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[MUSIC PLAYING]
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ALAN OPPENHEIM:
Well, last time we
00:00:59.370 --> 00:01:02.640
had discussed, in a very
general way, the field
00:01:02.640 --> 00:01:06.180
of digital signal
processing, and hopefully
00:01:06.180 --> 00:01:08.400
one of the things that
you were convinced of
00:01:08.400 --> 00:01:12.480
was the importance of
this set of techniques
00:01:12.480 --> 00:01:14.230
in a general sense.
00:01:14.230 --> 00:01:17.400
What we'd like to
begin on now is
00:01:17.400 --> 00:01:21.780
a discussion of some of the
details of digital signal
00:01:21.780 --> 00:01:23.250
processing.
00:01:23.250 --> 00:01:26.620
And in particular,
in this lecture,
00:01:26.620 --> 00:01:31.740
I would like to introduce the
class of discrete time signals,
00:01:31.740 --> 00:01:36.070
and also the class of
discrete time systems.
00:01:36.070 --> 00:01:42.330
Well, last time, I
reminded you that I
00:01:42.330 --> 00:01:49.050
will be assuming a familiarity
with continuous time signals
00:01:49.050 --> 00:01:50.460
and systems.
00:01:50.460 --> 00:01:53.580
And what you'll see as we
go through this lecture--
00:01:53.580 --> 00:01:56.460
and in fact, as we go through
all of these lectures--
00:01:56.460 --> 00:02:01.710
is a very strong similarity
between the results that you're
00:02:01.710 --> 00:02:05.110
familiar with in the
continuous time case,
00:02:05.110 --> 00:02:09.690
and results that will develop
for the discrete time case.
00:02:09.690 --> 00:02:14.380
Obviously, of course, there
will also be some differences.
00:02:14.380 --> 00:02:16.500
And these differences
are important,
00:02:16.500 --> 00:02:18.840
and they will be
important to focus on.
00:02:18.840 --> 00:02:23.340
So there will be a great
deal of similarity, also
00:02:23.340 --> 00:02:24.580
a few differences.
00:02:24.580 --> 00:02:28.440
And these few differences
are very important.
00:02:28.440 --> 00:02:31.200
In the discrete
time domain, we're
00:02:31.200 --> 00:02:36.150
concerned with processing
signals that are sequences.
00:02:36.150 --> 00:02:39.870
That is, these signals are
functions of an integer
00:02:39.870 --> 00:02:43.650
variable, which we'll call n.
00:02:43.650 --> 00:02:47.730
Typically, will depict
sequences graphically,
00:02:47.730 --> 00:02:54.570
as I've illustrated here on
this first viewgraph, where I've
00:02:54.570 --> 00:02:57.120
denoted a general sequence--
00:02:57.120 --> 00:02:59.910
which I call x of n--
00:02:59.910 --> 00:03:01.890
a function of the
energy variable
00:03:01.890 --> 00:03:08.730
n, and the sequence values,
I've represented by a bar graph
00:03:08.730 --> 00:03:11.820
with the height of
each bar corresponding
00:03:11.820 --> 00:03:14.340
to the sequence value.
00:03:14.340 --> 00:03:18.705
So we have, then,
for example, at n
00:03:18.705 --> 00:03:23.495
equals 0, a line of
height x of 0 at n
00:03:23.495 --> 00:03:28.380
equals 1 a line of
height x of 1, et cetera.
00:03:28.380 --> 00:03:31.740
Of course, as I've drawn
this horizontal axis,
00:03:31.740 --> 00:03:35.670
I've drawn it as
a continuous line,
00:03:35.670 --> 00:03:41.550
but it's important to keep in
mind that the sequences only
00:03:41.550 --> 00:03:42.600
are defined--
00:03:42.600 --> 00:03:44.400
or only make sense--
00:03:44.400 --> 00:03:47.530
for integer values
of the argument.
00:03:47.530 --> 00:03:51.940
If n is an integer,
then x of n makes sense.
00:03:51.940 --> 00:03:57.010
If n is not an integer,
it's not just that x of n
00:03:57.010 --> 00:04:02.200
is 0 or something like that,
it's simply that x of n
00:04:02.200 --> 00:04:03.470
is not defined.
00:04:03.470 --> 00:04:05.950
In other words, x
of n makes sense,
00:04:05.950 --> 00:04:09.550
or is defined for n an integer,
and does not make sense,
00:04:09.550 --> 00:04:13.940
or is not defined, if
n is not an integer.
00:04:13.940 --> 00:04:18.279
So this, then, is a
graphical representation
00:04:18.279 --> 00:04:21.579
of a general sequence.
00:04:21.579 --> 00:04:25.430
And just as in the
continuous time case,
00:04:25.430 --> 00:04:29.680
we have a number of
important basic sequences
00:04:29.680 --> 00:04:33.930
that we would like to focus on.
00:04:33.930 --> 00:04:36.480
The first of these that
I'd like to introduce
00:04:36.480 --> 00:04:41.390
is the unit sample,
or impulse sequence,
00:04:41.390 --> 00:04:45.850
which we'll denote
by delta of n.
00:04:45.850 --> 00:04:48.540
The unit sample
sequence is a sequence
00:04:48.540 --> 00:04:54.870
whose value is unity at n
equals 0, and it's equal to 0
00:04:54.870 --> 00:04:55.800
otherwise.
00:04:55.800 --> 00:05:00.840
That is delta of n equals 1
at n equals 0, and delta of n
00:05:00.840 --> 00:05:05.660
is equal to 0 for
n not equal to 0.
00:05:05.660 --> 00:05:09.230
The unit sample sequence
plays the same role
00:05:09.230 --> 00:05:13.280
in the discrete time case
that the unit impulse time
00:05:13.280 --> 00:05:16.400
function plays in the
continuous time case.
