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[MUSIC PLAYING]
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ALAN V. OPPENHEIM:
In the last lecture,
00:00:54.630 --> 00:00:59.600
we introduced the class
of discrete time systems,
00:00:59.600 --> 00:01:03.980
and in particular, imposed the
conditions of linearity first
00:01:03.980 --> 00:01:08.360
of all, and second, the
property or constraint of shift
00:01:08.360 --> 00:01:10.880
invariance.
00:01:10.880 --> 00:01:14.810
And those constraints led
us to the convolution sum
00:01:14.810 --> 00:01:16.880
representation.
00:01:16.880 --> 00:01:20.420
In today's lecture,
there are several issues
00:01:20.420 --> 00:01:22.470
that I'd like to focus on.
00:01:22.470 --> 00:01:29.660
The first is the introduction
of two additional constraints
00:01:29.660 --> 00:01:33.530
that it's sometimes useful to
impose, or at least consider,
00:01:33.530 --> 00:01:35.450
for discrete time systems--
00:01:35.450 --> 00:01:37.820
namely the constraints
or conditions
00:01:37.820 --> 00:01:41.630
of causality and stability.
00:01:41.630 --> 00:01:46.770
Second of all, I would like to
talk about a particular class,
00:01:46.770 --> 00:01:48.680
or at least introduce
a particular class
00:01:48.680 --> 00:01:51.800
of linear shift
invariant systems, namely
00:01:51.800 --> 00:01:54.920
those representable by linear
constant coefficient difference
00:01:54.920 --> 00:01:56.280
equations.
00:01:56.280 --> 00:01:59.990
And finally, I'd
like to introduce
00:01:59.990 --> 00:02:05.180
a representation of linear
shift invariant systems
00:02:05.180 --> 00:02:09.380
that is an alternative to the
convolution sum representation,
00:02:09.380 --> 00:02:11.270
and in particular,
that representation
00:02:11.270 --> 00:02:14.720
corresponds to the
representation in terms
00:02:14.720 --> 00:02:17.210
of a frequency response.
00:02:17.210 --> 00:02:20.630
Well, let's begin
with the notions
00:02:20.630 --> 00:02:25.280
of stability and causality,
reminding you, first of all,
00:02:25.280 --> 00:02:28.370
that, as we talked
about last time,
00:02:28.370 --> 00:02:31.610
we can consider a
general system--
00:02:31.610 --> 00:02:34.640
inputs and outputs
are sequences--
00:02:34.640 --> 00:02:37.820
and in general, the system
just simply corresponds
00:02:37.820 --> 00:02:40.730
to some transformation
from the input sequence
00:02:40.730 --> 00:02:43.310
to the output sequence.
00:02:43.310 --> 00:02:47.390
When we impose the conditions of
linearity and shift invariance,
00:02:47.390 --> 00:02:53.780
both conditions, then we can
express the output sequence
00:02:53.780 --> 00:02:58.580
in this form where h
of n is the response
00:02:58.580 --> 00:03:01.070
of the system to a unit sample.
00:03:01.070 --> 00:03:05.930
And this can also be
rearranged in the form
00:03:05.930 --> 00:03:08.900
that I've indicated here,
essentially interchanging
00:03:08.900 --> 00:03:12.620
the role of the unit sample
response and the input.
00:03:12.620 --> 00:03:16.430
The sum expressed in
either of these two forms,
00:03:16.430 --> 00:03:20.360
we referred to last time
as the convolution sum.
00:03:20.360 --> 00:03:24.080
So the convolution
sum comes out of--
00:03:24.080 --> 00:03:25.910
or is a consequence of--
00:03:25.910 --> 00:03:30.210
linearity and shift invariance.
00:03:30.210 --> 00:03:35.060
Two additional conditions
are stability and causality.
00:03:35.060 --> 00:03:39.290
Stability of a
system, in general,
00:03:39.290 --> 00:03:46.070
corresponds to the statement
that if the input sequence
00:03:46.070 --> 00:03:47.780
is bounded--
00:03:47.780 --> 00:03:51.800
in other words, if x of
n, essentially if x of n
00:03:51.800 --> 00:03:57.290
is finite for all n, including
as n goes to infinity,
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then a system is
said to be stable
00:04:00.260 --> 00:04:06.170
if the output sequence,
y of n, is also bounded.
00:04:06.170 --> 00:04:11.900
In other words, the magnitude
of y of n is finite for all n.
00:04:11.900 --> 00:04:15.590
If that's true for any
bounded input sequence
00:04:15.590 --> 00:04:17.570
that the output
sequence is bounded,
00:04:17.570 --> 00:04:20.420
the system is said to be stable.
00:04:20.420 --> 00:04:22.460
Now, that's a
statement that applies
00:04:22.460 --> 00:04:24.920
to general discrete
time systems.
00:04:24.920 --> 00:04:28.220
For the specific class of
systems that we'll be dealing
00:04:28.220 --> 00:04:32.000
with in this set of lessons,
namely the class of linear
00:04:32.000 --> 00:04:37.130
shift invariant systems, you
can show that an equivalent
00:04:37.130 --> 00:04:42.350
statement of stability-- or
a necessary and sufficient
00:04:42.350 --> 00:04:44.450
condition for this to be true--
00:04:44.450 --> 00:04:48.830
is simply that the unit
sample response of the system
00:04:48.830 --> 00:04:50.990
be absolutely summable.
00:04:50.990 --> 00:04:53.690
In other words, that the
sum of the magnitudes
00:04:53.690 --> 00:04:58.620
of the values in the unit
sample response are finite.
00:04:58.620 --> 00:05:02.280
For example, if we had a
unit sample response which
00:05:02.280 --> 00:05:07.410
was 2 to the n times a unit
step, 2 to the n, of course,
00:05:07.410 --> 00:05:09.780
as n increases--
00:05:09.780 --> 00:05:12.500
as n goes from 0 to infinity--
00:05:12.500 --> 00:05:16.020
2 the n grows
exponentially, in fact.
00:05:16.020 --> 00:05:20.430
And so, obviously, this is
not absolutely summable.
00:05:20.430 --> 00:05:26.120
So this is an example of a
system that would be unstable.
00:05:29.310 --> 00:05:34.110
Whereas, if h of n were a
1/2 to the n times u of n,
00:05:34.110 --> 00:05:36.570
so that for n greater than 0--
00:05:36.570 --> 00:05:38.970
and less than 0, of course,
both of these are 0--
00:05:38.970 --> 00:05:42.600
for n greater than 0, this
were decaying exponentially,
00:05:42.600 --> 00:05:46.260
if you sum up these
values from 0 to infinity,
00:05:46.260 --> 00:05:48.610
it converges to a finite number.
00:05:48.610 --> 00:05:55.122
So that this, in fact, would
correspond to a stable system.
00:05:55.122 --> 00:05:56.580
Now, we have the
option, of course,
00:05:56.580 --> 00:06:01.030
of talking about unstable
systems or stable systems.
00:06:01.030 --> 00:06:05.610
Generally, it is
true that stability
00:06:05.610 --> 00:06:10.410
is a condition on a system
that it's useful to impose.
00:06:10.410 --> 00:06:13.320
In other words, generally,
we would like our systems
00:06:13.320 --> 00:06:15.900
to be stable, although
there are actually
00:06:15.900 --> 00:06:18.540
some cases where
unstable systems are
00:06:18.540 --> 00:06:21.670
useful to talk about.
