WEBVTT
00:00:00.090 --> 00:00:02.490
The following content is
provided under a Creative
00:00:02.490 --> 00:00:04.030
Commons license.
00:00:04.030 --> 00:00:06.330
Your support will help
MIT OpenCourseWare
00:00:06.330 --> 00:00:10.720
continue to offer high quality
educational resources for free.
00:00:10.720 --> 00:00:13.320
To make a donation or
view additional materials
00:00:13.320 --> 00:00:17.280
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.280 --> 00:00:18.450
at ocw.mit.edu.
00:00:25.516 --> 00:00:28.492
[MUSIC PLAYING]
00:00:55.772 --> 00:00:57.770
ALAN OPPENHEIM: Hi.
00:00:57.770 --> 00:01:00.680
In this lecture,
there are two sets
00:01:00.680 --> 00:01:04.790
of ideas that I'd like
to discuss, both of which
00:01:04.790 --> 00:01:09.200
are related to our topic of the
last several lectures, namely,
00:01:09.200 --> 00:01:11.540
the Z-transform.
00:01:11.540 --> 00:01:15.290
The first topic that
we will focus on
00:01:15.290 --> 00:01:20.840
is what I'll referred to as
the geometric interpretation
00:01:20.840 --> 00:01:23.330
of the frequency response.
00:01:23.330 --> 00:01:28.320
Now, you recall that in
lecture 5 when I introduced
00:01:28.320 --> 00:01:33.350
the Z-transform, we
observed the relationship
00:01:33.350 --> 00:01:37.760
between the Z-transform
and the Fourier transform.
00:01:37.760 --> 00:01:41.330
The notion behind the geometric
interpretation of the frequency
00:01:41.330 --> 00:01:47.870
response is to relate, in
terms of a geometric picture,
00:01:47.870 --> 00:01:53.300
the rough characteristics of the
frequency response of a system
00:01:53.300 --> 00:01:56.720
and the pole-zero pattern
for the system function
00:01:56.720 --> 00:01:58.650
for the system.
00:01:58.650 --> 00:02:02.870
The second topic that we'll
focus on and eventually tie
00:02:02.870 --> 00:02:05.870
back into this
geometric interpretation
00:02:05.870 --> 00:02:12.410
is the general issue of
some of the properties
00:02:12.410 --> 00:02:14.420
of the Z-transform.
00:02:14.420 --> 00:02:16.850
And we'll illustrate
some of these properties
00:02:16.850 --> 00:02:19.010
with some examples.
00:02:19.010 --> 00:02:22.160
But first of all, let's
turn our attention
00:02:22.160 --> 00:02:26.990
to the subject of the geometric
interpretation of the frequency
00:02:26.990 --> 00:02:28.880
response.
00:02:28.880 --> 00:02:33.680
And let me begin
by reminding you
00:02:33.680 --> 00:02:40.280
of one of the important
properties of the Z-transform
00:02:40.280 --> 00:02:44.870
for a linear shift invariant
system, namely the fact
00:02:44.870 --> 00:02:50.240
that if we have a linear shift
invariant system with a unit
00:02:50.240 --> 00:02:57.020
sample response, H of n, input
x of n, and output y of n,
00:02:57.020 --> 00:03:00.560
then of course, y of n is
the convolution of H of n
00:03:00.560 --> 00:03:02.120
with x of n.
00:03:02.120 --> 00:03:05.420
And the Z-transform
of the output
00:03:05.420 --> 00:03:08.720
is the product of the
Z-transform of the input
00:03:08.720 --> 00:03:13.730
and H of Z, the Z-transform of
the unit sample response, which
00:03:13.730 --> 00:03:18.090
is what we've been referring
to as the system function.
00:03:18.090 --> 00:03:23.070
And in first developing
the Z-transform,
00:03:23.070 --> 00:03:27.420
we tie together the notion of
the Z-transform and the Fourier
00:03:27.420 --> 00:03:31.920
transform, in particular,
the important interpretation
00:03:31.920 --> 00:03:37.200
that the Fourier transform
is the Z-transform evaluated
00:03:37.200 --> 00:03:38.740
on the unit circle.
00:03:38.740 --> 00:03:44.070
So in terms of the
system function H of z,
00:03:44.070 --> 00:03:48.160
if we evaluate that
on the unit circle
00:03:48.160 --> 00:03:53.800
for z equal to e to
the j omega, the result
00:03:53.800 --> 00:03:59.530
is the frequency response of the
system, H of e to the j omega.
00:03:59.530 --> 00:04:01.720
Simply a statement
that the Z-transform
00:04:01.720 --> 00:04:05.050
evaluated on the unit circle
is the frequency response
00:04:05.050 --> 00:04:07.130
of the system.
00:04:07.130 --> 00:04:11.090
Well, in terms of
a simple example,
00:04:11.090 --> 00:04:15.270
we drag out our
usual simple example.
00:04:15.270 --> 00:04:21.620
The system function is 1 over
1 minus az to the minus 1 or z
00:04:21.620 --> 00:04:24.880
divided by z minus a.
00:04:24.880 --> 00:04:30.110
And in terms of the z
plane representation,
00:04:30.110 --> 00:04:33.800
or pole-zero representation
in the z-plane,
00:04:33.800 --> 00:04:39.380
we then have a representation of
this rational function in terms
00:04:39.380 --> 00:04:40.820
of one pole--
00:04:40.820 --> 00:04:42.485
the pole at z equals a--
00:04:42.485 --> 00:04:47.910
and one zero, that is
the zero at z equals 0.
00:04:47.910 --> 00:04:51.300
Well, to get the frequency
response associated
00:04:51.300 --> 00:04:53.700
with that system
function, we want
00:04:53.700 --> 00:04:59.780
to look at the Z-transform
evaluated on the unit circle.
00:04:59.780 --> 00:05:04.700
We can do that geometrically
by interpreting
00:05:04.700 --> 00:05:09.140
this complex number as
a vector in the z plane
00:05:09.140 --> 00:05:14.090
and this complex number as a
vector in the z plane, in which
00:05:14.090 --> 00:05:18.020
case the magnitude
of H of z will
00:05:18.020 --> 00:05:22.280
be the ratio of the
magnitudes of those vectors.
00:05:22.280 --> 00:05:27.620
And the angle of H of z will
be the angle of this vector
00:05:27.620 --> 00:05:30.240
minus the angle of this vector.
00:05:30.240 --> 00:05:34.270
In other words, suppose that
we want to look at H of z,
00:05:34.270 --> 00:05:35.780
at some value of z.
00:05:35.780 --> 00:05:40.130
And let's say, for the moment,
not on the unit circle,
00:05:40.130 --> 00:05:42.110
or it could be on
the unit circle.
00:05:42.110 --> 00:05:47.630
Let's just pick a general
value z equals z1.
00:05:47.630 --> 00:05:52.730
We have a vector corresponding
to this complex number, which
00:05:52.730 --> 00:05:57.170
is the vector, or
complex number, z.
00:05:57.170 --> 00:06:03.170
And that's a vector going from
the origin out to this point.
00:06:03.170 --> 00:06:10.750
We have one vector, which
is the vector z or z1
00:06:10.750 --> 00:06:14.320
because we're evaluating
this at z equals z1.
00:06:14.320 --> 00:06:17.830
So this is a vector z1.
00:06:17.830 --> 00:06:24.440
And the vector z corresponding
to the complex number z minus a
00:06:24.440 --> 00:06:30.840
is the vector z or z1
minus the vector a.
00:06:30.840 --> 00:06:34.900
The vector z1 is this vector.
00:06:34.900 --> 00:06:39.390
The vector minus
a is this vector.
00:06:39.390 --> 00:06:41.900
And the sum of those
two, the vector
00:06:41.900 --> 00:06:45.800
that corresponds to the
complex number z minus a,
00:06:45.800 --> 00:06:49.460
is the vector whose
tail is at this pole
00:06:49.460 --> 00:06:52.370
and whose head is at the
value of z, at which we
00:06:52.370 --> 00:06:55.280
want to evaluate H of z.
00:06:55.280 --> 00:07:02.610
And so we end up then
with an interpretation
00:07:02.610 --> 00:07:07.390
of the value of H
of z in polar form
00:07:07.390 --> 00:07:10.990
with its magnitude
equal to the magnitude
00:07:10.990 --> 00:07:16.630
of this vector divided by the
magnitude or length of the pole
00:07:16.630 --> 00:07:18.070
vector.
00:07:18.070 --> 00:07:22.232
And the angle of this
complex number, H of z at z
00:07:22.232 --> 00:07:26.410
equals z1 is the
angle of the 0 vector
00:07:26.410 --> 00:07:30.010
minus the angle of
the pole vector.
00:07:30.010 --> 00:07:32.430
That's a general statement.
00:07:32.430 --> 00:07:36.500
And z1 can be any
place in the z plane.
00:07:36.500 --> 00:07:42.310
So we have then the statement
as I've just made it,
00:07:42.310 --> 00:07:49.030
which we would now like to
apply to the interpretation
00:07:49.030 --> 00:07:56.060
or the generation of the
frequency response of a system.
