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PROFESSOR: OK.
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I want to review zero-mean
jointly
00:00:25.250 --> 00:00:27.600
Gaussian random variables.
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And I want to review a
couple of the other
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things I did last time.
00:00:32.180 --> 00:00:36.290
Because when I get into
questions of stationarity and
00:00:36.290 --> 00:00:41.630
things like that today, I think
it will be helpful to
00:00:41.630 --> 00:00:45.220
have a little better sense of
what all that was about, or it
00:00:45.220 --> 00:00:47.100
will just look like
a big mess.
00:00:47.100 --> 00:00:49.900
And there are sort of a small
number of critical things
00:00:49.900 --> 00:00:52.200
there that we have
to understand.
00:00:52.200 --> 00:00:57.380
One of them is that if you have
a non-singular covariance
00:00:57.380 --> 00:01:01.580
matrix, OK, you have a bunch
of random variables.
00:01:01.580 --> 00:01:04.780
This bunch of random variables,
each one of them
00:01:04.780 --> 00:01:06.640
has a has a covariance.
00:01:06.640 --> 00:01:11.000
Namely, z sub i and z sub j have
an expected value of z
00:01:11.000 --> 00:01:13.210
sub i and z sub j.
00:01:13.210 --> 00:01:16.110
I'm just talking about
zero mean here.
00:01:16.110 --> 00:01:18.860
Anytime I don't say whether
there's a mean or not, I mean
00:01:18.860 --> 00:01:20.670
there isn't a mean.
00:01:20.670 --> 00:01:23.990
But I think the notes are
usually pretty careful.
00:01:23.990 --> 00:01:27.480
If the covariance matrix is
non-singular, then the
00:01:27.480 --> 00:01:29.480
following things happen.
00:01:29.480 --> 00:01:32.910
The vector, z, is jointly
Gaussian.
00:01:32.910 --> 00:01:35.830
And our first definition
was, if you can
00:01:35.830 --> 00:01:38.120
represent it as a --
00:01:38.120 --> 00:01:43.560
all the components of it each as
linear combinations of IID
00:01:43.560 --> 00:01:46.920
normal Gaussian random
variables, namely independent
00:01:46.920 --> 00:01:50.070
Gaussian random variables -- so
they're all just built up
00:01:50.070 --> 00:01:53.770
from this common set
of independent
00:01:53.770 --> 00:01:55.400
Gaussian random variables.
00:01:55.400 --> 00:01:59.810
Which sort of matches
our idea of why
00:01:59.810 --> 00:02:02.520
noise should be Gaussian.
00:02:02.520 --> 00:02:06.980
Namely, it comes from a
collection of a whole bunch of
00:02:06.980 --> 00:02:11.540
independent things, which all
get added up in various ways.
00:02:11.540 --> 00:02:15.090
And we're trying to look at
processes, noise processes.
00:02:15.090 --> 00:02:19.090
The thing that's going to happen
then is that, at each
00:02:19.090 --> 00:02:23.570
epoch in time, the noise is
going to be some linear
00:02:23.570 --> 00:02:26.830
combination of all of these
noise effects, but is going to
00:02:26.830 --> 00:02:28.050
be filtered a little bit.
00:02:28.050 --> 00:02:30.430
And because it's being filtered
a little bit, the
00:02:30.430 --> 00:02:34.690
noise at one time and the noise
at another time are all
00:02:34.690 --> 00:02:38.020
going to depend on common
sets of variables.
00:02:38.020 --> 00:02:41.240
So that's the intuitive
idea here.
00:02:41.240 --> 00:02:44.630
Starting with that idea, last
time we derived what the
00:02:44.630 --> 00:02:47.100
probability density had to be.
00:02:47.100 --> 00:02:51.560
And if you remember, this
probability density came out
00:02:51.560 --> 00:02:52.430
rather simply.
00:02:52.430 --> 00:02:56.540
The only thing that was involved
here was this idea of
00:02:56.540 --> 00:03:01.490
going from little cubes in
the IID noise domain into
00:03:01.490 --> 00:03:04.180
parallelograms in
the z domain.
00:03:04.180 --> 00:03:06.170
Which is what happens when
you go through a linear
00:03:06.170 --> 00:03:07.530
transformation.
00:03:07.530 --> 00:03:11.280
And because of that, this is the
density that has to come
00:03:11.280 --> 00:03:12.310
out of there.
00:03:12.310 --> 00:03:15.030
By knowing a little bit
about matrices --
00:03:15.030 --> 00:03:17.750
which is covered in the
appendix, or you can just take
00:03:17.750 --> 00:03:20.660
it on faith if you want to --
00:03:20.660 --> 00:03:24.900
you can break up this matrix
into eigenvalues and
00:03:24.900 --> 00:03:26.150
eigenvectors.
00:03:27.620 --> 00:03:31.060
If the matrix is
nonsingular --
00:03:31.060 --> 00:03:34.100
well, if the matrix is singular
or nonsingular, if
00:03:34.100 --> 00:03:38.380
it's a covariance matrix,
or it's --
00:03:38.380 --> 00:03:42.630
a non-negative definite matrix,
is what these matrices
00:03:42.630 --> 00:03:45.930
are -- then you can always
break this matrix up into
00:03:45.930 --> 00:03:48.060
eigenvalues and eigenvectors.
00:03:48.060 --> 00:03:50.050
The eigenvectors
span the space.
00:03:50.050 --> 00:03:51.760
The eigenvectors can
be taken to be
00:03:51.760 --> 00:03:54.210
orthonormal to each other.
00:03:54.210 --> 00:03:58.200
And then when you express this
in those terms, this
00:03:58.200 --> 00:04:05.340
probability density becomes
just a product of normal
00:04:05.340 --> 00:04:09.790
Gaussian random variable
densities, where the
00:04:09.790 --> 00:04:12.830
particular Gaussian random
variables are these inner
00:04:12.830 --> 00:04:17.810
products of the noise vector
with these various
00:04:17.810 --> 00:04:18.610
eigenvectors.
00:04:18.610 --> 00:04:24.050
In other words, you're taking
this set of random variables
00:04:24.050 --> 00:04:27.300
which you're expressing in some
reference system, where
00:04:27.300 --> 00:04:33.940
z1 up to z sub k are
the different
00:04:33.940 --> 00:04:35.450
components to the vector.
00:04:35.450 --> 00:04:41.800
When you express this in a
different reference system,
00:04:41.800 --> 00:04:47.070
you're rotating that
space around.
00:04:47.070 --> 00:04:50.100
What this says is, you can
always rotate the space in
00:04:50.100 --> 00:04:59.830
such a way that what you will
find is orthogonal random
00:04:59.830 --> 00:05:02.850
variables, which are independent
of each other.
00:05:02.850 --> 00:05:06.140
So that this is the density
that you wind up having.
00:05:06.140 --> 00:05:10.760
And what this means is, if you
look at the region of equal
00:05:10.760 --> 00:05:20.950
probability density over this
set of random variables, what
00:05:20.950 --> 00:05:23.640
you find is it's a bunch
of ellipsoids.
00:05:23.640 --> 00:05:26.730
And the axes of the ellipsoids
are simply these
00:05:26.730 --> 00:05:28.270
eigenvectors here.
00:05:28.270 --> 00:05:29.990
OK, so this is the third way.
00:05:29.990 --> 00:05:34.410
If you can represent the noise
in this way, again, it has to
00:05:34.410 --> 00:05:39.340
be jointly Gaussian.
00:05:39.340 --> 00:05:42.770
Finally, if all linear
combinations of this random
00:05:42.770 --> 00:05:47.090
vector are Gaussian, that's
probably the simplest one.
00:05:47.090 --> 00:05:54.240
But it's the hardest one
to verify, in a sense.
00:05:54.240 --> 00:05:56.170
And it's the hardest one
to get all these
00:05:56.170 --> 00:05:58.150
other results from.
00:05:58.150 --> 00:06:01.810
But if all linear combinations
of these random variables are
00:06:01.810 --> 00:06:08.940
all Gaussian, then in fact,
again, the variables have to
00:06:08.940 --> 00:06:10.320
be jointly Gaussian.
00:06:10.320 --> 00:06:13.550
Again, it's important to
understand that jointly
00:06:13.550 --> 00:06:17.335
Gaussian means more than just
individually Gaussian.
00:06:17.335 --> 00:06:20.700
It doesn't mean what you would
think from the words jointly
00:06:20.700 --> 00:06:24.300
Gaussian, as saying that each of
the variables are Gaussian.
00:06:24.300 --> 00:06:27.630
It means a whole lot
more than that.
00:06:27.630 --> 00:06:31.580
In the problem set, what you've
done in one of the
00:06:31.580 --> 00:06:37.630
problems is to create a couple
of examples of two random
00:06:37.630 --> 00:06:41.210
variables which are each
Gaussian random variables, but
00:06:41.210 --> 00:06:45.660
which jointly are not
jointly Gaussian.
00:06:45.660 --> 00:06:49.440
OK, finally, if you have a
singular covariance matrix, z
00:06:49.440 --> 00:06:52.560
is jointly Gaussian if
a basis of these
00:06:52.560 --> 00:06:54.910
zi's are jointly Gaussian.
00:06:54.910 --> 00:06:55.210
OK?
00:06:55.210 --> 00:06:58.000
In other words, you throw out
the random variables which are
00:06:58.000 --> 00:07:01.390
just linear combinations of the
others, because you don't
00:07:01.390 --> 00:07:04.220
even want to think of those as
random variables in a sense.
00:07:04.220 --> 00:07:07.630
I mean, technically they are
random variables, because
00:07:07.630 --> 00:07:09.570
they're defined in
the sample space.
00:07:09.570 --> 00:07:12.500
But they're all just defined in
terms of other things, so
00:07:12.500 --> 00:07:13.900
who cares about them?
00:07:13.900 --> 00:07:16.690
So after you throw those
out, the others have
00:07:16.690 --> 00:07:18.820
to be jointly Gaussian.
00:07:18.820 --> 00:07:23.730
So in fact, if you have two
random variables, z1 and z2,
00:07:23.730 --> 00:07:26.490
and z2 equals z1 --
00:07:26.490 --> 00:07:30.130
OK, in other words the
probability density is on a
00:07:30.130 --> 00:07:33.230
straight diagonal line --
00:07:33.230 --> 00:07:36.920
and z1 is Gaussian, z1
and z2 are jointly
00:07:36.920 --> 00:07:38.690
Gaussian in that case.
00:07:38.690 --> 00:07:44.040
This is not an example like
the ones you did in this
00:07:44.040 --> 00:07:47.060
problem set that you're handing
in today, where you in
00:07:47.060 --> 00:07:49.950
fact have things like a
probability density which
00:07:49.950 --> 00:07:52.970
doesn't exist because it's
impulsive on this line, and
00:07:52.970 --> 00:07:55.040
also impulsive on this line.
00:07:55.040 --> 00:07:58.060
Or something which looks
Gaussian in two of the
00:07:58.060 --> 00:08:01.600
quadrants and is zero in both
the other quadrants, and
00:08:01.600 --> 00:08:03.730
really bizarre things
like that.
00:08:03.730 --> 00:08:04.320
OK?
00:08:04.320 --> 00:08:08.560
So please try to understand what
jointly Gaussian means.
00:08:08.560 --> 00:08:12.590
Because everything about noise
that we do is based on that.
00:08:12.590 --> 00:08:13.920
It really is.
00:08:13.920 --> 00:08:16.850
And if you don't know what
jointly Gaussian means, you're
00:08:16.850 --> 00:08:19.370
not going to understand
anything about noise
00:08:19.370 --> 00:08:23.560
detection, or anything
from this point on.
00:08:23.560 --> 00:08:25.930
I know last year I kept telling
people that, and in
00:08:25.930 --> 00:08:29.050
the final exam there were still
four or five people who
00:08:29.050 --> 00:08:34.600
didn't have the foggiest idea
what jointly Gaussian meant.
00:08:34.600 --> 00:08:36.720
And you know, you're not going
to understand the rest of
00:08:36.720 --> 00:08:39.260
this, you're not going to
understand why these things
00:08:39.260 --> 00:08:42.560
are happening, if you don't
understand that.
00:08:42.560 --> 00:08:44.470
OK.
00:08:44.470 --> 00:08:47.710
So the next thing we
said is Z of t --
00:08:47.710 --> 00:08:50.610
I use those little curly
brackets around something just
00:08:50.610 --> 00:08:53.690
as a shorthand way of saying
that what I'm interested in
00:08:53.690 --> 00:08:57.430
now is the random process, not
the random variable at a
00:08:57.430 --> 00:08:59.200
particular value of t.