00:06:21.670 --> 00:06:29.550
The second property or condition
that it is useful sometimes
00:06:29.550 --> 00:06:33.210
to consider, is the
condition of causality.
00:06:33.210 --> 00:06:38.190
And causality, first of
all, for a general system,
00:06:38.190 --> 00:06:42.240
is a statement basically
that the system doesn't
00:06:42.240 --> 00:06:44.950
respond before you kick it.
00:06:44.950 --> 00:06:52.200
In other words, if we have
an input, x of n, the output,
00:06:52.200 --> 00:06:58.250
y of n, for some value
of n-- let's say n 1--
00:06:58.250 --> 00:07:04.130
depends only on x of n
for previous values of n.
00:07:04.130 --> 00:07:09.830
So for any n 1, the
statement is that y of n
00:07:09.830 --> 00:07:16.100
only depends on x of n for
previous values of x of n.
00:07:16.100 --> 00:07:19.610
In other words, the system
can't anticipate the values
00:07:19.610 --> 00:07:23.240
that are going to be coming in.
00:07:23.240 --> 00:07:26.330
For a linear shift
invariant system,
00:07:26.330 --> 00:07:30.800
we can show that a necessary
and sufficient condition
00:07:30.800 --> 00:07:36.080
for causality is that the unit
sample response of the system
00:07:36.080 --> 00:07:38.720
be 0 for n less than 0.
00:07:38.720 --> 00:07:41.930
In other words, if the unit
sample response of the system
00:07:41.930 --> 00:07:44.510
is 0 for n less
than 0, the system
00:07:44.510 --> 00:07:47.120
is guaranteed to be causal.
00:07:47.120 --> 00:07:50.870
If it's not 0 for n less
than 0, then the system
00:07:50.870 --> 00:07:54.790
is guaranteed not to be causal.
00:07:54.790 --> 00:07:58.090
Just for example, if we had a
unit sample response which was
00:07:58.090 --> 00:08:01.130
2 to the n times u of minus n--
00:08:01.130 --> 00:08:04.830
in other words, 0 for n
greater than 0, and 2 the n
00:08:04.830 --> 00:08:07.240
for n less than 0,
this, of course,
00:08:07.240 --> 00:08:13.780
would correspond to
a non-causal system,
00:08:13.780 --> 00:08:17.320
since the unit sample
response has non-zero values
00:08:17.320 --> 00:08:20.460
for negative values of n.
00:08:20.460 --> 00:08:23.610
And we could also
examine stability.
00:08:23.610 --> 00:08:29.280
It would turn out that this
corresponds to a stable system,
00:08:29.280 --> 00:08:33.350
although in the previous
example, we had talked about 2
00:08:33.350 --> 00:08:36.210
to the n for n
positive corresponds
00:08:36.210 --> 00:08:37.750
to an unstable system.
00:08:37.750 --> 00:08:39.960
The point is that if
you look at 2 to the n
00:08:39.960 --> 00:08:45.380
as the index n runs
negative, then that is--
00:08:45.380 --> 00:08:48.240
as n runs negative,
it's an exponential
00:08:48.240 --> 00:08:51.090
that's decaying in
negative values of n.
00:08:51.090 --> 00:08:53.610
So then, in fact, this
is absolutely summable--
00:08:53.610 --> 00:08:57.870
that makes it stable, but
it's obviously non-causal.
00:08:57.870 --> 00:08:59.550
Now, I want to stress--
00:08:59.550 --> 00:09:02.790
it's a very important
point to stress--
00:09:02.790 --> 00:09:06.180
that causality is, of
course, a useful thing
00:09:06.180 --> 00:09:09.870
to inquire about about a
system, it's useful to ask,
00:09:09.870 --> 00:09:12.570
is the system causal
or is it not causal?
00:09:12.570 --> 00:09:17.340
But generally, it is
not necessarily true
00:09:17.340 --> 00:09:20.190
that causality is a
condition that we'll
00:09:20.190 --> 00:09:22.410
want to impose on the system.
00:09:22.410 --> 00:09:27.750
There are many examples of
useful, non-causal systems,
00:09:27.750 --> 00:09:30.510
and in many
instances, we'll want
00:09:30.510 --> 00:09:33.540
to talk about systems
which are non-causal.
00:09:33.540 --> 00:09:35.970
So again, it's useful
to inquire about
00:09:35.970 --> 00:09:38.250
whether a system is
causal or not causal,
00:09:38.250 --> 00:09:42.990
but it is not generally useful
to constrain ourselves to talk
00:09:42.990 --> 00:09:46.120
only about causal systems.
00:09:46.120 --> 00:09:48.720
The story is somewhat different
for stability in the sense
00:09:48.720 --> 00:09:53.220
that an unstable system
is somewhat less useful
00:09:53.220 --> 00:09:56.590
than a non-causal system.
00:09:56.590 --> 00:10:01.440
So these are two additional
conditions, or properties,
00:10:01.440 --> 00:10:05.160
that we'll sometimes want
to inquire about when
00:10:05.160 --> 00:10:08.670
we talk about a system.
00:10:08.670 --> 00:10:12.480
In general, for linear
shift invariant systems,
00:10:12.480 --> 00:10:16.500
there is a wide latitude
in terms of the description
00:10:16.500 --> 00:10:18.060
of those systems.
00:10:18.060 --> 00:10:21.360
And that is a similar
type of statement
00:10:21.360 --> 00:10:24.420
that applies in the
continuous time case, also.
00:10:24.420 --> 00:10:26.550
Just as in the
continuous time case,
00:10:26.550 --> 00:10:31.440
it is useful to
concentrate, in many cases,
00:10:31.440 --> 00:10:35.670
on systems that can
be implemented with
00:10:35.670 --> 00:10:36.960
r's, l's, and c's--
00:10:36.960 --> 00:10:38.580
and consequently
are representable
00:10:38.580 --> 00:10:41.940
by linear constant coefficient
differential equations.
00:10:41.940 --> 00:10:44.130
In the discrete time
case, it's often
00:10:44.130 --> 00:10:47.700
useful to concentrate
on systems that
00:10:47.700 --> 00:10:53.010
are describable by linear
constant coefficient difference
00:10:53.010 --> 00:10:54.540
equations.
00:10:54.540 --> 00:10:59.430
So I would like to
just briefly introduce
00:10:59.430 --> 00:11:03.450
the class of systems
which are representable
00:11:03.450 --> 00:11:07.230
by linear constant coefficient
difference equations.
00:11:07.230 --> 00:11:11.850
And the discussion
in this lecture
00:11:11.850 --> 00:11:14.700
is only a brief introduction,
and we'll, in fact,
00:11:14.700 --> 00:11:16.830
be returning to this
class of systems
00:11:16.830 --> 00:11:19.030
several times over
this set of lessons.
00:11:22.380 --> 00:11:25.290
We refer to an Nth order--
00:11:25.290 --> 00:11:28.290
linear constant coefficient
difference equation--
00:11:28.290 --> 00:11:31.990
as being in the form
that I've indicated here.
00:11:31.990 --> 00:11:37.470
And what it consists of
is a linear combination
00:11:37.470 --> 00:11:44.850
of the delayed output sequences
equal to a linear combination
00:11:44.850 --> 00:11:47.310
of delayed input sequences.