00:07:56.060 --> 00:08:01.880
So we have, then, our example
that is one pole at equals a
00:08:01.880 --> 00:08:06.920
and 0 at z equals 0.
00:08:06.920 --> 00:08:11.930
We have the 0 vector
and the pole vector.
00:08:11.930 --> 00:08:16.460
And as we generate the
frequency response for z1
00:08:16.460 --> 00:08:21.860
equal to e to the j
omega, we have the point
00:08:21.860 --> 00:08:26.000
on the unit circle at
which we're evaluating
00:08:26.000 --> 00:08:29.240
the Z-transform changing.
00:08:29.240 --> 00:08:33.350
If we think of angular
distance around the unit circle
00:08:33.350 --> 00:08:37.940
as omega, then as
omega varies and we
00:08:37.940 --> 00:08:44.570
consider the relative behavior
of these two vectors, what
00:08:44.570 --> 00:08:47.090
we essentially trace
out is the frequency
00:08:47.090 --> 00:08:50.430
response of the system.
00:08:50.430 --> 00:08:57.510
So let's look at this and for
this example see if we can get,
00:08:57.510 --> 00:09:01.850
for this example, a rough
idea of what the frequency
00:09:01.850 --> 00:09:04.540
response should look like.
00:09:04.540 --> 00:09:07.000
Well, first of all we observe.
00:09:07.000 --> 00:09:10.140
And this is an important
observation, since it's
00:09:10.140 --> 00:09:13.230
a point that comes
up frequently,
00:09:13.230 --> 00:09:17.160
is that as we travel
around the unit
00:09:17.160 --> 00:09:24.190
circle, the vector from
the 0 to the unit circle
00:09:24.190 --> 00:09:27.670
changes in angle, of course.
00:09:27.670 --> 00:09:30.250
But it doesn't change in length.
00:09:30.250 --> 00:09:33.040
The 0 vector for this
example, or in fact
00:09:33.040 --> 00:09:36.220
any vector from the origin
out to the unit circle,
00:09:36.220 --> 00:09:41.210
as we vary omega and, therefore,
trace around the unit circle,
00:09:41.210 --> 00:09:43.520
this vector doesn't
change in length.
00:09:43.520 --> 00:09:47.470
So if we're interested in, say,
the magnitude of the frequency
00:09:47.470 --> 00:09:51.880
response, any poles or zeros
at the origin, of course,
00:09:51.880 --> 00:09:54.430
have no effect on the magnitude.
00:09:54.430 --> 00:09:56.590
The only effect
that they have is
00:09:56.590 --> 00:09:59.800
on the phase of the
frequency response
00:09:59.800 --> 00:10:02.950
that is on the angle of H of z.
00:10:02.950 --> 00:10:06.640
And in fact, as we sweep
around the unit circle,
00:10:06.640 --> 00:10:08.470
what happens to this angle?
00:10:08.470 --> 00:10:12.760
Well, that angle is just
simply equal to the value omega
00:10:12.760 --> 00:10:16.660
that we're looking at that
is the frequency value omega.
00:10:16.660 --> 00:10:20.270
So then, in fact, poles or zeros
at the origin as, of course,
00:10:20.270 --> 00:10:24.610
we could see from a
strictly algebraic argument.
00:10:24.610 --> 00:10:26.240
But from a geometric
argument, it
00:10:26.240 --> 00:10:30.630
should be clear that poles or
zeros at the origin introduce,
00:10:30.630 --> 00:10:35.020
in the frequency response,
a linear phase term.
00:10:35.020 --> 00:10:37.510
And they have no
effect on the magnitude
00:10:37.510 --> 00:10:40.540
since the length of a vector
from the origin to the unit
00:10:40.540 --> 00:10:43.690
circle is obviously 1.
00:10:43.690 --> 00:10:45.530
As a consequence of
that, by the way,
00:10:45.530 --> 00:10:49.390
often when talking about
the frequency response
00:10:49.390 --> 00:10:55.050
or equivalently when looking
at pole-zero patterns,
00:10:55.050 --> 00:10:59.050
it's common to
minimize or ignore
00:10:59.050 --> 00:11:01.970
the presence of zeros
or poles at the origin
00:11:01.970 --> 00:11:06.460
since they just correspond
to a linear phase term.
00:11:06.460 --> 00:11:10.540
Now, let's look at what the
behavior of the pole vector is.
00:11:10.540 --> 00:11:16.750
The pole vector as we start,
say, at omega equals 0,
00:11:16.750 --> 00:11:19.720
we have a vector
going from the pole
00:11:19.720 --> 00:11:24.620
to this point on
the unit circle.
00:11:24.620 --> 00:11:29.050
And as we sweep
around in frequency
00:11:29.050 --> 00:11:33.970
until we get to omega equals pi
halfway around the unit circle,
00:11:33.970 --> 00:11:38.830
the length of this vector
is monotonically increasing.
00:11:38.830 --> 00:11:42.880
Well, what does that mean about
the magnitude of the frequency
00:11:42.880 --> 00:11:44.050
response?
00:11:44.050 --> 00:11:47.710
It means that the magnitude
of the frequency response,
00:11:47.710 --> 00:11:50.890
which should be the
magnitude of that vector--
00:11:50.890 --> 00:11:53.350
the magnitude of the
frequency response
00:11:53.350 --> 00:11:57.460
is monotonically decreasing
as we sweep from omega
00:11:57.460 --> 00:12:02.810
equals zero around
to omega equals pi.
00:12:02.810 --> 00:12:08.430
At omega equal to pi, we have
exactly the reverse situation.
00:12:08.430 --> 00:12:10.325
The length of the
vector from the pole--
00:12:10.325 --> 00:12:12.600
and let me draw that vector--
00:12:12.600 --> 00:12:15.990
from the pole to
omega equals pi as we
00:12:15.990 --> 00:12:17.880
sweep around back
to omega equals
00:12:17.880 --> 00:12:20.970
0, the length of that
vector decreases.
00:12:20.970 --> 00:12:25.390
So the magnitude of the
frequency response increases.
00:12:25.390 --> 00:12:29.220
So for this particular
example, by just observing
00:12:29.220 --> 00:12:32.340
what happens to the
length of this pole vector
00:12:32.340 --> 00:12:34.920
as we trace around
the unit circle,
00:12:34.920 --> 00:12:39.240
we can see that the frequency
response starts at some value.
00:12:39.240 --> 00:12:41.150
That's obvious.
00:12:41.150 --> 00:12:49.470
And as it sweeps around to pi,
it monotonically decreases.
00:12:49.470 --> 00:12:54.110
And when we come back
from pi around to 0,
00:12:54.110 --> 00:12:59.160
the frequency response
increases again.
00:12:59.160 --> 00:13:03.870
And I don't know if I've drawn
it so that it looks clearly
00:13:03.870 --> 00:13:04.910
this way.
00:13:04.910 --> 00:13:11.990
But in fact, this piece is
just this piece reflected over.
00:13:11.990 --> 00:13:16.160
There's another
important point that we
00:13:16.160 --> 00:13:17.870
can observe geometrically.
00:13:17.870 --> 00:13:21.290
It's a point about the frequency
response that, of course,
00:13:21.290 --> 00:13:24.860
we've emphasized several
times in several ways.
00:13:24.860 --> 00:13:27.950
This just offers an
opportunity to re-emphasise it
00:13:27.950 --> 00:13:30.380
in yet another way.
00:13:30.380 --> 00:13:38.270
As the frequency variable
goes from 0 around to 2pi,
00:13:38.270 --> 00:13:41.000
we trace out a
magnitude and then,
00:13:41.000 --> 00:13:44.420
of course, also a phase
for the frequency response.
00:13:44.420 --> 00:13:51.710
When we get back to 0 or 2pi,
if we go from 2pi around to 4pi,
00:13:51.710 --> 00:13:54.200
what should we see for
the frequency response?
00:13:54.200 --> 00:13:56.570
Well, we should see
exactly the same thing
00:13:56.570 --> 00:14:00.140
that we saw before because
what we're doing literally,
00:14:00.140 --> 00:14:03.020
actually, is going
around in circles.
00:14:03.020 --> 00:14:05.960
We started here, went
around there to get
00:14:05.960 --> 00:14:08.600
the 2pi, omega equals 2pi.
00:14:08.600 --> 00:14:11.860
For omega from 2pi to
4pi, we go around again.
00:14:11.860 --> 00:14:14.780
For 4pi to 6pi, we go
around again, et cetera.
00:14:14.780 --> 00:14:18.500
So just geometrically
by looking at
00:14:18.500 --> 00:14:22.460
this geometric interpretation
of the frequency response,
00:14:22.460 --> 00:14:24.440
it gives us an
opportunity to emphasize
00:14:24.440 --> 00:14:28.040
once again that the frequency
response is a periodic function
00:14:28.040 --> 00:14:30.690
of frequency.
00:14:30.690 --> 00:14:35.200
Now, this is for one
specific example.