00:08:59.200 --> 00:09:02.970
In other words, if I say Z of
t, what I'm usually talking
00:09:02.970 --> 00:09:07.690
about is a random variable,
which is the random variable
00:09:07.690 --> 00:09:11.340
corresponding to one particular
epoch of this
00:09:11.340 --> 00:09:12.550
random process.
00:09:12.550 --> 00:09:14.910
Here I'm talking about
the whole process.
00:09:14.910 --> 00:09:16.940
And it's a Gaussian process.
00:09:16.940 --> 00:09:22.610
If Z of t1 to Z of tk are
jointly Gaussian for all k and
00:09:22.610 --> 00:09:27.370
for all sets of t sub i.
00:09:27.370 --> 00:09:31.820
And if the process can be
represented as a linear
00:09:31.820 --> 00:09:38.290
combination of just ordinary
random variables multiplied by
00:09:38.290 --> 00:09:43.370
some set of orthonormal
functions, and these Z sub i
00:09:43.370 --> 00:09:48.550
are independent, and if the sum
of the variances of these
00:09:48.550 --> 00:09:52.320
random variables sum up to
something less than infinity.
00:09:57.250 --> 00:10:00.600
So you have finite energy in
these sample functions,
00:10:00.600 --> 00:10:02.370
effectively is what this says.
00:10:02.370 --> 00:10:05.930
Then it says the sample
functions of Z of t are L2
00:10:05.930 --> 00:10:08.210
with probability one.
00:10:08.210 --> 00:10:09.740
OK.
00:10:09.740 --> 00:10:14.230
You have almost proven that
in the problem set.
00:10:14.230 --> 00:10:17.380
If you don't believe that you've
almost proven it, you
00:10:17.380 --> 00:10:20.210
really have.
00:10:20.210 --> 00:10:22.120
And anyway, it's true.
00:10:22.120 --> 00:10:25.510
One of the things that we're
trying to do when we're trying
00:10:25.510 --> 00:10:31.100
to deal with these wave forms
that we transmit, the only way
00:10:31.100 --> 00:10:34.230
we can deal with them very
carefully is to know that
00:10:34.230 --> 00:10:35.330
they're L2 functions.
00:10:35.330 --> 00:10:37.280
Which means they have
Fourier transforms.
00:10:37.280 --> 00:10:40.020
We can do all this stuff
we've been doing.
00:10:40.020 --> 00:10:43.090
And you don't need anything more
mathematically than that.
00:10:43.090 --> 00:10:46.840
So it's important to be dealing
with L2 functions.
00:10:46.840 --> 00:10:50.660
We're now getting into random
processes, and it's important
00:10:50.660 --> 00:10:54.810
to know that the sample
functions are L2.
00:10:54.810 --> 00:10:58.330
Because so long as the sample
functions are L2, then you can
00:10:58.330 --> 00:11:01.250
do all of these things we've
done before and just put it
00:11:01.250 --> 00:11:02.800
together and say well,
you take a big
00:11:02.800 --> 00:11:04.370
ensemble of these things.
00:11:04.370 --> 00:11:06.960
They're all well defined,
they're all L2.
00:11:06.960 --> 00:11:08.850
We can take Fourier transforms,
we can do
00:11:08.850 --> 00:11:10.800
everything else we want to do.
00:11:10.800 --> 00:11:13.690
OK, so that's the game
we're playing.
00:11:13.690 --> 00:11:16.980
OK, I'm just going to assume
that sample functions are L2
00:11:16.980 --> 00:11:20.600
from now on, except in a couple
of bizarre cases that
00:11:20.600 --> 00:11:22.320
we look at.
00:11:22.320 --> 00:11:26.930
Then a linear functional is a
random variable given by the
00:11:26.930 --> 00:11:30.590
random variable is equal to
the integral of the noise
00:11:30.590 --> 00:11:33.410
process z of t, multiplied
by an ordinary
00:11:33.410 --> 00:11:35.600
function, g of t, dt.
00:11:35.600 --> 00:11:39.360
And we talked about that
a lot last time.
00:11:39.360 --> 00:11:42.990
This is the convolution of a
process -- well, it's not the
00:11:42.990 --> 00:11:43.430
convolution.
00:11:43.430 --> 00:11:49.180
It's just the inner product of
a process with a function.
00:11:49.180 --> 00:11:53.090
And we interpret this last time
in terms of the sample
00:11:53.090 --> 00:11:56.160
functions of the process and
the sample values of the
00:11:56.160 --> 00:11:58.030
random variable.
00:11:58.030 --> 00:12:00.230
And then since we could do that,
we could really talk
00:12:00.230 --> 00:12:02.620
about the random
variable here.
00:12:02.620 --> 00:12:08.760
This means that for all of the
sample values in the sample
00:12:08.760 --> 00:12:16.120
space with probability one,
the sample values of the
00:12:16.120 --> 00:12:21.340
random variable v are equal to
the integral of the sample
00:12:21.340 --> 00:12:24.940
values of the process
times g of t dt.
00:12:24.940 --> 00:12:27.730
OK, in other words,
this isn't really
00:12:27.730 --> 00:12:31.450
something unusual and new.
00:12:31.450 --> 00:12:37.285
I mean, students have the
capacity of looking at this in
00:12:37.285 --> 00:12:40.250
two ways, and I've seen
it happen for years.
00:12:40.250 --> 00:12:45.250
The first time you see this,
you say this is trivial.
00:12:45.250 --> 00:12:48.210
Because it just looks like the
kind of integration you're
00:12:48.210 --> 00:12:49.800
doing all your life.
00:12:49.800 --> 00:12:52.560
You work with it for a while,
and then at some point you
00:12:52.560 --> 00:12:55.780
wake up and you say, oh,
but my god, this is
00:12:55.780 --> 00:12:56.950
not a function here.
00:12:56.950 --> 00:12:58.990
This is a random process.
00:12:58.990 --> 00:13:01.720
And you say, what the
heck does this mean?
00:13:01.720 --> 00:13:04.250
And suddenly you're way
out in left field.
00:13:04.250 --> 00:13:06.350
Well, this says what it means.
00:13:06.350 --> 00:13:09.310
Once you know what it means, we
go back to this and we use
00:13:09.310 --> 00:13:10.380
this from now on.
00:13:10.380 --> 00:13:11.630
OK?
00:13:16.770 --> 00:13:20.800
If we have a zero-mean Gaussian
process z of t, if
00:13:20.800 --> 00:13:24.240
you have L2 sample functions,
like you had when you take a
00:13:24.240 --> 00:13:29.930
process and make it up as a
linear combination of random
00:13:29.930 --> 00:13:36.160
variables times orthonormal
functions, and if you have a
00:13:36.160 --> 00:13:41.350
bunch of different L2 functions,
g1 up to g sub j 0,
00:13:41.350 --> 00:13:43.710
then each of these random
variables, each of these
00:13:43.710 --> 00:13:48.730
linear functionals, are
going to be Gaussian.
00:13:48.730 --> 00:13:50.170
And in fact the whole
set of them
00:13:50.170 --> 00:13:51.880
together are jointly Gaussian.
00:13:51.880 --> 00:13:53.740
And we showed that last time.
00:13:53.740 --> 00:13:56.250
I'm not going to bother
to show it again.
00:13:59.030 --> 00:14:02.030
And we also found what the
covariance was between these
00:14:02.030 --> 00:14:03.380
random variables.
00:14:03.380 --> 00:14:05.850
And it was just this
expression here.
00:14:05.850 --> 00:14:06.370
OK.
00:14:06.370 --> 00:14:09.910
And this follows just by taking
vi, which is this kind
00:14:09.910 --> 00:14:13.780
of integral, times vj, and
interchanging the order of
00:14:13.780 --> 00:14:17.570
integration, taking the expected
value of z of t times
00:14:17.570 --> 00:14:20.450
z of tau, which gives
you this quantity.
00:14:20.450 --> 00:14:24.410
And you have the g i of t and
the g j of t stuck in there
00:14:24.410 --> 00:14:26.670
with an integral around it.
00:14:26.670 --> 00:14:31.350
And it's hard to understand what
those integrals mean and
00:14:31.350 --> 00:14:33.790
whether they really exist or
not, but we'll get to that a
00:14:33.790 --> 00:14:35.040
little later.
00:14:38.660 --> 00:14:42.260
We then talked about linear
filtering of processes.
00:14:42.260 --> 00:14:46.550
You have a process coming into
a filter, and the output is
00:14:46.550 --> 00:14:48.990
some other stochastic process.
00:14:48.990 --> 00:14:50.220
Or at least we hope it's a well
00:14:50.220 --> 00:14:51.840
defined stochastic process.
00:14:51.840 --> 00:14:55.530
And we talked a little bit
about this last time.
00:14:55.530 --> 00:15:02.850
And for every value of tau,
namely for each random
00:15:02.850 --> 00:15:08.260
variable in this output random
process, V of tau is just a
00:15:08.260 --> 00:15:09.880
linear functional.
00:15:09.880 --> 00:15:12.950
It's a linear functional
corresponding to the
00:15:12.950 --> 00:15:17.770
particular function h
of that tau minus t.
00:15:17.770 --> 00:15:21.970
So the linear functional is a
function of t here for the
00:15:21.970 --> 00:15:24.480
particular value of tau that
you have over here.
00:15:24.480 --> 00:15:27.170
So this is just a linear
functional like the ones we've
00:15:27.170 --> 00:15:29.240
been talking about.
00:15:29.240 --> 00:15:33.830
OK, if z of t is a zero-mean
Gaussian process, then you
00:15:33.830 --> 00:15:36.620
have a bunch of different linear
functionals here for
00:15:36.620 --> 00:15:40.800
any set of times tau
1 up to tau sub k.
00:15:40.800 --> 00:15:44.530
And those are jointly Gaussian
from what we just said.
00:15:44.530 --> 00:15:51.330
And if all of these sets
of k sample --
00:15:51.330 --> 00:15:58.490
If based a random process, v of
tau, is a Gaussian random
00:15:58.490 --> 00:16:05.490
process, if for all k sets of
epochs, v, tau 1, tau 2, up to
00:16:05.490 --> 00:16:11.900
tau sub k, if this set of
random variables are all
00:16:11.900 --> 00:16:13.080
jointly Gaussian.
00:16:13.080 --> 00:16:15.000
So that's what we have here.
00:16:15.000 --> 00:16:24.590
So the V of tau is actually a
Gaussian process if Z of t is
00:16:24.590 --> 00:16:27.080
a Gaussian random process
to start with.
00:16:27.080 --> 00:16:30.390
And the covariance function is
just this quantity that we
00:16:30.390 --> 00:16:32.190
talked about before.
00:16:32.190 --> 00:16:34.110
OK, so we have a covariance
function.
00:16:34.110 --> 00:16:37.300
We also have a covariance
function for the process we
00:16:37.300 --> 00:16:38.530
started with.
00:16:38.530 --> 00:16:41.650
And if it's Gaussian, all you
need to know is what the
00:16:41.650 --> 00:16:44.140
covariance function is.
00:16:44.140 --> 00:16:47.400
So that's all rather nice.
00:16:47.400 --> 00:16:49.580
OK, as we said, we're going
to start talking about
00:16:49.580 --> 00:16:52.500
stationarity today.
00:16:52.500 --> 00:16:55.460
I really want to talk about
two ideas of stationarity.
00:16:55.460 --> 00:16:59.750
One is the idea that you
have probably seen as
00:16:59.750 --> 00:17:04.940
undergraduates one place or
another, which is simple
00:17:04.940 --> 00:17:11.140
computationally, but is almost
impossible to understand when
00:17:11.140 --> 00:17:15.400
you try to say something
precise about it.
00:17:15.400 --> 00:17:18.230
And the other is something
called effectively stationary,
00:17:18.230 --> 00:17:20.690
which we're going to
talk about today.
00:17:20.690 --> 00:17:23.940
And I'll show you why
that makes sense.
00:17:23.940 --> 00:17:29.100
So we say that a process Z of
t is stationary if, for all
00:17:29.100 --> 00:17:36.930
integers k, all shifts tau, all
epochs t1 to t sub k, and
00:17:36.930 --> 00:17:43.840
all of values of z1 to zk, this
joint density here is
00:17:43.840 --> 00:17:47.620
equal to the joint
density shifted.
00:17:47.620 --> 00:17:50.590
In other words, what we're doing
is we're taking a set of
00:17:50.590 --> 00:17:55.220
times over here, t1,
t2, t3, t4, t5.