00:11:47.310 --> 00:11:49.980
A differential equation, of
course, in continuous time
00:11:49.980 --> 00:11:52.830
involves linear
combinations of derivatives.
00:11:52.830 --> 00:11:57.690
The corresponding situation
in the discrete time case,
00:11:57.690 --> 00:12:01.870
is a linear combination
of differences.
00:12:01.870 --> 00:12:08.520
So this is, then, a
general Nth order, linear--
00:12:08.520 --> 00:12:11.010
in the sense that it's
a linear combination--
00:12:11.010 --> 00:12:16.170
constant coefficient-- meaning
that these are constant
00:12:16.170 --> 00:12:18.780
numbers, as opposed to
being functions of n--
00:12:18.780 --> 00:12:21.450
difference equation--
meaning it involves
00:12:21.450 --> 00:12:23.505
differences of the input
and output sequence.
00:12:26.070 --> 00:12:28.620
The order of the
difference equation
00:12:28.620 --> 00:12:33.780
generally is used to refer to
the number of delays required
00:12:33.780 --> 00:12:35.790
in the output sequence.
00:12:35.790 --> 00:12:40.020
In general, the number of
delays in the input sequence, M,
00:12:40.020 --> 00:12:43.620
does not have to be equal to N.
But it's generally convenient
00:12:43.620 --> 00:12:45.960
to refer to the order
as corresponding
00:12:45.960 --> 00:12:48.855
to the number of delays
involved in the output sequence.
00:12:51.390 --> 00:12:56.550
For the 0th order
difference equation,
00:12:56.550 --> 00:13:00.450
the solution if it corresponds
to a representation
00:13:00.450 --> 00:13:02.710
of a linear shift
and invariant system,
00:13:02.710 --> 00:13:05.310
the solution is trivial--
00:13:05.310 --> 00:13:06.910
very straightforward.
00:13:06.910 --> 00:13:09.810
Particular, let's examine
what this difference equation
00:13:09.810 --> 00:13:13.620
reduces to, if N is equal to 0.
00:13:13.620 --> 00:13:16.800
And just for
convenience, we'll choose
00:13:16.800 --> 00:13:18.870
the coefficients
to be normalized
00:13:18.870 --> 00:13:21.240
so that a sub 0 is equal to 1.
00:13:21.240 --> 00:13:24.600
Obviously anything I
say is straightforwardly
00:13:24.600 --> 00:13:27.620
generalized for a
0 not equal to 1,
00:13:27.620 --> 00:13:33.520
but that's just a convenient
normalization to impose.
00:13:33.520 --> 00:13:36.970
All right, if N is equal to
0, and a 0 is equal to 1,
00:13:36.970 --> 00:13:39.120
then what this
equation looks like
00:13:39.120 --> 00:13:43.290
is just on the left hand side,
y of n, and the right hand side
00:13:43.290 --> 00:13:44.880
as it is.
00:13:44.880 --> 00:13:50.510
So we have y of n is
equal to this sum.
00:13:50.510 --> 00:13:54.960
And that looks suspiciously
like the convolution sum--
00:13:54.960 --> 00:14:00.040
it involves a linear combination
only of delayed input
00:14:00.040 --> 00:14:01.540
sequences.
00:14:01.540 --> 00:14:07.340
And so, in fact, this is
identical to the convolution
00:14:07.340 --> 00:14:08.110
sum.
00:14:08.110 --> 00:14:13.000
If we thought of this, say,
as h of r, h of r, then,
00:14:13.000 --> 00:14:14.920
is equal to b sub r.
00:14:14.920 --> 00:14:19.720
So the unit sample response
corresponding to the 0th order
00:14:19.720 --> 00:14:25.660
difference equation is just
this set of coefficients,
00:14:25.660 --> 00:14:31.120
and, of course, r
runs only from 0 to M,
00:14:31.120 --> 00:14:36.850
so it's the coefficients b
sub n for n between 0 and M.
00:14:36.850 --> 00:14:39.440
And it's equal to 0, otherwise.
00:14:39.440 --> 00:14:40.891
So that's very straightforward.
00:14:40.891 --> 00:14:42.640
We can pick off the
solution, essentially,
00:14:42.640 --> 00:14:46.090
by recognizing this as
the convolution sum.
00:14:46.090 --> 00:14:52.500
Obviously, another way of
obtaining the unit sample
00:14:52.500 --> 00:14:54.990
response corresponding
to this system
00:14:54.990 --> 00:14:58.500
is simply to plug in a
unit sample for x of n.
00:14:58.500 --> 00:15:00.810
And you'll see, in fact,
that what comes rolling out
00:15:00.810 --> 00:15:04.260
are these coefficients
b sub r, or b sub n.
00:15:04.260 --> 00:15:09.120
So if this is used to describe
a linear shift invariant system,
00:15:09.120 --> 00:15:11.970
the unit sample response
corresponds simply
00:15:11.970 --> 00:15:16.050
to the coefficients in
the difference equation.
00:15:16.050 --> 00:15:19.500
For n not equal to 0, it's
not quite as straightforward
00:15:19.500 --> 00:15:20.910
as that.
00:15:20.910 --> 00:15:27.150
First of all, let me point
out that for N not equal to 0,
00:15:27.150 --> 00:15:31.650
the left hand side of this
equation involves y of n,
00:15:31.650 --> 00:15:34.290
y of n minus 1, y of
minus 2, et cetera,
00:15:34.290 --> 00:15:38.070
up to y of little
n minus capital N.
00:15:38.070 --> 00:15:43.880
We can take all of the terms
except the one involving y of n
00:15:43.880 --> 00:15:47.600
over to the right hand
side of the equation,
00:15:47.600 --> 00:15:50.150
and end up with
an equation which
00:15:50.150 --> 00:15:55.640
expresses y of n as a linear
combination of delayed inputs,
00:15:55.640 --> 00:16:01.440
and a linear combination
of past output sequences.
00:16:01.440 --> 00:16:04.830
If we rewrite this
equation in this form--
00:16:04.830 --> 00:16:06.860
and again, just for
convenience, we'll
00:16:06.860 --> 00:16:13.250
choose a 0 equal to 1, simply
normalizing that coefficient.
00:16:13.250 --> 00:16:16.820
Then, the difference
equation that we end up
00:16:16.820 --> 00:16:22.070
with is in the form that
I've indicated here, y of n
00:16:22.070 --> 00:16:27.980
is a linear combination of
delayed input sequences minus--
00:16:27.980 --> 00:16:31.130
because we brought that from
the other side of the equation--
00:16:31.130 --> 00:16:33.920
minus a linear
combination of past--
00:16:33.920 --> 00:16:35.960
of the previous output values.
00:16:38.710 --> 00:16:43.960
What that says is
that we presumably
00:16:43.960 --> 00:16:48.610
can iterate this equation,
or equivalently, what it says
00:16:48.610 --> 00:16:51.430
is that the difference
equation corresponds
00:16:51.430 --> 00:16:56.320
to an explicit input/output
relationship for the system,
00:16:56.320 --> 00:17:00.940
because if somehow we could
get the equation going,
00:17:00.940 --> 00:17:05.319
then we can solve for y of
n-- we can solve for y of n
00:17:05.319 --> 00:17:09.880
if we have the right
previous values for y of n.
