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[MUSIC PLAYING]
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PROFESSOR: Last time, we began
the discussion of the
11
00:00:58,610 --> 00:01:01,240
z-transform.
12
00:01:01,240 --> 00:01:04,110
As with the Laplace transform
in continuous time, we
13
00:01:04,110 --> 00:01:06,190
developed it as a
generalization
14
00:01:06,190 --> 00:01:09,000
of the Fourier transform.
15
00:01:09,000 --> 00:01:14,230
The expression that we got for
the z-transform is the sum
16
00:01:14,230 --> 00:01:17,050
that I indicate here.
17
00:01:17,050 --> 00:01:20,750
We also briefly talked about
an inverse z-transform
18
00:01:20,750 --> 00:01:25,320
integral and some other informal
methods of computing
19
00:01:25,320 --> 00:01:27,660
the inverse z-transform.
20
00:01:27,660 --> 00:01:32,220
But we focused in particular
on the relationship between
21
00:01:32,220 --> 00:01:36,090
the z-transform and the Fourier
transform, pointing
22
00:01:36,090 --> 00:01:41,800
out first of all that the
z-transform, when we choose
23
00:01:41,800 --> 00:01:44,590
the magnitude of
z equal to 1--
24
00:01:44,590 --> 00:01:48,440
so the magnitude of z of the
form e to the j omega--
25
00:01:48,440 --> 00:01:52,150
just simply reduces
to the Fourier
26
00:01:52,150 --> 00:01:55,210
transform of the sequence.
27
00:01:55,210 --> 00:02:00,160
Then, in addition, we explored
the z-transform for z, a more
28
00:02:00,160 --> 00:02:02,640
general complex number.
29
00:02:02,640 --> 00:02:06,390
In the discrete time z-transform
case, we expressed
30
00:02:06,390 --> 00:02:12,130
that complex number in polar
form as r e to the j omega,
31
00:02:12,130 --> 00:02:17,800
and recognize that the
z-transform expression in fact
32
00:02:17,800 --> 00:02:22,900
corresponds to the Fourier
transform of the sequence
33
00:02:22,900 --> 00:02:25,400
exponentially weighted.
34
00:02:25,400 --> 00:02:30,610
Because of the exponential
weighting, the z-transform
35
00:02:30,610 --> 00:02:35,100
converges for some values of
r corresponding to some
36
00:02:35,100 --> 00:02:39,320
exponential waiting, and perhaps
not for others, and
37
00:02:39,320 --> 00:02:44,890
that led to a notion which
corresponded to the region of
38
00:02:44,890 --> 00:02:49,180
convergence associated
with the z-transform.
39
00:02:49,180 --> 00:02:51,680
We talked some about properties
of the region of
40
00:02:51,680 --> 00:02:55,230
convergence, particularly
in relation to
41
00:02:55,230 --> 00:02:57,900
the pole-zero pattern.
42
00:02:57,900 --> 00:03:03,090
Now, they z-transform has a
number of important and useful
43
00:03:03,090 --> 00:03:06,780
properties, just as the Laplace
transform does.
44
00:03:06,780 --> 00:03:10,580
As one part of this lecture,
what we'll want to do is
45
00:03:10,580 --> 00:03:14,440
exploit some of these properties
in the context of
46
00:03:14,440 --> 00:03:16,930
systems described by linear
constant coefficient
47
00:03:16,930 --> 00:03:19,360
difference equations.
48
00:03:19,360 --> 00:03:23,660
These particular properties that
play an important role in
49
00:03:23,660 --> 00:03:25,280
that context are the properties
50
00:03:25,280 --> 00:03:27,570
that I indicate here.
51
00:03:27,570 --> 00:03:29,500
In particular, there is--
52
00:03:29,500 --> 00:03:31,780
as with continuous time--
53
00:03:31,780 --> 00:03:36,500
a linearity property that tells
us that the z-transform
54
00:03:36,500 --> 00:03:40,870
of a linear combination of
sequences is the same linear
55
00:03:40,870 --> 00:03:45,140
combination of the z-transforms,
a shifting
56
00:03:45,140 --> 00:03:50,580
property that indicates that
the z-transform of x of n
57
00:03:50,580 --> 00:03:56,740
shifted is the z transform of x
of n multiplied by a factor
58
00:03:56,740 --> 00:03:59,780
z to the minus n 0.
59
00:03:59,780 --> 00:04:06,120
Then the convolution property
for which the z-transform of a
60
00:04:06,120 --> 00:04:11,500
convolution of sequences is the
product of the associated
61
00:04:11,500 --> 00:04:14,250
z-transforms.
62
00:04:14,250 --> 00:04:17,769
With all of these properties, of
course, there is again the
63
00:04:17,769 --> 00:04:21,760
issue of what the associated
region of convergence is in
64
00:04:21,760 --> 00:04:23,840
comparison with the region
of convergence of
65
00:04:23,840 --> 00:04:26,500
the original sequences.
66
00:04:26,500 --> 00:04:30,040
That is an issue that's
addressed somewhat more in the
67
00:04:30,040 --> 00:04:31,730
text and let's not
go into it here.
68
00:04:31,730 --> 00:04:34,240
69
00:04:34,240 --> 00:04:40,890
With the convolution property,
the convolution property as in
70
00:04:40,890 --> 00:04:45,220
continuous time, of course,
provides a mechanism for
71
00:04:45,220 --> 00:04:48,750
dealing with linear time
invariant systems.
72
00:04:48,750 --> 00:04:52,640
In particular, in the time
domain a linear time invariant
73
00:04:52,640 --> 00:04:55,980
system is described through
convolution--
74
00:04:55,980 --> 00:05:00,610
namely, the output is the
convolution of the input and
75
00:05:00,610 --> 00:05:02,770
the impulse response.
76
00:05:02,770 --> 00:05:06,810
Because of the convolution
property associated with the
77
00:05:06,810 --> 00:05:11,590
z-transform, the z-transform
of the output is the
78
00:05:11,590 --> 00:05:15,650
z-transform of the input
times the z-transform
79
00:05:15,650 --> 00:05:17,580
of the impulse response.
80
00:05:17,580 --> 00:05:22,410
Again, very much the same as
what we had in continuous time
81
00:05:22,410 --> 00:05:26,590
and also what we had in the
context of the discussion with
82
00:05:26,590 --> 00:05:29,410
the Fourier transform.
83
00:05:29,410 --> 00:05:32,290
In fact, because of the
relationship between the
84
00:05:32,290 --> 00:05:37,280
z-transform and the Fourier
transform, the z-transform of
85
00:05:37,280 --> 00:05:41,690
the impulse response evaluated
on the unit circle--
86
00:05:41,690 --> 00:05:44,570
in other words, for the
magnitude of z equal to 1--
87
00:05:44,570 --> 00:05:47,620
in fact corresponds
to the frequency
88
00:05:47,620 --> 00:05:50,060
response of the system.
89
00:05:50,060 --> 00:05:53,400
More generally, when we talk
about the z-transform of the
90
00:05:53,400 --> 00:05:58,300
impulse response, we will refer
to it as the system
91
00:05:58,300 --> 00:06:00,225
function associated
with the system.
92
00:06:00,225 --> 00:06:03,130
93
00:06:03,130 --> 00:06:06,030
Now, the convolution property
and these other properties, as
94
00:06:06,030 --> 00:06:09,740
I indicated, we will find
useful in talking about
95
00:06:09,740 --> 00:06:13,260
systems which are described by
linear constant coefficient
96
00:06:13,260 --> 00:06:18,460
difference equations, and in
fact, we'll do that shortly.
97
00:06:18,460 --> 00:06:25,320
But first what I'd like to do is
to continue to focus on the
98
00:06:25,320 --> 00:06:29,270
system function for linear time
invariant systems, and
99
00:06:29,270 --> 00:06:33,350
make a couple of comments that
tie back to some things that
100
00:06:33,350 --> 00:06:38,280
we said in the last lecture
relating to the relationship
101
00:06:38,280 --> 00:06:43,510
between the region of
convergence of a system, or--
102
00:06:43,510 --> 00:06:47,080
I'm sorry, the region of
convergence of a z-transform--
103
00:06:47,080 --> 00:06:54,040
and the issue of where that is
in relation to the poles of
104
00:06:54,040 --> 00:06:55,290
the z-transform.