00:17:55.220 --> 00:17:58.040
We're looking at the joint
distribution function for
00:17:58.040 --> 00:18:01.200
those random variables
here, and then we're
00:18:01.200 --> 00:18:03.020
shifting it by tau.
00:18:03.020 --> 00:18:06.210
And we're saying if the process
is stationery, the
00:18:06.210 --> 00:18:10.260
joint distribution function here
has to be the same as the
00:18:10.260 --> 00:18:12.440
joint distribution
function there.
00:18:15.830 --> 00:18:18.710
You might think that all you
need is the distribution
00:18:18.710 --> 00:18:21.630
function here has to be the
same as the distribution
00:18:21.630 --> 00:18:22.380
function there.
00:18:22.380 --> 00:18:25.610
But you really want the same
relationship to hold through
00:18:25.610 --> 00:18:31.140
here as there if you want to
call process stationary.
00:18:31.140 --> 00:18:32.390
OK.
00:18:34.580 --> 00:18:39.420
If we have zero-mean Gauss,
that's just equivalent to
00:18:39.420 --> 00:18:44.320
saying that the covariance
function at ti and t sub j is
00:18:44.320 --> 00:18:47.360
the same as the covariance
function at ti plus
00:18:47.360 --> 00:18:49.980
tau and tj plus tau.
00:18:49.980 --> 00:18:53.330
And that's true for
t1 up to t sub k.
00:18:53.330 --> 00:18:58.060
Which really just means this
has to be true for all tau,
00:18:58.060 --> 00:19:02.020
for all t sub i and
for all t sub j.
00:19:02.020 --> 00:19:04.620
So you don't need to worry about
the k at all once you're
00:19:04.620 --> 00:19:08.700
dealing with a Gaussian
process.
00:19:08.700 --> 00:19:11.200
Because all you need
to worry about is
00:19:11.200 --> 00:19:12.700
the covariance function.
00:19:12.700 --> 00:19:15.700
And the covariance function
is only a function of two
00:19:15.700 --> 00:19:20.705
variables, the process at one
epoch and the process at
00:19:20.705 --> 00:19:22.770
another epoch.
00:19:22.770 --> 00:19:27.480
And this is equivalent, even
more simply, to saying that
00:19:27.480 --> 00:19:32.710
the covariance function at t1
and t2 has to be equal to the
00:19:32.710 --> 00:19:37.480
covariance function at
t1 minus t2 and zero.
00:19:37.480 --> 00:19:41.210
Can you see why that is?
00:19:41.210 --> 00:19:46.970
If I start out with this, and I
know that this is true, then
00:19:46.970 --> 00:19:49.740
the thing that I can do is take
this and I can shift it
00:19:49.740 --> 00:19:52.570
by any amount that I want to.
00:19:52.570 --> 00:19:56.540
So if I shift this by adding tau
here, then what I wind up
00:19:56.540 --> 00:20:03.840
with is kz of t1 minus t2 plus
tau, comma tau, is equal to kz
00:20:03.840 --> 00:20:06.650
of t1 plus tau, t2 plus tau.
00:20:06.650 --> 00:20:08.540
And if I change variables
around a little
00:20:08.540 --> 00:20:11.280
bit, I come to this.
00:20:11.280 --> 00:20:15.110
So this is the condition we need
for a Gaussian process to
00:20:15.110 --> 00:20:17.550
be stationary.
00:20:17.550 --> 00:20:21.350
I would defy any of you to ever
show in any way that a
00:20:21.350 --> 00:20:25.350
process satisfies all
of these conditions.
00:20:25.350 --> 00:20:29.310
I mean, if you don't have nice
structural properties in the
00:20:29.310 --> 00:20:32.300
process, like a Gaussian
process, which says that all
00:20:32.300 --> 00:20:36.670
you need to define it is this,
this is something that you
00:20:36.670 --> 00:20:39.070
just can't deal with
very well.
00:20:39.070 --> 00:20:40.320
So we have this.
00:20:43.660 --> 00:20:51.980
And then we say, OK, this
covariance is so easy
00:20:51.980 --> 00:20:53.200
to work with --
00:20:53.200 --> 00:20:56.690
I mean no, it's not easy to work
with, but it's one hell
00:20:56.690 --> 00:21:00.010
of a lot easier to work
with than that.
00:21:00.010 --> 00:21:02.935
And therefore if you want to
start talking about processes,
00:21:02.935 --> 00:21:06.070
and you don't really want to
go into all the detail of
00:21:06.070 --> 00:21:09.880
these joint distribution
functions, you will say to
00:21:09.880 --> 00:21:16.120
yourselves that one thing I
might ask for in a process is
00:21:16.120 --> 00:21:20.960
the question of whether the
covariance function satisfies
00:21:20.960 --> 00:21:21.910
this property.
00:21:21.910 --> 00:21:25.460
I mean, for a Gaussian process
this is all you need to make
00:21:25.460 --> 00:21:27.070
the process stationary.
00:21:27.070 --> 00:21:30.210
For other processes you need
more, but at least it's an
00:21:30.210 --> 00:21:32.220
interesting question to ask.
00:21:32.220 --> 00:21:37.580
Is the process partly
stationary, in the sense that
00:21:37.580 --> 00:21:40.810
the covariance function at
least is stationary?
00:21:40.810 --> 00:21:43.780
Namely, the covariance function
here looks the same
00:21:43.780 --> 00:21:46.280
as the covariance
function there.
00:21:46.280 --> 00:21:48.880
So we say that a zero-mean
process is wide-sense
00:21:48.880 --> 00:21:53.050
stationary if it satisfies
this condition.
00:21:53.050 --> 00:21:57.110
So a Gaussian process then is
stationary if and only if it's
00:21:57.110 --> 00:21:58.990
wide-sense stationary.
00:22:01.690 --> 00:22:05.720
And a random process with a
mean, or with a mean that
00:22:05.720 --> 00:22:09.850
isn't necessarily zero, is
going to be stationary or
00:22:09.850 --> 00:22:13.570
wide-sense stationary if the
mean is constant and the
00:22:13.570 --> 00:22:16.060
fluctuation is stationary.
00:22:16.060 --> 00:22:18.510
Or wide-sense stationary,
as the case may be.
00:22:18.510 --> 00:22:20.190
So you want both of these
properties there.
00:22:23.400 --> 00:22:26.660
So as before, we're just going
to throw out the mean and not
00:22:26.660 --> 00:22:27.470
worry about it.
00:22:27.470 --> 00:22:30.230
Because if we're thinking of it
as noise, it's not going to
00:22:30.230 --> 00:22:31.210
have a mean.
00:22:31.210 --> 00:22:33.710
Because of it has a mean, it's
not part of the noise.
00:22:33.710 --> 00:22:35.740
It's just something we
know, and we might
00:22:35.740 --> 00:22:38.490
as well remove it.
00:22:38.490 --> 00:22:41.550
OK, interesting example here.
00:22:41.550 --> 00:22:44.720
Let's look at a process,
V of t, which is
00:22:44.720 --> 00:22:47.440
defined in this way here.
00:22:47.440 --> 00:22:52.600
It's the sum of a set of random
variables, V sub k,
00:22:52.600 --> 00:22:56.630
times the sinc function for
some sampling interval,
00:22:56.630 --> 00:22:59.970
capital T. And I'm assuming
that the v sub k are
00:22:59.970 --> 00:23:06.090
zero-mean, and that, at least
as far as second moments are
00:23:06.090 --> 00:23:09.220
concerned, they have
this stationarity
00:23:09.220 --> 00:23:12.440
property between them.
00:23:12.440 --> 00:23:17.880
Namely, they are uncorrelated
from one j to another k.
00:23:17.880 --> 00:23:21.700
Expected value of vj squared
is sigma squared.
00:23:21.700 --> 00:23:27.140
Expected value of vj vk for
k unequal to j is zero.
00:23:27.140 --> 00:23:32.180
So they're uncorrelated; they
all have the same variance.
00:23:32.180 --> 00:23:35.660
Then I claim that the process
V of t is wide-sense
00:23:35.660 --> 00:23:36.910
stationary.
00:23:41.340 --> 00:23:46.760
And I claim that this covariance
function is going
00:23:46.760 --> 00:23:53.430
to be sigma squared times sinc
of t minus tau over t.
00:23:53.430 --> 00:23:56.060
Now for how many of you
is this an obvious
00:23:56.060 --> 00:23:57.310
consequence of that?
00:24:01.580 --> 00:24:04.680
How many of you would
guess this if you
00:24:04.680 --> 00:24:05.930
had to guess something?
00:24:09.920 --> 00:24:12.830
Well, if you wouldn't guess
it, you should.
00:24:12.830 --> 00:24:15.110
Any time you don't know
something, the best thing to
00:24:15.110 --> 00:24:17.670
do is to make a guess and try
to see whether your guess is
00:24:17.670 --> 00:24:18.540
right or not.
00:24:18.540 --> 00:24:21.410
Because otherwise you never
know what to do.
00:24:21.410 --> 00:24:23.310
So if you have a function
which is
00:24:23.310 --> 00:24:27.600
defined in this way --
00:24:27.600 --> 00:24:29.030
think of it this way.
00:24:29.030 --> 00:24:33.490
Suppose these V sub k's were
not random variables, but
00:24:33.490 --> 00:24:35.570
instead suppose they were
all just constants.
00:24:35.570 --> 00:24:37.590
Suppose they were all 1.
00:24:37.590 --> 00:24:41.140
Suppose you're looking at the
function V of t, which is the
00:24:41.140 --> 00:24:44.230
sum of all of the
sinc functions.
00:24:44.230 --> 00:24:46.520
And think of what you'd get
when you add up a bunch of
00:24:46.520 --> 00:24:49.880
sinc functions which are
all exactly the same.
00:24:49.880 --> 00:24:51.130
What do you get?
00:24:53.490 --> 00:24:56.070
I mean, you have to get
a constant, right?
00:24:56.070 --> 00:24:58.450
You're just interpolating
between all these points, and
00:24:58.450 --> 00:25:00.590
all the points are the same.
00:25:00.590 --> 00:25:02.980
So it'd be very bizarre
if you didn't get
00:25:02.980 --> 00:25:06.180
something which was constant.
00:25:06.180 --> 00:25:09.250
Well the same thing
happens here.
00:25:09.250 --> 00:25:14.100
The derivation of this is one of
those derivations which is
00:25:14.100 --> 00:25:19.700
very, very simple and very, very
slick and very elegant.
00:25:19.700 --> 00:25:23.000
And I was going to go through it
in class and I thought, no.
00:25:23.000 --> 00:25:25.700
You just can't follow
this in real time.
00:25:25.700 --> 00:25:27.710
It's something you have
to sit down and think
00:25:27.710 --> 00:25:30.350
about for five minutes.
00:25:30.350 --> 00:25:34.960
So I urge you all to read this,
because this is a trick
00:25:34.960 --> 00:25:37.950
that you need to use
all the time.
00:25:37.950 --> 00:25:42.620
And you should understand it,
because various problems as we
00:25:42.620 --> 00:25:47.290
go along are going to use this
idea in various ways.
00:25:47.290 --> 00:25:50.540
But anyway, when you have a
process defined this way, it
00:25:50.540 --> 00:25:52.460
has to be wide-sense
stationary.
00:25:55.250 --> 00:26:01.290
And the covariance function is
just this function here.
00:26:01.290 --> 00:26:10.320
Now if these V sub k up here are
jointly Gaussian, and they
00:26:10.320 --> 00:26:15.290
all have the same variance, so
they're all IID, then what
00:26:15.290 --> 00:26:19.510
this means is you have a
Gaussian random process.
00:26:19.510 --> 00:26:23.340
And it's a stationary Gaussian
random process.
00:26:23.340 --> 00:26:26.910
In other words, it's a process
where, if you look at the
00:26:26.910 --> 00:26:30.690
random variable v of tau at
any given tau, you get the
00:26:30.690 --> 00:26:34.140
same thing for all tau.
00:26:34.140 --> 00:26:36.840
In other words, it's a little
peculiar, in the sense that
00:26:36.840 --> 00:26:40.720
when you look at this,
it looks like time 0
00:26:40.720 --> 00:26:42.370
was a little special.
00:26:42.370 --> 00:26:46.860
Because time 0 is where you
specified the process.
00:26:46.860 --> 00:26:49.140
Time t you specified
the process.
00:26:49.140 --> 00:26:51.810
Time 2t you specified
the process.
00:26:51.810 --> 00:26:54.580
But in fact this
is saying, no.
00:26:54.580 --> 00:26:56.770
Time 0 is not special at all.
00:26:56.770 --> 00:26:59.800
Which is why we say that the
process is wide-sense
00:26:59.800 --> 00:27:00.640
stationary.