00:17:09.880 --> 00:17:12.671
And then we can solve
for y of n plus 1, y of n
00:17:12.671 --> 00:17:13.420
plus 2, et cetera.
00:17:13.420 --> 00:17:17.960
In other words, we can continue
to iterate the equation.
00:17:17.960 --> 00:17:21.010
What's required to
do that are that we
00:17:21.010 --> 00:17:22.140
have to know the input--
00:17:22.140 --> 00:17:24.339
and we assume, of course,
that we know that--
00:17:24.339 --> 00:17:28.750
and we have to know some
previous output values.
00:17:28.750 --> 00:17:32.960
And to know the
previous output values,
00:17:32.960 --> 00:17:37.010
then, means that there is some
additional information that we
00:17:37.010 --> 00:17:40.100
have to specify, and
those we'll generally
00:17:40.100 --> 00:17:41.960
refer to as the
boundary conditions,
00:17:41.960 --> 00:17:43.970
or the initial conditions.
00:17:43.970 --> 00:17:46.550
For example, let's
see how this equation
00:17:46.550 --> 00:17:51.950
would work if we focused
on the first order case.
00:17:51.950 --> 00:17:57.600
So the first order case,
capital N is equal to 1.
00:17:57.600 --> 00:18:00.740
We have the equation, y of
n minus a, y of n minus 1
00:18:00.740 --> 00:18:03.110
equals x of n.
00:18:03.110 --> 00:18:07.950
And let's find the unit sample
response for the system--
00:18:07.950 --> 00:18:12.110
in other words, choosing x
of n equal to a unit sample.
00:18:12.110 --> 00:18:15.590
And for the initial
conditions, let's assume
00:18:15.590 --> 00:18:19.640
that, for a unit sample
input, the output is
00:18:19.640 --> 00:18:22.730
equal to 0 for n less than 0.
00:18:22.730 --> 00:18:26.780
Now, when we do that, we're
making a specific assumption
00:18:26.780 --> 00:18:28.250
about causality--
00:18:28.250 --> 00:18:31.850
we're imposing the
boundary conditions
00:18:31.850 --> 00:18:35.810
that the unit sample response
has to be 0 for n less than 0--
00:18:35.810 --> 00:18:38.600
that's exactly the necessary
and sufficient condition
00:18:38.600 --> 00:18:40.410
that we need for causality.
00:18:40.410 --> 00:18:44.570
So basically, we're saying that
we're imposing on the solution
00:18:44.570 --> 00:18:47.720
that we're about to generate,
the additional condition
00:18:47.720 --> 00:18:49.880
of causality.
00:18:49.880 --> 00:18:53.440
All right, well now,
rewriting this equation
00:18:53.440 --> 00:18:55.990
by taking this term over
to the right hand side,
00:18:55.990 --> 00:18:59.050
and taking account of
the fact that we're
00:18:59.050 --> 00:19:01.660
considering the input
to be a unit sample,
00:19:01.660 --> 00:19:04.960
we have the output
sequences unit
00:19:04.960 --> 00:19:09.100
sample plus a times the
output sequence delayed.
00:19:09.100 --> 00:19:14.810
And now let's run through an
iteration of this equation,
00:19:14.810 --> 00:19:18.070
obtaining some
successive values.
00:19:18.070 --> 00:19:22.270
y of minus 1 we can
get simply by referring
00:19:22.270 --> 00:19:24.250
to the initial condition.
00:19:24.250 --> 00:19:27.400
We stated there y of n
is 0 for n less than 0,
00:19:27.400 --> 00:19:33.090
and so that means, of course,
that y of minus 1 is 0.
00:19:33.090 --> 00:19:38.510
y of 0 is equal to delta of
0, but that's equal to 1,
00:19:38.510 --> 00:19:42.360
plus a times y of
minus 1, which was 0.
00:19:42.360 --> 00:19:46.644
So y of 0 is equal to 1.
00:19:46.644 --> 00:19:50.720
y of 1 is equal to
delta of 1, which is 0--
00:19:50.720 --> 00:19:54.660
the unit sample is only
non-0 at n equals 0.
00:19:54.660 --> 00:19:57.940
So that 0 plus a times y of 0--
00:19:57.940 --> 00:19:59.880
y of 0 is equal to 1--
00:19:59.880 --> 00:20:06.090
so y of 1 is equal
to a times 1, or a.
00:20:06.090 --> 00:20:11.030
y of 2, if we follow that
through, is equal to a squared.
00:20:11.030 --> 00:20:13.100
And if you run through
a few more of those,
00:20:13.100 --> 00:20:17.410
you'll see rather quickly that
what we get for the unit sample
00:20:17.410 --> 00:20:22.750
response, imposing the
condition of causality,
00:20:22.750 --> 00:20:27.850
is that the unit sample
response is a to the N for n
00:20:27.850 --> 00:20:29.470
greater than 0.
00:20:29.470 --> 00:20:32.080
Of course, it's 0 for
n less than 0, because
00:20:32.080 --> 00:20:33.800
of our initial condition.
00:20:33.800 --> 00:20:39.220
So it's a to N times u of n.
00:20:39.220 --> 00:20:43.630
And it's obviously causal,
because that's what we imposed.
00:20:43.630 --> 00:20:47.360
It may or may not be stable,
depending on the value of a.
00:20:47.360 --> 00:20:52.850
In particular, if the
magnitude of a is less than 1,
00:20:52.850 --> 00:20:56.200
then this sequence will
be decaying exponentially
00:20:56.200 --> 00:20:57.890
as n increases.
00:20:57.890 --> 00:21:00.850
So for the magnitude
of a less than 1,
00:21:00.850 --> 00:21:06.590
this corresponds
to a stable system.
00:21:06.590 --> 00:21:09.920
Now this isn't the
only initial condition
00:21:09.920 --> 00:21:14.750
that we can impose on
the system and still
00:21:14.750 --> 00:21:18.500
have it correspond to a
linear shift invariant system.
00:21:18.500 --> 00:21:21.560
We could, alternatively,
impose a different set
00:21:21.560 --> 00:21:25.310
of initial conditions,
which is the--
00:21:25.310 --> 00:21:28.010
or boundary conditions,
really, because they're not,
00:21:28.010 --> 00:21:30.140
for this case,
initial conditions--
00:21:30.140 --> 00:21:33.500
boundary conditions
that state instead
00:21:33.500 --> 00:21:36.470
of the statement that
the system is causal,
00:21:36.470 --> 00:21:39.890
the statement that the
system is totally non-causal.
00:21:39.890 --> 00:21:42.890
What I mean is, let's
take the same example--
00:21:42.890 --> 00:21:45.830
the same difference equation--
00:21:45.830 --> 00:21:50.360
let's again choose the
input to be a unit sample,
00:21:50.360 --> 00:21:57.140
but let's impose another
condition on the solution,
00:21:57.140 --> 00:21:58.580
boundary condition.
00:21:58.580 --> 00:22:01.160
Namely that the
unit sample response
00:22:01.160 --> 00:22:04.700
is 0 for n greater
than 0, rather than
00:22:04.700 --> 00:22:10.750
the boundary condition that says
that it's 0 for n less than 0.