105
00:06:55,290 --> 00:06:56,980
106
00:06:56,980 --> 00:07:00,260
In particular, we can draw some
conclusions tying back to
107
00:07:00,260 --> 00:07:05,780
that discussion about the pole
locations of the system
108
00:07:05,780 --> 00:07:11,810
function in relation to whether
the system is stable
109
00:07:11,810 --> 00:07:15,510
and whether the system
is causal.
110
00:07:15,510 --> 00:07:19,960
In particular, recall from one
of the early lectures way back
111
00:07:19,960 --> 00:07:25,830
when that stability for a system
corresponded to the
112
00:07:25,830 --> 00:07:28,500
statement that the
impulse response
113
00:07:28,500 --> 00:07:31,450
is absolutely summable.
114
00:07:31,450 --> 00:07:35,640
Furthermore, when we talked
about the Fourier transform,
115
00:07:35,640 --> 00:07:42,220
the Fourier transform of a
sequence converges if the
116
00:07:42,220 --> 00:07:44,500
sequence is absolutely
summable.
117
00:07:44,500 --> 00:07:50,630
So in fact, the condition for
stability of a system and the
118
00:07:50,630 --> 00:07:53,540
condition for convergence of the
Fourier transform of its
119
00:07:53,540 --> 00:07:58,820
impulse response are the same
condition, namely absolute
120
00:07:58,820 --> 00:08:00,900
summability.
121
00:08:00,900 --> 00:08:02,750
Now what does this mean?
122
00:08:02,750 --> 00:08:07,870
What it means is that if the
Fourier transform converges,
123
00:08:07,870 --> 00:08:11,140
that means that the z-transform
converges on the
124
00:08:11,140 --> 00:08:13,640
unit circle.
125
00:08:13,640 --> 00:08:18,200
Consequently, if the system
is stable, then the system
126
00:08:18,200 --> 00:08:22,500
function, the z-transform or
the impulse response, must
127
00:08:22,500 --> 00:08:25,210
also converge on the
unit circle.
128
00:08:25,210 --> 00:08:28,610
In other words, the impulse
response must have a Fourier
129
00:08:28,610 --> 00:08:31,160
transform that converges.
130
00:08:31,160 --> 00:08:36,549
So for a stable system, then,
the region of convergence of
131
00:08:36,549 --> 00:08:39,450
the system function must
include the unit
132
00:08:39,450 --> 00:08:42,620
circle in the z-plane.
133
00:08:42,620 --> 00:08:50,080
So we see how stability relates
to the location of the
134
00:08:50,080 --> 00:08:54,450
region of convergence, and we
can also relate causality to
135
00:08:54,450 --> 00:08:55,670
the region of convergence.
136
00:08:55,670 --> 00:09:01,850
In particular, we know that if
a system is causal, then the
137
00:09:01,850 --> 00:09:05,980
impulse response
is right-sided.
138
00:09:05,980 --> 00:09:08,250
For a sequence that's
right-sided, the region of
139
00:09:08,250 --> 00:09:12,060
convergence of its z-transform
must be outside
140
00:09:12,060 --> 00:09:15,110
the outermost pole.
141
00:09:15,110 --> 00:09:18,800
So for causality, the region of
convergence of the system
142
00:09:18,800 --> 00:09:21,710
function must be outside
the outermost pole.
143
00:09:21,710 --> 00:09:25,350
For stability, the region of
convergence must include the
144
00:09:25,350 --> 00:09:30,150
unit circle, and we can also
then draw from that the
145
00:09:30,150 --> 00:09:34,840
conclusion that if we have a
system that's causal unstable,
146
00:09:34,840 --> 00:09:39,430
then all poles must be inside
the unit circle because of the
147
00:09:39,430 --> 00:09:41,130
fact that the poles must--
148
00:09:41,130 --> 00:09:44,420
because of the fact that the
region of convergence must be
149
00:09:44,420 --> 00:09:47,480
outside the outermost
pole and has to also
150
00:09:47,480 --> 00:09:49,910
include the unit circle.
151
00:09:49,910 --> 00:09:55,780
So for example, if we had let's
say a system with a
152
00:09:55,780 --> 00:10:05,840
system function as I indicate
here with the algebraic
153
00:10:05,840 --> 00:10:09,930
expression for the system
function being the expression
154
00:10:09,930 --> 00:10:13,810
that I indicate here with the
pole at z equals a third and
155
00:10:13,810 --> 00:10:18,290
another pole at z equals 2
and a zero at the origin.
156
00:10:18,290 --> 00:10:24,210
If, in fact, the system was
causal, corresponding to an
157
00:10:24,210 --> 00:10:28,050
impulse response that's
right-sided, this then would
158
00:10:28,050 --> 00:10:34,050
be the region of convergence
of the system function.
159
00:10:34,050 --> 00:10:38,220
Alternatively, if I knew that
the system was stable, then I
160
00:10:38,220 --> 00:10:40,910
know that the region of
convergence must include the
161
00:10:40,910 --> 00:10:45,080
unit circle, and so this would
be then the region of
162
00:10:45,080 --> 00:10:46,640
convergence.
163
00:10:46,640 --> 00:10:53,690
And what you might now want to
ask yourselves is if instead
164
00:10:53,690 --> 00:10:57,980
the region of convergence for
the system function is this,
165
00:10:57,980 --> 00:11:00,570
then is the system causal?
166
00:11:00,570 --> 00:11:02,390
That's the first question.
167
00:11:02,390 --> 00:11:06,340
The second question is
is the system stable?
168
00:11:06,340 --> 00:11:08,840
Remembering that for causality,
the region of
169
00:11:08,840 --> 00:11:12,150
convergence must be outside the
outermost pole, and for
170
00:11:12,150 --> 00:11:16,540
stability it must include
the unit circle.
171
00:11:16,540 --> 00:11:25,200
Now what I'd like to do is look
at the properties of the
172
00:11:25,200 --> 00:11:29,430
z-transform, and in particular
exploit these properties in
173
00:11:29,430 --> 00:11:33,850
the context of systems that
are described by linear
174
00:11:33,850 --> 00:11:36,930
constant coefficient difference
equations.
175
00:11:36,930 --> 00:11:40,230
The three basic properties that
play a key role in that
176
00:11:40,230 --> 00:11:44,980
discussion are the linearity
property, the shifting
177
00:11:44,980 --> 00:11:49,860
property, and the convolution
property.
178
00:11:49,860 --> 00:11:52,430
These, then, are the properties
179
00:11:52,430 --> 00:11:55,380
that I want to exploit.
180
00:11:55,380 --> 00:11:59,740
So let's do that by first
looking at a first order
181
00:11:59,740 --> 00:12:02,480
difference equation.
182
00:12:02,480 --> 00:12:06,310
In the case of a first order
difference equation, which
183
00:12:06,310 --> 00:12:09,170
I've written as I
indicate here--
184
00:12:09,170 --> 00:12:11,930
no terms on the right hand side,
but we could have terms
185
00:12:11,930 --> 00:12:13,860
of course, in general--
186
00:12:13,860 --> 00:12:17,900
y of n minus ay of n minus
1 equals x of n.
187
00:12:17,900 --> 00:12:24,630
We can use the linearity
property, so that if we take
188
00:12:24,630 --> 00:12:28,570
the z-transform of both sides of
this expression, that will
189
00:12:28,570 --> 00:12:32,110
then be the z-transform of this
term plus the z-transform
190
00:12:32,110 --> 00:12:34,180
of this term.
191
00:12:34,180 --> 00:12:39,790
So using those properties and
together with the shifting
192
00:12:39,790 --> 00:12:44,010
property, the property that
tells us that the z-transform
193
00:12:44,010 --> 00:12:49,110
of y of n minus 1 is z to the
minus 1 times the z-transform
194
00:12:49,110 --> 00:12:53,520
of y of n, we then convert the
difference equation to an
195
00:12:53,520 --> 00:12:55,400
algebraic expression.
196
00:12:55,400 --> 00:12:58,940
And we can solve this algebraic
expression for the
197
00:12:58,940 --> 00:13:02,600
z-transform of the output
in terms of the
198
00:13:02,600 --> 00:13:06,100
z-transform of the input.
199
00:13:06,100 --> 00:13:11,820
Now what we know from the
convolution property is that
200
00:13:11,820 --> 00:13:15,670
for a system, the z-transform
of the output is the system
201
00:13:15,670 --> 00:13:18,440
function times the z-transform
of the input.