00:27:00.640 --> 00:27:03.180
All times look the same.
00:27:03.180 --> 00:27:08.020
On the other hand, if you take
these V sub k here to be
00:27:08.020 --> 00:27:11.550
binary random variables which
take one the value plus 1 or
00:27:11.550 --> 00:27:14.970
minus 1 with equal probability,
00:27:14.970 --> 00:27:18.270
what do you have then?
00:27:18.270 --> 00:27:22.900
You look at the sum here, and at
the sample points -- namely
00:27:22.900 --> 00:27:27.240
at 0, at t, at 2t, and so forth
-- what values can the
00:27:27.240 --> 00:27:28.490
process take away?
00:27:31.580 --> 00:27:33.670
At time 0?
00:27:33.670 --> 00:27:39.050
If v sub 0 is either plus
1 or minus 1, what
00:27:39.050 --> 00:27:41.780
is v of 0 over here?
00:27:41.780 --> 00:27:44.100
It's plus 1 or minus 1.
00:27:44.100 --> 00:27:46.970
It can only be those
two things.
00:27:46.970 --> 00:27:51.220
If you look at V of capital T
over 2, namely if you look
00:27:51.220 --> 00:27:57.230
halfway between these two sample
points, you then ask,
00:27:57.230 --> 00:28:01.720
what are the different
values that v of T
00:28:01.720 --> 00:28:05.270
over 2 can take on?
00:28:05.270 --> 00:28:07.660
It's an awful mess.
00:28:07.660 --> 00:28:11.270
It's a very bizarre
random variable.
00:28:11.270 --> 00:28:13.540
But anyway, it's not a random
variable which is
00:28:13.540 --> 00:28:16.570
plus or minus 1.
00:28:16.570 --> 00:28:20.370
Because it's really a sum of an
infinite number of terms.
00:28:20.370 --> 00:28:24.840
So you're taking an infinite
number of binary random
00:28:24.840 --> 00:28:30.190
variables, each with arbitrary
multipliers.
00:28:30.190 --> 00:28:32.100
So you're adding them all up.
00:28:32.100 --> 00:28:34.070
So you get something that's
not differentiable.
00:28:34.070 --> 00:28:37.370
It's not anything nice.
00:28:37.370 --> 00:28:38.460
I don't care about that.
00:28:38.460 --> 00:28:42.060
The only thing that I care about
is that v of capital T
00:28:42.060 --> 00:28:47.030
over 2 is not a binary random
variable anymore.
00:28:47.030 --> 00:28:50.900
And therefore this process is
not stationary anymore.
00:28:50.900 --> 00:28:54.500
Here's an example of a
wide-sense stationary process
00:28:54.500 --> 00:28:57.600
which is not stationary.
00:28:57.600 --> 00:29:00.770
So it's a nice example of where
you have wide-sense
00:29:00.770 --> 00:29:04.350
stationarity, but you don't
have stationarity.
00:29:04.350 --> 00:29:08.200
So all kinds of questions about
power and things like
00:29:08.200 --> 00:29:12.970
that, this process works
very, very well.
00:29:12.970 --> 00:29:17.440
Because questions about power
you answer only in terms of
00:29:17.440 --> 00:29:20.010
covariance functions.
00:29:20.010 --> 00:29:23.420
Questions of individual
possible values and
00:29:23.420 --> 00:29:26.520
probability distributions you
can't answer the very well.
00:29:30.080 --> 00:29:31.870
One more thing.
00:29:31.870 --> 00:29:35.540
If these variables are Gaussian,
and if you actually
00:29:35.540 --> 00:29:39.390
believe me that this is a
stationary Gaussian process,
00:29:39.390 --> 00:29:44.740
and it really is a stationary
Gaussian process, what we have
00:29:44.740 --> 00:29:53.640
is a way of creating a broad
category of random processes.
00:29:53.640 --> 00:29:56.750
Because if I look at the sample
functions here of this
00:29:56.750 --> 00:30:01.710
process, each sample function
is bandlimited.
00:30:01.710 --> 00:30:03.150
Baseband limited.
00:30:03.150 --> 00:30:08.760
And it's baseband limited to 1
over 2 times capital T. 1 over
00:30:08.760 --> 00:30:13.780
the quantity 2 times capital T.
So by choosing capital T to
00:30:13.780 --> 00:30:18.250
be different things, I can make
these sample functions
00:30:18.250 --> 00:30:21.760
have large bandwidth or small
bandwidth so I can look at a
00:30:21.760 --> 00:30:24.340
large variety of different
things.
00:30:24.340 --> 00:30:28.410
All of them have Fourier
transforms, which look sort of
00:30:28.410 --> 00:30:31.320
flat in magnitude.
00:30:31.320 --> 00:30:34.270
And we'll talk about
that as we go on.
00:30:34.270 --> 00:30:39.430
And when I pass these things
through linear filters, what
00:30:39.430 --> 00:30:44.020
we're going to find is we can
create any old Gaussian random
00:30:44.020 --> 00:30:46.120
process we want to.
00:30:46.120 --> 00:30:47.480
So that's why they're nice.
00:30:47.480 --> 00:30:47.750
Yes?
00:30:47.750 --> 00:30:50.280
AUDIENCE: [INAUDIBLE]
00:30:50.280 --> 00:30:53.970
PROFESSOR: Can we make
V of t white noise?
00:30:53.970 --> 00:30:55.740
In a practical sense, yes.
00:30:55.740 --> 00:30:58.200
In a mathematical sense, no.
00:30:58.200 --> 00:31:00.060
In a mathematical sense,
you can't make
00:31:00.060 --> 00:31:01.800
anything white noise.
00:31:01.800 --> 00:31:04.130
In a mathematical sense, white
noise is something
00:31:04.130 --> 00:31:06.580
which does not exist.
00:31:06.580 --> 00:31:08.890
I'm going to get to
that later today.
00:31:08.890 --> 00:31:09.950
And it's a good question.
00:31:09.950 --> 00:31:15.140
But the answer is yes and no.
00:31:22.220 --> 00:31:26.430
OK, the trouble with stationary
processes is that
00:31:26.430 --> 00:31:29.950
the sample functions
aren't L2.
00:31:29.950 --> 00:31:33.940
That's not the serious problem
with them, because all of us
00:31:33.940 --> 00:31:37.770
as engineers are willing to say,
well, whether it's L2 or
00:31:37.770 --> 00:31:39.540
not I don't care.
00:31:39.540 --> 00:31:43.270
I'm just going to use it because
it looks like a nice
00:31:43.270 --> 00:31:44.280
random process.
00:31:44.280 --> 00:31:46.870
It's not going to burn anything
else or anything.
00:31:46.870 --> 00:31:48.950
So, so what?
00:31:48.950 --> 00:31:53.480
But the serious problem here is
that it's very difficult to
00:31:53.480 --> 00:31:56.520
view stationary processes as
00:31:56.520 --> 00:32:00.020
approximations to real processes.
00:32:00.020 --> 00:32:03.980
I mean, we've already said that
you can't have a random
00:32:03.980 --> 00:32:06.530
process which is running
merrily away
00:32:06.530 --> 00:32:09.780
before the Big Bang.
00:32:09.780 --> 00:32:12.360
And you can't have something
that's going to keep on
00:32:12.360 --> 00:32:17.650
running along merrily after we
destroy ourselves, which is
00:32:17.650 --> 00:32:20.020
probably sooner in the
future than the Big
00:32:20.020 --> 00:32:23.370
Bang was in the past.
00:32:23.370 --> 00:32:29.530
But anyway, this definition of
stationarity does not give us
00:32:29.530 --> 00:32:32.390
any way to approximate this.
00:32:32.390 --> 00:32:35.520
Namely, if a process is
stationary, it is either
00:32:35.520 --> 00:32:40.140
identically zero, or every
sample function with
00:32:40.140 --> 00:32:42.600
probability one has infinite
energy in it.
00:32:45.610 --> 00:32:47.170
In other words, they keep
running on forever.
00:32:47.170 --> 00:32:50.230
They keep building up
energy as they go.
00:32:50.230 --> 00:32:53.670
And as far as the definitions
current is concerned, there's
00:32:53.670 --> 00:32:59.400
no way to say large negative
times and large positive times
00:32:59.400 --> 00:33:01.840
are unimportant.
00:33:01.840 --> 00:33:04.680
If you're going to use
mathematical things, though
00:33:04.680 --> 00:33:07.780
you're using them as
approximations, you really
00:33:07.780 --> 00:33:11.780
have to consider what they're
approximations to.
00:33:11.780 --> 00:33:14.130
So that's why I'm going to
develop this idea of
00:33:14.130 --> 00:33:17.750
effectively wide-sense
stationary or effectively
00:33:17.750 --> 00:33:20.120
stationary.
00:33:20.120 --> 00:33:20.890
OK.
00:33:20.890 --> 00:33:25.880
So a zero-mean process is
effectively stationary or
00:33:25.880 --> 00:33:28.720
effectively wide-sense
stationary, which is what I'm
00:33:28.720 --> 00:33:34.260
primarily interested in, within
two intervals, within
00:33:34.260 --> 00:33:38.210
some wide interval of time,
minus T0 to plus T0.
00:33:38.210 --> 00:33:42.440
I'm thinking here of choosing
T0 to be enormous.
00:33:42.440 --> 00:33:47.310
Namely if you build a piece of
equipment, you would have
00:33:47.310 --> 00:33:51.820
minus T0 to plus T0 include the
amount of time that the
00:33:51.820 --> 00:33:54.400
equipment was running.
00:33:54.400 --> 00:33:59.200
So we'll say it's effectively
stationary within these limits
00:33:59.200 --> 00:34:02.960
if the joint probability
assignment, or the covariance
00:34:02.960 --> 00:34:05.600
matrix if we're talking about
effectively wide-sense
00:34:05.600 --> 00:34:11.690
stationary, for t1 up to t sub
k is the same as that for t1
00:34:11.690 --> 00:34:15.410
plus tau up to tk plus tau.
00:34:15.410 --> 00:34:23.310
In other words I have this big
interval, and I have a bunch
00:34:23.310 --> 00:34:24.560
of times here.
00:34:29.710 --> 00:34:30.440
t sub k.
00:34:30.440 --> 00:34:37.280
And I have a bunch of times over
here, t1 plus tau up to
00:34:37.280 --> 00:34:41.230
tk plus tau.
00:34:41.230 --> 00:34:44.600
And this set of times don't have
to be disjoint from that
00:34:44.600 --> 00:34:44.940
set of times.
00:34:44.940 --> 00:34:48.830
And I have this time over here,
minus t0, and I have
00:34:48.830 --> 00:34:52.500
this time way out here
which is plus t0.
00:34:52.500 --> 00:34:55.160
And what I'm saying is, I'm
going to call this function
00:34:55.160 --> 00:34:59.460
wide-sense stationery if, when
we truncate it to minus t0 to
00:34:59.460 --> 00:35:04.300
plus t0, it is stationary
as far as all
00:35:04.300 --> 00:35:06.480
those times are concerned.
00:35:06.480 --> 00:35:09.790
In other words, we just want to
ignore what happens before
00:35:09.790 --> 00:35:13.410
minus t0 and what happens
after plus t0.
00:35:13.410 --> 00:35:14.980
You have to be able
to do that.
00:35:14.980 --> 00:35:18.930
Because if you can't talk about
the process in that way,
00:35:18.930 --> 00:35:22.510
you can't talk about
the process at all.
00:35:22.510 --> 00:35:26.750
The only thing you can do is
view the noise over some
00:35:26.750 --> 00:35:28.760
finite interval.
00:35:28.760 --> 00:35:30.090
OK.
00:35:30.090 --> 00:35:33.880
And as far as covariance matrix
for wide-sense and
00:35:33.880 --> 00:35:39.120
stationary, well it's
the same definition.
00:35:39.120 --> 00:35:42.590
So we're going to truncate the
process and deal with that.
00:35:45.590 --> 00:35:53.800
For effectively stationary
or effectively wide-sense
00:35:53.800 --> 00:35:57.690
stationary, I want to view the
process as being truncated to
00:35:57.690 --> 00:35:59.740
minus t0 to plus t0.
00:35:59.740 --> 00:36:01.780
We have this process which
might or might not be
00:36:01.780 --> 00:36:02.780
stationary.
00:36:02.780 --> 00:36:05.150
I just truncate it and I say I'm
only going to look at this
00:36:05.150 --> 00:36:07.910
finite segment of it.
00:36:07.910 --> 00:36:13.410
I'm going to define this one
variable covariance function
00:36:13.410 --> 00:36:18.260
as kz of t1 and t2.