00:05:16.400 --> 00:05:20.720
Let me just mention, however,
that the unit sample sequence
00:05:20.720 --> 00:05:23.610
is very easily defined.
00:05:23.610 --> 00:05:26.900
There's no issue here
about the definition.
00:05:26.900 --> 00:05:30.050
In contrast with the usual
mathematical problems
00:05:30.050 --> 00:05:32.550
with a continuous time impulse--
00:05:32.550 --> 00:05:37.310
which is infinitely big at the
origin, and 0 every place else,
00:05:37.310 --> 00:05:39.650
but has some area, et cetera.
00:05:39.650 --> 00:05:43.790
So there's a very precise
definition here of the unit
00:05:43.790 --> 00:05:47.570
sample, or unit
impulse sequence,
00:05:47.570 --> 00:05:51.380
and it will play a role similar
to the unit impulse time
00:05:51.380 --> 00:05:55.430
function in the
continuous time case.
00:05:55.430 --> 00:06:01.310
Another important basic sequence
is the unit step sequence,
00:06:01.310 --> 00:06:05.180
which I'm denoting by u of n.
00:06:05.180 --> 00:06:10.550
And the unit step sequence
is equal to unity for n
00:06:10.550 --> 00:06:15.080
greater than or equal
to 0, and is equal to 0
00:06:15.080 --> 00:06:18.980
for n less than 0.
00:06:18.980 --> 00:06:24.680
So the unit step sequence,
then, is 0 for n less than 0,
00:06:24.680 --> 00:06:30.890
it's unity four n greater
than or equal to 0.
00:06:30.890 --> 00:06:33.560
This shouldn't be, incidentally,
less than or equal to 0,
00:06:33.560 --> 00:06:36.590
this should be n less than 0.
00:06:36.590 --> 00:06:44.300
And it is-- again, does not have
the difficulties in definition
00:06:44.300 --> 00:06:49.170
that a continuous time
unit step normally has.
00:06:49.170 --> 00:06:53.420
It's the change in value
from n equals minus 1 to n
00:06:53.420 --> 00:06:57.590
equals 0 is precisely
and easily defined.
00:06:57.590 --> 00:07:01.520
Here the unit step is 0,
here the unit step is 1.
00:07:01.520 --> 00:07:04.850
There's no issue, as there is
in the continuous time case,
00:07:04.850 --> 00:07:06.020
about discontinuity.
00:07:08.680 --> 00:07:11.100
Just as in the
continuous time case,
00:07:11.100 --> 00:07:16.810
there is a simple relationship
between the unit sample,
00:07:16.810 --> 00:07:20.020
or a unit impulse,
and the unit step.
00:07:20.020 --> 00:07:22.420
In a discrete time
case, there is likewise
00:07:22.420 --> 00:07:26.650
a simple relationship between
the unit sample sequence,
00:07:26.650 --> 00:07:29.530
and the unit step sequence.
00:07:29.530 --> 00:07:33.220
In particular, the
unit sample sequence
00:07:33.220 --> 00:07:37.210
can be obtained from
the unit step sequence
00:07:37.210 --> 00:07:41.370
by constructing the
first difference.
00:07:41.370 --> 00:07:44.410
The unit sample
sequence, delta of n,
00:07:44.410 --> 00:07:48.310
is equal to u of
n, the unit step,
00:07:48.310 --> 00:07:52.690
minus the unit step
delayed by 1 sample.
00:07:52.690 --> 00:07:55.030
And we can see
that that obviously
00:07:55.030 --> 00:07:59.740
is the case, if we look at the
unit step sequence, u of n.
00:07:59.740 --> 00:08:04.000
Here we have the unit step
sequence delayed by 1--
00:08:04.000 --> 00:08:08.132
so this occurs at n equals
0, this is n equals 1.
00:08:08.132 --> 00:08:11.520
And clearly, the difference
between those two
00:08:11.520 --> 00:08:15.910
will generate a sample at
n equals 0, 0, of course,
00:08:15.910 --> 00:08:21.250
for n less than 0, and 0,
likewise, for n greater than 0,
00:08:21.250 --> 00:08:24.640
because these sequence
values will be canceled out
00:08:24.640 --> 00:08:26.740
by those sequence values.
00:08:26.740 --> 00:08:32.230
So the unit sample sequence is
the unit step minus the delayed
00:08:32.230 --> 00:08:35.110
unit step, or the
unit sample sequence
00:08:35.110 --> 00:08:40.150
is equal to the first difference
of the unit step sequence.
00:08:40.150 --> 00:08:43.929
Similarly, we can obtain
the unit step sequence
00:08:43.929 --> 00:08:50.190
from the unit sample sequence
by constructing a running sum.
00:08:50.190 --> 00:08:53.950
In particular, the
unit step sequence
00:08:53.950 --> 00:08:59.980
is equal to a running sum
of the unit sample sequence.
00:08:59.980 --> 00:09:02.650
In other words, we're
constructing here a sum
00:09:02.650 --> 00:09:08.750
from k equals minus infinity
to the independent variable n
00:09:08.750 --> 00:09:12.280
of the sequence, delta of k.
00:09:12.280 --> 00:09:16.060
Here we have the
sequence, delta of k.
00:09:16.060 --> 00:09:22.000
If n is less than 0, then
we're adding up sequence values
00:09:22.000 --> 00:09:27.160
from minus infinity to
some negative value of k.
00:09:27.160 --> 00:09:29.500
And the only values
that we collect when we
00:09:29.500 --> 00:09:32.230
do that are values equal to 0.
00:09:32.230 --> 00:09:38.770
So for n less than 0, we see
that we obtain 0 for this sum.
00:09:38.770 --> 00:09:44.140
If n is greater than 0, as I've
indicated here, then the sum,
00:09:44.140 --> 00:09:48.490
as we collect sequence values
from minus infinity up to n,
00:09:48.490 --> 00:09:53.440
we collect 1 non-0 sequence
value which is equal to 1.