00:14:35.200 --> 00:14:37.750
The more general
statement, then,
00:14:37.750 --> 00:14:45.310
is that the magnitude of
the frequency response
00:14:45.310 --> 00:14:53.180
is equal to the product of
the length of the 0 vectors--
00:14:53.180 --> 00:14:57.890
the vectors from the
zeros to the unit circle--
00:14:57.890 --> 00:15:02.330
divided by the product of the
length of the pole vectors--
00:15:02.330 --> 00:15:06.380
the vectors from the
poles to the unit circle.
00:15:06.380 --> 00:15:09.740
And the angle of the
frequency response
00:15:09.740 --> 00:15:14.150
is the sum of the
angles of the 0 vectors
00:15:14.150 --> 00:15:18.050
minus the sum of the
angles of the pole vectors.
00:15:18.050 --> 00:15:21.230
Incidentally, it usually is
the case-- or it is for me,
00:15:21.230 --> 00:15:22.280
anyway--
00:15:22.280 --> 00:15:29.270
that often, it's possible to
get a rough picture of what
00:15:29.270 --> 00:15:31.730
the magnitude of the
frequency response
00:15:31.730 --> 00:15:34.790
is like by looking at
this geometric picture.
00:15:34.790 --> 00:15:38.570
It's usually somewhat more
difficult except, perhaps,
00:15:38.570 --> 00:15:40.820
for linear phase
terms, et cetera,
00:15:40.820 --> 00:15:47.450
to get a very clear picture
of what the phase looks like.
00:15:47.450 --> 00:15:51.770
Well let's just look at
one more example, which
00:15:51.770 --> 00:15:56.990
is a kind of example that we
haven't discussed explicitly
00:15:56.990 --> 00:15:57.980
up to this point.
00:15:57.980 --> 00:16:00.950
And this also provides
us with an opportunity
00:16:00.950 --> 00:16:04.580
to introduce this idea.
00:16:04.580 --> 00:16:09.050
Let's consider the
case of a Z-transform
00:16:09.050 --> 00:16:13.220
which has a pole-zero
pattern consisting
00:16:13.220 --> 00:16:17.240
of a pair of complex conjugate
poles in the z-plane.
00:16:17.240 --> 00:16:20.090
These are complex
conjugate poles
00:16:20.090 --> 00:16:22.940
with a radius equal
to some value,
00:16:22.940 --> 00:16:28.440
say r, and an angular
spacing equal to omega 0.
00:16:28.440 --> 00:16:31.520
So this pole is at omega 0.
00:16:31.520 --> 00:16:36.410
This pole is at an
angle of minus omega 0.
00:16:36.410 --> 00:16:39.830
And using this notion of
interpreting the frequency
00:16:39.830 --> 00:16:44.780
response geometrically,
let's just sketch out
00:16:44.780 --> 00:16:47.630
roughly what we would
expect the frequency
00:16:47.630 --> 00:16:51.370
response to look like.
00:16:51.370 --> 00:16:52.980
Well, let's see.
00:16:52.980 --> 00:16:57.540
If we start at
omega equals zero,
00:16:57.540 --> 00:17:02.550
then we have a
vector from this pole
00:17:02.550 --> 00:17:06.119
and a vector from this pole.
00:17:06.119 --> 00:17:08.670
The resulting value of
the frequency response
00:17:08.670 --> 00:17:12.090
is 1 over the product
of those two vectors.
00:17:14.660 --> 00:17:22.010
As we move around the unit
circle from omega equals 0,
00:17:22.010 --> 00:17:28.580
let's say, to some point
that's closer to omega 0,
00:17:28.580 --> 00:17:33.670
then this vector gets
changed to that one.
00:17:33.670 --> 00:17:38.590
And this vector gets
changed to this one.
00:17:38.590 --> 00:17:41.980
Well, we can see what's
happening to the vector
00:17:41.980 --> 00:17:43.210
lengths.
00:17:43.210 --> 00:17:48.490
This vector is getting
shorter as we approach
00:17:48.490 --> 00:17:51.160
omega equal to omega 0.
00:17:51.160 --> 00:17:53.680
This vector is getting longer.
00:17:53.680 --> 00:17:56.320
But you can imagine-- and I
think it should be relatively
00:17:56.320 --> 00:17:57.880
clear from the picture--
00:17:57.880 --> 00:18:00.880
that this vector is getting
shorter faster than this one
00:18:00.880 --> 00:18:04.010
is getting longer.
00:18:04.010 --> 00:18:08.570
It seems relatively clear
from a geometric picture
00:18:08.570 --> 00:18:12.440
that the product of the
lengths of these two vectors
00:18:12.440 --> 00:18:18.170
is smaller when omega is closer
to omega 0 than, let's say,
00:18:18.170 --> 00:18:21.710
when omega is equal to 0.
00:18:21.710 --> 00:18:26.240
So consequently, we would expect
the magnitude of the frequency
00:18:26.240 --> 00:18:30.200
response to start at some value.
00:18:30.200 --> 00:18:34.160
As we get in the vicinity
of omega equal to omega 0,
00:18:34.160 --> 00:18:37.640
we would expect that
frequency response to peak.
00:18:37.640 --> 00:18:41.090
As we pass that pole,
the frequency response
00:18:41.090 --> 00:18:43.640
will now begin to decrease.
00:18:43.640 --> 00:18:53.160
And clearly, when we are
at omega equal to pi,
00:18:53.160 --> 00:18:57.030
the product of the lengths
of these two vectors
00:18:57.030 --> 00:18:59.610
is considerably longer
than the products
00:18:59.610 --> 00:19:02.970
of the lengths of the
vectors at omega equals 0.
00:19:02.970 --> 00:19:05.460
So on the basis
of that argument,
00:19:05.460 --> 00:19:09.240
we can see that roughly
what we expect the frequency
00:19:09.240 --> 00:19:13.950
response to do is
begin at some value,
00:19:13.950 --> 00:19:19.170
increase as we approach the
pole at omega equals omega 0,
00:19:19.170 --> 00:19:25.740
decrease as we pass the
pole heading toward pi.
00:19:25.740 --> 00:19:30.270
And then from pi around to
pi, or equivalently from 0
00:19:30.270 --> 00:19:34.530
around to minus pi, we would
expect the reciprocal behavior.
00:19:34.530 --> 00:19:38.130
That is, the frequency
response would increase,
00:19:38.130 --> 00:19:41.850
peak in the vicinity of
this pole in the lower
00:19:41.850 --> 00:19:46.260
half of the z-plane, and
then return to the same value
00:19:46.260 --> 00:19:47.610
that it started from.
00:19:47.610 --> 00:19:50.280
And of course, it'll
be periodic as we
00:19:50.280 --> 00:19:53.070
run around the unit circle.
00:19:53.070 --> 00:19:57.180
Well, this is a
resonant characteristic,
00:19:57.180 --> 00:20:01.410
reminiscent in the analog
case of what we would expect
00:20:01.410 --> 00:20:05.700
from a complex conjugate
pole pair close to the j
00:20:05.700 --> 00:20:08.760
omega axis that's
in the s-plane.
00:20:08.760 --> 00:20:11.130
And in fact, of
course, in the s-plane,
00:20:11.130 --> 00:20:14.490
we have exactly the same
kinds of geometrical arguments
00:20:14.490 --> 00:20:18.570
to allow us to roughly sketch
out the frequency response.
00:20:18.570 --> 00:20:20.880
The only real
important difference
00:20:20.880 --> 00:20:23.220
is that in the
discrete time case,
00:20:23.220 --> 00:20:26.760
it's the unit circle, which
is the locus in the z-plane
00:20:26.760 --> 00:20:28.020
that we're looking at.
00:20:28.020 --> 00:20:30.090
In the continuous
time case, it's
00:20:30.090 --> 00:20:33.120
the vertical, or j omega
axis in the s-plane
00:20:33.120 --> 00:20:35.230
that we're looking at.
00:20:35.230 --> 00:20:38.400
Well, this shouldn't be
particularly evident from what
00:20:38.400 --> 00:20:39.270
we've done here.
00:20:39.270 --> 00:20:45.120
But by inference or by
carrying your intuition
00:20:45.120 --> 00:20:48.670
from the continuous time
to the discrete time,
00:20:48.670 --> 00:20:53.770
can you guess at what you
would expect the unit sample
00:20:53.770 --> 00:21:00.460
response of a system with this
pole-zero pattern to look like?
00:21:00.460 --> 00:21:03.510
Well, it would look
like the discrete time
00:21:03.510 --> 00:21:07.110
counterpart of what happens
in the continuous time case.
00:21:07.110 --> 00:21:12.810
That is, it would look like
a damped sinusoidal sequence
00:21:12.810 --> 00:21:17.640
with the damping influenced
by the distance of these poles
00:21:17.640 --> 00:21:19.230
from the unit circle.
00:21:19.230 --> 00:21:21.990
The closer the pole
is to the unit circle,
00:21:21.990 --> 00:21:23.700
the sharper this
resonant peak will
00:21:23.700 --> 00:21:26.760
be and the less damping
on the sinusoid.