00:36:18.260 --> 00:36:28.330
And kz of t1 and t2 is
going to be -- blah.
00:36:28.330 --> 00:36:33.930
The single variable covariance
function is defined as kz if
00:36:33.930 --> 00:36:38.460
t1 minus t2 is equal to the
actual covariance function
00:36:38.460 --> 00:36:41.640
evaluated with one argument
at t1 and the
00:36:41.640 --> 00:36:43.120
other argument at t2.
00:36:43.120 --> 00:36:46.080
And I want this to hold
true for all t1 and
00:36:46.080 --> 00:36:48.650
all t2 in this interval.
00:36:48.650 --> 00:36:51.130
And this square here gives
a picture of what
00:36:51.130 --> 00:36:53.200
we're talking about.
00:36:53.200 --> 00:36:56.310
If you look at the square in the
notes, unfortunately all
00:36:56.310 --> 00:37:00.870
the diagonal lines do not appear
because of the bizarre
00:37:00.870 --> 00:37:03.400
characteristics of LaTeX.
00:37:03.400 --> 00:37:07.600
LaTeX is better than most
programs, but the graphics in
00:37:07.600 --> 00:37:09.670
it are awful.
00:37:09.670 --> 00:37:13.790
But anyway, here it
is drawn properly.
00:37:13.790 --> 00:37:20.200
And along this line, t minus
tau is constant.
00:37:20.200 --> 00:37:24.860
So this is the line over which
we're insisting that kz of t1
00:37:24.860 --> 00:37:26.960
and t2 be constant.
00:37:26.960 --> 00:37:33.420
So for a random process to be
wide-sense stationary, what
00:37:33.420 --> 00:37:37.540
were insisting is that within
these limits, minus t0 to plus
00:37:37.540 --> 00:37:43.840
t0, the covariance function is
constant along each of these
00:37:43.840 --> 00:37:45.970
lines along here.
00:37:45.970 --> 00:37:49.620
So it doesn't depend
on both t1 and t2.
00:37:49.620 --> 00:37:52.560
It only depends on the
difference between them.
00:37:52.560 --> 00:37:56.260
Which is what we require for
stationary to start with.
00:37:56.260 --> 00:37:58.630
If you don't like the idea
of effectively wide-sense
00:37:58.630 --> 00:38:02.360
stationary, just truncate the
process and say, well if it's
00:38:02.360 --> 00:38:05.100
stationary, this has
to be satisfied.
00:38:05.100 --> 00:38:08.990
It has to be constant
along these lines.
00:38:08.990 --> 00:38:12.600
The peculiar thing about this
is that the single variable
00:38:12.600 --> 00:38:18.960
covariance function does not run
from minus t0 to plus t0.
00:38:18.960 --> 00:38:24.350
It runs from minus
2T0 to plus 2T0.
00:38:24.350 --> 00:38:28.580
So that's a little bizarre
and a little unpleasant.
00:38:28.580 --> 00:38:32.280
And it's unfortunately
the way things are.
00:38:32.280 --> 00:38:36.270
And it also says that this one
variable covariance function
00:38:36.270 --> 00:38:41.920
is not always the same as the
covariance evaluated at t
00:38:41.920 --> 00:38:44.100
minus tau and 0.
00:38:44.100 --> 00:38:47.960
Namely, it's not the same
because t minus tau might be
00:38:47.960 --> 00:38:51.420
considerably larger than t0.
00:38:51.420 --> 00:38:55.470
And therefore, this covariance
is zero, if we truncated the
00:38:55.470 --> 00:38:58.420
process, and this covariance
is not zero.
00:38:58.420 --> 00:39:03.440
In other words, for these points
up here, and in fact
00:39:03.440 --> 00:39:08.150
this point in particular --
well, I guess the best thing
00:39:08.150 --> 00:39:12.200
to look at is points along
here, for example.
00:39:12.200 --> 00:39:21.560
Points along here, what we
insist on is that t minus tau,
00:39:21.560 --> 00:39:27.970
k sub z of t and tau is constant
along this line.
00:39:27.970 --> 00:39:34.570
But we don't insist on kz of
t minus tau, which is some
00:39:34.570 --> 00:39:41.910
quantity bigger than t0 along
this line, being the same as
00:39:41.910 --> 00:39:44.220
this quantity here.
00:39:44.220 --> 00:39:49.120
OK, so aside from that little
peculiarity, this is the same
00:39:49.120 --> 00:39:52.290
thing that you would expect from
just taking a function
00:39:52.290 --> 00:39:54.040
and truncating it.
00:39:54.040 --> 00:39:57.660
There's nothing else that's
peculiar going on there.
00:40:00.830 --> 00:40:05.180
OK, so let's see if we can do
anything with this idea.
00:40:05.180 --> 00:40:08.150
And the first thing we want to
do is to say, how about these
00:40:08.150 --> 00:40:11.810
linear functionals we've
been talking about?
00:40:11.810 --> 00:40:13.850
Why do I keep talking
about linear
00:40:13.850 --> 00:40:16.770
functionals and filtering?
00:40:16.770 --> 00:40:24.230
Because any time you take a
noise process and you receive
00:40:24.230 --> 00:40:26.920
it, you're going to start
filtering it.
00:40:26.920 --> 00:40:28.410
You're going to take
start taking linear
00:40:28.410 --> 00:40:30.280
functionals of it.
00:40:30.280 --> 00:40:33.830
Namely all the processing you
do is going to start with
00:40:33.830 --> 00:40:37.390
finding random variables
in this particular way.
00:40:37.390 --> 00:40:39.650
And when you look at the
covariance between two of
00:40:39.650 --> 00:40:45.660
them, what you find is the same
thing we found last time.
00:40:45.660 --> 00:40:52.140
Expected value of Vi times Vj is
equal to the integral of gi
00:40:52.140 --> 00:40:56.690
of t times the covariance
function evaluated at t and
00:40:56.690 --> 00:41:00.460
tau, times gj of tau.
00:41:00.460 --> 00:41:03.620
All of this is for
real processes
00:41:03.620 --> 00:41:05.510
and for real functions.
00:41:05.510 --> 00:41:11.060
Because noise random processes
are in fact real.
00:41:11.060 --> 00:41:14.870
I keep trying to alternate
between making all of this
00:41:14.870 --> 00:41:17.980
complex and making it real.
00:41:17.980 --> 00:41:19.600
Both have their advantages.
00:41:19.600 --> 00:41:23.020
But at least in the present
version of the notes,
00:41:23.020 --> 00:41:27.590
everything is real, concerned
with random processes.
00:41:27.590 --> 00:41:29.420
So I have this function here.
00:41:29.420 --> 00:41:35.625
If gj of t is zero for t greater
than T0, what's going
00:41:35.625 --> 00:41:37.420
to happen here then?
00:41:37.420 --> 00:41:40.100
I'm evaluating this where
this is a double
00:41:40.100 --> 00:41:45.310
integral over what region?
00:41:45.310 --> 00:41:49.810
It's a double integral over
the region where t is
00:41:49.810 --> 00:41:54.550
constrained to minus T0
to plus T0, and tau is
00:41:54.550 --> 00:41:58.380
constrained to the interval
minus t0 to plus T0.
00:41:58.380 --> 00:42:02.700
In other words, I can evaluate
this expected value, this
00:42:02.700 --> 00:42:07.060
covariance, simply by looking
at this box here.
00:42:07.060 --> 00:42:09.860
If I know what the random
process is doing within this
00:42:09.860 --> 00:42:14.480
box, I can evaluate
this thing.
00:42:14.480 --> 00:42:18.120
So that if the process is
effectively stationary within
00:42:18.120 --> 00:42:23.450
minus t0 to plus t0, I don't
need to know anything else to
00:42:23.450 --> 00:42:27.740
evaluate all of these
linear functionals.
00:42:27.740 --> 00:42:30.280
But that's exactly the way
that were choosing this
00:42:30.280 --> 00:42:32.680
quantity, T0.
00:42:32.680 --> 00:42:36.290
Namely, we're choosing T0 to
be so large that everything
00:42:36.290 --> 00:42:39.670
we're interested in happens
inside of there.
00:42:39.670 --> 00:42:42.410
So we're saying that all of
these linear functionals for
00:42:42.410 --> 00:42:46.300
effective stationarity are just
defined for what happens
00:42:46.300 --> 00:42:47.550
inside this interval.
00:42:51.970 --> 00:42:55.310
So if Z of t is effectively
wide-sense stationary within
00:42:55.310 --> 00:42:58.400
this interval, you make the
intervals large as you want or
00:42:58.400 --> 00:42:59.420
as small as you want.
00:42:59.420 --> 00:43:03.260
The real quantity is that these
functions g have to be
00:43:03.260 --> 00:43:07.820
constrained within minus
T0 to plus T0.
00:43:07.820 --> 00:43:12.340
Then Vi, Vj, are jointly
Gaussian.
00:43:12.340 --> 00:43:17.470
In other words, you can talk
about jointly Gaussian without
00:43:17.470 --> 00:43:20.850
talking at all about whether
the process is really
00:43:20.850 --> 00:43:22.730
stationary or not.
00:43:22.730 --> 00:43:25.770
You can evaluate this for
everything within these
00:43:25.770 --> 00:43:29.220
limits, strictly in terms
of what's going on
00:43:29.220 --> 00:43:30.470
within those limits.
00:43:37.190 --> 00:43:38.460
That one was easy.
00:43:38.460 --> 00:43:39.880
The next one is a
little harder.
00:43:42.680 --> 00:43:45.050
We have our linear filter now.
00:43:45.050 --> 00:43:49.050
We have the noise process going
into a linear filter.
00:43:49.050 --> 00:43:50.910
And we have some
kind of process
00:43:50.910 --> 00:43:52.800
coming out of the filter.
00:43:52.800 --> 00:43:56.450
Again, we would like to say we
couldn't care less about what
00:43:56.450 --> 00:44:01.950
this process is doing outside
of these humongous limits.
00:44:01.950 --> 00:44:06.830
And the way we do that
is to say --
00:44:06.830 --> 00:44:10.580
well, let's forget about that
for the time being.
00:44:10.580 --> 00:44:15.510
Let us do what you did as an
undergraduate, before you
00:44:15.510 --> 00:44:20.810
acquired wisdom, and say let's
just integrate and not worry
00:44:20.810 --> 00:44:24.170
at all about changing orders of
00:44:24.170 --> 00:44:26.370
integration or anything else.
00:44:26.370 --> 00:44:31.330
Let's just run along and
do everything as
00:44:31.330 --> 00:44:34.530
carelessly as we want to.
00:44:34.530 --> 00:44:39.370
The covariance function for what
comes out of this filter
00:44:39.370 --> 00:44:42.690
is something we already
evaluated.
00:44:42.690 --> 00:44:47.650
Namely, what you want to look
at is v of t and v of tau. v
00:44:47.650 --> 00:44:52.620
of t is this integral here. v
of tau is this integral here
00:44:52.620 --> 00:44:55.870
with tau substituted for t.
00:44:55.870 --> 00:44:58.360
So when you put both of these
together and you take the
00:44:58.360 --> 00:45:02.220
expected value, and then you
bring the expected value in
00:45:02.220 --> 00:45:04.410
through the integrals --
00:45:04.410 --> 00:45:07.870
who knows whether you can do
that or not, but we'll do it
00:45:07.870 --> 00:45:11.740
anyway -- what we're going to
get is a double integral of
00:45:11.740 --> 00:45:16.350
the impulse response, evaluated
at t minus t1, times
00:45:16.350 --> 00:45:21.410
the covariance function
evaluated at t1 and t2, times
00:45:21.410 --> 00:45:25.970
the function that we're dealing
with, h, evaluated at
00:45:25.970 --> 00:45:27.550
tau minus t2.
00:45:27.550 --> 00:45:31.710
This is what you got just by
taking the expected value of v
00:45:31.710 --> 00:45:34.680
of t times v of tau.
00:45:34.680 --> 00:45:36.250
And you just write it out.
00:45:36.250 --> 00:45:39.580
You write out the expected value
of these two integrals.
00:45:39.580 --> 00:45:41.240
It's what we did last time.
00:45:41.240 --> 00:45:44.250
It's what's in the notes if you
don't see how to do it.
00:45:44.250 --> 00:45:47.220
You take the expected
value inside, and
00:45:47.220 --> 00:45:49.350
this is what you get.
00:45:49.350 --> 00:45:52.510
Well now we start playing
these games.
00:45:52.510 --> 00:45:57.620
And the first game to play is,
we would like to use our
00:45:57.620 --> 00:46:00.000
notion of stationarity.