00:09:53.440 --> 00:09:56.110
Consequently for
n greater than 0,
00:09:56.110 --> 00:10:00.790
we obtain, in this running
sum, u of n equal to 1.
00:10:00.790 --> 00:10:03.130
This plays a role,
this running sum
00:10:03.130 --> 00:10:07.690
plays a role, similar to the
integral in the continuous time
00:10:07.690 --> 00:10:08.720
case.
00:10:08.720 --> 00:10:11.020
And so now we have
a relationship
00:10:11.020 --> 00:10:12.970
between the unit
sample sequence,
00:10:12.970 --> 00:10:16.250
and the unit step sequence.
00:10:16.250 --> 00:10:18.490
There are other
basic sequences that
00:10:18.490 --> 00:10:21.430
will play an important
role, just as they
00:10:21.430 --> 00:10:25.660
do in the continuous time case.
00:10:25.660 --> 00:10:32.110
In particular, we have the
real exponential sequence.
00:10:32.110 --> 00:10:38.560
The sequence x of n is
equal to alpha to the n.
00:10:38.560 --> 00:10:42.040
I've depicted the
exponential sequence here
00:10:42.040 --> 00:10:48.040
for the case in which alpha is
positive but less than unity,
00:10:48.040 --> 00:10:53.710
so that as n increases,
the exponential decreases.
00:10:53.710 --> 00:10:57.910
So alpha to the n, if
alpha is between 0 and 1,
00:10:57.910 --> 00:11:00.280
decreases with increasing n.
00:11:00.280 --> 00:11:03.910
If alpha is greater than 1,
then the exponential sequence
00:11:03.910 --> 00:11:07.660
grows, of course, exponentially.
00:11:07.660 --> 00:11:12.640
We have, also, the
sinusoidal sequence.
00:11:12.640 --> 00:11:15.220
The general form of
the sinusoidal sequence
00:11:15.220 --> 00:11:21.730
is x of n is equal to a
cosine omega 0 n plus phi--
00:11:21.730 --> 00:11:27.100
phi a phase angle, omega 0 we'll
refer to as the frequency, a,
00:11:27.100 --> 00:11:29.500
of course, is the amplitude.
00:11:29.500 --> 00:11:33.640
I've illustrated a
sinusoidal sequence here
00:11:33.640 --> 00:11:35.890
for a particular
choice of omega 0
00:11:35.890 --> 00:11:39.310
and phi, namely omega
0 equal to pi over 4,
00:11:39.310 --> 00:11:42.580
and phi equal minus pi over 8.
00:11:42.580 --> 00:11:49.990
And in looking at this, we see
that we get a periodic sequence
00:11:49.990 --> 00:11:52.420
that is, for this
choice of parameters,
00:11:52.420 --> 00:11:55.390
this sequence is periodic.
00:11:55.390 --> 00:11:57.250
However, an
important distinction
00:11:57.250 --> 00:12:01.330
between the continuous time
case and the discrete time case,
00:12:01.330 --> 00:12:06.310
is that a discrete time
sinusoidal signal is not
00:12:06.310 --> 00:12:09.070
necessarily periodic.
00:12:09.070 --> 00:12:11.260
Particular, I've
illustrated here
00:12:11.260 --> 00:12:15.130
another sinusoidal sequence
with a different choice
00:12:15.130 --> 00:12:17.560
for omega 0 and phi.
00:12:17.560 --> 00:12:22.840
In this case, omega 0
equal to 3 pi over 7.
00:12:22.840 --> 00:12:25.120
With that choice
of omega 0, these
00:12:25.120 --> 00:12:27.760
are the sequence
values we obtain.
00:12:27.760 --> 00:12:31.990
And it should be obvious
by examining this sequence
00:12:31.990 --> 00:12:35.170
that this sequence is
no longer periodic,
00:12:35.170 --> 00:12:37.930
whereas this sequence was.
00:12:37.930 --> 00:12:42.190
So the sinusoidal sequence
may or may not be periodic.
00:12:42.190 --> 00:12:46.780
Of course, let me remind you
that if n was allowed to vary
00:12:46.780 --> 00:12:49.780
continuously-- if it was
not an integer variable--
00:12:49.780 --> 00:12:52.290
x of n would be periodic.
00:12:52.290 --> 00:12:56.800
The periodicity is lost
because n is now constrained
00:12:56.800 --> 00:12:59.710
to be an integer variable.
00:12:59.710 --> 00:13:02.440
And that is one of the
important distinctions
00:13:02.440 --> 00:13:06.310
between continuous time signals
and systems, and discrete time
00:13:06.310 --> 00:13:08.500
signals and systems.
00:13:08.500 --> 00:13:13.120
Another point which
I'll suggest now,
00:13:13.120 --> 00:13:17.020
although it's a point
that I'll want to refer to
00:13:17.020 --> 00:13:21.400
in more detail in some
of the future lectures,
00:13:21.400 --> 00:13:31.480
is that the sinusoidal sequence
is only distinguishable
00:13:31.480 --> 00:13:37.240
as omega 0 runs from
over the range 0 to 2 pi
00:13:37.240 --> 00:13:40.060
or minus pi to pi.
00:13:40.060 --> 00:13:45.340
If omega 0-- if we were to
replace omega 0 by omega 0
00:13:45.340 --> 00:13:50.830
plus 2 pi, then we would have
in this argument omega 0 n plus
00:13:50.830 --> 00:13:52.180
2 pi n.
00:13:52.180 --> 00:13:54.370
The 2 pi n, of course,
would have no effect,
00:13:54.370 --> 00:13:56.480
and we'd end up with
the same sequence.