202
00:13:18,440 --> 00:13:24,720
So this factor that we have,
then, must correspond to the
203
00:13:24,720 --> 00:13:29,260
system function, or equivalently
the z-transform
204
00:13:29,260 --> 00:13:32,360
of the impulse response
of the system.
205
00:13:32,360 --> 00:13:36,200
In fact, then, if we have this
z-transform, we could figure
206
00:13:36,200 --> 00:13:41,560
out what the impulse response of
the system is by computing
207
00:13:41,560 --> 00:13:45,730
or determining what the inverse
z-transform, except
208
00:13:45,730 --> 00:13:52,590
for the fact that expression is
an algebraic expression and
209
00:13:52,590 --> 00:13:55,940
doesn't yet totally specify
the z-transform because we
210
00:13:55,940 --> 00:14:00,210
don't yet know what the region
of convergence is.
211
00:14:00,210 --> 00:14:02,670
How do we get the region
of convergence?
212
00:14:02,670 --> 00:14:05,320
Well, we have the same issue
here as we had with the
213
00:14:05,320 --> 00:14:06,980
Laplace transform--
214
00:14:06,980 --> 00:14:13,440
namely, the point that the
difference equation tells us,
215
00:14:13,440 --> 00:14:17,340
in essence, what the algebraic
expression is for the system
216
00:14:17,340 --> 00:14:22,080
function, but doesn't specify
the region of convergence.
217
00:14:22,080 --> 00:14:26,910
That is specified by either
explicitly, because one way or
218
00:14:26,910 --> 00:14:30,590
another we know what the
impulse response is, or
219
00:14:30,590 --> 00:14:33,940
implicitly, because we know
certain properties of the
220
00:14:33,940 --> 00:14:40,130
system, such as causality
and or stability.
221
00:14:40,130 --> 00:14:46,030
If I, let's say, imposed on this
system, in addition to
222
00:14:46,030 --> 00:14:55,580
the difference equation, the
condition of causality, then
223
00:14:55,580 --> 00:14:59,530
what that requires is that
the impulse response be
224
00:14:59,530 --> 00:15:03,700
right-sided or the region of
convergence be outside the
225
00:15:03,700 --> 00:15:05,330
outermost pole.
226
00:15:05,330 --> 00:15:09,350
So, for this example that would
require, then, that the
227
00:15:09,350 --> 00:15:12,940
region of convergence correspond
to the magnitude of
228
00:15:12,940 --> 00:15:18,190
z greater than the
magnitude of a.
229
00:15:18,190 --> 00:15:21,670
What you might think about is
whether I would get the same
230
00:15:21,670 --> 00:15:25,960
condition if I required
instead that
231
00:15:25,960 --> 00:15:29,310
the system be stable.
232
00:15:29,310 --> 00:15:31,500
Furthermore, you could think
about the question of whether
233
00:15:31,500 --> 00:15:35,480
I could specify or impose on
this system that it be both
234
00:15:35,480 --> 00:15:37,090
stable and causal.
235
00:15:37,090 --> 00:15:39,760
The real issue-- and let me
just kind of point to it--
236
00:15:39,760 --> 00:15:43,970
is that the answer to those
questions relate to whether
237
00:15:43,970 --> 00:15:46,730
the magnitude of a is less than
1 or the magnitude of a
238
00:15:46,730 --> 00:15:48,160
is greater than 1.
239
00:15:48,160 --> 00:15:50,810
If the magnitude of a is less
than 1 and I specify
240
00:15:50,810 --> 00:15:57,320
causality, that will also mean
that the system is stable.
241
00:15:57,320 --> 00:16:02,170
In any case, given this region
of convergence, then the
242
00:16:02,170 --> 00:16:06,340
impulse response is the inverse
transform of that
243
00:16:06,340 --> 00:16:08,780
z-transform, which is a
to the n times u of n.
244
00:16:08,780 --> 00:16:11,490
245
00:16:11,490 --> 00:16:15,680
Now let's look at a second order
equation, and there's a
246
00:16:15,680 --> 00:16:17,470
very similar strategy.
247
00:16:17,470 --> 00:16:21,430
For the second order equation,
the one that I've picked is of
248
00:16:21,430 --> 00:16:24,700
this particular form, and I've
written the coefficients
249
00:16:24,700 --> 00:16:28,270
parametrically as I indicate
here for a specific reason,
250
00:16:28,270 --> 00:16:29,880
which we'll see shortly.
251
00:16:29,880 --> 00:16:32,870
Again, I can apply the
z-transform to this
252
00:16:32,870 --> 00:16:38,580
expression, and I've skipped an
algebraic step or two here.
253
00:16:38,580 --> 00:16:42,970
When I do this, then, again I
use the linearity property and
254
00:16:42,970 --> 00:16:46,790
the shifting property, and I
end up with this algebraic
255
00:16:46,790 --> 00:16:47,890
expression.
256
00:16:47,890 --> 00:16:54,240
If I now solve that to express
y of z in terms of x of z and
257
00:16:54,240 --> 00:16:58,790
a function of z, then
this is what I get
258
00:16:58,790 --> 00:17:00,040
for the system function.
259
00:17:00,040 --> 00:17:02,430
260
00:17:02,430 --> 00:17:06,970
So this is now a second order
system function, and we'll
261
00:17:06,970 --> 00:17:12,960
have two zeroes at the origin
and it will have two poles.
262
00:17:12,960 --> 00:17:17,339
Again, there's the question of
what we assume about the
263
00:17:17,339 --> 00:17:18,349
region of convergence--
264
00:17:18,349 --> 00:17:20,240
I haven't specified that yet.
265
00:17:20,240 --> 00:17:25,990
But if we, let's say, assume
that the system is causal,
266
00:17:25,990 --> 00:17:29,050
which I will tend to do, then
that means that the region of
267
00:17:29,050 --> 00:17:32,830
convergence is outside
the outermost pole.
268
00:17:32,830 --> 00:17:34,080
Now, where are the poles?
269
00:17:34,080 --> 00:17:38,920
Well, let me just kind
of indicate that if--
270
00:17:38,920 --> 00:17:40,010
and you can verify this
271
00:17:40,010 --> 00:17:43,250
algebraically at your leisure--
272
00:17:43,250 --> 00:17:52,680
that if the cosine theta term is
less than 1, then the roots
273
00:17:52,680 --> 00:17:56,110
of this polynomial
will be complex.
274
00:17:56,110 --> 00:18:00,570
And in fact, the poles
are at r e to the
275
00:18:00,570 --> 00:18:02,740
plus or minus j theta.
276
00:18:02,740 --> 00:18:08,970
So for cosine theta less than 1,
then the poles are complex,
277
00:18:08,970 --> 00:18:13,960
and the complex poles at an
angle theta and with a
278
00:18:13,960 --> 00:18:18,070
distance from the origin equal
to the parameter r.
279
00:18:18,070 --> 00:18:21,550
In fact, let's just
look at that.
280
00:18:21,550 --> 00:18:27,640
What I show here is the
pole zero pattern.
281
00:18:27,640 --> 00:18:34,490
Assuming that r is less than
1, and that cosine theta is
282
00:18:34,490 --> 00:18:40,300
less than 1, and we have a
complex pole pair shown here--
283
00:18:40,300 --> 00:18:44,210
now if we assume that the system
was causal, that means
284
00:18:44,210 --> 00:18:48,240
that the region of convergence
is outside these poles.
285
00:18:48,240 --> 00:18:51,300
That would then include the unit
circle, which means that
286
00:18:51,300 --> 00:18:54,290
the system is also stable.
287
00:18:54,290 --> 00:18:57,370
In fact, as long as the reach
of convergence includes the
288
00:18:57,370 --> 00:19:02,850
unit circle, we can also talk
about the frequency response
289
00:19:02,850 --> 00:19:03,680
of the system--
290
00:19:03,680 --> 00:19:05,240
namely, we can evaluate
the system
291
00:19:05,240 --> 00:19:08,500
function on the unit circle.
292
00:19:08,500 --> 00:19:13,270
We, in fact, evaluated the
Fourier transform associated
293
00:19:13,270 --> 00:19:16,660
with this pole zero
pattern last time.
294
00:19:16,660 --> 00:19:21,210
Recall that the frequency
response, then, is one that
295
00:19:21,210 --> 00:19:26,620
has a resonant character with
the resonant peak being
296
00:19:26,620 --> 00:19:33,010
roughly in the vicinity of the
angle of the pole location.