00:46:00.000 --> 00:46:04.790
So in place of kz of t1 and t2,
we want to substitute the
00:46:04.790 --> 00:46:11.880
single variable form, kz
tilde of t1 minus t2.
00:46:11.880 --> 00:46:15.520
But we don't like the t1 minus
t2 in there, so we substitute
00:46:15.520 --> 00:46:18.060
phi for t1 and t2.
00:46:18.060 --> 00:46:22.460
And then what we get is
the double integral
00:46:22.460 --> 00:46:25.000
of kz of phi now.
00:46:25.000 --> 00:46:29.760
And we have gotten rid of t1
by this substitution, so we
00:46:29.760 --> 00:46:33.530
wind up with the integral
over phi and over t2.
00:46:36.900 --> 00:46:38.530
Next thing we want to
do, we want to get
00:46:38.530 --> 00:46:40.850
rid of the t2 also.
00:46:40.850 --> 00:46:45.880
So we're going to let mu
equal t2 minus tau.
00:46:45.880 --> 00:46:49.610
When we were starting here, we
had four variables, t and tau
00:46:49.610 --> 00:46:52.460
and t1 and t2.
00:46:52.460 --> 00:46:54.840
We're not getting rid of the t
and we're not getting rid of
00:46:54.840 --> 00:46:58.440
the tau, because that's what
we're trying to calculate.
00:46:58.440 --> 00:47:01.290
But we can play around with the
t1 and t2 as much as we
00:47:01.290 --> 00:47:06.180
want, and we can substitute
variables of integration here.
00:47:06.180 --> 00:47:09.110
So mu is going to
be t2 minus tau.
00:47:09.110 --> 00:47:11.640
That's going to let us
get rid of the t2.
00:47:11.640 --> 00:47:14.670
And we wind up with this form
here when we do that
00:47:14.670 --> 00:47:15.480
substitution.
00:47:15.480 --> 00:47:20.080
It's just that h of tau minus
t2 becomes h of minus mu
00:47:20.080 --> 00:47:22.170
instead of h of plus mu.
00:47:22.170 --> 00:47:23.670
I don't know what that means.
00:47:23.670 --> 00:47:25.350
I don't care what it means.
00:47:25.350 --> 00:47:29.350
The only thing I'm interested
in now is, you look at where
00:47:29.350 --> 00:47:32.950
the t's and the taus are, and
the only place that t's and
00:47:32.950 --> 00:47:35.260
taus occur are here.
00:47:35.260 --> 00:47:37.360
And they occur together.
00:47:37.360 --> 00:47:41.530
And the only thing this is a
function of is t minus tau.
00:47:41.530 --> 00:47:44.100
I don't know what kind of
function it is, but the only
00:47:44.100 --> 00:47:46.200
thing we're dealing with
is t minus tau.
00:47:46.200 --> 00:47:48.780
I don't know whether these
intervals exists or not.
00:47:48.780 --> 00:47:51.960
I can't make any very good
argument that they can.
00:47:51.960 --> 00:47:55.780
But if they do exist, it's a
function only of t minus tau.
00:47:55.780 --> 00:47:58.180
So aside from the
pseudo-mathematics we've been
00:47:58.180 --> 00:48:02.670
using, V of t is wide-sense
stationary.
00:48:02.670 --> 00:48:05.210
OK?
00:48:05.210 --> 00:48:07.620
Now if you didn't follow
all this integration,
00:48:07.620 --> 00:48:09.680
don't worry about it.
00:48:09.680 --> 00:48:11.600
It's all just substituting
integrals
00:48:11.600 --> 00:48:14.670
and integrating away.
00:48:14.670 --> 00:48:18.680
And it's something you can do.
00:48:18.680 --> 00:48:22.230
Some people just take five
minutes to do it, some people
00:48:22.230 --> 00:48:23.480
10 minutes to do it.
00:48:26.240 --> 00:48:29.430
Now we want to put the effective
stationarity in it.
00:48:29.430 --> 00:48:31.950
Because again, we don't
know what this means.
00:48:31.950 --> 00:48:34.050
We can't interpret what
it means in terms of
00:48:34.050 --> 00:48:36.390
stationarity.
00:48:36.390 --> 00:48:40.660
So we're going to assume that
our filter, h of t, has a
00:48:40.660 --> 00:48:43.740
finite duration impulse
response.
00:48:43.740 --> 00:48:48.340
So I'm going to assume that it's
zero when the magnitude
00:48:48.340 --> 00:48:51.550
of t is greater than A. Again,
I'm assuming a filter which
00:48:51.550 --> 00:48:55.680
starts before zero and runs
until after zero.
00:48:55.680 --> 00:48:58.930
Because again, what I'm assuming
is the receiver
00:48:58.930 --> 00:49:01.530
timing is different from
the transmitter timing.
00:49:01.530 --> 00:49:03.850
And I've set the receiver
timing at
00:49:03.850 --> 00:49:05.610
some convenient place.
00:49:05.610 --> 00:49:09.460
So my filter starts doing things
before anything hits
00:49:09.460 --> 00:49:13.190
it, just because of this
change of time.
00:49:13.190 --> 00:49:17.630
So v of t, then, is equal to the
integral of the process Z
00:49:17.630 --> 00:49:21.370
of t1 times h of t minus t1.
00:49:21.370 --> 00:49:25.360
This is a linear functional
again.
00:49:25.360 --> 00:49:30.250
This depends only on what Z of
t1 is for t minus A less than
00:49:30.250 --> 00:49:35.470
or equal to t1, less than or
equal to t plus A. Because I'm
00:49:35.470 --> 00:49:40.070
doing this integration here,
this function here is 0
00:49:40.070 --> 00:49:42.160
outside of that interval.
00:49:42.160 --> 00:49:45.400
I'm assuming that h of t is
equal to 0 outside of this
00:49:45.400 --> 00:49:47.350
finite interval.
00:49:47.350 --> 00:49:52.870
And this is just a shift
of t1 on it.
00:49:52.870 --> 00:49:58.510
So this integral is going to be
equal to zero from when t
00:49:58.510 --> 00:50:05.170
is equal to t1 minus A to when t
is equal to t1 plus A. Every
00:50:05.170 --> 00:50:09.930
place else, this is
equal to zero.
00:50:09.930 --> 00:50:17.580
So V of t, the whole process
depends -- and if we only want
00:50:17.580 --> 00:50:22.780
to evaluate it within minus T0
plus A to T0 minus A -- in
00:50:22.780 --> 00:50:29.990
other words we look at 10 to
the eighth years and we
00:50:29.990 --> 00:50:33.170
subtract off a microsecond
from that region.
00:50:33.170 --> 00:50:36.890
So now we're saying we want to
see whether this is wide-sense
00:50:36.890 --> 00:50:41.020
stationary within 10 to the
eighth years minus a
00:50:41.020 --> 00:50:44.980
microsecond, minus that
to plus that.
00:50:44.980 --> 00:50:48.420
So that's what we're
dealing with here.
00:50:48.420 --> 00:50:54.770
V of t in this region depends
only on Z of t for t in the
00:50:54.770 --> 00:50:57.580
interval minus T0 to plus T0.
00:50:57.580 --> 00:51:00.520
In other words, when you're
calculating V of t for any
00:51:00.520 --> 00:51:04.520
time in this interval, you're
only interested in Z of t,
00:51:04.520 --> 00:51:08.900
which is diddling within plus
or minus A of that.
00:51:08.900 --> 00:51:10.410
Because that's the only
place where the
00:51:10.410 --> 00:51:13.820
filter is doing anything.
00:51:13.820 --> 00:51:17.840
So V of t depends
only on Z of t.
00:51:17.840 --> 00:51:20.790
And here, I'm going to assume
that Z of t is wide-sense
00:51:20.790 --> 00:51:23.700
stationary within
those limits.
00:51:23.700 --> 00:51:26.340
And therefore I don't have to
worry about what Z of t is
00:51:26.340 --> 00:51:28.190
doing outside of those limits.
00:51:32.090 --> 00:51:37.140
So if the sample functions of Z
of t are L2 within minus T0
00:51:37.140 --> 00:51:42.030
to plus T0, then the sample
functions of V of t are L2
00:51:42.030 --> 00:51:46.010
within minus T0 plus
A to T0 minus A.
00:51:46.010 --> 00:51:48.610
Now this is not obvious.
00:51:48.610 --> 00:51:50.860
And there's a proof in the
notes of this, which goes
00:51:50.860 --> 00:51:54.550
through all this L1 stuff and
L2 stuff that you've been
00:51:54.550 --> 00:51:56.030
struggling with.
00:51:56.030 --> 00:52:00.370
It's not a difficult proof,
but if you don't dig L2
00:52:00.370 --> 00:52:04.810
theory, be my guest and
don't worry about it.
00:52:04.810 --> 00:52:08.120
If you do want to really
understand exactly what's
00:52:08.120 --> 00:52:12.050
going on here, be my guest and
do worry about it, and go
00:52:12.050 --> 00:52:12.910
through it.
00:52:12.910 --> 00:52:15.040
But it's not really
an awful argument.
00:52:15.040 --> 00:52:18.110
But this is what happens.
00:52:18.110 --> 00:52:21.470
So what this says is we can
now view wide-sense
00:52:21.470 --> 00:52:27.750
stationarity and stationarity,
at least for Gaussian
00:52:27.750 --> 00:52:30.960
processes, as a limit of
effectively wide-sense
00:52:30.960 --> 00:52:33.920
stationary processes.
00:52:33.920 --> 00:52:38.000
In other words, so long as
I deal with filters whose
00:52:38.000 --> 00:52:42.180
impulse response is constrained
in time, the only
00:52:42.180 --> 00:52:46.550
effect of that filtering is to
reduce the interval over which
00:52:46.550 --> 00:52:51.200
the process is wide-sense
stationary by the quantity A.
00:52:51.200 --> 00:52:54.870
And the quantity A is going to
be very much smaller than T0,
00:52:54.870 --> 00:52:57.230
because we're going to choose
T0 to be as large as
00:52:57.230 --> 00:52:58.760
we want it to be.
00:52:58.760 --> 00:53:01.170
Namely, that's always the thing
that we do when we try
00:53:01.170 --> 00:53:03.620
to go through limiting
arguments.
00:53:03.620 --> 00:53:07.030
So what we're saying here
is, by using effective
00:53:07.030 --> 00:53:12.320
stationarity, we have managed
to find a tool that lets us
00:53:12.320 --> 00:53:16.690
look at stationary as a limit
of effectively stationary
00:53:16.690 --> 00:53:21.690
processes as you let T0 become
larger and larger.
00:53:21.690 --> 00:53:23.560
And we have our cake and
we eat it too, here.
00:53:23.560 --> 00:53:27.150
Because at this point, all these
functions we're dealing
00:53:27.150 --> 00:53:29.420
with, we can assume
that they're L2.
00:53:29.420 --> 00:53:33.590
Because they're L2 for every
T0 that we want to look at.
00:53:33.590 --> 00:53:36.640
So we don't have to worry
about that anymore.
00:53:36.640 --> 00:53:38.130
So we have these processes.
00:53:38.130 --> 00:53:41.140
Now we will just call them
stationary or wide-sense
00:53:41.140 --> 00:53:42.310
stationary.
00:53:42.310 --> 00:53:45.630
Because we know that what we're
really talking about is
00:53:45.630 --> 00:53:49.040
a limit as T0 goes to infinity
of things that
00:53:49.040 --> 00:53:50.290
make perfect sense.
00:53:54.750 --> 00:53:59.760
So suddenly the mystery, or the
pseudo-mystery has been
00:53:59.760 --> 00:54:01.570
taken out of this, I hope.
00:54:05.100 --> 00:54:09.340
So you have a wide-sense
stationary process with a
00:54:09.340 --> 00:54:13.480
covariance now of
k sub z of tau.
00:54:13.480 --> 00:54:16.140
In other words, it's effectively
stationary over
00:54:16.140 --> 00:54:20.680
some very large interval, and
I now have this covariance
00:54:20.680 --> 00:54:24.100
function, which is a function
of one variable.
00:54:24.100 --> 00:54:30.770
I want to define the spectral
density of this process as a
00:54:30.770 --> 00:54:33.740
Fourier transform of k sub z.
00:54:33.740 --> 00:54:38.810
In other words, as soon as I get
a one variable covariance
00:54:38.810 --> 00:54:42.460
function, I can talk about
its Fourier transform.
00:54:42.460 --> 00:54:45.880
At least assuming that
it's L2 or something.
00:54:45.880 --> 00:54:49.270
And let's forget about that for
the time being, because
00:54:49.270 --> 00:54:50.360
that all works.