00:13:56.480 --> 00:14:01.590
So in fact as, we varied omega
0 between minus pi and plus pi,
00:14:01.590 --> 00:14:06.820
you will have seen all of
the sinusoidal sequences
00:14:06.820 --> 00:14:08.830
with this amplitude
and this phase
00:14:08.830 --> 00:14:12.820
that we can possibly generate.
00:14:12.820 --> 00:14:20.020
One of the important of
this set of basic signals
00:14:20.020 --> 00:14:24.850
is that they can be used to
represent a more general class
00:14:24.850 --> 00:14:28.880
of signals, just as in
the continuous time case,
00:14:28.880 --> 00:14:33.040
we use impulses, or we
use complex exponentials,
00:14:33.040 --> 00:14:37.570
or we use sinusoids to
represent very general signals,
00:14:37.570 --> 00:14:41.500
we can develop similar
representations here.
00:14:41.500 --> 00:14:47.050
Let me illustrate this
with an example where
00:14:47.050 --> 00:14:55.440
I consider a general
sequence here, x of n.
00:14:55.440 --> 00:15:01.180
The sequence values are x of
0, x of 1, x of 2, et cetera.
00:15:01.180 --> 00:15:05.860
And I mean to suggest here
a very general sequence.
00:15:05.860 --> 00:15:10.360
Now, I can decompose
this sequence
00:15:10.360 --> 00:15:16.590
into a linear combination
of weighted, delayed unit
00:15:16.590 --> 00:15:20.920
samples, unit sample
sequences, by simply
00:15:20.920 --> 00:15:24.910
extracting individual
sequence values out of x of n.
00:15:24.910 --> 00:15:29.420
For example, let's
consider the sequence
00:15:29.420 --> 00:15:34.340
x of 0 times delta of n,
as I've illustrated here.
00:15:34.340 --> 00:15:38.890
X of 0 times delta of n is
equal to the original sequence
00:15:38.890 --> 00:15:43.600
x of n at n equals 0, and
it's equal to 0 otherwise.
00:15:46.290 --> 00:15:50.860
A second sequence, x of 1
times delta of n minus 1,
00:15:50.860 --> 00:15:55.510
is a unit sample
sequence, delayed by 1
00:15:55.510 --> 00:15:58.600
and having an amplitude
equal to x of 1.
00:15:58.600 --> 00:16:03.100
So this sequence is equal
to this one at n equals 1,
00:16:03.100 --> 00:16:05.320
and it's equal to 0, otherwise.
00:16:05.320 --> 00:16:07.960
The sum of these
two, of course, is
00:16:07.960 --> 00:16:14.020
equal to this x of n at these
two values, and equal to 0,
00:16:14.020 --> 00:16:15.100
otherwise.
00:16:15.100 --> 00:16:19.660
Similarly, we can consider
x of minus 1 times
00:16:19.660 --> 00:16:21.890
delta of n plus 1.
00:16:21.890 --> 00:16:24.820
That's a unit sample
at n equals minus 1,
00:16:24.820 --> 00:16:28.070
with an amplitude
of x of minus 1,
00:16:28.070 --> 00:16:31.060
which picks up this
sequence value.
00:16:31.060 --> 00:16:37.600
x of minus 2 times delta of n
plus 2 is a unit sample at n
00:16:37.600 --> 00:16:42.650
equals minus 2 multiplied by
an amplitude x of minus 2,
00:16:42.650 --> 00:16:47.450
which picks up this
sequence value, et cetera.
00:16:47.450 --> 00:16:52.870
So I think you can see, then,
that as we add these up,
00:16:52.870 --> 00:16:57.710
what we'll generate is this
arbitrary sequence, x of n.
00:16:57.710 --> 00:17:03.490
In other words, we can construct
x of n, an arbitrary sequence,
00:17:03.490 --> 00:17:09.060
as a linear combination of
weighted delayed unit samples--
00:17:09.060 --> 00:17:15.700
x of 0 times delta of n plus x
of 1 times delta of n minus 1
00:17:15.700 --> 00:17:19.060
plus x of minus 1 delta
n plus 1, et cetera.
00:17:19.060 --> 00:17:22.710
Or, more generally,
the sum from k
00:17:22.710 --> 00:17:27.640
equals minus infinity to
plus infinity of x of k times
00:17:27.640 --> 00:17:30.010
delta of n minus k.
00:17:30.010 --> 00:17:36.490
This, then, corresponds to
a general representation
00:17:36.490 --> 00:17:41.320
of an arbitrary sequence in
terms of weighted delayed unit
00:17:41.320 --> 00:17:42.740
samples.
00:17:42.740 --> 00:17:46.600
And it's a representation that
we'll want to refer back to
00:17:46.600 --> 00:17:47.980
in a few minutes.
00:17:47.980 --> 00:17:51.400
It's not, as I've indicated,
the only representation
00:17:51.400 --> 00:17:53.490
of arbitrary sequences--
00:17:53.490 --> 00:17:56.230
we'll also be developing
representations
00:17:56.230 --> 00:17:58.450
in terms of complex
exponentials,
00:17:58.450 --> 00:18:03.700
or real exponentials, or in
terms of sines and cosines.
00:18:03.700 --> 00:18:12.760
OK, that is an introduction
to the basic class of signals
00:18:12.760 --> 00:18:14.800
and the notion of sequences.
00:18:14.800 --> 00:18:19.540
What I'd like to do now is focus
on the class of discrete time
00:18:19.540 --> 00:18:23.800
systems, and then refer back
to some of these results
00:18:23.800 --> 00:18:25.840
to develop a general
representation
00:18:25.840 --> 00:18:28.555
for a special class of
discrete time systems.
00:18:35.160 --> 00:18:41.560
First of all, let me begin
with a general system, which
00:18:41.560 --> 00:18:47.540
is a discrete time system
that has an input--
00:18:47.540 --> 00:18:50.200
which is a sequence x of n.