00:22:10.750 --> 00:22:14.950
In that case, it's convenient
to generate the solution
00:22:14.950 --> 00:22:19.450
iteratively by running the
difference equation backwards,
00:22:19.450 --> 00:22:23.270
in other words, by
expressing y of n minus 1,
00:22:23.270 --> 00:22:26.950
in terms of y of n and x
of n-- this is just simply
00:22:26.950 --> 00:22:30.260
another rearrangement of
the difference equation.
00:22:30.260 --> 00:22:36.580
And with this initial condition,
if we look at y of 1--
00:22:36.580 --> 00:22:40.210
of course, y of 1 is equal to
0 by virtue of the boundary
00:22:40.210 --> 00:22:45.130
condition that we've imposed,
so this is equal to 0--
00:22:45.130 --> 00:22:49.790
y of 0 corresponds to n
equal to 1 in this equation.
00:22:49.790 --> 00:22:55.990
So n equal to 1 here, we have
1 over a times y of 1, which is
00:22:55.990 --> 00:22:58.690
0 minus delta of 1 which is 0.
00:22:58.690 --> 00:23:01.970
So y of 0 is again equal to 0.
00:23:01.970 --> 00:23:07.030
y of minus 1 corresponds
to n equal to 0,
00:23:07.030 --> 00:23:11.140
so we have 1 over a times
y of 0, which was 0,
00:23:11.140 --> 00:23:13.790
minus delta of 0, which is 1.
00:23:13.790 --> 00:23:18.980
So we get minus a
minus to the minus 1.
00:23:18.980 --> 00:23:22.736
y of minus 2, we would
substitute n equals minus 1.
00:23:22.736 --> 00:23:28.430
y of minus 1 we had as
minus a to the minus 1.
00:23:28.430 --> 00:23:33.980
And delta of minus 1 is
0, so we have here minus a
00:23:33.980 --> 00:23:39.530
to the minus 2, and it
continues on that way.
00:23:39.530 --> 00:23:41.780
And if you generated
some more of these,
00:23:41.780 --> 00:23:44.270
what you'd see rather
quickly is that this
00:23:44.270 --> 00:23:51.480
is of the form minus a to the
n times u of minus n minus 1.
00:23:51.480 --> 00:23:56.150
In other words, it is an
exponential to form a to the n,
00:23:56.150 --> 00:23:59.600
as we found for the
causal solution, also.
00:23:59.600 --> 00:24:02.960
But it's 0 for n greater
than 0, and it's of the form
00:24:02.960 --> 00:24:05.600
a to the n for n less than 0.
00:24:05.600 --> 00:24:07.280
What that means,
in particular, is
00:24:07.280 --> 00:24:10.550
that if we again
inquired as to whether it
00:24:10.550 --> 00:24:15.160
was stable or unstable, for
the magnitude of a less than 1,
00:24:15.160 --> 00:24:17.330
what we would
find, in this case,
00:24:17.330 --> 00:24:22.530
is that it's unstable
rather than stable.
00:24:22.530 --> 00:24:25.110
So what we've
seen, then, is that
00:24:25.110 --> 00:24:29.940
a linear constant coefficient
difference equation, by itself,
00:24:29.940 --> 00:24:32.670
doesn't specify
uniquely a system--
00:24:32.670 --> 00:24:36.810
it requires a set of
initial conditions.
00:24:36.810 --> 00:24:39.780
And depending on the initial
conditions or boundary
00:24:39.780 --> 00:24:42.960
conditions that are
imposed, it may correspond
00:24:42.960 --> 00:24:45.030
to a causal system,
or it may correspond
00:24:45.030 --> 00:24:46.920
to a non-causal system.
00:24:46.920 --> 00:24:50.400
And in some situations,
we might want
00:24:50.400 --> 00:24:54.530
it to correspond to
either one of those two.
00:24:54.530 --> 00:24:57.890
Something that I haven't
stated explicitly,
00:24:57.890 --> 00:25:02.180
but there is some discussion
of this in the text,
00:25:02.180 --> 00:25:07.890
is the fact that it is not
for every set of boundary
00:25:07.890 --> 00:25:11.190
conditions that a linear
constant coefficient difference
00:25:11.190 --> 00:25:16.200
equation corresponds to a
linear shift invariant system,
00:25:16.200 --> 00:25:21.000
but it does in particular
for the two sets of boundary
00:25:21.000 --> 00:25:22.980
conditions that
are imposed here.
00:25:22.980 --> 00:25:26.090
More generally, what's required
of the boundary conditions,
00:25:26.090 --> 00:25:29.340
so that the difference equation
corresponds to a linear shift
00:25:29.340 --> 00:25:32.700
invariant system, is that
the boundary conditions
00:25:32.700 --> 00:25:36.330
have to be consistent
with the statement
00:25:36.330 --> 00:25:43.680
that if the input, x of
n were 0, 0 for all n,
00:25:43.680 --> 00:25:46.600
that the output would
also be 0 for all n.
00:25:46.600 --> 00:25:49.860
And there is some additional
discussion of this in the text.
00:25:49.860 --> 00:25:56.040
I indicate also that we'll be
returning from time to time
00:25:56.040 --> 00:25:59.730
to further discussion of linear
constant coefficient difference
00:25:59.730 --> 00:26:03.420
equations, and discussing,
in particular, other ways
00:26:03.420 --> 00:26:08.310
of solving this
class of equations.
00:26:08.310 --> 00:26:10.680
For the remainder
of the lecture,
00:26:10.680 --> 00:26:15.840
I'd like to focus on an
alternative representation
00:26:15.840 --> 00:26:18.180
of linear shift
invariant systems,
00:26:18.180 --> 00:26:23.430
alternative to the time
domain, or convolution sum
00:26:23.430 --> 00:26:28.350
representation, that we dealt
with in the last lecture.
00:26:28.350 --> 00:26:31.890
In particular, the
very useful alternative
00:26:31.890 --> 00:26:36.330
is a description of linear
shift invariant systems in terms
00:26:36.330 --> 00:26:38.850
of its frequency response.
00:26:38.850 --> 00:26:40.710
In other words, in
terms of its response
00:26:40.710 --> 00:26:43.530
either to sinusoidal
excitations,
00:26:43.530 --> 00:26:48.330
or to complex
exponential excitations.
00:26:48.330 --> 00:26:54.720
Well, the basic notion
behind the frequency response
00:26:54.720 --> 00:26:58.740
description of linear
shift invariant systems
00:26:58.740 --> 00:27:03.670
is the fact that
complex exponentials
00:27:03.670 --> 00:27:07.930
are eigenfunctions of linear
shift invariant systems.
00:27:07.930 --> 00:27:11.980
What I mean by that is that for
linear shift invariant systems,
00:27:11.980 --> 00:27:14.350
if you put in a
complex exponential,
00:27:14.350 --> 00:27:17.230
you get out a
complex exponential.
00:27:17.230 --> 00:27:21.970
And the only change is
in the complex amplitude
00:27:21.970 --> 00:27:25.180
of the complex exponential--
the functional form is
00:27:25.180 --> 00:27:27.730
the same, so complex
exponential in gives you
00:27:27.730 --> 00:27:30.370
a complex exponential out.
