
which of course are discrete, as
Fourier series coefficients
744
00:48:56,670 --> 00:49:01,960
are, and which periodically
repeat because of the fact
745
00:49:01,960 --> 00:49:04,290
that the associated complex
exponentials
746
00:49:04,290 --> 00:49:05,540
periodically repeat.
747
00:49:07,960 --> 00:49:11,030
We then used an argument similar
to the continuous-time
748
00:49:11,030 --> 00:49:14,840
case for going from periodic
time functions to aperiodic
749
00:49:14,840 --> 00:49:16,290
time functions.
750
00:49:16,290 --> 00:49:20,190
And we ended up with a
relationship describing a
751
00:49:20,190 --> 00:49:25,710
representation for aperiodic
discrete-time signals in which
752
00:49:25,710 --> 00:49:30,460
now the synthesis equation went
from a summation to an
753
00:49:30,460 --> 00:49:32,620
integration, since the
frequencies are now
754
00:49:32,620 --> 00:49:36,770
infinitesimally close, involving
frequencies only
755
00:49:36,770 --> 00:49:40,500
over a 2 pi interval,
and for which the
756
00:49:40,500 --> 00:49:42,800
amplitude factor X of omega--
757
00:49:42,800 --> 00:49:45,140
well, the amplitude factor
is X of omega d omega
758
00:49:45,140 --> 00:49:47,190
divided by 2 pi.
759
00:49:47,190 --> 00:49:51,060
And this term, X of omega,
which is the Fourier
760
00:49:51,060 --> 00:49:57,120
transform, is given by this
summation, and of course
761
00:49:57,120 --> 00:50:01,500
involves all of the
values of x of n.
762
00:50:01,500 --> 00:50:05,420
And so the important difference
between the
763
00:50:05,420 --> 00:50:08,390
continuous-time and
discrete-time case kind of
764
00:50:08,390 --> 00:50:11,460
arose, in part, out of the fact
that discrete time is
765
00:50:11,460 --> 00:50:14,830
discrete time, continuous time
is continuous time, and the
766
00:50:14,830 --> 00:50:18,770
fact that complex exponentials
are periodic in discrete time.
767
00:50:18,770 --> 00:50:21,910
The harmonically-related ones
periodically repeat whereas
768
00:50:21,910 --> 00:50:25,470
they don't in continuous time.
769
00:50:25,470 --> 00:50:28,830
Now this, among other things,
has an important consequence
770
00:50:28,830 --> 00:50:30,270
for duality.
771
00:50:30,270 --> 00:50:34,540
And let's go back again and look
at this equation, this
772
00:50:34,540 --> 00:50:35,660
pair of equations.
773
00:50:35,660 --> 00:50:38,800
And clearly there is no duality
774
00:50:38,800 --> 00:50:41,040
between these two equations.
775
00:50:41,040 --> 00:50:44,230
This involves a summation, this
involves an integration.
776
00:50:44,230 --> 00:50:50,810
And so, in fact, if we make
reference to duality, there
777
00:50:50,810 --> 00:50:52,810
isn't duality in the
778
00:50:52,810 --> 00:50:55,900
continuous-time Fourier series.
779
00:50:55,900 --> 00:50:58,910
However, for the continuous-time
Fourier
780
00:50:58,910 --> 00:51:03,910
transform, we're talking about
aperiodic time functions and
781
00:51:03,910 --> 00:51:06,310
aperiodic frequency functions.
782
00:51:06,310 --> 00:51:09,350
And, in fact, when we look at
these two equations, we see
783
00:51:09,350 --> 00:51:11,550
very definitely a duality.
784
00:51:11,550 --> 00:51:15,530
In other words, the time
function effectively is the
785
00:51:15,530 --> 00:51:18,340
Fourier transform of the
Fourier transform.
786
00:51:18,340 --> 00:51:20,660
There's a little time reversal
in there, but basically that's
787
00:51:20,660 --> 00:51:21,780
the result.
788
00:51:21,780 --> 00:51:26,890
And, in fact, we had exploited
that duality property when we
789
00:51:26,890 --> 00:51:27,620
talked about the
790
00:51:27,620 --> 00:51:32,020
continuous-time Fourier transform.
791
00:51:32,020 --> 00:51:40,370
With the discrete-time Fourier
series, we have a duality
792
00:51:40,370 --> 00:51:44,590
indicated by the fact that we
have a periodic time function
793
00:51:44,590 --> 00:51:48,910
and a sequence which
is periodic in
794
00:51:48,910 --> 00:51:50,170
the frequency domain.
795
00:51:50,170 --> 00:51:53,150
And in fact, if you look at
these two expressions, you see
796
00:51:53,150 --> 00:51:55,200
the duality very clearly.
797
00:51:55,200 --> 00:51:58,630
And so it's the discrete-time
Fourier
798
00:51:58,630 --> 00:52:02,630
series that has a duality.
799
00:52:02,630 --> 00:52:07,080
And finally the discrete-time
Fourier transform loses the
800
00:52:07,080 --> 00:52:11,150
duality because of the fact,
among other things, that in
801
00:52:11,150 --> 00:52:15,040
the time domain things are
inherently discrete whereas in
802
00:52:15,040 --> 00:52:18,580
the frequency domain they're
inherently continuous.