00:18:50.200 --> 00:18:53.680
It has an output, which
is a sequence y of n.
00:18:53.680 --> 00:18:56.150
And it has a system
transformation,
00:18:56.150 --> 00:19:00.760
which I've denoted by T.
Of course, there isn't much
00:19:00.760 --> 00:19:03.850
that you can do with
a general system--
00:19:03.850 --> 00:19:07.960
the difficulty with trying
to describe a general system
00:19:07.960 --> 00:19:10.450
in general is that
there are no properties
00:19:10.450 --> 00:19:12.920
that you can take advantage of.
00:19:12.920 --> 00:19:16.240
So always, in
characterizing systems,
00:19:16.240 --> 00:19:19.630
it's useful to specialize
the class of systems--
00:19:19.630 --> 00:19:22.060
that is, impose properties
on the system which
00:19:22.060 --> 00:19:25.330
you can exploit to
represent the system,
00:19:25.330 --> 00:19:28.930
or to implement the
system, et cetera.
00:19:28.930 --> 00:19:31.900
The special class that
we'll want to consider
00:19:31.900 --> 00:19:36.250
is the class of systems which
are, first of all, linear.
00:19:36.250 --> 00:19:40.210
And second of all,
shift invariant.
00:19:40.210 --> 00:19:44.650
And this class corresponds,
in the continuous time case,
00:19:44.650 --> 00:19:47.140
to the class of systems
that we normally
00:19:47.140 --> 00:19:50.470
refer to as linear
and time invariant.
00:19:50.470 --> 00:19:53.700
We'll tend to refer to these
as linear and shift invariant,
00:19:53.700 --> 00:20:00.700
and for a shorthand notation,
just express this as well LSI.
00:20:00.700 --> 00:20:03.610
When I refer to an LSI
system, what I mean
00:20:03.610 --> 00:20:07.180
is a system that is linear
and shift invariant.
00:20:07.180 --> 00:20:09.740
Well, let's define these terms.
00:20:09.740 --> 00:20:11.530
First of all, linearity.
00:20:14.170 --> 00:20:18.690
Linearity states
that if I excite
00:20:18.690 --> 00:20:24.750
the system with a sequence x 1
of n, and I get at the output
00:20:24.750 --> 00:20:30.120
a sequence y 1 of n, and
if I excite with x 2 of n,
00:20:30.120 --> 00:20:37.470
and get at the output y 2 of n,
then the condition of linearity
00:20:37.470 --> 00:20:43.530
is that the linear combination,
a x 1 of n plus b x 2 of n,
00:20:43.530 --> 00:20:48.570
produces at the output a
y1 of n plus b y 2 of n.
00:20:48.570 --> 00:20:53.630
That is, the response of the
system to a sum of inputs,
00:20:53.630 --> 00:20:55.860
or a linear
combination of inputs,
00:20:55.860 --> 00:21:00.130
is a linear combination of
the corresponding outputs.
00:21:00.130 --> 00:21:04.230
Now, by repeated application
of just this statement,
00:21:04.230 --> 00:21:07.120
we can make a more
general statement,
00:21:07.120 --> 00:21:14.910
which is that the sum of a sub
k times x sub k of n produces,
00:21:14.910 --> 00:21:24.090
as an output, the sum of a
sub k times y sub k of n.
00:21:24.090 --> 00:21:28.050
Linearity is stated here,
of course, for two inputs,
00:21:28.050 --> 00:21:32.730
but we can obviously extend
that to an arbitrary number
00:21:32.730 --> 00:21:36.360
of inputs so that the statement
of linearity, as we'll
00:21:36.360 --> 00:21:39.300
refer to it, will
generally be that
00:21:39.300 --> 00:21:43.260
a general linear combination of
inputs produces, at the output,
00:21:43.260 --> 00:21:47.820
the same linear combination
of the corresponding outputs.
00:21:47.820 --> 00:21:51.630
So that's the
condition of linearity.
00:21:51.630 --> 00:21:53.460
The second condition
that we want
00:21:53.460 --> 00:21:56.040
to impose on our
class of systems
00:21:56.040 --> 00:21:58.880
is the condition of
shift invariance.
00:21:58.880 --> 00:22:05.630
What shift invariance says,
simply, is that if x of n
00:22:05.630 --> 00:22:10.680
produces y of n,
then x of n minus n 0
00:22:10.680 --> 00:22:13.710
produces y of n minus n 0.
00:22:13.710 --> 00:22:17.160
It's basically a statement that
the system doesn't particularly
00:22:17.160 --> 00:22:20.460
care what you call n equals 0.
00:22:20.460 --> 00:22:24.120
In other words, if we
shift the input in n,
00:22:24.120 --> 00:22:25.830
we shift the output in n.
00:22:25.830 --> 00:22:28.740
It's exactly like the
condition of time invariance
00:22:28.740 --> 00:22:30.000
in the continuous time case.
00:22:32.570 --> 00:22:37.920
For example, if we excited
the system with a unit sample,
00:22:37.920 --> 00:22:40.985
to get the unit
sample response, h
00:22:40.985 --> 00:22:46.490
of n, then the response of the
system, if the system is shift
00:22:46.490 --> 00:22:49.340
invariant, to a
delayed unit sample
00:22:49.340 --> 00:22:54.150
is the delayed version of
the unit sample response.
00:22:54.150 --> 00:22:56.390
So this is shift invariance.
00:22:56.390 --> 00:22:58.460
We had the condition
of linearity.
00:22:58.460 --> 00:23:00.810
These are two
independent conditions
00:23:00.810 --> 00:23:03.020
which we'll now
want to put together
00:23:03.020 --> 00:23:07.310
to develop a general
representation for linear shift
00:23:07.310 --> 00:23:10.200
invariant systems.