297
00:19:33,010 --> 00:19:36,490
As the parameter r varies--
let's say, as r gets smaller--
298
00:19:36,490 --> 00:19:38,710
this peak tends to broaden.
299
00:19:38,710 --> 00:19:41,980
As r gets closer to
1, the resonance
300
00:19:41,980 --> 00:19:43,230
tends to get sharper.
301
00:19:43,230 --> 00:19:49,020
302
00:19:49,020 --> 00:19:55,100
This is now a look at the
z-transform, and we see very
303
00:19:55,100 --> 00:19:59,000
strong parallels to the
Laplace transform.
304
00:19:59,000 --> 00:20:02,920
In fact, throughout the
course, I've tried to
305
00:20:02,920 --> 00:20:03,700
emphasize--
306
00:20:03,700 --> 00:20:05,590
and it just naturally
happens--
307
00:20:05,590 --> 00:20:10,370
that there are very strong
relationships and parallels
308
00:20:10,370 --> 00:20:14,890
between continuous time
and discrete time.
309
00:20:14,890 --> 00:20:19,940
In fact, at one point we
specifically mapped from
310
00:20:19,940 --> 00:20:22,960
continuous time to discrete
time when we talked about
311
00:20:22,960 --> 00:20:27,460
discrete time processing of
continuous time signals.
312
00:20:27,460 --> 00:20:32,760
What I'd like to do now is turn
our attention to another
313
00:20:32,760 --> 00:20:38,000
very important reason for
mapping from continuous time
314
00:20:38,000 --> 00:20:41,600
to discrete time, and in the
process of doing this, what
315
00:20:41,600 --> 00:20:46,670
we'll need to do is exploit
fairly heavily the insight,
316
00:20:46,670 --> 00:20:51,210
intuition, and procedures that
we've developed for the
317
00:20:51,210 --> 00:20:55,150
Laplace transform and
the z-transform.
318
00:20:55,150 --> 00:20:59,540
Specifically, what I would like
to begin is a discussion
319
00:20:59,540 --> 00:21:06,530
relating to mapping continuous
time filters to discrete time
320
00:21:06,530 --> 00:21:12,260
filters, or continuous time
system functions to discrete
321
00:21:12,260 --> 00:21:14,410
time system functions.
322
00:21:14,410 --> 00:21:17,170
Now, why would we
want to do that?
323
00:21:17,170 --> 00:21:21,130
Well, there are at least several
reasons for wanting to
324
00:21:21,130 --> 00:21:25,140
map continuous time filters
to discrete time filters.
325
00:21:25,140 --> 00:21:29,940
One, of course, is the fact that
in some situations what
326
00:21:29,940 --> 00:21:34,140
we're interested in doing is
processing continuous time
327
00:21:34,140 --> 00:21:38,220
signals with discrete
time systems--
328
00:21:38,220 --> 00:21:45,050
or, said another way, simulate
continuous time systems with
329
00:21:45,050 --> 00:21:46,690
discrete time systems.
330
00:21:46,690 --> 00:21:49,530
So it would be natural in a
setting like that to think of
331
00:21:49,530 --> 00:21:53,500
mapping the desired continuous
time filter to a
332
00:21:53,500 --> 00:21:55,840
discrete time filter.
333
00:21:55,840 --> 00:21:58,480
So that's one very important
context.
334
00:21:58,480 --> 00:22:02,880
There's another very important
context in which this is done,
335
00:22:02,880 --> 00:22:11,650
and that is in the context or
for the purpose of exploiting
336
00:22:11,650 --> 00:22:16,180
established design procedures
for continuous time filters.
337
00:22:16,180 --> 00:22:18,440
The point is the following.
338
00:22:18,440 --> 00:22:22,240
We may or may not be processing
a sample continuous
339
00:22:22,240 --> 00:22:25,260
time signal with our discrete
time filter-- it may just be
340
00:22:25,260 --> 00:22:28,630
discrete time signals that
we're working with.
341
00:22:28,630 --> 00:22:34,460
But in that situation, still,
we need to design the
342
00:22:34,460 --> 00:22:38,280
appropriate discrete
time filter.
343
00:22:38,280 --> 00:22:46,300
Historically, there is a very
rich history associated with
344
00:22:46,300 --> 00:22:50,290
design of continuous
time filters.
345
00:22:50,290 --> 00:22:54,860
In many cases, it's possible
and very worthwhile and
346
00:22:54,860 --> 00:23:00,790
efficient to take those designs
and map them to
347
00:23:00,790 --> 00:23:04,880
discrete time designs to use
them as discrete time filters.
348
00:23:04,880 --> 00:23:08,650
So, another very important
reason for talking about the
349
00:23:08,650 --> 00:23:13,120
kinds of mappings that we will
be going into is to simply
350
00:23:13,120 --> 00:23:16,800
take advantage of what has been
done historically in the
351
00:23:16,800 --> 00:23:18,050
continuous time case.
352
00:23:18,050 --> 00:23:21,960
353
00:23:21,960 --> 00:23:26,160
Now, if we want to map
continuous time filters to
354
00:23:26,160 --> 00:23:31,330
discrete time filters, then in
continuous time, we're talking
355
00:23:31,330 --> 00:23:34,820
about a system function
and an associated
356
00:23:34,820 --> 00:23:37,840
differential equation.
357
00:23:37,840 --> 00:23:43,230
In discrete time, there is the
corresponding system function
358
00:23:43,230 --> 00:23:46,105
and the corresponding
difference equation.
359
00:23:46,105 --> 00:23:49,170
360
00:23:49,170 --> 00:23:53,960
Basically what we want to do is
generate from a continuous
361
00:23:53,960 --> 00:23:59,640
time system in some way a
discrete time system that
362
00:23:59,640 --> 00:24:05,650
meets an associated set of
desired specifications.
363
00:24:05,650 --> 00:24:09,530
Now, there are certain
constraints that it's
364
00:24:09,530 --> 00:24:13,410
reasonable and important to
impose on whatever kinds of
365
00:24:13,410 --> 00:24:15,940
mappings we use.
366
00:24:15,940 --> 00:24:19,230
Obviously, we want a mapping
that will take our continuous
367
00:24:19,230 --> 00:24:21,900
time system function
and map it to a
368
00:24:21,900 --> 00:24:24,780
discrete time system function.
369
00:24:24,780 --> 00:24:28,210
Correspondingly in the time
domain, there is a continuous
370
00:24:28,210 --> 00:24:32,520
time impulse response that
maps to the associated
371
00:24:32,520 --> 00:24:35,610
discrete time impulse
response.
372
00:24:35,610 --> 00:24:37,710
These are more or
less natural.
373
00:24:37,710 --> 00:24:41,310
The two that are important and
sometimes easy to lose sight
374
00:24:41,310 --> 00:24:45,450
of are the two that
I indicate here.
375
00:24:45,450 --> 00:24:50,970
In particular, if we are mapping
a continuous time
376
00:24:50,970 --> 00:24:55,130
filter with, let's say, a
desired or desirable frequency
377
00:24:55,130 --> 00:24:59,680
response to a discrete time
filter and we would like to
378
00:24:59,680 --> 00:25:03,360
preserve the good qualities of
that frequency response as we
379
00:25:03,360 --> 00:25:09,460
look at the discrete time
frequency response, then it's
380
00:25:09,460 --> 00:25:14,680
important what happens in the
s-plane for the continuous
381
00:25:14,680 --> 00:25:19,860
time filter along the j omega
axis relate in a nice way to
382
00:25:19,860 --> 00:25:24,310
what happens in the z-plane
around the unit circle,
383
00:25:24,310 --> 00:25:29,170
because it's this over here that
represents the frequency
384
00:25:29,170 --> 00:25:33,440
response in continuous time
and this contour over here
385
00:25:33,440 --> 00:25:36,570
that represents the frequency
response in discrete time.
386
00:25:36,570 --> 00:25:38,090
So that's an important
property.
387
00:25:38,090 --> 00:25:41,100
We want to kind of the
j omega axis to
388
00:25:41,100 --> 00:25:44,370
map to the unit circle.
389
00:25:44,370 --> 00:25:49,160
Another more or less natural
condition to impose is a
390
00:25:49,160 --> 00:25:53,960
condition that if we are assured
in some way that our
391
00:25:53,960 --> 00:25:59,400
continuous time filter is
stable, then we would like to
392
00:25:59,400 --> 00:26:02,510
concentrate on design procedures
that more or less
393
00:26:02,510 --> 00:26:07,550
preserve that and will give
us stable digital filters.