00:21:29.710 --> 00:21:35.200
One additional point
to remind you of--
00:21:35.200 --> 00:21:38.350
I made reference to this
in an earlier lecture.
00:21:38.350 --> 00:21:43.500
But notice that in talking
about the Z-transform poles
00:21:43.500 --> 00:21:47.520
and zeros, et cetera
here, I didn't say what
00:21:47.520 --> 00:21:49.560
the region of convergence was.
00:21:49.560 --> 00:21:51.060
Or at least, I
didn't say explicitly
00:21:51.060 --> 00:21:53.220
what the region of
convergence was.
00:21:53.220 --> 00:21:57.060
But did I say
implicitly what it is?
00:21:57.060 --> 00:22:01.800
Well, sure I did because what I
said or what I've been assuming
00:22:01.800 --> 00:22:05.100
is that the systems that
we've been talking about
00:22:05.100 --> 00:22:08.640
can be described in terms
of a frequency response.
00:22:08.640 --> 00:22:10.740
In other words, the
unit sample response
00:22:10.740 --> 00:22:13.190
has a Fourier transform.
00:22:13.190 --> 00:22:16.200
Well for that to be the case,
the region of convergence
00:22:16.200 --> 00:22:18.850
has to include the unit circle.
00:22:18.850 --> 00:22:20.700
And then, we can use
all those other rules
00:22:20.700 --> 00:22:25.170
of regions of convergences
to allow us to figure out
00:22:25.170 --> 00:22:30.144
from there how far on both
sides of the unit circle
00:22:30.144 --> 00:22:31.560
the region of
convergence extends.
00:22:34.390 --> 00:22:39.370
All right, well this is, then,
a geometrical interpretation
00:22:39.370 --> 00:22:41.320
of the frequency response.
00:22:41.320 --> 00:22:43.930
It's often useful
as a rough guide
00:22:43.930 --> 00:22:47.590
in getting a general picture
of what the frequency
00:22:47.590 --> 00:22:49.240
response might look like.
00:22:49.240 --> 00:22:52.720
Although, for complicated
cases, it's often
00:22:52.720 --> 00:22:55.930
difficult to get precise
details about the frequency
00:22:55.930 --> 00:23:01.210
response, which of course, we
could also get algebraically.
00:23:01.210 --> 00:23:06.300
Well now, I'd like to,
at least temporarily,
00:23:06.300 --> 00:23:15.390
change gears or topics and talk
about the issue of properties
00:23:15.390 --> 00:23:19.920
of the Z-transform in
talking about the properties
00:23:19.920 --> 00:23:22.950
and working some
examples to illustrate
00:23:22.950 --> 00:23:25.620
the use of the properties
of the Z-transform.
00:23:25.620 --> 00:23:29.400
In fact, I'll have occasion
to make reference back
00:23:29.400 --> 00:23:33.600
to this geometric interpretation
of the frequency response
00:23:33.600 --> 00:23:37.180
to help with at least
one of the examples.
00:23:37.180 --> 00:23:41.490
So now I'd like to turn
our attention, then,
00:23:41.490 --> 00:23:46.969
to the question or the
topic of the properties
00:23:46.969 --> 00:23:47.760
of the Z-transform.
00:23:47.760 --> 00:23:52.070
Well, first of
all, why do we want
00:23:52.070 --> 00:23:53.360
properties of the Z-transform?
00:23:53.360 --> 00:23:55.680
The Z-transform has properties.
00:23:55.680 --> 00:23:57.780
Why do we want these properties?
00:23:57.780 --> 00:23:58.830
One of the reasons--
00:23:58.830 --> 00:24:01.100
it's a very practical reason--
00:24:01.100 --> 00:24:04.760
is that the properties
of the Z-transform
00:24:04.760 --> 00:24:09.290
help us in calculating
Z-transforms and inverse
00:24:09.290 --> 00:24:11.210
Z-transforms.
00:24:11.210 --> 00:24:15.590
And they also, obviously,
provide a certain amount
00:24:15.590 --> 00:24:18.260
of intuition and
insight with regard
00:24:18.260 --> 00:24:21.500
to generally dealing
with Z-transforms
00:24:21.500 --> 00:24:24.500
and their inverses.
00:24:24.500 --> 00:24:26.670
Well, there are a
lot of properties.
00:24:26.670 --> 00:24:28.820
In fact, there are
trivial properties.
00:24:28.820 --> 00:24:31.370
There are very
complicated properties.
00:24:31.370 --> 00:24:34.850
There are properties
that we more commonly
00:24:34.850 --> 00:24:36.650
tend to carry around.
00:24:36.650 --> 00:24:41.900
And I've listed a few that
shouldn't be considered
00:24:41.900 --> 00:24:44.930
to be exhaustive
but generally tend
00:24:44.930 --> 00:24:49.280
to be the properties that
turn out to be the handiest.
00:24:49.280 --> 00:24:51.170
That is, these are
the properties,
00:24:51.170 --> 00:24:53.780
at a minimum, that you
should carry around
00:24:53.780 --> 00:24:56.820
in your back pocket.
00:24:56.820 --> 00:24:58.380
Well, let's see.
00:24:58.380 --> 00:25:02.700
We're talking about a sequence,
x of n with a Z-transform,
00:25:02.700 --> 00:25:04.580
x of z.
00:25:04.580 --> 00:25:09.150
One of the properties that
we've taken advantage of already
00:25:09.150 --> 00:25:12.120
throughout the entire
discussion of the Z-transform
00:25:12.120 --> 00:25:15.060
is the fact that
the Z-transform maps
00:25:15.060 --> 00:25:19.310
convolution to multiplication.
00:25:19.310 --> 00:25:23.390
If I have the convolution
of two sequences,
00:25:23.390 --> 00:25:28.610
then the resulting
Z-transform is the product
00:25:28.610 --> 00:25:31.650
of the Z-transforms.
00:25:31.650 --> 00:25:35.640
The second property, which
is often very useful,
00:25:35.640 --> 00:25:40.500
is referred to as the shifting
property, which says that if I
00:25:40.500 --> 00:25:44.130
shift x of n by an amount n0--
00:25:46.800 --> 00:25:49.980
and you should think, by the
way, of if n0 is positive,
00:25:49.980 --> 00:25:53.500
does that mean shifting to the
right or shifting to the left?
00:25:53.500 --> 00:25:55.080
Well, I'll let you
think about that.
00:25:55.080 --> 00:25:57.990
It's something that
you should nail down.
00:25:57.990 --> 00:26:02.400
If I shift the sequence x of n
by replacing the argument by n
00:26:02.400 --> 00:26:06.480
plus n0, then the
resulting Z-transform
00:26:06.480 --> 00:26:09.750
is z to the n0 times x of z.
00:26:09.750 --> 00:26:14.040
That is, shifting corresponds
to multiplying by z to the n0.
00:26:16.690 --> 00:26:21.700
Another useful property has
to do with taking a sequence
00:26:21.700 --> 00:26:24.100
and turning it around in n.
00:26:24.100 --> 00:26:28.060
That is, replacing n by minus n.
00:26:28.060 --> 00:26:30.640
The resulting effect
on the Z-transform
00:26:30.640 --> 00:26:36.140
is to replace z by 1 over z.
00:26:36.140 --> 00:26:39.500
Another useful
property is the result
00:26:39.500 --> 00:26:44.570
of multiplying a sequence x of
n by an exponential, a to the n.
00:26:44.570 --> 00:26:45.740
a might be complex.
00:26:45.740 --> 00:26:47.360
Or it might be real.
00:26:47.360 --> 00:26:52.850
And the result there is that
the z transform is x of 8
00:26:52.850 --> 00:26:55.610
to the minus 1 times z.
00:26:55.610 --> 00:26:59.180
Another one which is
useful is multiplication
00:26:59.180 --> 00:27:04.040
of a sequence by n which
results in a z transform, which
00:27:04.040 --> 00:27:08.690
is minus z times the
derivative of x of z.
00:27:08.690 --> 00:27:10.940
And then I've indicated--
00:27:10.940 --> 00:27:12.890
tried to be somewhat explicit--
00:27:12.890 --> 00:27:15.170
that that's not the
end of the list.
00:27:15.170 --> 00:27:17.360
There are lots of
other properties--
00:27:17.360 --> 00:27:20.490
a number of others that
are presented in the text,
00:27:20.490 --> 00:27:23.150
others besides that that aren't
presented in the text, ones
00:27:23.150 --> 00:27:25.640
that you can dream up yourself,
ones that your friends
00:27:25.640 --> 00:27:29.040
know that you don't, et cetera.
00:27:29.040 --> 00:27:33.930
Now, we could, of course, go
through the proof of all these.
00:27:33.930 --> 00:27:39.300
The proofs of
properties tend to all
00:27:39.300 --> 00:27:43.200
be in somewhat of a similar
vein, as a matter of fact.