00:23:10.200 --> 00:23:14.990
Well, let me remind you
from the last viewgraph
00:23:14.990 --> 00:23:19.820
that we had looked at, that we
had a general representation
00:23:19.820 --> 00:23:25.040
for a sequence in terms of
weighted delayed units samples.
00:23:25.040 --> 00:23:30.180
That is, previously we had
developed a representation,
00:23:30.180 --> 00:23:35.510
x of n, of this form, which
expresses an arbitrary
00:23:35.510 --> 00:23:40.370
sequence in terms of weighted
delayed unit samples.
00:23:40.370 --> 00:23:43.370
Well, this, then,
says that x of n
00:23:43.370 --> 00:23:48.260
is expressed as a linear
combination of basic inputs,
00:23:48.260 --> 00:23:53.210
and if we restrict
ourselves to linear systems,
00:23:53.210 --> 00:23:59.870
then the output must correspond
to the same linear combination
00:23:59.870 --> 00:24:02.000
of the corresponding outputs.
00:24:02.000 --> 00:24:07.910
Well, delta of n minus k into
a linear shift invariant system
00:24:07.910 --> 00:24:13.160
will produce, at the
output, by virtue
00:24:13.160 --> 00:24:18.380
of the property of shift
invariance, the sequence
00:24:18.380 --> 00:24:21.500
h of n minus k.
00:24:21.500 --> 00:24:25.840
And the linear combination
of these delayed unit
00:24:25.840 --> 00:24:30.380
samples will produce at the
output, by virtue of linearity,
00:24:30.380 --> 00:24:38.000
the same linear combination
of the responses
00:24:38.000 --> 00:24:40.130
to delta of n minus k.
00:24:40.130 --> 00:24:45.200
So if we consider a linear shift
invariant system, and because
00:24:45.200 --> 00:24:48.110
of this representation,
we can say
00:24:48.110 --> 00:24:52.190
that the response due
to an arbitrary input
00:24:52.190 --> 00:24:58.550
is equal to a linear combination
of delayed unit sample
00:24:58.550 --> 00:24:59.900
responses.
00:24:59.900 --> 00:25:02.067
Where, in general,
this is a sum from k
00:25:02.067 --> 00:25:05.990
equals minus infinity
to plus infinity.
00:25:05.990 --> 00:25:08.450
Well, this is the key result--
00:25:08.450 --> 00:25:11.480
this is a statement
that says that if we
00:25:11.480 --> 00:25:14.750
talk about linear shift
invariant systems,
00:25:14.750 --> 00:25:17.480
then all that we need
to know about the system
00:25:17.480 --> 00:25:21.980
to characterize it is its
response to a unit sample.
00:25:21.980 --> 00:25:24.410
If we have its response
to a unit sample,
00:25:24.410 --> 00:25:27.620
we can construct y
of n in general--
00:25:27.620 --> 00:25:30.980
that is, we can construct
the output for an arbitrary
00:25:30.980 --> 00:25:32.000
input, x of k.
00:25:34.670 --> 00:25:40.150
This is generally referred
to as the convolution sum.
00:25:40.150 --> 00:25:44.750
In analogy with the convolution
integral in the continuous time
00:25:44.750 --> 00:25:49.880
case, this is one way of
writing the convolution sum.
00:25:49.880 --> 00:25:53.450
There is an
alternative, which is
00:25:53.450 --> 00:25:57.350
interesting in that it suggests
some important properties
00:25:57.350 --> 00:25:59.810
of linear shift
invariant systems,
00:25:59.810 --> 00:26:01.790
and it's obtained
simply by applying
00:26:01.790 --> 00:26:05.120
a substitution of variables
to this expression.
00:26:05.120 --> 00:26:10.970
In particular, suppose
that we replace n minus k
00:26:10.970 --> 00:26:16.880
by the variable r,
or equivalently, k
00:26:16.880 --> 00:26:20.510
is equal to n minus r.
00:26:20.510 --> 00:26:27.050
Then what we obtain is y of
n is now equal to a sum--
00:26:27.050 --> 00:26:29.480
and this is now a sum on r.
00:26:29.480 --> 00:26:31.400
For k, we have n minus r.
00:26:31.400 --> 00:26:36.416
So this is x of
n minus r times h
00:26:36.416 --> 00:26:41.030
of n minus k, which
is equal to r.
00:26:41.030 --> 00:26:44.060
So this h of r.
00:26:44.060 --> 00:26:46.790
Well, it's a simple
step to take,
00:26:46.790 --> 00:26:52.730
but in fact, what it says
is that the system doesn't
00:26:52.730 --> 00:26:57.560
particularly care what you
call the input to the system,
00:26:57.560 --> 00:27:01.780
and what you call the unit
sample response of the system.
00:27:01.780 --> 00:27:07.760
Said another way, it says that
convolution is commutative--
00:27:07.760 --> 00:27:12.380
that is, if we represent
the convolution of x of n
00:27:12.380 --> 00:27:16.220
with h of n with
an asterisk, then,
00:27:16.220 --> 00:27:19.730
because of the fact that
we were able to interchange
00:27:19.730 --> 00:27:23.120
the roles of x of n and
h of n, that implies,
00:27:23.120 --> 00:27:27.020
essentially, that x of
n convolved with h o n
00:27:27.020 --> 00:27:31.940
is the same thing as h of
n convolved with x of n.
00:27:31.940 --> 00:27:35.450
That implies, as I
indicated, that if we
00:27:35.450 --> 00:27:39.890
had a system with an impulse
response or unit sample
00:27:39.890 --> 00:27:46.670
response, h of n, and input
x of n, and output y of n,
00:27:46.670 --> 00:27:50.840
that if I call this the input,
and I call this the unit sample
00:27:50.840 --> 00:27:51.950
response--
00:27:51.950 --> 00:27:53.690
as I've done here--
00:27:53.690 --> 00:27:55.220
we obtain the same output.