394
00:26:07,550 --> 00:26:11,150
So these are kind of reasonable
conditions to
395
00:26:11,150 --> 00:26:13,150
impose on the procedure.
396
00:26:13,150 --> 00:26:16,960
What I'd like to do in the
remainder of this lecture is
397
00:26:16,960 --> 00:26:22,040
look at two common procedures
for mapping continuous time
398
00:26:22,040 --> 00:26:25,370
filters to discrete
time filters.
399
00:26:25,370 --> 00:26:28,020
The first one that I want to
talk about is one that, in
400
00:26:28,020 --> 00:26:35,210
fact, is very frequently used,
and also one that, as we'll
401
00:26:35,210 --> 00:26:40,990
see for a variety of reasons,
is highly undesirable.
402
00:26:40,990 --> 00:26:44,930
The second is one that is also
frequently used, and as we'll
403
00:26:44,930 --> 00:26:50,280
see is, in certain situations,
very desirable.
404
00:26:50,280 --> 00:26:53,830
The first one that I want to
talk about is the more or less
405
00:26:53,830 --> 00:27:00,260
intuitive simple procedure
of mapping a differential
406
00:27:00,260 --> 00:27:04,790
equation to a difference
equation by simply replacing
407
00:27:04,790 --> 00:27:07,265
derivatives by differences.
408
00:27:07,265 --> 00:27:09,890
409
00:27:09,890 --> 00:27:13,870
The idea is that a derivative is
more or less a difference,
410
00:27:13,870 --> 00:27:16,390
and there's some dummy parameter
capital T that I've
411
00:27:16,390 --> 00:27:19,720
thrown in here, which I won't
focus too much on.
412
00:27:19,720 --> 00:27:24,870
But in any case, this seems
to have some plausibility.
413
00:27:24,870 --> 00:27:27,730
If we take the differential
equation and do this with all
414
00:27:27,730 --> 00:27:32,410
the derivatives, both in terms
of y of t and x of t, what
415
00:27:32,410 --> 00:27:36,560
we'll end up with is a
difference equation.
416
00:27:36,560 --> 00:27:42,540
Now what we can use are the
properties of the Laplace
417
00:27:42,540 --> 00:27:45,920
transform and the z-transform
to see what this means in
418
00:27:45,920 --> 00:27:47,440
terms of a mapping--
419
00:27:47,440 --> 00:27:50,940
in particular, using the
differentiation property for
420
00:27:50,940 --> 00:27:52,390
Laplace transforms.
421
00:27:52,390 --> 00:27:56,970
In the Laplace transform domain,
we would have this.
422
00:27:56,970 --> 00:28:02,400
Using the properties for the
z-transform, the z-transform
423
00:28:02,400 --> 00:28:04,740
of this expression
would be this.
424
00:28:04,740 --> 00:28:09,950
So, in effect, what it says is
that every place in the system
425
00:28:09,950 --> 00:28:12,370
function or in the differential
equation that we
426
00:28:12,370 --> 00:28:16,460
would be multiplying by s when
Laplace transformed.
427
00:28:16,460 --> 00:28:18,000
In the difference equation,
we would be
428
00:28:18,000 --> 00:28:21,070
multiplying by this factor.
429
00:28:21,070 --> 00:28:26,840
In fact, what this means is
that the mapping from
430
00:28:26,840 --> 00:28:32,000
continuous time to discrete time
corresponds to taking the
431
00:28:32,000 --> 00:28:38,200
system function and replacing
s wherever we see it by 1
432
00:28:38,200 --> 00:28:43,110
minus z to the minus 1 over
capital T. So if we have a
433
00:28:43,110 --> 00:28:47,400
system function in continuous
time and we map it to a
434
00:28:47,400 --> 00:28:51,220
discrete time system function
this way by replacing
435
00:28:51,220 --> 00:28:55,700
derivatives by differences,
then that corresponds to
436
00:28:55,700 --> 00:29:01,390
replacing s by 1 minus z to the
minus 1 over capital T.
437
00:29:01,390 --> 00:29:06,190
Now we'll see shortly what this
mapping actually means
438
00:29:06,190 --> 00:29:09,980
more specifically in relating
the s-plane to the z-plane.
439
00:29:09,980 --> 00:29:13,370
Let me just quickly, because I
want to refer to this, also
440
00:29:13,370 --> 00:29:16,670
point to another procedure
very much like backward
441
00:29:16,670 --> 00:29:20,910
differences which corresponds
to replacing derivatives not
442
00:29:20,910 --> 00:29:23,770
by the backward differences
that I just showed, but by
443
00:29:23,770 --> 00:29:25,640
forward differences.
444
00:29:25,640 --> 00:29:29,600
In that case, then, the
mapping corresponds to
445
00:29:29,600 --> 00:29:36,030
replacing s by z to the minus
1 over capital T. It looks
446
00:29:36,030 --> 00:29:39,270
very similar to the
previous case.
447
00:29:39,270 --> 00:29:43,330
So there the relationship
between these system functions
448
00:29:43,330 --> 00:29:44,860
is what I indicate here.
449
00:29:44,860 --> 00:29:47,800
450
00:29:47,800 --> 00:29:53,590
Let's just take a look at what
those mappings correspond to
451
00:29:53,590 --> 00:29:56,730
when we look at this
specifically in the s-plane
452
00:29:56,730 --> 00:29:59,250
and in the z-plane.
453
00:29:59,250 --> 00:30:04,920
What I show here is the s-plane,
and of course it's
454
00:30:04,920 --> 00:30:07,350
things on the left half of the
s-plane, poles on the left
455
00:30:07,350 --> 00:30:10,350
half of the s-plane that would
guarantee stability.
456
00:30:10,350 --> 00:30:14,340
It's the j omega axis the tells
us about the frequency
457
00:30:14,340 --> 00:30:21,630
response, and in the z-plane
it's the unit circle that
458
00:30:21,630 --> 00:30:24,490
tells us about the frequency
response.
459
00:30:24,490 --> 00:30:27,560
Things inside the unit circle,
or poles inside the unit
460
00:30:27,560 --> 00:30:30,170
circle, that guarantee
stability.
461
00:30:30,170 --> 00:30:35,310
Now the mapping from s-plane to
the z-plane corresponding
462
00:30:35,310 --> 00:30:40,660
to replacing derivatives by
backward differences in fact
463
00:30:40,660 --> 00:30:48,140
can be shown to correspond to
mapping the j omega axis not
464
00:30:48,140 --> 00:30:51,780
to the unit circle, but to the
little circle that I show
465
00:30:51,780 --> 00:30:55,230
here, which is inside
the unit circle.
466
00:30:55,230 --> 00:30:58,550
The left half of the
s-plane maps to the
467
00:30:58,550 --> 00:31:00,430
inside of that circle.
468
00:31:00,430 --> 00:31:01,210
What does that mean?
469
00:31:01,210 --> 00:31:06,500
That means that if we have a
really good frequency response
470
00:31:06,500 --> 00:31:11,150
characteristic along this
contour in the s-plane, we'll
471
00:31:11,150 --> 00:31:13,270
see that same frequency
response
472
00:31:13,270 --> 00:31:14,910
along this little circle.
473
00:31:14,910 --> 00:31:17,030
That's not the one that
we want, though--
474
00:31:17,030 --> 00:31:20,420
we would like to see that same
frequency response around the
475
00:31:20,420 --> 00:31:22,080
unit circle.
476
00:31:22,080 --> 00:31:26,960
To emphasize this point even
more-- suppose, for example,
477
00:31:26,960 --> 00:31:33,230
that we had a pair of poles in
our continuous time system
478
00:31:33,230 --> 00:31:36,000
function that looked
like this.
479
00:31:36,000 --> 00:31:42,610
Then, where they're likely to
end up in the z-plane is
480
00:31:42,610 --> 00:31:45,050
inside the unit circle,
of course.
481
00:31:45,050 --> 00:31:50,080
But if the poles here are close
to the j omega axis,
482
00:31:50,080 --> 00:31:53,930
that means that these poles will
be close to this circle,
483
00:31:53,930 --> 00:31:58,320
but in fact might be very far
away from the unit circle.