00:27:43.200 --> 00:27:47.790
And the style, once you see
what the trick is or roughly
00:27:47.790 --> 00:27:49.374
how you go about
proving properties,
00:27:49.374 --> 00:27:50.790
then you can just
prove properties
00:27:50.790 --> 00:27:52.740
and prove properties.
00:27:52.740 --> 00:27:56.070
And we won't do that
in this lecture,
00:27:56.070 --> 00:27:59.430
with the exception
of illustrating
00:27:59.430 --> 00:28:04.480
the style of proving properties
with a couple of examples.
00:28:04.480 --> 00:28:08.340
And the two examples that I've
picked, somewhat arbitrarily
00:28:08.340 --> 00:28:11.940
as a matter of fact, is
the Shifting Property,
00:28:11.940 --> 00:28:16.260
that is Property 2, and the
result of multiplication
00:28:16.260 --> 00:28:21.420
by an exponential,
which is Property 4.
00:28:21.420 --> 00:28:25.230
But this is only to
illustrate the style
00:28:25.230 --> 00:28:27.090
of proving properties.
00:28:27.090 --> 00:28:30.060
And you can guess
where, actually, you
00:28:30.060 --> 00:28:34.030
get a chance to see the
proof of some of the others.
00:28:34.030 --> 00:28:36.820
All right, let's take
a look at Property 2,
00:28:36.820 --> 00:28:39.960
that is the Shifting Property.
00:28:39.960 --> 00:28:46.110
Well, to prove that
replacing n by n plus n0
00:28:46.110 --> 00:28:49.830
results in z to the
n0 times x of z,
00:28:49.830 --> 00:28:57.150
we want to consider a sequence,
x1 of n equal to x of n plus n0
00:28:57.150 --> 00:29:00.500
so that its
Z-transform, x1 of z,
00:29:00.500 --> 00:29:05.990
is the sum of x of n plus
n0 times z to the minus n.
00:29:05.990 --> 00:29:11.030
Well, a simple idea here is
a substitution of variables.
00:29:11.030 --> 00:29:18.560
Let's replace n plus n0
by a new variable, m, or n
00:29:18.560 --> 00:29:22.890
is equal to m minus
n0, in which case,
00:29:22.890 --> 00:29:28.430
we can rewrite this
expression as x1 of z
00:29:28.430 --> 00:29:35.300
is the sum on m of x of
m, because this is now m.
00:29:35.300 --> 00:29:38.570
n is replaced by m minus n0.
00:29:38.570 --> 00:29:45.590
So we have z to the n0
times z to the minus m.
00:29:45.590 --> 00:29:54.750
Well, the z to the n0
can come outside the sum.
00:29:54.750 --> 00:29:56.540
The limits on the
sum, incidentally,
00:29:56.540 --> 00:29:59.370
are m equals minus
infinity to plus infinity
00:29:59.370 --> 00:30:04.910
because if n0 is finite, as
n runs from minus infinity
00:30:04.910 --> 00:30:08.510
to plus infinity, so does m.
00:30:08.510 --> 00:30:11.150
That comes outside the sum.
00:30:11.150 --> 00:30:16.040
And what's left, then, is the
sum of x of m z to the minus m,
00:30:16.040 --> 00:30:17.930
which is just x of z.
00:30:17.930 --> 00:30:22.910
So consequently, then, what
we end up with is that x of z
00:30:22.910 --> 00:30:30.030
is z to the n0 times x of z.
00:30:30.030 --> 00:30:33.630
x1 of z is z to the
n0 times x of z,
00:30:33.630 --> 00:30:37.650
which is, of course,
the way I advertised.
00:30:37.650 --> 00:30:41.010
Well, there are lots of times
when, in fact, the Shifting
00:30:41.010 --> 00:30:44.770
Property comes into play.
00:30:44.770 --> 00:30:47.430
One thing that we can
use it for immediately
00:30:47.430 --> 00:30:51.390
is to tie together
a couple of things
00:30:51.390 --> 00:30:54.330
that have been floating, more
or less, in the background.
00:30:57.090 --> 00:31:01.850
I have alluded several
times to the fact
00:31:01.850 --> 00:31:09.330
that systems whose system
function is rational
00:31:09.330 --> 00:31:13.230
correspond to systems
that are characterized
00:31:13.230 --> 00:31:16.140
by linear constant coefficient
difference equations.
00:31:16.140 --> 00:31:18.600
That is, linear constant
coefficient difference
00:31:18.600 --> 00:31:21.690
equations, those
are the systems that
00:31:21.690 --> 00:31:25.770
end up with system functions
that are rational functions.
00:31:25.770 --> 00:31:30.360
And in fact, we can see that
in a straightforward way
00:31:30.360 --> 00:31:34.950
by just simply applying
the Shifting Property.
00:31:34.950 --> 00:31:42.320
Well, let's consider a linear
constant coefficient difference
00:31:42.320 --> 00:31:48.190
equation of the general form is
the sum from k equals 0 to n.
00:31:48.190 --> 00:31:54.860
a sub k y of n minus k is equal
to a sum of b sub k times x
00:31:54.860 --> 00:31:55.850
of n minus k.
00:31:59.250 --> 00:32:05.640
If we take the Z-transform of
both sides of this equation
00:32:05.640 --> 00:32:08.430
and use the fact-- incidentally,
this is another property
00:32:08.430 --> 00:32:12.000
that I've more or
less been using,
00:32:12.000 --> 00:32:13.710
although I've never
stated explicitly--
00:32:13.710 --> 00:32:17.430
that the Z-transform of a sum
is the sum of the Z-transforms.
00:32:17.430 --> 00:32:21.480
Taking, then, the
Z-transform of this equation,
00:32:21.480 --> 00:32:26.590
y of n minus k using
the Shifting Property
00:32:26.590 --> 00:32:34.170
will give us z to the
minus k times y of z.
00:32:34.170 --> 00:32:44.770
And x of n minus k will give us
z to the minus k times x of z.
00:32:44.770 --> 00:32:50.320
Consequently, the sum of a
sub k z to the minus k y of z
00:32:50.320 --> 00:32:55.960
is equal to the sum of b sub
k z to the minus k x of z.
00:32:55.960 --> 00:32:59.170
If we solve that
equation for y of z
00:32:59.170 --> 00:33:03.730
over x of z, which is
the system function,
00:33:03.730 --> 00:33:07.570
then we end up with
the sum of b sub k z
00:33:07.570 --> 00:33:13.090
to the minus k divided by
the sum of a sub k times z
00:33:13.090 --> 00:33:14.770
to the minus k.
00:33:14.770 --> 00:33:17.210
And there are a couple of
important observations.
00:33:17.210 --> 00:33:21.130
One is we ended up with a
rational function, which
00:33:21.130 --> 00:33:26.090
is what we expected, or what
I said we were going to get.
00:33:26.090 --> 00:33:32.680
And a second is that the
coefficients in the numerator
00:33:32.680 --> 00:33:37.540
polynomial are exactly the
same as the coefficients
00:33:37.540 --> 00:33:41.230
on the right hand side of
the difference equation.
00:33:41.230 --> 00:33:44.530
And the coefficients in
the denominator polynomial
00:33:44.530 --> 00:33:48.580
are exactly the same as the
coefficients on the left hand
00:33:48.580 --> 00:33:51.070
side of the difference equation.
00:33:51.070 --> 00:33:54.730
This by the way, is exactly
consistent, or analogous,
00:33:54.730 --> 00:33:59.800
with what happens when we
apply the Laplace transform
00:33:59.800 --> 00:34:02.620
to linear constant coefficient
differential equation.
00:34:02.620 --> 00:34:05.530
Exactly the same thing happens.
00:34:05.530 --> 00:34:07.950
It should be clear
then, incidentally,
00:34:07.950 --> 00:34:10.420
that if I give you a
system function that's
00:34:10.420 --> 00:34:14.260
a rational function of z,
that you could construct,
00:34:14.260 --> 00:34:17.139
in a straightforward way,
the difference equation that
00:34:17.139 --> 00:34:19.389
characterizes that
system because you
00:34:19.389 --> 00:34:22.420
can pick the coefficients
off from the numerator.
00:34:22.420 --> 00:34:24.500
Those are on the
right hand side.
00:34:24.500 --> 00:34:26.380
And you can pick
the coefficients off
00:34:26.380 --> 00:34:27.730
from the denominator.
00:34:27.730 --> 00:34:32.980
Those are on the left hand side
of the difference equation.
00:34:32.980 --> 00:34:37.750
The Shifting Property is a
property that arises very often
00:34:37.750 --> 00:34:39.989
and, in fact, is a very,
very useful property.
00:34:42.690 --> 00:34:50.360
Now, the second property that
I want to outline the proof for
00:34:50.360 --> 00:34:58.490
is the property related to
multiplication of the sequence
00:34:58.490 --> 00:35:03.890
by an exponential a to the n.
00:35:03.890 --> 00:35:09.230
I am forming a new sequence x1
of n, which is a to the n times
00:35:09.230 --> 00:35:11.530
x of n.