803
00:52:18,580 --> 00:52:21,395
So, in fact, here there
is no duality.
804
00:52:25,690 --> 00:52:30,280
OK, now that says that there's
a difference in the duality,
805
00:52:30,280 --> 00:52:32,510
continuous time and
discrete time.
806
00:52:32,510 --> 00:52:37,780
And there's one more very
important piece to the duality
807
00:52:37,780 --> 00:52:39,610
relationships.
808
00:52:39,610 --> 00:52:44,400
And we can see that first
algebraically by comparing the
809
00:52:44,400 --> 00:52:49,280
continuous-time Fourier
series and the
810
00:52:49,280 --> 00:52:51,435
discrete-time Fourier transform.
811
00:52:54,240 --> 00:52:59,210
The continuous-time Fourier
series in the time domain is a
812
00:52:59,210 --> 00:53:03,580
periodic continuous function, in
the frequency domain is an
813
00:53:03,580 --> 00:53:06,930
aperiodic sequence.
814
00:53:06,930 --> 00:53:13,680
In the discrete-time case, in
the time domain we have an
815
00:53:13,680 --> 00:53:19,850
aperiodic sequence, and in the
frequency domain we have a
816
00:53:19,850 --> 00:53:21,990
function of a continuous
variable
817
00:53:21,990 --> 00:53:24,480
which we know is periodic.
818
00:53:24,480 --> 00:53:28,100
And so in fact we have,
in the time domain
819
00:53:28,100 --> 00:53:30,370
here, aperiodic sequence.
820
00:53:30,370 --> 00:53:32,300
In the frequency domain
we have a
821
00:53:32,300 --> 00:53:34,990
continuous periodic function.
822
00:53:34,990 --> 00:53:38,600
And in fact, if you look at the
relationship between these
823
00:53:38,600 --> 00:53:48,390
two, then what we see in fact
is a duality between the
824
00:53:48,390 --> 00:53:53,370
continuous-time Fourier
series and the
825
00:53:53,370 --> 00:53:56,500
discrete-time Fourier transform.
826
00:53:56,500 --> 00:54:00,170
One way of thinking of that is
to kind of think, and this is
827
00:54:00,170 --> 00:54:02,820
a little bit of a tongue twister
which you might want
828
00:54:02,820 --> 00:54:08,240
to get straightened out slowly,
but the Fourier
829
00:54:08,240 --> 00:54:10,000
transform in discrete time is a
830
00:54:10,000 --> 00:54:12,270
periodic function of frequency.
831
00:54:12,270 --> 00:54:17,190
That periodic function has a
Fourier series representation.
832
00:54:17,190 --> 00:54:19,540
What is this Fourier series?
833
00:54:19,540 --> 00:54:21,170
What are the Fourier series
coefficients of
834
00:54:21,170 --> 00:54:22,730
that periodic function?
835
00:54:22,730 --> 00:54:26,030
Well in fact, except for an
issue of time reversal, what
836
00:54:26,030 --> 00:54:29,920
it is the original sequence
for which
837
00:54:29,920 --> 00:54:31,670
that's the Fourier transform.
838
00:54:31,670 --> 00:54:35,610
And that is the duality that I'm
trying to emphasize here.
839
00:54:38,600 --> 00:54:42,910
OK, well, so what we see is
that these four sets of
840
00:54:42,910 --> 00:54:47,110
relationships all tie together
in a whole variety of ways.
841
00:54:47,110 --> 00:54:51,070
And we will be exploiting as
the discussion goes on the
842
00:54:51,070 --> 00:54:52,690
inner-connections and
relationships
843
00:54:52,690 --> 00:54:55,000
that I've talked about.
844
00:54:55,000 --> 00:54:58,730
Also, as we've talked about the
Fourier transform, both
845
00:54:58,730 --> 00:55:02,750
continuous time and discrete
time, two important properties
846
00:55:02,750 --> 00:55:06,570
that we focused on, among many
of the properties, are the
847
00:55:06,570 --> 00:55:09,950
convolution property and the
modulation property.
848
00:55:09,950 --> 00:55:14,670
We've also shown that the
convolution property leads to
849
00:55:14,670 --> 00:55:19,020
a very important concept,
namely filtering.
850
00:55:19,020 --> 00:55:22,700
The modulation property leads
to an important concept,
851
00:55:22,700 --> 00:55:24,380
namely modulation.
852
00:55:24,380 --> 00:55:31,200
We've also very briefly
indicated how these properties
853
00:55:31,200 --> 00:55:35,000
and how these concepts have
practical implications.
854
00:55:35,000 --> 00:55:37,700
In the next several lectures,
we'll focus in more
855
00:55:37,700 --> 00:55:41,100
specifically first on filtering,
and then on
856
00:55:41,100 --> 00:55:42,570
modulation.
857
00:55:42,570 --> 00:55:49,000
And as we'll see the filtering
and modulation concepts form
858
00:55:49,000 --> 00:55:51,750
really the cornerstone
of many, many
859
00:55:51,750 --> 00:55:53,530
signal processing ideas.
860
00:55:53,530 --> 00:55:54,780
Thank you.