484
00:31:58,320 --> 00:32:01,530
What would happen, then, is
that if we saw in the
485
00:32:01,530 --> 00:32:06,220
continuous time filter a very
sharp resonance, the discrete
486
00:32:06,220 --> 00:32:08,900
time filter in fact might very
well have that resonance
487
00:32:08,900 --> 00:32:12,660
broadened considerably because
the poles are so far away from
488
00:32:12,660 --> 00:32:14,500
the unit circle.
489
00:32:14,500 --> 00:32:19,530
Now, one plus with this method,
and it's about the
490
00:32:19,530 --> 00:32:25,570
only one, is the fact that the
left half of the s-plane maps
491
00:32:25,570 --> 00:32:30,110
inside the unit circle-- in
fact, inside a circle inside
492
00:32:30,110 --> 00:32:32,250
the unit circle, and so
stability is always
493
00:32:32,250 --> 00:32:33,870
guaranteed.
494
00:32:33,870 --> 00:32:36,170
Let me just quickly mention,
and you'll have a chance to
495
00:32:36,170 --> 00:32:38,910
look at this a little more
carefully in the video course
496
00:32:38,910 --> 00:32:42,960
manual, that for forward
differences instead of
497
00:32:42,960 --> 00:32:51,120
backward differences, this
contour in the s-plane maps to
498
00:32:51,120 --> 00:32:56,380
a line in the z-plane, which is
a line tangent to the unit
499
00:32:56,380 --> 00:33:01,250
circle, and in fact is the
line that I showed there.
500
00:33:01,250 --> 00:33:05,890
So not only are forward
differences equally bad in
501
00:33:05,890 --> 00:33:09,270
terms of the issue of whether
they map from the j omega axis
502
00:33:09,270 --> 00:33:13,440
to the unit circle, but they
have a further difficulty
503
00:33:13,440 --> 00:33:19,360
associated with them, which is
the difficulty they may not
504
00:33:19,360 --> 00:33:21,505
and generally don't guarantee
stability.
505
00:33:21,505 --> 00:33:25,070
506
00:33:25,070 --> 00:33:29,840
Now, that's one method, and one,
as I indicated, that's
507
00:33:29,840 --> 00:33:35,600
often used partly
because it seems
508
00:33:35,600 --> 00:33:37,510
so intuitively plausible.
509
00:33:37,510 --> 00:33:40,960
What you can see is that by
understanding carefully the
510
00:33:40,960 --> 00:33:44,500
issues and the techniques of
Laplace and z-transforms, you
511
00:33:44,500 --> 00:33:48,280
can begin to see what some of
the difficulties with those
512
00:33:48,280 --> 00:33:50,450
methods are.
513
00:33:50,450 --> 00:33:54,080
The next method that I'd like
to talk about is a method
514
00:33:54,080 --> 00:33:57,000
that, in fact, is very
commonly used.
515
00:33:57,000 --> 00:34:01,360
It's a very important, useful
method, which kind of can be
516
00:34:01,360 --> 00:34:06,330
motivated by thinking along
the lines of mapping the
517
00:34:06,330 --> 00:34:11,630
continuous time filter to a
discrete time filter in such a
518
00:34:11,630 --> 00:34:15,580
way that the shape of the
impulse response is
519
00:34:15,580 --> 00:34:16,620
preserved--
520
00:34:16,620 --> 00:34:20,989
and, in fact, more specifically
so that the
521
00:34:20,989 --> 00:34:26,590
discrete time impulse response
corresponds to samples of the
522
00:34:26,590 --> 00:34:28,960
continuous time impulse
response.
523
00:34:28,960 --> 00:34:32,800
And this is a method that's
referred to as impulse
524
00:34:32,800 --> 00:34:35,260
invariance.
525
00:34:35,260 --> 00:34:40,790
So what impulse invariance
corresponds to is designing
526
00:34:40,790 --> 00:34:46,100
the filter in such a way that
the discrete time filter
527
00:34:46,100 --> 00:34:51,550
impulse response is simply
a sample version of the
528
00:34:51,550 --> 00:34:57,200
continuous time filter impulse
response with a sampling
529
00:34:57,200 --> 00:35:01,450
period which I denote here as
capital T. That will turn into
530
00:35:01,450 --> 00:35:06,350
a slightly confusing parameter
shortly, and perhaps carried
531
00:35:06,350 --> 00:35:08,270
over into the next lecture.
532
00:35:08,270 --> 00:35:11,900
Hopefully, we'll get that
straighted out, though, within
533
00:35:11,900 --> 00:35:14,030
those two lectures.
534
00:35:14,030 --> 00:35:18,680
Remembering the issues of
sampling, the discrete time
535
00:35:18,680 --> 00:35:22,970
frequency response, then since
the frequency responses the
536
00:35:22,970 --> 00:35:27,180
Fourier transform of the impulse
response is related to
537
00:35:27,180 --> 00:35:31,950
the continuous time impulse
response as I indicate here,
538
00:35:31,950 --> 00:35:41,130
what this says is that it
is the superposition of
539
00:35:41,130 --> 00:35:46,600
replications of the continuous
time frequency response,
540
00:35:46,600 --> 00:35:51,440
linearly scaled in frequency
and shifted and
541
00:35:51,440 --> 00:35:53,120
added to each other.
542
00:35:53,120 --> 00:35:56,830
It's the same old sort of
shifting, adding, or aliasing
543
00:35:56,830 --> 00:35:58,430
issue-- the same sampling
issues that
544
00:35:58,430 --> 00:36:01,740
we've addressed before.
545
00:36:01,740 --> 00:36:04,690
This equation will help us
understand what the frequency
546
00:36:04,690 --> 00:36:06,290
response looks like.
547
00:36:06,290 --> 00:36:11,560
But in terms of an analytical
procedure for mapping the
548
00:36:11,560 --> 00:36:14,540
continuous time system function
to a discrete time
549
00:36:14,540 --> 00:36:20,020
system function, we can see
that and develop it in the
550
00:36:20,020 --> 00:36:22,200
following way.
551
00:36:22,200 --> 00:36:27,000
Let's consider the continuous
time system function expanded
552
00:36:27,000 --> 00:36:30,250
in a partial fraction
expansion.
553
00:36:30,250 --> 00:36:33,320
And just for convenience, I'm
picking first order poles--
554
00:36:33,320 --> 00:36:44,540
this can be generalized multiple
order poles, and we
555
00:36:44,540 --> 00:36:45,730
won't do that here.
556
00:36:45,730 --> 00:36:48,200
The same basic strategy
applies.
557
00:36:48,200 --> 00:36:54,120
If we expand this in a partial
fraction expansion, and we
558
00:36:54,120 --> 00:36:57,210
look at the impulse response
associated with this-- we know
559
00:36:57,210 --> 00:36:59,940
how to take the inverse of
Laplace transform of this,
560
00:36:59,940 --> 00:37:03,280
where I'm just naturally
assuming causality throughout
561
00:37:03,280 --> 00:37:05,490
the discussion--
562
00:37:05,490 --> 00:37:09,070
the continuous time impulse
response, then, is the sum of
563
00:37:09,070 --> 00:37:14,610
exponentials with these
amplitudes and at these
564
00:37:14,610 --> 00:37:17,910
complex exponential
frequencies.
565
00:37:17,910 --> 00:37:21,570
Now, impulse invariance
corresponds to sampling this,
566
00:37:21,570 --> 00:37:26,090
and so the discrete time impulse
response is simply a
567
00:37:26,090 --> 00:37:29,700
sampled version of this.
568
00:37:29,700 --> 00:37:33,720
The A sub k, of course,
carries down.
569
00:37:33,720 --> 00:37:35,440
We have the exponential--
570
00:37:35,440 --> 00:37:39,340
we're sampling at t equals
n capital T, and so we've
571
00:37:39,340 --> 00:37:43,550
replaced that here, and then
the unit step to truncate
572
00:37:43,550 --> 00:37:46,550
things for negative time.
573
00:37:46,550 --> 00:37:50,360
Let's manipulate this further,
and eventually what we want to
574
00:37:50,360 --> 00:37:52,530
get is a relationship--
575
00:37:52,530 --> 00:37:53,230
a mapping--
576
00:37:53,230 --> 00:37:58,360
from the continuous time to
the discrete time filter.