00:54:50.360 --> 00:54:52.860
We want to take the Fourier
transform of that.
00:54:52.860 --> 00:54:56.230
I'm going to call that
spectral density.
00:54:56.230 --> 00:55:00.235
Now if I call that spectral
density, the thing you ought
00:55:00.235 --> 00:55:04.010
to want to know is, why am I
calling that spectral density?
00:55:04.010 --> 00:55:06.680
What does it have to do with
anything you might be
00:55:06.680 --> 00:55:08.700
interested in?
00:55:08.700 --> 00:55:11.530
Well, we look at it and we say
well, what might it have to do
00:55:11.530 --> 00:55:13.360
with anything?
00:55:13.360 --> 00:55:17.270
Well, the first thing we're
going to try is to say, what
00:55:17.270 --> 00:55:19.460
happens to these linear
functionals we've
00:55:19.460 --> 00:55:20.760
been talking about?
00:55:20.760 --> 00:55:24.460
We have a linear functional, V.
It's an integral of g of t
00:55:24.460 --> 00:55:26.740
times Z of t dt.
00:55:26.740 --> 00:55:30.020
We would like to be able to talk
about the expected value
00:55:30.020 --> 00:55:31.320
of V squared.
00:55:31.320 --> 00:55:33.700
Talk about zero-mean things,
so we don't care about the
00:55:33.700 --> 00:55:35.370
mean at zero.
00:55:35.370 --> 00:55:38.910
So we'll just talk about the
variance of any linear
00:55:38.910 --> 00:55:41.380
functional of this form.
00:55:41.380 --> 00:55:43.460
If I can talk about the
variance of any linear
00:55:43.460 --> 00:55:47.310
functional for any g of t that
I'm interested in, I can
00:55:47.310 --> 00:55:51.810
certainly say a lot about what
this process is doing.
00:55:51.810 --> 00:55:54.640
So I want to find
that variance.
00:55:54.640 --> 00:55:57.300
If I write this out, it's
same thing we've been
00:55:57.300 --> 00:55:58.510
writing out all along.
00:55:58.510 --> 00:56:03.360
It's the integral of g of t
times this one variable
00:56:03.360 --> 00:56:10.230
covariance function now, times
g of tau d tau dt.
00:56:10.230 --> 00:56:13.820
When you look at this, and
you say OK, what is this?
00:56:13.820 --> 00:56:19.130
It's an integral of g of t
times some function of t.
00:56:19.130 --> 00:56:23.010
And that function of
t is really the
00:56:23.010 --> 00:56:24.830
convolution of k sub z.
00:56:24.830 --> 00:56:26.910
This is a function now.
00:56:26.910 --> 00:56:29.800
It's not a random process
anymore; it's just a function.
00:56:29.800 --> 00:56:36.250
That's a convolution of the
function k tilde with g.
00:56:36.250 --> 00:56:41.770
So I can rewrite this as the
integral of g of t times this
00:56:41.770 --> 00:56:43.100
convolution of t.
00:56:43.100 --> 00:56:46.500
And then I'm going to call the
convolution theta of t, just
00:56:46.500 --> 00:56:47.870
to give it a name.
00:56:47.870 --> 00:56:54.180
So this variance is equal to g
of t times this function theta
00:56:54.180 --> 00:57:00.050
of t, which is just the
convolution of k with g.
00:57:00.050 --> 00:57:02.820
Well now I say, OK, I
can use I can use
00:57:02.820 --> 00:57:05.620
Parseval's relation on that.
00:57:05.620 --> 00:57:08.770
And what I get is this integral,
if I look in the
00:57:08.770 --> 00:57:13.350
frequency domain now, as the
integral of the Fourier
00:57:13.350 --> 00:57:18.330
transform of g of t, times the
complex conjugate of theta
00:57:18.330 --> 00:57:21.240
star of f times df.
00:57:21.240 --> 00:57:25.570
I've cheated it you just a
little bit there because what
00:57:25.570 --> 00:57:28.290
I really want to talk
about is --
00:57:28.290 --> 00:57:32.060
I mean, Parseval's theorem
relates this quantity to this
00:57:32.060 --> 00:57:35.700
quantity, where theta is theta
complex conjugate.
00:57:35.700 --> 00:57:38.690
But fortunately,
theta is real.
00:57:38.690 --> 00:57:42.380
Because g is real
and k is real.
00:57:42.380 --> 00:57:45.205
Because we're only dealing with
real processes here, and
00:57:45.205 --> 00:57:47.050
we're only dealing with
real filters.
00:57:47.050 --> 00:57:48.860
So this is real, this is real.
00:57:48.860 --> 00:57:53.700
So this integral is equal to
this in the frequency domain.
00:57:53.700 --> 00:58:01.030
And now, what is the Fourier
transform of --
00:58:01.030 --> 00:58:04.950
OK, theta of t is this
convolution.
00:58:04.950 --> 00:58:08.790
When I take the Fourier
transform of theta, what I get
00:58:08.790 --> 00:58:12.940
is the product of the Fourier
transform of k times the
00:58:12.940 --> 00:58:14.540
Fourier transform of g.
00:58:14.540 --> 00:58:20.040
So in fact what I get is g hat
of f times theta star of f.
00:58:20.040 --> 00:58:26.780
Which is g complex conjugate
of that times k complex
00:58:26.780 --> 00:58:30.420
conjugate -- times -- it doesn't
make any difference
00:58:30.420 --> 00:58:32.210
whether it's a complex
conjugate or not,
00:58:32.210 --> 00:58:34.490
because it's real.
00:58:34.490 --> 00:58:38.710
So I wind up with the integral
of the magnitude squared of g
00:58:38.710 --> 00:58:41.740
times this spectral density.
00:58:41.740 --> 00:58:44.160
Well at this point, we can
interpret all sorts
00:58:44.160 --> 00:58:46.420
of things from this.
00:58:46.420 --> 00:58:49.140
And we now know that we can
interpret this also in terms
00:58:49.140 --> 00:58:52.460
of effectively stationary
things.
00:58:52.460 --> 00:58:54.900
So we don't have any problem
with things going to infinity
00:58:54.900 --> 00:58:55.940
or anything.
00:58:55.940 --> 00:58:59.270
I mean, you can go through this
argument carefully for
00:58:59.270 --> 00:59:03.880
effectively stationary things,
and everything works out fine.
00:59:03.880 --> 00:59:08.610
So long as g of t is constrained
in time.
00:59:08.610 --> 00:59:13.245
Well now, I can choose
this function any
00:59:13.245 --> 00:59:15.980
way that I want to.
00:59:15.980 --> 00:59:18.550
In other words, I can choose
my function g of t in
00:59:18.550 --> 00:59:20.070
any way I want to.
00:59:20.070 --> 00:59:22.990
I can choose this function
in any way I want to.
00:59:22.990 --> 00:59:25.910
So long as its inverse
transform is real.
00:59:25.910 --> 00:59:32.110
Which means g hat of f has to
be equal to g hat of minus f
00:59:32.110 --> 00:59:34.480
complex conjugate.
00:59:34.480 --> 00:59:38.340
But aside from that, I can
choose this to be anything.
00:59:38.340 --> 00:59:42.240
And if I choose this to be very,
very narrow band, what
00:59:42.240 --> 00:59:43.840
is this doing?
00:59:43.840 --> 00:59:48.410
It's saying that if I take a
very narrow band g of t, like
00:59:48.410 --> 00:59:53.000
a sinusoid truncated out to
some very wide region, I
00:59:53.000 --> 00:59:59.420
multiply that by this process, I
integrate it out, and I look
00:59:59.420 --> 01:00:00.620
at the variance.
01:00:00.620 --> 01:00:03.210
What am I doing when
I'm doing that?
01:00:03.210 --> 01:00:08.320
I'm effectively filtering my
process with a very, very
01:00:08.320 --> 01:00:12.470
narrow band filter, and I'm
asking, what's the variance of
01:00:12.470 --> 01:00:14.260
the output?
01:00:14.260 --> 01:00:18.250
So the variance of the output
really is related to the
01:00:18.250 --> 01:00:23.110
amount of energy in this process
at that frequency.
01:00:26.110 --> 01:00:29.050
That's exactly what the
mathematics says here.
01:00:29.050 --> 01:00:36.670
It says that s sub z of f is
in fact the amount of noise
01:00:36.670 --> 01:00:41.280
power per unit bandwidth
at this frequency.
01:00:41.280 --> 01:00:44.790
It's the only way you
can interpret it.
01:00:44.790 --> 01:00:49.470
Because I can make this as
narrow as I want to, so that
01:00:49.470 --> 01:00:53.800
if there is any interpretation
of something in this process
01:00:53.800 --> 01:00:57.365
as being at one frequency rather
than another frequency
01:00:57.365 --> 01:01:00.330
and the only way I can interpret
that is by filtering
01:01:00.330 --> 01:01:02.010
the process.
01:01:02.010 --> 01:01:04.850
This is saying that when you
filter the process, what you
01:01:04.850 --> 01:01:10.070
see at that bandwidth is how
much power there is in the
01:01:10.070 --> 01:01:12.770
process at that bandwidth.
01:01:12.770 --> 01:01:17.300
So this is simply giving us an
interpretation of spectral
01:01:17.300 --> 01:01:20.300
density as power per
unit bandwidth.
01:01:22.960 --> 01:01:32.390
OK, let's go on with this.
01:01:32.390 --> 01:01:35.830
If this spectral density is
constant over all frequencies
01:01:35.830 --> 01:01:39.200
of interest, we say
that it's white.
01:01:39.200 --> 01:01:41.650
Here's the answer to your
question: what is white
01:01:41.650 --> 01:01:44.660
Gaussian noise?
01:01:44.660 --> 01:01:50.410
White Gaussian noise is a
Gaussian process which has a
01:01:50.410 --> 01:01:53.720
constant spectral density over
the frequencies that we're
01:01:53.720 --> 01:01:55.530
interested in.
01:01:55.530 --> 01:01:59.900
So we now have this looking
at it in a certain band of
01:01:59.900 --> 01:02:02.520
frequencies.
01:02:02.520 --> 01:02:06.330
You know, if you think about
this in more or less practical
01:02:06.330 --> 01:02:09.260
terms, suppose you're building
a wireless network, and it's
01:02:09.260 --> 01:02:14.100
going to operate, say, at
five or six gigahertz.
01:02:17.170 --> 01:02:20.380
And suppose your bandwidth
is maybe 100
01:02:20.380 --> 01:02:21.430
megahertz or something.
01:02:21.430 --> 01:02:24.870
Or maybe it's 10 megahertz,
or maybe it's 1 megahertz.
01:02:24.870 --> 01:02:29.920
In terms of this frequency of
many gigahertz, you're talking
01:02:29.920 --> 01:02:32.560
about very narrowband
communication.
01:02:32.560 --> 01:02:35.330
People might call it wideband
communication, but it's really
01:02:35.330 --> 01:02:36.580
pretty narrowband.
01:02:38.980 --> 01:02:43.270
Now suppose you have Gaussian
noise, which is caused by all
01:02:43.270 --> 01:02:49.060
of these small tiny noise
affects all over the place.
01:02:49.060 --> 01:02:51.890
And you ask, is this going
to be flat or is it
01:02:51.890 --> 01:02:52.770
not going to be flat?
01:02:52.770 --> 01:02:56.050
Well, you might look at it and
say, well, if there's no noise
01:02:56.050 --> 01:02:58.150
in this little band and there's
a lot of noise in this
01:02:58.150 --> 01:03:02.220
band, I'm going to use this
little band here.
01:03:02.220 --> 01:03:04.560
But after you get all done
playing those games, what
01:03:04.560 --> 01:03:06.330
you're saying is, well,
this noise is sort of
01:03:06.330 --> 01:03:08.710
uniform over this band.
01:03:08.710 --> 01:03:11.810
And therefore if I'm only
transmitting in that band, if
01:03:11.810 --> 01:03:15.510
I never transmit anything
outside of that band, there's
01:03:15.510 --> 01:03:20.630
no way you can tell what the
noise is outside of that band.
01:03:20.630 --> 01:03:28.980
You all know that the noise
you experience if you're
01:03:28.980 --> 01:03:32.930
dealing with a carrier frequency
in the kilohertz
01:03:32.930 --> 01:03:36.860
band is very different than what
you see in the megahertz
01:03:36.860 --> 01:03:40.140
band, is very different from
what you see at 100 megahertz,
01:03:40.140 --> 01:03:43.240
is very different from what you
see in the gigahertz band,
01:03:43.240 --> 01:03:46.090
is very different from what you
see in the optical bands.