00:27:30.370 --> 00:27:34.750
We can see that rather
easily by referring
00:27:34.750 --> 00:27:39.370
to the convolution sum
description of linear shift
00:27:39.370 --> 00:27:42.280
invariant systems, as
I've indicated here.
00:27:42.280 --> 00:27:46.660
In particular, suppose that we
choose an input sequence which
00:27:46.660 --> 00:27:50.370
is a complex exponential.
00:27:50.370 --> 00:27:54.650
And let's substitute this
into this expression,
00:27:54.650 --> 00:27:58.500
so that the output is
then the sum of h of k
00:27:58.500 --> 00:28:02.140
e to the j omega n minus k.
00:28:02.140 --> 00:28:09.050
This term can be decomposed into
a product of two exponentials,
00:28:09.050 --> 00:28:17.920
e to the j omega n, and
e to the minus j omega k.
00:28:17.920 --> 00:28:22.960
And since e to the j omega n
doesn't depend on the index k,
00:28:22.960 --> 00:28:28.450
we can take that
piece outside the sum,
00:28:28.450 --> 00:28:31.750
and what we're
left with is y of n
00:28:31.750 --> 00:28:36.330
is e to the j omega n
times the sum of h of k
00:28:36.330 --> 00:28:41.770
e to the minus j omega k,
as I've indicated up here.
00:28:41.770 --> 00:28:45.220
So we started with a
complex exponential sequence
00:28:45.220 --> 00:28:47.710
going into the system.
00:28:47.710 --> 00:28:50.350
None of this stuff
over here depends on n,
00:28:50.350 --> 00:28:53.470
so when we sum all that
up, that's just a number--
00:28:53.470 --> 00:28:55.090
it's a function
of omega, depends
00:28:55.090 --> 00:28:57.850
on what complex
frequency we've put in.
00:28:57.850 --> 00:29:00.250
But it doesn't depend on n.
00:29:00.250 --> 00:29:05.860
And, in fact,
notationally, we'll
00:29:05.860 --> 00:29:09.790
refer to this as the
H of e to the j omega.
00:29:09.790 --> 00:29:14.680
So consequently,
the output sequence,
00:29:14.680 --> 00:29:17.560
due to a complex
exponential input,
00:29:17.560 --> 00:29:24.625
is this number, or
function of omega, times
00:29:24.625 --> 00:29:26.910
e of the j omega n.
00:29:26.910 --> 00:29:29.520
The change in the
complex amplitude, then,
00:29:29.520 --> 00:29:33.720
is H of e to the j omega,
but the functional form
00:29:33.720 --> 00:29:36.820
of the output is the same as the
functional form of the input.
00:29:36.820 --> 00:29:38.490
And that's what we
mean when we say
00:29:38.490 --> 00:29:41.730
that the complex exponential
is an eigenfunction
00:29:41.730 --> 00:29:44.860
of a linear shift
invariant system.
00:29:44.860 --> 00:29:49.080
So finally, H of e to the j
omega is given by this sum--
00:29:49.080 --> 00:29:54.810
I've just changed the
index of summation
00:29:54.810 --> 00:29:56.970
from what I had up here,
but obviously that's
00:29:56.970 --> 00:29:59.470
no important change.
00:29:59.470 --> 00:30:04.680
And we will refer to this,
or to H of e to the j omega,
00:30:04.680 --> 00:30:09.680
as the frequency
response of the system.
00:30:09.680 --> 00:30:14.150
One of the reasons why
the frequency response
00:30:14.150 --> 00:30:15.330
is important--
00:30:15.330 --> 00:30:17.720
of course, one of the facts
that you can see right away
00:30:17.720 --> 00:30:21.620
is that it's easily
obtained directly
00:30:21.620 --> 00:30:24.230
from the unit sample response.
00:30:24.230 --> 00:30:29.000
One of the reasons why the
frequency response is useful
00:30:29.000 --> 00:30:33.470
is because it's essentially
the frequency response that
00:30:33.470 --> 00:30:39.230
allows us to obtain quite easily
the response of the system
00:30:39.230 --> 00:30:43.340
to sinusoidal excitations.
00:30:43.340 --> 00:30:46.820
And, as we'll see
in later lectures,
00:30:46.820 --> 00:30:49.730
starting actually
with the next lecture,
00:30:49.730 --> 00:30:53.810
essentially arbitrary sequences
can be represented either
00:30:53.810 --> 00:30:57.110
as linear combinations
of complex exponentials,
00:30:57.110 --> 00:31:01.910
or as linear combinations
of sinusoidal sequences.
00:31:01.910 --> 00:31:04.820
And so, if we know
what the response is
00:31:04.820 --> 00:31:08.360
to a complex exponential
or to sinusoidal sequences,
00:31:08.360 --> 00:31:10.970
we can, in effect,
describe the response
00:31:10.970 --> 00:31:12.350
to arbitrary sequences.
00:31:12.350 --> 00:31:16.100
We'll see all of that coming
up in the next lecture.
00:31:16.100 --> 00:31:20.300
But to see how this
frequency response relates
00:31:20.300 --> 00:31:24.980
to the sinusoidal response,
the relation essentially
00:31:24.980 --> 00:31:28.580
pops out from the
fact that if we
00:31:28.580 --> 00:31:35.150
have a sinusoidal excitation,
a sinusoidal excitation
00:31:35.150 --> 00:31:39.290
can be represented as a
linear combination of two
00:31:39.290 --> 00:31:41.270
complex exponentials--
00:31:41.270 --> 00:31:43.420
as I've indicated here.
00:31:43.420 --> 00:31:47.720
So a over 2 to the j
phi, e to j omega 0 n,
00:31:47.720 --> 00:31:51.560
this is one complex exponential
with a complex frequency omega
00:31:51.560 --> 00:31:57.440
0, and a complex amplitude,
a over 2 e to j phi.
00:31:57.440 --> 00:32:00.380
Then its complex
conjugate term, those two
00:32:00.380 --> 00:32:04.980
added up give us a
complex exponential.
00:32:04.980 --> 00:32:11.790
We can find the response
to each of these simply
00:32:11.790 --> 00:32:13.200
from the frequency response.
00:32:13.200 --> 00:32:15.870
We know that all that happens
to a complex exponential
00:32:15.870 --> 00:32:19.260
is that its complex
amplitude gets multiplied
00:32:19.260 --> 00:32:21.610
by the frequency response.
00:32:21.610 --> 00:32:25.170
And so this is one complex
exponential, and another one.
00:32:25.170 --> 00:32:27.460
We're talking about
linear systems,
00:32:27.460 --> 00:32:31.330
so the response of
the sum of these two
00:32:31.330 --> 00:32:33.510
is the sum of the responses.
00:32:33.510 --> 00:32:40.320
And so if we express the
frequency response in a polar
00:32:40.320 --> 00:32:47.850
form, as I've indicated here,
in terms of magnitude and phase,
00:32:47.850 --> 00:32:52.050
and if you track through what
happens when you multiply each
00:32:52.050 --> 00:32:56.580
of these complex exponentials
by the appropriate frequency
00:32:56.580 --> 00:32:59.610
response-- this one at
a frequency omega 0,
00:32:59.610 --> 00:33:02.630
this one at a frequency
minus omega 0--
00:33:02.630 --> 00:33:05.310
and add the terms
back together again,
00:33:05.310 --> 00:33:08.340
the resulting
output sequence has
00:33:08.340 --> 00:33:14.130
a change in real amplitude
given by, or dictated
00:33:14.130 --> 00:33:18.030
by the magnitude of
the frequency response,
00:33:18.030 --> 00:33:24.780
and a change in phase dictated
by the angle of the frequency
00:33:24.780 --> 00:33:25.340
response.