00:35:11.530 --> 00:35:17.440
And to derive the relationship
between x1 of z and x of n,
00:35:17.440 --> 00:35:22.750
again, we can look at
the Z-transform, x1 of z,
00:35:22.750 --> 00:35:28.330
which is the sum of a to the n,
x of n, times z to the minus n.
00:35:28.330 --> 00:35:30.640
Well, it's
straightforward to rewrite
00:35:30.640 --> 00:35:41.590
what's inside the sum as x of
n times a to the minus 1 times
00:35:41.590 --> 00:35:44.440
z to the minus n.
00:35:47.340 --> 00:35:53.070
x1 of z is the sum of x of
n times a to the minus 1z--
00:35:53.070 --> 00:35:56.180
all that raised to the minus n.
00:35:56.180 --> 00:35:59.390
Well, that looks just like
the Z-transform of x of n
00:35:59.390 --> 00:36:03.200
but with z replaced by a
to the minus 1 times z.
00:36:03.200 --> 00:36:11.270
So this says, consequently,
that the Z-transform of x1 of n
00:36:11.270 --> 00:36:16.700
is equal to the
Z-transform of x of n
00:36:16.700 --> 00:36:21.800
but with z replaced by a
to the minus 1 times z.
00:36:21.800 --> 00:36:25.850
So we stick in here a
to the minus 1 times z.
00:36:25.850 --> 00:36:28.250
And that then relates
the Z-transform
00:36:28.250 --> 00:36:30.650
of the original sequence
and the Z-transform
00:36:30.650 --> 00:36:35.740
of the sequence multiplied
by a decaying exponential.
00:36:35.740 --> 00:36:43.590
Well, it's interesting to look
at what this property implies
00:36:43.590 --> 00:36:49.290
in terms of the movement of the
poles and zeros in the z-plane.
00:36:49.290 --> 00:36:54.120
That is, a useful
notion or a useful fact
00:36:54.120 --> 00:36:57.900
to have, again, stored
away in your hip pocket
00:36:57.900 --> 00:37:03.240
is the effect on the poles
and zeros of a system function
00:37:03.240 --> 00:37:08.580
or a Z-transform of
multiplying the sequence
00:37:08.580 --> 00:37:11.670
by an exponential-- maybe
a complex exponential,
00:37:11.670 --> 00:37:14.530
maybe a real exponential.
00:37:14.530 --> 00:37:19.520
Well, let's focus
on a pole or a 0.
00:37:19.520 --> 00:37:22.670
And the result that
we get by considering
00:37:22.670 --> 00:37:25.790
just a simple pole
or 0 will, of course,
00:37:25.790 --> 00:37:28.580
generalize to all
the poles and zeros.
00:37:28.580 --> 00:37:32.390
So let's consider x
of z to have a factor
00:37:32.390 --> 00:37:35.390
either in the numerator
and denominator of the form
00:37:35.390 --> 00:37:38.500
z minus z0.
00:37:38.500 --> 00:37:42.880
Well, x1 of z will
then have a factor
00:37:42.880 --> 00:37:47.140
derived from that one, but
for which z is replaced
00:37:47.140 --> 00:37:51.220
by 8 to the minus 1 times z.
00:37:51.220 --> 00:37:53.630
And then we have the minus z0.
00:37:53.630 --> 00:37:57.440
Or if we pull the a
to the minus outside,
00:37:57.440 --> 00:38:04.950
we have a to the minus 1
times z minus a times z0.
00:38:04.950 --> 00:38:06.780
So there are two effects here.
00:38:06.780 --> 00:38:11.040
One effect-- if we think of x of
z as a product of zeros divided
00:38:11.040 --> 00:38:12.930
by a product of poles--
00:38:12.930 --> 00:38:16.200
one effect is that there
is a constant that,
00:38:16.200 --> 00:38:19.380
perhaps, collects out in front.
00:38:19.380 --> 00:38:23.730
But we never see that constant
anyway in the pole-0 pattern.
00:38:23.730 --> 00:38:27.840
The more important point
is that whereas here we
00:38:27.840 --> 00:38:36.210
had, say, 0 at z equals z0, that
0 is now shifted to a times z0.
00:38:36.210 --> 00:38:41.910
So the 0 at z0 gets replaced
by a 0 at eight times z0.
00:38:41.910 --> 00:38:46.670
A pole at z0 gets replaced
by a pole at a times z0.
00:38:52.230 --> 00:38:57.330
Well, more specifically,
then, here's
00:38:57.330 --> 00:39:01.920
our polar 0 which gets converted
to that polar 0 multiplied
00:39:01.920 --> 00:39:04.915
by a.
00:39:04.915 --> 00:39:10.860
And if we write
this or think of it
00:39:10.860 --> 00:39:15.930
in polar form, as r0
e to the j theta 0,
00:39:15.930 --> 00:39:21.930
then the result is a times
r0 times e to the j theta 0.
00:39:21.930 --> 00:39:28.090
Well obviously then, if this
number a is a real number,
00:39:28.090 --> 00:39:33.750
then the only effect on
the location of the pole
00:39:33.750 --> 00:39:38.990
is to change its radial value
and not change its angle.
00:39:38.990 --> 00:39:44.540
More generally, if a is complex,
then to write the resulting
00:39:44.540 --> 00:39:48.980
polar zero in polar
form, we would
00:39:48.980 --> 00:39:54.560
replace this by its magnitude
and add to the phase--
00:39:54.560 --> 00:39:57.750
the phase that corresponds
to that number a.
00:40:00.270 --> 00:40:04.880
Well consequently, first of
all, if a is real and positive,
00:40:04.880 --> 00:40:08.450
actually, if a is real
and positive, then
00:40:08.450 --> 00:40:12.380
if we had a pole
of x of n say here,
00:40:12.380 --> 00:40:17.460
here being any place,
then if a is real,
00:40:17.460 --> 00:40:25.080
what happens to that pole is
that it moves either in or out.
00:40:25.080 --> 00:40:30.180
But it moves radially as
we vary the value of a.
00:40:30.180 --> 00:40:33.615
So that's for a real.
00:40:36.900 --> 00:40:41.890
What's the movement of the
pole if a is pure imaginary?
00:40:41.890 --> 00:40:48.840
Well, if a is pure imaginary,
then the magnitude of a is--
00:40:48.840 --> 00:40:50.490
I'm sorry.
00:40:50.490 --> 00:40:54.120
Not if a is pure imaginary,
but if the magnitude of a
00:40:54.120 --> 00:40:58.920
is equal to 1 and it has only
a phase component, in that case
00:40:58.920 --> 00:41:01.290
there's no effect
on the radial value.
00:41:01.290 --> 00:41:03.660
There's only an
effect on the angle.
00:41:03.660 --> 00:41:06.990
And in that case, the
movement of the pole
00:41:06.990 --> 00:41:12.660
is such that the radial
value stays the same.
00:41:12.660 --> 00:41:16.420
But the angle of
the pole changes.
00:41:16.420 --> 00:41:19.456
So in general, of
course, if a is complex,
00:41:19.456 --> 00:41:21.330
the poles can have a
little movement that way
00:41:21.330 --> 00:41:24.150
and also a little
movement that way.
00:41:24.150 --> 00:41:30.090
In some cases, if
a is pure real,
00:41:30.090 --> 00:41:32.470
the movement will
just be this way.
00:41:32.470 --> 00:41:35.580
And if the magnitude
of a is equal to 1,
00:41:35.580 --> 00:41:39.700
the movement of the pole
will just be that way.
00:41:39.700 --> 00:41:43.420
All right, so we have first
of all, a list of properties.
00:41:43.420 --> 00:41:45.940
But in particular,
there are two that we've
00:41:45.940 --> 00:41:49.080
spent a little time on.
00:41:49.080 --> 00:41:52.690
Finally, let's
look at, actually,
00:41:52.690 --> 00:41:56.880
one example or one
and a half examples
00:41:56.880 --> 00:41:59.460
to see how some
of the properties
00:41:59.460 --> 00:42:05.900
might be useful in obtaining
the frequency response,
00:42:05.900 --> 00:42:09.050
or the Z-transform of a system.
00:42:09.050 --> 00:42:12.740
And the sequence that
I want to focus on
00:42:12.740 --> 00:42:17.810
is a sequence that will
play an important role
00:42:17.810 --> 00:42:19.940
throughout digital
signal processing
00:42:19.940 --> 00:42:22.440
and in particular, in
some lectures coming up,
00:42:22.440 --> 00:42:26.920
which is a sequence that I'll
refer to as a boxcar sequence.
00:42:26.920 --> 00:42:32.000
It's the sequence which is
equal to unity for n between 0
00:42:32.000 --> 00:42:34.160
and capital N minus 1.
00:42:34.160 --> 00:42:37.250
And it's equal to 0 otherwise.
00:42:37.250 --> 00:42:41.300
It's basically a
rectangular sequence.
00:42:41.300 --> 00:42:44.790
0 for n negative, 0
for n greater than
00:42:44.790 --> 00:42:47.820
or equal to capital N
and unity otherwise.