01:03:46.090 --> 01:03:49.400
And all of that stuff.
01:03:49.400 --> 01:03:55.330
So it doesn't make any sense
to model the noise as being
01:03:55.330 --> 01:04:00.300
uniform spectral density over
all of that region.
01:04:00.300 --> 01:04:03.690
So the only thing we would ever
want to do is to model
01:04:03.690 --> 01:04:08.390
the noise as having a uniform
density over this narrowband
01:04:08.390 --> 01:04:09.640
that we're interested in.
01:04:12.040 --> 01:04:14.900
And that's how we define
white Gaussian noise.
01:04:14.900 --> 01:04:18.870
We say the noise is white
Gaussian noise if, in fact,
01:04:18.870 --> 01:04:23.150
when we go out and measure it
we can measure it by passing
01:04:23.150 --> 01:04:26.010
it through a filter by finding
this variance, which is
01:04:26.010 --> 01:04:30.170
exactly what we're talking about
here, if what we get at
01:04:30.170 --> 01:04:32.550
one frequency is the
same as what we
01:04:32.550 --> 01:04:33.840
get at another frequency.
01:04:33.840 --> 01:04:37.300
So long as we're talking about
the frequencies of interest,
01:04:37.300 --> 01:04:40.370
then we say it's white.
01:04:40.370 --> 01:04:43.580
Now how long do we have to
make this measurement?
01:04:43.580 --> 01:04:46.460
Well, we don't make it forever,
because all the
01:04:46.460 --> 01:04:49.690
filters we're using and
everything else are filters
01:04:49.690 --> 01:04:54.420
which only have a duration over
a certain amount of time.
01:04:54.420 --> 01:04:57.090
We only use our device over
a certain amount of time.
01:04:57.090 --> 01:05:01.530
So what we're interested in is
looking at the noise over some
01:05:01.530 --> 01:05:06.130
large effective time from
minus T0 to plus T0.
01:05:06.130 --> 01:05:09.630
We want the noise to be
effectively stationary within
01:05:09.630 --> 01:05:13.950
minus T0 to plus T0.
01:05:13.950 --> 01:05:16.030
And then what's the next
step in the argument?
01:05:16.030 --> 01:05:18.700
You want the noise to be
effectively stationary between
01:05:18.700 --> 01:05:21.610
these very broad limits.
01:05:21.610 --> 01:05:23.610
And then we think about
it for a little bit,
01:05:23.610 --> 01:05:24.800
and we say, but listen.
01:05:24.800 --> 01:05:28.990
I'm only using this thing in a
very small fraction of that
01:05:28.990 --> 01:05:30.860
time region.
01:05:30.860 --> 01:05:34.410
And therefore as far as my
model is concerned, I
01:05:34.410 --> 01:05:37.850
shouldn't be bothered
with T0 at all.
01:05:37.850 --> 01:05:40.950
I should just say
mathematically, this process
01:05:40.950 --> 01:05:45.180
is going to be stationary, and
I forget about the T0.
01:05:45.180 --> 01:05:48.740
And I look at it in frequency,
and I say I'm going to use
01:05:48.740 --> 01:05:52.640
this over my 10 megahertz or
100 megahertz, or whatever
01:05:52.640 --> 01:05:54.930
frequency band I'm
interested in.
01:05:54.930 --> 01:05:56.470
And I'm only interested
in what the
01:05:56.470 --> 01:05:58.430
noise is in that band.
01:05:58.430 --> 01:06:02.050
I don't want to specify
what the bandwidth is.
01:06:02.050 --> 01:06:05.470
And therefore I say, I will just
model it as being uniform
01:06:05.470 --> 01:06:08.450
over all frequencies.
01:06:08.450 --> 01:06:11.450
So what white noise is, is
you have effectively
01:06:11.450 --> 01:06:13.340
gotten rid of the T0.
01:06:13.340 --> 01:06:17.310
You've effectively gotten
rid of the w0.
01:06:17.310 --> 01:06:20.540
And after you've gotten rid of
both of these things, you have
01:06:20.540 --> 01:06:25.070
noise which has constant
spectral density over all
01:06:25.070 --> 01:06:30.100
frequencies, and noise way
which has constant
01:06:30.100 --> 01:06:32.460
power over all time.
01:06:32.460 --> 01:06:35.700
And you look at it,
and what happens?
01:06:35.700 --> 01:06:40.880
If you take the inverse Fourier
transform of sz of f,
01:06:40.880 --> 01:06:45.340
and you assume that sz of f is
just non-zero within a certain
01:06:45.340 --> 01:06:48.400
frequency band, what you get
when you take the Fourier
01:06:48.400 --> 01:06:53.240
transform is a little kind
of wiggle around zero.
01:06:53.240 --> 01:06:55.760
And that's all very
interesting.
01:06:55.760 --> 01:06:59.100
If you then say well, I
don't care about that.
01:06:59.100 --> 01:07:01.950
I just like to assume that
it's uniform over all
01:07:01.950 --> 01:07:03.570
frequencies.
01:07:03.570 --> 01:07:05.610
You then take the Fourier
transform, and what you've got
01:07:05.610 --> 01:07:06.860
is an impulse function.
01:07:09.510 --> 01:07:13.190
And what the impulse function
tells you is, when you pass
01:07:13.190 --> 01:07:17.920
the noise through a filter, what
comes out of the filter,
01:07:17.920 --> 01:07:22.520
just like any time you deal
with impulse functions --
01:07:22.520 --> 01:07:24.810
I mean, the impulse response
is in fact the
01:07:24.810 --> 01:07:27.640
response to an impulse.
01:07:27.640 --> 01:07:30.190
So that as far as the output
from the filter goes,
01:07:30.190 --> 01:07:31.470
it's all very fine.
01:07:31.470 --> 01:07:37.650
It only cares about what the
integral is of that pulse.
01:07:37.650 --> 01:07:41.910
And the integral of the pulse
doesn't depend very much on
01:07:41.910 --> 01:07:45.650
what happens at enormously large
frequencies or anything.
01:07:45.650 --> 01:07:48.980
So all of that's well behaved,
so long as you go through some
01:07:48.980 --> 01:07:52.020
kind of filtering first.
01:07:52.020 --> 01:07:56.180
Unfortunately, as soon as you
start talking about a
01:07:56.180 --> 01:07:59.140
covariance function which
is an impulse,
01:07:59.140 --> 01:08:00.250
you're in real trouble.
01:08:00.250 --> 01:08:03.730
Because the covariance function
evaluated at zero is
01:08:03.730 --> 01:08:05.240
the power in the process.
01:08:08.070 --> 01:08:10.810
And the power in the process
is then infinite.
01:08:10.810 --> 01:08:15.760
So you wind up with this process
which is easy to work
01:08:15.760 --> 01:08:18.230
with any time you filter it.
01:08:18.230 --> 01:08:21.250
It's easy to work with because
you don't have these constants
01:08:21.250 --> 01:08:25.120
capital T0 and capital
W0 stuck in them.
01:08:25.120 --> 01:08:27.820
Which you don't really care
about, because what you're
01:08:27.820 --> 01:08:31.390
assuming is you can wander
around as much as you want in
01:08:31.390 --> 01:08:33.420
frequency, subject to
the antennas and
01:08:33.420 --> 01:08:35.660
so on that you have.
01:08:35.660 --> 01:08:38.280
And you want to be able to
wander around as much in time
01:08:38.280 --> 01:08:41.650
as you want to and assume that
things are uniform over all
01:08:41.650 --> 01:08:44.390
that region.
01:08:44.390 --> 01:08:48.060
But you then have this problem
that you have a noise process
01:08:48.060 --> 01:08:51.010
which just doesn't make
any sense at all.
01:08:51.010 --> 01:08:53.400
Because it's infinite
everywhere.
01:08:53.400 --> 01:08:56.270
You look at any little frequency
band of it and it
01:08:56.270 --> 01:09:00.130
has infinite energy if you
integrate over all time.
01:09:00.130 --> 01:09:04.910
So you really want to somehow
use these ideas of being
01:09:04.910 --> 01:09:11.750
effectively stationary and of
being effectively bandlimited,
01:09:11.750 --> 01:09:14.380
and say what I want
is noise which is
01:09:14.380 --> 01:09:16.780
flat over those regions.
01:09:16.780 --> 01:09:20.980
Now what we're going to do after
the quiz, which is on
01:09:20.980 --> 01:09:24.420
Wednesday, is we're going to
start talking about how you
01:09:24.420 --> 01:09:27.920
actually detect signals in
the presence of noise.
01:09:27.920 --> 01:09:30.960
And what we're going to find
out is, when the noise is
01:09:30.960 --> 01:09:34.710
white in the sense -- namely,
when it behaves the same over
01:09:34.710 --> 01:09:38.150
all the degrees of freedom that
we're looking at -- then
01:09:38.150 --> 01:09:41.040
it doesn't matter where
you put your signal.
01:09:41.040 --> 01:09:43.470
You can put your signal anywhere
in this huge time
01:09:43.470 --> 01:09:48.490
space, in this huge bandwidth
that we're talking about.
01:09:48.490 --> 01:09:50.500
And we somehow want
to find out how to
01:09:50.500 --> 01:09:52.150
detect signals there.
01:09:52.150 --> 01:09:56.330
And we find out that the
detection process is
01:09:56.330 --> 01:10:00.300
independent of what time we're
looking at and what frequency
01:10:00.300 --> 01:10:01.320
we're looking at.
01:10:01.320 --> 01:10:04.340
So what we have to focus on is
this relatively small interval
01:10:04.340 --> 01:10:08.280
of time and relatively small
interval of bandwidth.
01:10:08.280 --> 01:10:09.730
So all of this works well.
01:10:09.730 --> 01:10:13.260
Which is why we assume
white Gaussian noise.
01:10:13.260 --> 01:10:17.120
But the white Gaussian noise
assumption really makes sense
01:10:17.120 --> 01:10:20.330
when you're looking at what the
noise looks like in these
01:10:20.330 --> 01:10:22.030
various degrees of freedom.
01:10:22.030 --> 01:10:25.070
Namely, what the noise looks
like when you pass the noise
01:10:25.070 --> 01:10:28.810
through a filter and look at
the output at specific
01:10:28.810 --> 01:10:30.580
instance of time.
01:10:30.580 --> 01:10:34.000
And that's where the modeling
assumption is come in.
01:10:34.000 --> 01:10:38.950
So this is really a very
sophisticated use of modeling.
01:10:38.950 --> 01:10:41.680
Did the engineers who created
this sense of modeling have
01:10:41.680 --> 01:10:44.190
any idea of what they
were doing?
01:10:44.190 --> 01:10:44.430
No.
01:10:44.430 --> 01:10:46.770
They didn't have the foggiest
idea of what they were doing,
01:10:46.770 --> 01:10:49.390
except they had common sense.
01:10:49.390 --> 01:10:52.320
And they had enough common
sense to realize that no
01:10:52.320 --> 01:10:55.490
matter where they put their
signals, this same noise was
01:10:55.490 --> 01:10:58.630
going to be affecting them.
01:10:58.630 --> 01:11:03.980
And because of that, what they
did is they created some kind
01:11:03.980 --> 01:11:08.410
of pseudo-theory, which said we
have noise which looks the
01:11:08.410 --> 01:11:09.930
same wherever it is.
01:11:09.930 --> 01:11:13.530
Mathematicians got a hold of
it, went through all this
01:11:13.530 --> 01:11:15.810
theory of generalized
functions,
01:11:15.810 --> 01:11:17.490
came back to the engineers.
01:11:17.490 --> 01:11:19.640
The engineers couldn't
understand any of that, and
01:11:19.640 --> 01:11:23.510
it's been going back
and forth forever.
01:11:23.510 --> 01:11:29.460
Where we are now, I think, is we
have a theory where we can
01:11:29.460 --> 01:11:33.130
actually look at finite time
intervals, finite frequency
01:11:33.130 --> 01:11:37.180
intervals, see what's going on
there, make mathematical sense
01:11:37.180 --> 01:11:40.580
out of it, and then say that
the results don't depend on
01:11:40.580 --> 01:11:45.590
what t0 is or w0 is, so
we can leave it out.
01:11:45.590 --> 01:11:47.850
And at that point we really
have our cake and
01:11:47.850 --> 01:11:49.200
we can eat it too.
01:11:49.200 --> 01:11:52.610
So we can do what the engineers
I've always been
01:11:52.610 --> 01:11:56.310
doing, but we really understand
it at this point.
01:11:56.310 --> 01:11:57.470
OK.
01:11:57.470 --> 01:11:59.440
I think I will stop
at that point.