00:33:25.340 --> 00:33:27.900
In other words, by
this theta of omega.
00:33:27.900 --> 00:33:30.840
So the frequency response,
when thought of in polar form--
00:33:30.840 --> 00:33:32.820
magnitude and angle--
00:33:32.820 --> 00:33:36.210
the magnitude represents the
change in the real amplitude
00:33:36.210 --> 00:33:39.600
of a sinusoidal excitation.
00:33:39.600 --> 00:33:43.110
And the angle, or
complex argument,
00:33:43.110 --> 00:33:45.190
represents the phase shift.
00:33:45.190 --> 00:33:47.250
And that is exactly
analogous, exactly
00:33:47.250 --> 00:33:52.260
identical to what we're used
to in the continuous time case.
00:33:52.260 --> 00:33:54.210
Well, let's just
look at an example
00:33:54.210 --> 00:33:56.850
of a simple linear
shift invariant system,
00:33:56.850 --> 00:33:59.795
and the resulting
frequency response.
00:34:02.600 --> 00:34:06.220
Let's return to the first
order difference equation
00:34:06.220 --> 00:34:09.969
that we talked about
just a short time ago.
00:34:09.969 --> 00:34:14.650
We saw that there
were two solutions
00:34:14.650 --> 00:34:16.725
that we could generate
for this, depending
00:34:16.725 --> 00:34:18.100
on whether we
assumed that it was
00:34:18.100 --> 00:34:22.030
a causal or non-causal system.
00:34:22.030 --> 00:34:24.790
Let's focus on the causal case.
00:34:24.790 --> 00:34:28.310
If the system was causal,
we generated, essentially,
00:34:28.310 --> 00:34:33.820
iteratively the solution that
the unit sample response is
00:34:33.820 --> 00:34:36.380
a to the n times u of n.
00:34:36.380 --> 00:34:39.580
And let's assume that we're
talking about a stable system,
00:34:39.580 --> 00:34:43.761
so that a is magnitude
of a is less than 1--
00:34:43.761 --> 00:34:45.969
of course, this could be
negative and still be stable
00:34:45.969 --> 00:34:47.320
between minus 1 and 0.
00:34:50.440 --> 00:34:54.230
I'll assume in the pictures
that I draw that a is, in fact,
00:34:54.230 --> 00:34:57.530
positive, and then
also less than 1.
00:35:00.500 --> 00:35:03.390
Now, the expression for
the frequency response
00:35:03.390 --> 00:35:07.410
of the system, from what
we derived just above,
00:35:07.410 --> 00:35:15.210
is the sum overall n of h of n
times e to the minus j omega n.
00:35:15.210 --> 00:35:20.010
Because of the unit step in
here, the limits on the sum
00:35:20.010 --> 00:35:23.940
change from 0 to infinity--
in other words, all of this
00:35:23.940 --> 00:35:28.260
is going to be 0 from minus
infinity up to 0, because
00:35:28.260 --> 00:35:30.100
of the unit step.
00:35:30.100 --> 00:35:32.980
For n greater than
0, we have a to the n
00:35:32.980 --> 00:35:39.610
times this exponential, so
we have a to the n times
00:35:39.610 --> 00:35:41.800
e to the--
00:35:41.800 --> 00:35:43.680
surely there's a
minus sign there--
00:35:43.680 --> 00:35:47.920
e to the minus j omega n.
00:35:47.920 --> 00:35:53.380
If we sum this, this is just
the sum of a geometric series.
00:35:53.380 --> 00:35:55.360
In other words,
it's of the form,
00:35:55.360 --> 00:35:57.610
the sum of alpha to the n--
00:35:57.610 --> 00:36:00.610
alpha equals 0 to infinity.
00:36:00.610 --> 00:36:05.990
Sums of this form are equal
to 1 over 1 minus alpha.
00:36:05.990 --> 00:36:10.180
And alpha, in this case, is
a e to the minus j omega.
00:36:10.180 --> 00:36:18.070
So that sum, then, is 1 over 1
minus a e to the minus j omega.
00:36:18.070 --> 00:36:19.540
Well, as we saw--
00:36:19.540 --> 00:36:21.640
at least for the
sinusoidal response--
00:36:21.640 --> 00:36:27.100
it's useful to look at the
magnitude and phase of this,
00:36:27.100 --> 00:36:29.570
in other words, to look
at it in polar form.
00:36:29.570 --> 00:36:35.170
So if we want the magnitude
of this frequency response,
00:36:35.170 --> 00:36:38.920
we can obtain that
by multiplying this
00:36:38.920 --> 00:36:41.560
by its complex conjugate.
00:36:41.560 --> 00:36:44.830
Consequently, the magnitude
of the frequency response
00:36:44.830 --> 00:36:48.220
is given by H of
e to the j omega--
00:36:48.220 --> 00:36:51.520
the minus sign is
conveniently there for us--
00:36:51.520 --> 00:36:53.980
multiplied by its
complex conjugate, which
00:36:53.980 --> 00:36:57.210
is 1 minus a e to the minus--
00:36:57.210 --> 00:37:02.530
I'm sorry, 1 over 1 minus
a 3 to the plus j omega.
00:37:02.530 --> 00:37:10.090
If we multiply these together,
then what we obtain is 1 over 1
00:37:10.090 --> 00:37:14.830
plus a squared minus 2
a times cosine omega.
00:37:14.830 --> 00:37:22.750
And the phase angle of this is
equal to the arctangent of--
00:37:22.750 --> 00:37:24.700
if you just simply
work this out--
00:37:24.700 --> 00:37:26.950
the arctangent of a
sine omega divided
00:37:26.950 --> 00:37:30.220
by 1 minus a cosine omega.
00:37:30.220 --> 00:37:33.490
I suspect, actually, that
because of that algebraic sign
00:37:33.490 --> 00:37:37.960
error I made, the
minus sign down here,
00:37:37.960 --> 00:37:41.060
actually I think that this
comes out with a minus sign.
00:37:41.060 --> 00:37:45.460
But I won't guarantee that
on the spot right now--
00:37:45.460 --> 00:37:47.440
you can simply
verify that, but I
00:37:47.440 --> 00:37:51.730
suspect that there should
be a minus sign there.
00:37:51.730 --> 00:37:56.590
OK, well this, then,
represents the phase shift
00:37:56.590 --> 00:38:00.520
that would be encountered
by a sinusoidal input going
00:38:00.520 --> 00:38:03.460
through the linear
shift invariant system.
00:38:03.460 --> 00:38:06.580
This would represent the
square of the magnitude
00:38:06.580 --> 00:38:11.030
change of a sinusoidal input.
00:38:11.030 --> 00:38:16.930
And if we were to sketch
this, then what we see
00:38:16.930 --> 00:38:21.205
is that at omega equal to
0, cosine omega of course
00:38:21.205 --> 00:38:23.620
at omega equal to 0 is 1.
00:38:23.620 --> 00:38:28.360
So this is 1 over 1 plus a
squared minus 2 a, or 1 over 1
00:38:28.360 --> 00:38:31.480
minus a quantity squared.