00:27:55.220 --> 00:27:59.960
That is, h of n into a system
with unit sample response, x
00:27:59.960 --> 00:28:05.600
of n, gives us at the
output, y of n, also.
00:28:05.600 --> 00:28:10.190
An implication of that is that
linear shift invariant systems
00:28:10.190 --> 00:28:15.050
in cascade don't
particularly care which order
00:28:15.050 --> 00:28:17.300
the systems are cascaded in.
00:28:17.300 --> 00:28:21.470
That is, if I have
a system h 1 of n--
00:28:21.470 --> 00:28:24.360
that is with unit sample
response, h 1 of n--
00:28:24.360 --> 00:28:30.710
and cascade with a system with
unit sample response h 2 of n,
00:28:30.710 --> 00:28:34.550
and input x of n,
the unit sample
00:28:34.550 --> 00:28:37.480
response of this
overall system is h 1
00:28:37.480 --> 00:28:41.090
of n convolved with h 2 of n.
00:28:41.090 --> 00:28:43.250
But since convolution
is commutative,
00:28:43.250 --> 00:28:46.700
we could just as easily
convolve h 2 with h 1,
00:28:46.700 --> 00:28:52.340
and that corresponds to a
cascade of the system h 2 of n
00:28:52.340 --> 00:28:54.380
with the system h 1 of n.
00:28:54.380 --> 00:28:58.400
So simply because of the
fact that convolution
00:28:58.400 --> 00:29:06.010
is commutative, that
implies that, for example,
00:29:06.010 --> 00:29:09.860
that linear shift invariant
systems in cascade
00:29:09.860 --> 00:29:13.720
don't particularly care in
which order their cascaded.
00:29:13.720 --> 00:29:16.420
There are lots of other
properties of convolution
00:29:16.420 --> 00:29:20.200
and simple properties of
linear shift invariant systems
00:29:20.200 --> 00:29:24.910
that are a direct
consequence of convolution,
00:29:24.910 --> 00:29:28.300
and some of the properties that
we've mentioned of convolution.
00:29:28.300 --> 00:29:32.470
And a number of these, we'll see
in the lectures as we go along,
00:29:32.470 --> 00:29:35.290
and also a number of
them you'll have a chance
00:29:35.290 --> 00:29:39.780
to wrestle with in
the study guide.
00:29:39.780 --> 00:29:45.790
OK, so this is a development
of the convolution sum.
00:29:45.790 --> 00:29:50.500
One of the important aspects
of the convolution sum
00:29:50.500 --> 00:29:54.970
is the steps involved in
actually computing it--
00:29:54.970 --> 00:29:59.650
that is, the manipulations
involved in forming this sum.
00:29:59.650 --> 00:30:03.410
And as the last point that I'd
like to make in this lecture,
00:30:03.410 --> 00:30:10.000
I'd like to illustrate, first
with a viewgraph, and then
00:30:10.000 --> 00:30:14.740
with a movie, the computation,
or the implementation,
00:30:14.740 --> 00:30:16.600
of the convolution sum.
00:30:16.600 --> 00:30:19.600
So let's return
to the viewgraph.
00:30:30.850 --> 00:30:36.550
And I've indicated here,
again, the convolution sum as--
00:30:36.550 --> 00:30:41.980
we've derived it-- y of n is the
sum of x of k h of n minus k.
00:30:41.980 --> 00:30:44.890
Now, in implementing
the convolution sum,
00:30:44.890 --> 00:30:50.590
let me stress that
it is for n, which
00:30:50.590 --> 00:30:52.540
is the independent
variable for which
00:30:52.540 --> 00:30:54.370
we are computing the output--
00:30:54.370 --> 00:30:58.090
that is, k is just
simply a dummy variable
00:30:58.090 --> 00:31:00.670
inside this summation.
00:31:00.670 --> 00:31:06.160
Well, I've illustrated here, for
a specific example, a sequence,
00:31:06.160 --> 00:31:12.070
x of k, which is a
constant from 0 to I
00:31:12.070 --> 00:31:18.610
guess n equals 10, equal to
unity for n equals 0 to 10,
00:31:18.610 --> 00:31:20.660
and 0 otherwise.
00:31:20.660 --> 00:31:24.310
And I've indicated the sequence
a little differently here than
00:31:24.310 --> 00:31:28.120
I had previously, since I
want to distinguish between
00:31:28.120 --> 00:31:30.610
the x's, h's, and the y's.
00:31:30.610 --> 00:31:35.020
The sequence x of k, I'm
denoting with little x's--
00:31:35.020 --> 00:31:38.350
the sequence h, I'll denote
note with little h's, and later
00:31:38.350 --> 00:31:42.190
on, the sequence y, we'll
denote with little y's.
00:31:42.190 --> 00:31:48.790
All right, so here we have the
sequence a, that's this one.
00:31:48.790 --> 00:31:53.530
And plotted as a function of k.
00:31:53.530 --> 00:31:57.130
Here, I have the
sequence, h of k,
00:31:57.130 --> 00:31:59.710
which I've chosen to
be an exponential for k
00:31:59.710 --> 00:32:03.460
greater than 0, and
0 for k less than 0.
00:32:03.460 --> 00:32:07.360
But it's not h of k that
we want in this sum.
00:32:07.360 --> 00:32:09.840
It's h of n minus k.
00:32:09.840 --> 00:32:10.930
And what's n?
00:32:10.930 --> 00:32:14.470
Well, n is whatever value
of the output sequence
00:32:14.470 --> 00:32:16.340
we're trying to compute.
00:32:16.340 --> 00:32:20.000
So if n was equal
to 0, for example,
00:32:20.000 --> 00:32:23.680
we would want the
sequence, h of 0 minus k.
00:32:23.680 --> 00:32:25.880
And that's this sequence.