00:42:47.820 --> 00:42:53.980
It's the counterpart of the
rectangular time function.
00:42:53.980 --> 00:42:57.660
Well, there are a couple of
ways of getting at Z-transform.
00:42:57.660 --> 00:43:01.560
Since we had some properties,
let's use one of them,
00:43:01.560 --> 00:43:04.320
in particular, the
Shifting Property.
00:43:04.320 --> 00:43:08.000
We can think of
a boxcar sequence
00:43:08.000 --> 00:43:10.310
as the sum of two sequences.
00:43:10.310 --> 00:43:13.580
One is a unit step.
00:43:13.580 --> 00:43:17.830
And the second is the negative
of a unit step starting
00:43:17.830 --> 00:43:20.540
at n equals capital
N to subtract off
00:43:20.540 --> 00:43:23.420
these other values.
00:43:23.420 --> 00:43:28.010
We can express x of n,
the boxcar sequence,
00:43:28.010 --> 00:43:34.460
as a unit step minus a unit
step delayed by capital N.
00:43:34.460 --> 00:43:38.750
Well, if you don't see this
graphical picture exactly,
00:43:38.750 --> 00:43:41.120
you can just see quickly
that this is true
00:43:41.120 --> 00:43:45.740
since for n greater than
or equal to capital N,
00:43:45.740 --> 00:43:48.530
both of these arguments
are non-negative.
00:43:48.530 --> 00:43:51.380
So the value of both of
these units steps is unity.
00:43:51.380 --> 00:43:52.780
And they subtract off to zero.
00:43:55.320 --> 00:43:58.140
All right, then to
get the Z-transform,
00:43:58.140 --> 00:44:02.390
we can add the Z-transform
or this piece and this piece.
00:44:02.390 --> 00:44:04.790
The Z-transform of a
unit step, well that's
00:44:04.790 --> 00:44:07.760
our old friend a to the
n times a unit step,
00:44:07.760 --> 00:44:10.340
except in this case, a equals 1.
00:44:10.340 --> 00:44:14.180
So the Z-transform of this
piece is 1 over 1 minus z
00:44:14.180 --> 00:44:17.550
to the minus 1.
00:44:17.550 --> 00:44:20.610
This one is this one shifted.
00:44:20.610 --> 00:44:25.110
So we can apply our Shifting
Property to multiply this by z
00:44:25.110 --> 00:44:29.960
to the minus capital N, since
that's the amount of our shift,
00:44:29.960 --> 00:44:33.890
so that this piece, then, has
a Z-transform z to the minus
00:44:33.890 --> 00:44:37.810
n over 1 minus z to the minus 1.
00:44:37.810 --> 00:44:45.860
Or if we add these two
together, we have 1 minus z
00:44:45.860 --> 00:44:53.110
to the minus capital N divided
by 1 minus z to the minus 1.
00:44:53.110 --> 00:44:56.500
Or we can rewrite that,
just to focus on something
00:44:56.500 --> 00:45:02.750
a little more clearly,
multiplying top and bottom by z
00:45:02.750 --> 00:45:04.620
to the n minus 1.
00:45:04.620 --> 00:45:07.010
We can rewrite that
in this form so
00:45:07.010 --> 00:45:09.260
that we have in the
numerator z to the capital
00:45:09.260 --> 00:45:13.180
N minus 1, in the
denominator, this factor times
00:45:13.180 --> 00:45:14.240
z to the minus 1.
00:45:17.030 --> 00:45:19.390
Well, let's look at
the pole-zero pattern.
00:45:22.860 --> 00:45:33.180
First of all, at z
equals 1, we have a pole.
00:45:33.180 --> 00:45:35.190
So there's that pole.
00:45:35.190 --> 00:45:39.990
Second of all, at z equals
0, we have n minus1 poles.
00:45:39.990 --> 00:45:43.710
Well, let's stick in
the n minus 1 poles.
00:45:43.710 --> 00:45:47.430
And let me just draw that with
an asterisk and an indication
00:45:47.430 --> 00:45:51.330
that that's n minus 1 poles.
00:45:51.330 --> 00:45:53.820
That's from that term.
00:45:53.820 --> 00:45:56.510
And the zeros, well,
where are the zeros?
00:45:56.510 --> 00:45:58.320
They're the roots of
the numerator, which
00:45:58.320 --> 00:46:03.680
are at the N roots of unity.
00:46:03.680 --> 00:46:04.750
Where are they?
00:46:04.750 --> 00:46:05.630
They're distributed.
00:46:05.630 --> 00:46:07.790
The N roots of unity
are distributed
00:46:07.790 --> 00:46:12.800
around the unit circle
equally spaced in angle
00:46:12.800 --> 00:46:16.070
starting at z equals 1.
00:46:16.070 --> 00:46:18.515
So there's a zero at z equals 1.
00:46:18.515 --> 00:46:21.170
But there's also a
pole it equals 1.
00:46:21.170 --> 00:46:24.500
So in fact at z equals 1,
there's neither 0 nor a pole
00:46:24.500 --> 00:46:27.570
because the two cancel out.
00:46:27.570 --> 00:46:30.330
If we look at the
other zeros, then
00:46:30.330 --> 00:46:33.900
let's take a specific
case, that is n equals 8.
00:46:33.900 --> 00:46:37.980
We'd expect to see eight
zeros, except for the 1 at z
00:46:37.980 --> 00:46:40.080
equals 1 that got canceled out.
00:46:40.080 --> 00:46:42.820
So there are seven left--
00:46:42.820 --> 00:46:52.160
one there, there, there,
there, here, here, and here.
00:46:52.160 --> 00:46:54.740
Then, there was the
one at the origin.
00:46:54.740 --> 00:47:00.240
Let me just indicate
that and the fact
00:47:00.240 --> 00:47:02.940
that it got canceled
out by a pole.
00:47:02.940 --> 00:47:06.330
So in fact, that
one isn't there.
00:47:06.330 --> 00:47:09.480
So the pole-zero pattern
for the boxcar sequence,
00:47:09.480 --> 00:47:13.830
then, is n minus 1
poles at the origin
00:47:13.830 --> 00:47:18.570
plus zeros equally spaced
in angle, but with the 1
00:47:18.570 --> 00:47:21.650
at z equals 1 missing.
00:47:21.650 --> 00:47:25.130
Now, what does this mean
in terms of the frequency
00:47:25.130 --> 00:47:26.630
response?
00:47:26.630 --> 00:47:30.620
Well, we can very quickly
generate the frequency
00:47:30.620 --> 00:47:34.760
response, or a rough idea
of the frequency response,
00:47:34.760 --> 00:47:39.290
geometrically by referring
back to the set of ideas
00:47:39.290 --> 00:47:42.740
that we introduced at the
beginning of the lecture
00:47:42.740 --> 00:47:48.110
and ask what the behavior
of the pole-zero vectors
00:47:48.110 --> 00:47:51.860
are as we go around
the unit circle.
00:47:51.860 --> 00:47:55.760
The pole vectors, first of all,
introduce only a linear phase
00:47:55.760 --> 00:47:58.160
term and have no effect
on the magnitude.
00:47:58.160 --> 00:48:01.520
We had agreed on that before.
00:48:01.520 --> 00:48:08.260
And so for the magnitude, we
only need focus on the zeros.
00:48:08.260 --> 00:48:11.110
Well, one thing is
obvious and that
00:48:11.110 --> 00:48:18.810
is that as we go
around the unit circle,
00:48:18.810 --> 00:48:20.820
the frequency
response is obviously
00:48:20.820 --> 00:48:25.110
0 when we hit each
one of these zeros.
00:48:25.110 --> 00:48:31.200
That corresponds to pi over
4, pi over 2, pi over 2
00:48:31.200 --> 00:48:33.480
plus pi over 4--
whatever that is,
00:48:33.480 --> 00:48:38.640
pi, and the next increment
of pi over 4, et cetera.
00:48:38.640 --> 00:48:41.970
Furthermore, you can see that
at least, it's not implausible.
00:48:41.970 --> 00:48:46.200
Actually, we really could
argue this somewhat precisely,
00:48:46.200 --> 00:48:52.080
that at omega equals 0,
that's the place where
00:48:52.080 --> 00:48:56.280
we're the farthest away
from all the zeros.
00:48:56.280 --> 00:48:59.040
As we start moving
around the unit circle,
00:48:59.040 --> 00:49:01.265
if we're in-between
two of these zeros,
00:49:01.265 --> 00:49:03.390
maybe we're a little farther
away from one of them,
00:49:03.390 --> 00:49:05.530
but we're closer to another one.
00:49:05.530 --> 00:49:08.850
And in terms of the product of
the length of the zero vectors,
00:49:08.850 --> 00:49:11.790
that will tend to stay
smaller than the product
00:49:11.790 --> 00:49:16.020
of the lengths of the zero
vectors at z equals 1.
00:49:16.020 --> 00:49:21.030
Consequently, the frequency
response starts at some value
00:49:21.030 --> 00:49:26.160
and of course, goes
down to 0 at pi over 4.