00:38:31.480 --> 00:38:34.220
We're assuming that
a is between 0 and 1,
00:38:34.220 --> 00:38:39.420
and so we're indicated
that by this value.
00:38:39.420 --> 00:38:43.410
When omega is equal
to pi, cosine omega
00:38:43.410 --> 00:38:45.300
is equal to minus 1.
00:38:45.300 --> 00:38:50.470
This, then, comes out to be 1
over 1 plus a quantity squared,
00:38:50.470 --> 00:38:54.030
which for a between 0 and
1 is less than this value.
00:38:54.030 --> 00:38:59.580
So the frequency response, then,
between minus pi and plus pi,
00:38:59.580 --> 00:39:03.700
would have the shape
that I've indicated here.
00:39:03.700 --> 00:39:07.880
Now, what happens if we
continue on in frequency-- omega
00:39:07.880 --> 00:39:12.900
can, of course, run from pi
to 2 pi, and on past that.
00:39:12.900 --> 00:39:15.660
You can verify in a
very straightforward way
00:39:15.660 --> 00:39:19.950
from this expression
that, from pi to 2 pi,
00:39:19.950 --> 00:39:23.250
the frequency response
will look exactly like it
00:39:23.250 --> 00:39:26.220
did from minus pi to pi.
00:39:26.220 --> 00:39:29.910
It will, then, look like that.
00:39:29.910 --> 00:39:33.070
And for minus pi
to minus 2 pi, it
00:39:33.070 --> 00:39:36.480
will look exactly as it
did from pi down to 0.
00:39:36.480 --> 00:39:39.850
So it will look like this.
00:39:39.850 --> 00:39:43.620
And in fact, it's
straightforward to verify
00:39:43.620 --> 00:39:48.090
for both the magnitude and
the phase that both of those
00:39:48.090 --> 00:39:53.970
are periodic in omega,
with a period of 2 pi.
00:39:53.970 --> 00:39:58.290
So this is one period,
say, from minus pi to pi,
00:39:58.290 --> 00:40:03.670
and then another period
would start from pi to 3 pi.
00:40:03.670 --> 00:40:06.550
And it would continue
on like that.
00:40:06.550 --> 00:40:08.980
So, in fact, if we
were to sketch this out
00:40:08.980 --> 00:40:11.350
over a wider range of
omega, we would see
00:40:11.350 --> 00:40:13.030
this periodically repeating.
00:40:13.030 --> 00:40:16.850
For this example, that's
straightforward to verify.
00:40:16.850 --> 00:40:23.630
So there are, then, two
properties of the frequency
00:40:23.630 --> 00:40:27.200
response which I would like
to call your attention to
00:40:27.200 --> 00:40:32.060
in preparation for our
discussion next time.
00:40:32.060 --> 00:40:34.220
One of them is the
fact, very important
00:40:34.220 --> 00:40:38.930
to keep in mind, that the
frequency response, as we're
00:40:38.930 --> 00:40:43.850
talking about it, is a function
of a continuous variable,
00:40:43.850 --> 00:40:44.960
omega.
00:40:44.960 --> 00:40:48.140
Omega, I hadn't stressed
this point previously,
00:40:48.140 --> 00:40:55.010
but omega is a variable
that changes continuously
00:40:55.010 --> 00:40:57.770
over whatever range
we're talking about,
00:40:57.770 --> 00:41:02.090
as opposed to sequences which
are functions, obviously,
00:41:02.090 --> 00:41:05.000
of a discrete variable.
00:41:05.000 --> 00:41:06.590
Here we're talking
about a function
00:41:06.590 --> 00:41:09.320
of a continuous variable, omega.
00:41:09.320 --> 00:41:14.300
As we saw, for our example,
the frequency response
00:41:14.300 --> 00:41:17.010
is a periodic function of omega.
00:41:17.010 --> 00:41:20.280
And the period is equal to 2 pi.
00:41:20.280 --> 00:41:22.970
Now, the reason that it's a
periodic function of omega
00:41:22.970 --> 00:41:27.080
is, in fact, somewhat obvious.
00:41:27.080 --> 00:41:31.490
Suppose that we take a complex
exponential, e to the j omega
00:41:31.490 --> 00:41:35.900
n, and inquire as to how
the complex exponential
00:41:35.900 --> 00:41:41.840
itself behaves if we change
omega over an interval of more
00:41:41.840 --> 00:41:43.130
than 2 pi.
00:41:43.130 --> 00:41:48.920
Suppose that we replace omega
by omega plus 2 pi times k,
00:41:48.920 --> 00:41:52.220
and now if we decompose
this into a product,
00:41:52.220 --> 00:41:55.910
we have e to the j omega
n, times e to the j
00:41:55.910 --> 00:41:59.540
2 pi k times n.
00:41:59.540 --> 00:42:02.990
This is an integer
multiple-- this exponent
00:42:02.990 --> 00:42:05.550
is an integer multiple of 2 pi.
00:42:05.550 --> 00:42:08.450
And so this is simply
equal to unity.
00:42:08.450 --> 00:42:13.460
Now what that says, is that
a complex exponential--
00:42:13.460 --> 00:42:19.130
once we've varied omega
over an interval of 2 pi,
00:42:19.130 --> 00:42:22.190
and we go past that,
there are no more
00:42:22.190 --> 00:42:24.170
no new complex
exponentials to see.
00:42:24.170 --> 00:42:26.990
We'll see the same ones over
and over and over again.
00:42:26.990 --> 00:42:30.260
And consequently,
the system response
00:42:30.260 --> 00:42:33.860
has to be periodic in
omega with period 2 pi,
00:42:33.860 --> 00:42:37.250
because we're putting in,
essentially, the same inputs
00:42:37.250 --> 00:42:39.810
over and over and over again.
00:42:39.810 --> 00:42:43.910
This is a point that I'll be
mentioning from time to time,
00:42:43.910 --> 00:42:49.100
and it's, in fact, somewhat
important to keep in mind.
00:42:49.100 --> 00:42:53.630
Also, it's discussed in some
detail again in the text.
00:42:53.630 --> 00:42:55.760
So these, then,
are some properties
00:42:55.760 --> 00:42:58.640
of the frequency response.
00:42:58.640 --> 00:43:02.720
There is a generalization of
the frequency response, which
00:43:02.720 --> 00:43:07.700
is, in fact, very important
for describing both signals
00:43:07.700 --> 00:43:08.900
and systems.
00:43:08.900 --> 00:43:13.370
The generalization is what
we'll refer to as the Fourier
00:43:13.370 --> 00:43:16.250
transform, which plays
the identical role
00:43:16.250 --> 00:43:19.820
in the discrete time case
that the Fourier transform did
00:43:19.820 --> 00:43:22.050
in the continuous time case.
00:43:22.050 --> 00:43:24.890
And so in the next
lecture, we'll
00:43:24.890 --> 00:43:29.240
be going on to a
discussion of the Fourier
00:43:29.240 --> 00:43:33.350
transform taking off
from the set of ideas
00:43:33.350 --> 00:43:38.540
that we've developed here, with
regard to frequency response.
00:43:38.540 --> 00:43:40.250
Thank you.
00:43:40.250 --> 00:43:42.400
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