00:32:25.880 --> 00:32:32.410
It's h of k, flipped around in
k, because of this minus sign,
00:32:32.410 --> 00:32:34.300
and not shifted one
way or the other
00:32:34.300 --> 00:32:36.890
simply because we have a 0 here.
00:32:36.890 --> 00:32:40.870
So here's h of k, but
that's not what you want.
00:32:40.870 --> 00:32:44.260
It's h of 0 minus k.
00:32:44.260 --> 00:32:50.140
And now to compute y of 0, we
would multiply this sequence
00:32:50.140 --> 00:32:54.400
by this one, and compute
the sum from minus infinity
00:32:54.400 --> 00:32:55.720
to plus infinity.
00:32:55.720 --> 00:32:58.810
That would give us
the value, y of 0.
00:32:58.810 --> 00:33:01.330
For a different
value of n, we would
00:33:01.330 --> 00:33:05.290
have to look at h of n minus
k, for whichever value of n
00:33:05.290 --> 00:33:06.960
we were computing this for.
00:33:06.960 --> 00:33:10.660
For example, if n
is equal to minus 4,
00:33:10.660 --> 00:33:14.830
the sequence that we want
is h of minus 4 minus k,
00:33:14.830 --> 00:33:19.930
and that is this sequence
shifted to the left by 4.
00:33:19.930 --> 00:33:24.190
So to compute y of
minus 4, we want
00:33:24.190 --> 00:33:28.060
this sequence
multiplied by this one,
00:33:28.060 --> 00:33:31.570
and the sum computed from minus
infinity to plus infinity.
00:33:31.570 --> 00:33:34.960
And you can see, obviously,
for that particular case--
00:33:34.960 --> 00:33:37.750
that is n equal to minus 4--
00:33:37.750 --> 00:33:40.300
the product of
this and this is 0,
00:33:40.300 --> 00:33:41.730
and the sum will be equal to 0.
00:33:41.730 --> 00:33:49.180
Well, let's look at this example
a little more dynamically,
00:33:49.180 --> 00:33:54.370
with a movie that was prepared
at Bell Telephone Laboratories
00:33:54.370 --> 00:33:55.610
by Dr. Ronald Shaefer.
00:34:06.710 --> 00:34:08.630
OK, the convolution
sum, then, we
00:34:08.630 --> 00:34:12.469
have is the sum of x
of k, h of n minus k.
00:34:12.469 --> 00:34:15.110
And so what we would
like to illustrate
00:34:15.110 --> 00:34:19.370
is the operation of
evaluating this sum.
00:34:19.370 --> 00:34:21.770
On the top, we have x of k.
00:34:21.770 --> 00:34:26.300
On the bottom we have h of k--
he of k being an exponential.
00:34:26.300 --> 00:34:31.909
And now, we see h of minus k,
namely h of k flipped over.
00:34:31.909 --> 00:34:35.489
As we shift h of k to the
left, corresponding to n
00:34:35.489 --> 00:34:42.969
equals minus 1, then back
to 0, and now to the right.
00:34:42.969 --> 00:34:45.929
So we have h of 1 minus k.
00:34:45.929 --> 00:34:51.570
And then to 4 y of n, we want
the product of h of minus k
00:34:51.570 --> 00:34:55.110
shifted with x of
k, that product sum
00:34:55.110 --> 00:34:58.360
from minus infinity
to plus infinity.
00:34:58.360 --> 00:35:02.880
So here, we see x of k
times h of 1 minus k, 2
00:35:02.880 --> 00:35:06.300
minus k, 3 minus k, et cetera.
00:35:06.300 --> 00:35:08.880
Those multiplied and then
summed from minus infinity
00:35:08.880 --> 00:35:12.980
to plus infinity, we
see during this portion
00:35:12.980 --> 00:35:14.400
that more and more
values of h are
00:35:14.400 --> 00:35:17.730
engaged with non-0 values
of x, and so y of n
00:35:17.730 --> 00:35:21.720
grows until we reach the
point where values fall off
00:35:21.720 --> 00:35:28.320
the end of the non-0 values
of x, where y of n decays.
00:35:28.320 --> 00:35:31.410
So this, then, is
an illustration
00:35:31.410 --> 00:35:36.660
of the linear convolution
of x of k with h of k.
00:35:40.830 --> 00:35:43.850
OK, this completes
our introduction
00:35:43.850 --> 00:35:47.150
to discrete time
signals and systems.
00:35:47.150 --> 00:35:50.480
There were a number
of important points
00:35:50.480 --> 00:35:53.010
that we've made
during this lecture.
00:35:53.010 --> 00:35:56.990
But the key result,
and the result
00:35:56.990 --> 00:36:00.290
that it's important to
develop some experience with,
00:36:00.290 --> 00:36:03.140
is the convolution sum.
00:36:03.140 --> 00:36:07.490
Next time, we'll introduce
some additional considerations,
00:36:07.490 --> 00:36:11.510
namely the considerations
of stability and causality
00:36:11.510 --> 00:36:13.760
for discrete time systems.
00:36:13.760 --> 00:36:17.960
And we'll also discuss, briefly,
the class of linear shift
00:36:17.960 --> 00:36:21.410
invariant systems
that are represented
00:36:21.410 --> 00:36:25.320
by linear constant coefficient
difference equations.
00:36:25.320 --> 00:36:29.220
Finally, we'll try to
tie some of this together
00:36:29.220 --> 00:36:33.770
and in particular, present what
will be called the frequency
00:36:33.770 --> 00:36:35.660
response of the systems.
00:36:35.660 --> 00:36:38.360
And this will eventually
lead into a discussion
00:36:38.360 --> 00:36:40.146
of Fourier transforms.
00:36:40.146 --> 00:36:41.138
Thank you.
00:36:42.626 --> 00:36:45.364
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