577
00:37:58,360 --> 00:38:04,380
We have this step, and we can
rewrite that now as I show
578
00:38:04,380 --> 00:38:08,860
here, just simply taking this
n outside, and we have e to
579
00:38:08,860 --> 00:38:13,770
the s sub k capital T. Now this
is of the form the sum of
580
00:38:13,770 --> 00:38:19,100
terms like A sub k times
is beta to the n.
581
00:38:19,100 --> 00:38:23,750
We can compute the z-transform
of this, and the z-transform
582
00:38:23,750 --> 00:38:26,840
that we get I show here--
583
00:38:26,840 --> 00:38:30,680
it's simply A sub k over 1 minus
e to the s sub k capital
584
00:38:30,680 --> 00:38:36,700
T z to the minus 1, simply
carrying this term or this
585
00:38:36,700 --> 00:38:39,720
parameter down.
586
00:38:39,720 --> 00:38:42,750
So we started with a continuous
time system
587
00:38:42,750 --> 00:38:46,800
function, which was a sum of
terms like A sub k over s
588
00:38:46,800 --> 00:38:49,730
minus s sub k-- the poles
were at s sub k.
589
00:38:49,730 --> 00:38:54,200
We now have the discrete time
filter in this form.
590
00:38:54,200 --> 00:38:57,110
Consequently, then, this
procedure of impulse
591
00:38:57,110 --> 00:39:03,400
invariance corresponds to
mapping the continuous time
592
00:39:03,400 --> 00:39:07,840
filter to a discrete time filter
by mapping the poles in
593
00:39:07,840 --> 00:39:10,080
the continuous time filter.
594
00:39:10,080 --> 00:39:15,870
According to this mapping, the
continuous time filter pole at
595
00:39:15,870 --> 00:39:20,530
s sub k gets mapped to a pole
e to the s sub k capital T,
596
00:39:20,530 --> 00:39:27,280
and the coefficients A
sub k are preserved.
597
00:39:27,280 --> 00:39:31,210
That, then, algebraically, is
what the procedure of impulse
598
00:39:31,210 --> 00:39:34,960
invariance corresponds to.
599
00:39:34,960 --> 00:39:39,630
Let's look at how we interpret
some of this in
600
00:39:39,630 --> 00:39:42,180
the frequency domain.
601
00:39:42,180 --> 00:39:49,020
In particular, we have the
expression that tells us how
602
00:39:49,020 --> 00:39:54,700
the discrete time frequency
response is related to the
603
00:39:54,700 --> 00:39:57,470
continuous time frequency
response.
604
00:39:57,470 --> 00:40:02,570
This is the expression that we
had previously when we had
605
00:40:02,570 --> 00:40:05,490
talked about issues
of sampling.
606
00:40:05,490 --> 00:40:07,910
So that means that we would
form the discrete time
607
00:40:07,910 --> 00:40:12,460
frequency response by taking the
continuous time 1, scaling
608
00:40:12,460 --> 00:40:17,100
it in frequency according to
this parameter capital T, and
609
00:40:17,100 --> 00:40:21,190
then adding replications
of that together.
610
00:40:21,190 --> 00:40:26,890
So if this is the continuous
time frequency response, just
611
00:40:26,890 --> 00:40:30,460
simply an ideal low-pass filter
with a cutoff frequency
612
00:40:30,460 --> 00:40:36,400
of omega sub c, then the
frequency scaling operation
613
00:40:36,400 --> 00:40:42,130
would keep the same basic shape
but linearly scale the
614
00:40:42,130 --> 00:40:47,600
frequency axis so that we now
have omega sub c times T. Then
615
00:40:47,600 --> 00:40:52,060
the discrete time frequency
response would be a
616
00:40:52,060 --> 00:40:57,400
superposition of these added
together at multiples of 2 pi
617
00:40:57,400 --> 00:41:00,190
in discrete time frequency.
618
00:41:00,190 --> 00:41:03,350
So that's what we have here--
619
00:41:03,350 --> 00:41:06,245
so this is the discrete time
frequency response.
620
00:41:06,245 --> 00:41:08,920
621
00:41:08,920 --> 00:41:11,810
This looks very nice-- it looks
like impulse invariance
622
00:41:11,810 --> 00:41:15,610
will take the continuous time
frequency response, just
623
00:41:15,610 --> 00:41:18,470
simply linearly scale the
frequency axis, and
624
00:41:18,470 --> 00:41:21,210
incidentally periodically
repeat it.
625
00:41:21,210 --> 00:41:24,360
We know that for an ideal
low-pass filter,
626
00:41:24,360 --> 00:41:25,670
that looks just fine.
627
00:41:25,670 --> 00:41:28,940
In fact, for a band-limited
frequency response,
628
00:41:28,940 --> 00:41:31,300
that looks just fine.
629
00:41:31,300 --> 00:41:35,355
But we know also that any time
that we're sampling--
630
00:41:35,355 --> 00:41:38,210
and here we're sampling
the impulse response--
631
00:41:38,210 --> 00:41:42,210
we have an effect in the
frequency domain or the
632
00:41:42,210 --> 00:41:44,440
potential for an affect
an effect that
633
00:41:44,440 --> 00:41:46,710
we refer to as aliasing.
634
00:41:46,710 --> 00:41:52,810
So in fact, if instead of the
ideal low-pass filter we had
635
00:41:52,810 --> 00:41:56,030
taken a filter that was an
approximation to a low-pass
636
00:41:56,030 --> 00:42:01,510
filter, then the corresponding
frequency scale version would
637
00:42:01,510 --> 00:42:03,590
look as I've shown here.
638
00:42:03,590 --> 00:42:08,390
And now as we add these
together, then what we will
639
00:42:08,390 --> 00:42:13,070
have is some potential for
distortion corresponding to
640
00:42:13,070 --> 00:42:17,530
the fact that these replications
overlap, and what
641
00:42:17,530 --> 00:42:19,660
that will lead to is aliasing.
642
00:42:19,660 --> 00:42:22,330
643
00:42:22,330 --> 00:42:26,720
So some things that we can say
about impulse invariance is
644
00:42:26,720 --> 00:42:29,270
that we have an algebraic
procedure--
645
00:42:29,270 --> 00:42:32,370
and I'll illustrate with another
example shortly--
646
00:42:32,370 --> 00:42:35,200
for taking a continuous time
system function and mapping it
647
00:42:35,200 --> 00:42:37,680
to a discrete time
system function.
648
00:42:37,680 --> 00:42:44,290
It has a very nice property in
terms of mapping, the mapping
649
00:42:44,290 --> 00:42:48,910
from the frequency axis in
continuous time due to the
650
00:42:48,910 --> 00:42:50,190
unit circle--
651
00:42:50,190 --> 00:42:53,750
namely, to a first
approximation.
652
00:42:53,750 --> 00:42:57,900
As long as there's no aliasing,
the mapping is just
653
00:42:57,900 --> 00:43:01,200
simply a linear scaling of the
frequency axis, although there
654
00:43:01,200 --> 00:43:02,910
may be some aliasing.
655
00:43:02,910 --> 00:43:06,640
That means, of course, that this
method can only be used
656
00:43:06,640 --> 00:43:09,810
if the frequency response that's
being mapped, or if the
657
00:43:09,810 --> 00:43:13,300
system that's being mapped,
has a frequency response
658
00:43:13,300 --> 00:43:15,300
that's approximately
low-pass--
659
00:43:15,300 --> 00:43:16,940
it has to be approximately
band-limited.
660
00:43:16,940 --> 00:43:20,520
661
00:43:20,520 --> 00:43:23,970
Then what we have is some of
potential distortion, which
662
00:43:23,970 --> 00:43:26,540
comes about because
of aliasing.
663
00:43:26,540 --> 00:43:30,660
Also because of the mapping, the
fact that poles at s sub k
664
00:43:30,660 --> 00:43:36,290
get mapped to poles at e to the
s sub k capital T, if the
665
00:43:36,290 --> 00:43:39,960
analog or continuous time
filter is stable--
666
00:43:39,960 --> 00:43:43,050
meaning that the real part
of s sub k is negative--
667
00:43:43,050 --> 00:43:47,240
then the discrete time filter
is guaranteed to be stable.
668
00:43:47,240 --> 00:43:51,220
In other words, the magnitude of
z sub k will be guaranteed
669
00:43:51,220 --> 00:43:53,060
to be less than 1.
670
00:43:53,060 --> 00:43:56,300
I'm assuming, of course, in that
discussion that we are
671
00:43:56,300 --> 00:43:58,730
always imposing causality
on the systems.