00:49:26.160 --> 00:49:29.870
That's because of this 0.
00:49:29.870 --> 00:49:34.610
Then, it comes back up
again, but not quite as far,
00:49:34.610 --> 00:49:38.780
and then goes back
down to 0 at pi over 2.
00:49:38.780 --> 00:49:42.610
Then it goes up again
and comes down again.
00:49:42.610 --> 00:49:44.450
It goes up not quite as far.
00:49:44.450 --> 00:49:47.720
And that's not particularly
obvious geometrically.
00:49:47.720 --> 00:49:52.850
And then, the same thing
again going to 0 at pi.
00:49:52.850 --> 00:49:55.370
And then of course as we come
back around the unit circle,
00:49:55.370 --> 00:49:58.450
we see the same thing,
the same type of behavior,
00:49:58.450 --> 00:49:59.784
repeated again.
00:50:04.220 --> 00:50:08.740
So roughly, we can get
a geometric picture
00:50:08.740 --> 00:50:13.090
of the frequency response
for a boxcar sequence
00:50:13.090 --> 00:50:17.410
by looking at the
location of the poles
00:50:17.410 --> 00:50:21.640
and zeros in the z-plane and
the behavior of the vectors
00:50:21.640 --> 00:50:25.230
as we travel around
the unit circle.
00:50:25.230 --> 00:50:31.500
Well finally, let's just
look at the Fourier transform
00:50:31.500 --> 00:50:35.640
of the boxcar somewhat more
formally because in fact, it's
00:50:35.640 --> 00:50:37.200
an important sequence.
00:50:37.200 --> 00:50:42.720
And it's important to
have a fairly complete
00:50:42.720 --> 00:50:45.510
precise statement of
the Fourier transform
00:50:45.510 --> 00:50:49.360
and a complete picture
of what it looks like.
00:50:49.360 --> 00:50:51.060
Well, let's see.
00:50:51.060 --> 00:50:55.120
We had the Z-transform
as 1 minus z
00:50:55.120 --> 00:50:59.192
to the minus n over 1
minus z to the minus 1.
00:50:59.192 --> 00:51:01.890
Substituting in z
equals z to the j omega,
00:51:01.890 --> 00:51:07.600
then we have x of e to
the j omega as this.
00:51:07.600 --> 00:51:11.560
We can factor out a factor e
to the minus j omega capital
00:51:11.560 --> 00:51:17.220
N over 2, leaving e to the j
omega capital N over 2 minus
00:51:17.220 --> 00:51:22.010
e to the minus j omega capital
N over 2, and a denominator
00:51:22.010 --> 00:51:26.840
factor e to the minus j omega
over 2 times e to the j omega--
00:51:26.840 --> 00:51:31.490
over 2 minus e to the
minus j omega over 2.
00:51:31.490 --> 00:51:40.890
This piece we recognize as 2j
sine omega capital N over to 2.
00:51:40.890 --> 00:51:50.430
And this piece we recognize as
2-j times sine omega over 2.
00:51:50.430 --> 00:51:53.910
And consequently, putting
these two terms together
00:51:53.910 --> 00:51:58.440
and inserting this substitution,
the Fourier transform
00:51:58.440 --> 00:52:03.210
is e to the minus j omega
capital N minus 1 over 2--
00:52:03.210 --> 00:52:06.780
that's a linear phase
term, by the way--
00:52:06.780 --> 00:52:13.980
times sine omega capital N over
2 divided by sine omega over 2.
00:52:13.980 --> 00:52:18.510
And this function,
sine omega capital N
00:52:18.510 --> 00:52:22.290
over 2 divided by
sine omega over 2,
00:52:22.290 --> 00:52:27.830
is the discrete time
counterpart of what we usually
00:52:27.830 --> 00:52:33.320
find in the continuous
time case as sine x over x.
00:52:33.320 --> 00:52:37.910
That is, this is a sine nx
over sine x kind of function.
00:52:37.910 --> 00:52:41.690
And it plays exactly the same
role in the discrete time case
00:52:41.690 --> 00:52:45.800
that the sine x over x function
plays in the continuous time
00:52:45.800 --> 00:52:47.610
case.
00:52:47.610 --> 00:52:49.880
And that's not
unreasonable, actually,
00:52:49.880 --> 00:52:54.810
because this arose by
looking at the Fourier
00:52:54.810 --> 00:52:58.530
transform of a
rectangular sequence,
00:52:58.530 --> 00:53:01.830
whereas the sine
x over x function
00:53:01.830 --> 00:53:05.970
arises by looking at the Fourier
transform of a continuous time
00:53:05.970 --> 00:53:08.020
rectangle.
00:53:08.020 --> 00:53:12.550
So the sine nx over
sine x function,
00:53:12.550 --> 00:53:17.280
which is what we have here,
is an extremely important
00:53:17.280 --> 00:53:21.540
function-- as important as sine
x over x in the continuous time
00:53:21.540 --> 00:53:22.780
case.
00:53:22.780 --> 00:53:25.590
And consequently,
let me just show you
00:53:25.590 --> 00:53:29.160
this function plotted out
a little more precisely
00:53:29.160 --> 00:53:31.170
than I would be able
to do at the board.
00:53:31.170 --> 00:53:40.220
In particular, let me show you
a viewgraph which illustrates
00:53:40.220 --> 00:53:44.030
the function sine
omega capital N over 2
00:53:44.030 --> 00:53:47.150
divided by sine omega over 2.
00:53:47.150 --> 00:53:52.160
And I've sketched this now
over more than just a 0 to 2 pi
00:53:52.160 --> 00:53:58.040
interval to, again, stress the
fact that this is periodic--
00:53:58.040 --> 00:54:00.320
as all Fourier transforms are--
00:54:00.320 --> 00:54:04.460
periodic with a period of 2 pi.
00:54:04.460 --> 00:54:06.770
The important
characteristics of it--
00:54:06.770 --> 00:54:08.144
some important characteristics.
00:54:08.144 --> 00:54:08.810
There are a lot.
00:54:08.810 --> 00:54:11.600
But some important
characteristics of it
00:54:11.600 --> 00:54:15.270
are that it has an envelope.
00:54:15.270 --> 00:54:18.620
It's basically a
sinusoidal function
00:54:18.620 --> 00:54:24.920
with an envelope that is the
reciprocal of a sinusoid.
00:54:24.920 --> 00:54:32.330
The period of this
sinusoid is from 0 to 2pi,
00:54:32.330 --> 00:54:34.790
whereas this one wiggles
faster, depending
00:54:34.790 --> 00:54:38.180
on the value of capital
N. I've sketched it here
00:54:38.180 --> 00:54:40.370
for n equals 15.
00:54:40.370 --> 00:54:44.120
But in fact, it has a
lot of the character
00:54:44.120 --> 00:54:48.950
of a sine x over x function.
00:54:48.950 --> 00:54:52.880
That is, it has a
big central lobe,
00:54:52.880 --> 00:54:56.330
decays down and wiggles and
gets smaller as it's decaying.
00:54:56.330 --> 00:54:59.180
But then, of course, the fact
that it has to be periodic
00:54:59.180 --> 00:55:01.970
is what distinguishes
it in the discrete time
00:55:01.970 --> 00:55:06.090
case from the sine x over x
function in the continuous time
00:55:06.090 --> 00:55:06.590
case.
00:55:13.380 --> 00:55:13.880
OK.
00:55:13.880 --> 00:55:20.450
Well, this concludes our
discussion of the Z-transform.
00:55:20.450 --> 00:55:23.094
We've now talked
about two transforms.
00:55:23.094 --> 00:55:24.760
We've talked about
the Fourier transform
00:55:24.760 --> 00:55:30.710
and the Z-transform spread
out over about five lectures.
00:55:30.710 --> 00:55:34.490
And in the next lecture,
the next set of two or three
00:55:34.490 --> 00:55:39.740
lectures, we'll be talking about
yet another transform, which
00:55:39.740 --> 00:55:42.980
is a transform that's
really somewhat special
00:55:42.980 --> 00:55:47.570
and linked very closely to
the notion of discrete time
00:55:47.570 --> 00:55:50.420
signals and discrete
time signal processing.
00:55:50.420 --> 00:55:54.260
That transform is the
Discrete Fourier transform.
00:55:54.260 --> 00:55:58.520
And besides being a
mathematical tool,
00:55:58.520 --> 00:56:02.230
as the Fourier transform and
the Z-transform have been,
00:56:02.230 --> 00:56:07.970
Discrete Fourier transform has
some important computational
00:56:07.970 --> 00:56:11.930
realizations and
computational implications
00:56:11.930 --> 00:56:14.990
that will be one of the
important things that
00:56:14.990 --> 00:56:18.110
will want to capitalize on
in applying digital signal
00:56:18.110 --> 00:56:20.650
processing to real problems.
00:56:20.650 --> 00:56:22.540
Thank you.
00:56:22.540 --> 00:56:25.590
[MUSIC PLAYING]