672
00:43:58,730 --> 00:44:01,700
673
00:44:01,700 --> 00:44:06,030
To just look at the algebraic
mapping a little more
674
00:44:06,030 --> 00:44:09,650
carefully, let's take
a simple example.
675
00:44:09,650 --> 00:44:15,130
Here is an example of a system,
a continuous time
676
00:44:15,130 --> 00:44:21,690
system, where I simply have a
resident pole pair with an
677
00:44:21,690 --> 00:44:26,000
imaginary part along the
imaginary axis of omega sub r
678
00:44:26,000 --> 00:44:28,660
and a real part of
minus alpha.
679
00:44:28,660 --> 00:44:34,650
And so the associated system
function then is just the
680
00:44:34,650 --> 00:44:39,110
expression which incorporates
the two poles, and I've put in
681
00:44:39,110 --> 00:44:42,850
a scale factor of
2 omega sub r.
682
00:44:42,850 --> 00:44:47,160
And now to design the discrete
time filter using impulse
683
00:44:47,160 --> 00:44:49,990
invariance, you would carry
out a partial fraction
684
00:44:49,990 --> 00:44:53,690
expansion of this, and that
partial fraction expansion is
685
00:44:53,690 --> 00:44:55,990
shown below.
686
00:44:55,990 --> 00:44:59,550
We have a pole at minus alpha
minus j omega r and at minus
687
00:44:59,550 --> 00:45:02,130
alpha plus j omega r.
688
00:45:02,130 --> 00:45:07,430
And to determine the discrete
time filter based on impulse
689
00:45:07,430 --> 00:45:10,640
invariance, we would map the
poles and preserve the
690
00:45:10,640 --> 00:45:14,480
coefficients a sub k, referred
to as the residues.
691
00:45:14,480 --> 00:45:18,740
And so the discrete time filter
that we would end up
692
00:45:18,740 --> 00:45:25,560
with as a system function,
which I indicate here--
693
00:45:25,560 --> 00:45:30,870
and we have, as I said,
preserved the residue, and the
694
00:45:30,870 --> 00:45:34,820
pole is now at a to the minus
alpha T, e to the minus j
695
00:45:34,820 --> 00:45:40,020
omega sub r T. That's one term,
and the other term in
696
00:45:40,020 --> 00:45:44,950
the sum has a pole at the
complex conjugate location.
697
00:45:44,950 --> 00:45:50,090
If we were to add these two
factors together, then what we
698
00:45:50,090 --> 00:45:57,420
would get is both poles
and a 0 at the origin.
699
00:45:57,420 --> 00:46:04,270
In fact, then, the pole is
defined by its angle, and this
700
00:46:04,270 --> 00:46:15,050
angle is e to the j omega sub
r capital T, and by its
701
00:46:15,050 --> 00:46:24,030
radius, and this radius is e to
the minus alpha capital T.
702
00:46:24,030 --> 00:46:28,310
Now we can look at the frequency
response associated
703
00:46:28,310 --> 00:46:31,600
with that, and let's
just do that.
704
00:46:31,600 --> 00:46:40,730
For the original continuous time
frequency response, what
705
00:46:40,730 --> 00:46:45,900
we have is simply a resonant
character, as I've shown here.
706
00:46:45,900 --> 00:46:50,730
And if we map this using impulse
invariance, which we
707
00:46:50,730 --> 00:46:56,920
just did, the resulting
frequency response that we get
708
00:46:56,920 --> 00:46:59,670
is the frequency response
which I indicate.
709
00:46:59,670 --> 00:47:03,920
We see that that's basically
identical to the continuous
710
00:47:03,920 --> 00:47:09,430
time frequency response, except
for a linear scaling in
711
00:47:09,430 --> 00:47:13,510
the frequency axis, if you just
compare the dimensions
712
00:47:13,510 --> 00:47:21,210
along which the frequency axis
is shown except for one minor
713
00:47:21,210 --> 00:47:25,500
issue, which is particularly
highlighted when we look at
714
00:47:25,500 --> 00:47:30,180
the frequency response at
higher frequencies.
715
00:47:30,180 --> 00:47:35,140
What's the reason why those two
curves don't quite follow
716
00:47:35,140 --> 00:47:38,980
each other at higher
frequencies?
717
00:47:38,980 --> 00:47:41,790
Well, the reason is aliasing.
718
00:47:41,790 --> 00:47:44,980
In other words, what's happened
is that in the
719
00:47:44,980 --> 00:47:50,320
process so applying impulse
invariance, the frequency
720
00:47:50,320 --> 00:47:53,770
response of the original
continuous time filter is
721
00:47:53,770 --> 00:47:57,540
approximately preserved, except
for some distortion,
722
00:47:57,540 --> 00:48:01,420
that distortion corresponding
to aliasing.
723
00:48:01,420 --> 00:48:06,000
Well, just for comparison,
let's look at what would
724
00:48:06,000 --> 00:48:08,490
happen if we took the
same example--
725
00:48:08,490 --> 00:48:10,710
and we're not going to work
it through here carefully.
726
00:48:10,710 --> 00:48:15,100
We're not work it through it
all, not even not carefully.
727
00:48:15,100 --> 00:48:19,560
If we took the same example and
mapped it to a discrete
728
00:48:19,560 --> 00:48:22,720
time filter by replacing
derivatives by backward
729
00:48:22,720 --> 00:48:28,200
differences, what happens in
that case is that we get a
730
00:48:28,200 --> 00:48:32,030
frequency response that
I indicate here.
731
00:48:32,030 --> 00:48:36,720
Notice that the resonance in the
original continuous time
732
00:48:36,720 --> 00:48:41,020
filter is totally lost.
733
00:48:41,020 --> 00:48:44,950
In fact, basically the character
of the continuous
734
00:48:44,950 --> 00:48:47,510
time frequency response
is lost.
735
00:48:47,510 --> 00:48:48,410
What's the reason?
736
00:48:48,410 --> 00:48:49,920
Well, the reason goes
back to the
737
00:48:49,920 --> 00:48:51,810
picture that I drew before.
738
00:48:51,810 --> 00:48:58,920
The continuous time filter had a
pair of resident poles close
739
00:48:58,920 --> 00:49:00,830
to the j omega axis.
740
00:49:00,830 --> 00:49:03,850
When those get mapped using
backward differences, they end
741
00:49:03,850 --> 00:49:07,850
up close to this little circle
that's inside the unit circle,
742
00:49:07,850 --> 00:49:10,700
but in fact for this example,
are far away
743
00:49:10,700 --> 00:49:11,950
from the unit circle.
744
00:49:11,950 --> 00:49:16,350
745
00:49:16,350 --> 00:49:20,320
So far we have one useful
technique for mapping
746
00:49:20,320 --> 00:49:23,600
continuous time filters to
discrete time filters.
747
00:49:23,600 --> 00:49:27,160
In part to highlight some
of the issues, I focused
748
00:49:27,160 --> 00:49:31,780
attention also on some not
so useful methods--
749
00:49:31,780 --> 00:49:35,410
namely, mapping derivatives
to forward or backward
750
00:49:35,410 --> 00:49:37,630
differences.
751
00:49:37,630 --> 00:49:43,360
Next time what I would like
to do is look at impulse
752
00:49:43,360 --> 00:49:46,470
invariance for another
example--
753
00:49:46,470 --> 00:49:51,360
namely, the design of a
Butterworth filter, and I'll
754
00:49:51,360 --> 00:49:54,970
talk more specifically about
what Butterworth filters are
755
00:49:54,970 --> 00:49:57,550
at the beginning of
that lecture.
756
00:49:57,550 --> 00:50:01,170
Then, in addition, what we'll
introduce is another very
757
00:50:01,170 --> 00:50:05,600
useful technique, which has
some difficulties which
758
00:50:05,600 --> 00:50:11,360
impulse invariance doesn't have,
but avoids the principal
759
00:50:11,360 --> 00:50:13,480
difficulty that impulse
invariance does have--
760
00:50:13,480 --> 00:50:14,950
namely, aliasing.
761
00:50:14,950 --> 00:50:19,880
That method is referred to the
bilinear transformation, which
762
00:50:19,880 --> 00:50:24,090
we will define and utilize
next time.
763
00:50:24,090 --> 00:50:25,340
Thank you.
764
00:50:25,340 --> 00:50:26,965