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PROFESSOR: --And go on with
lectures 8 to 10.
00:00:30.530 --> 00:00:33.090
First I want to briefly
review what we said
00:00:33.090 --> 00:00:36.320
about measurable functions.
00:00:36.320 --> 00:00:40.580
Again, I encourage you if you
hate this material and you
00:00:40.580 --> 00:00:43.700
think it's only for
mathematicians,
00:00:43.700 --> 00:00:45.950
please let me know.
00:00:45.950 --> 00:00:48.190
I don't know whether it's
appropriate to cover it in
00:00:48.190 --> 00:00:50.530
this class either.
00:00:50.530 --> 00:00:54.810
I think it probably is because
you need to know a little more
00:00:54.810 --> 00:01:00.340
mathematics when you deal with
these Fourier transforms and
00:01:00.340 --> 00:01:01.900
Fourier series.
00:01:04.490 --> 00:01:08.510
When you're dealing with
communication theory, then you
00:01:08.510 --> 00:01:10.330
need to know for signal
processing
00:01:10.330 --> 00:01:12.100
and things like that.
00:01:12.100 --> 00:01:15.750
When you get into learning a
little more, my sense at this
00:01:15.750 --> 00:01:21.070
point is it's much easier to
learn a good deal more than it
00:01:21.070 --> 00:01:23.660
is to learn just a little bit
more, because a little bit
00:01:23.660 --> 00:01:27.280
more you're always faced with
all of these questions of what
00:01:27.280 --> 00:01:29.610
does this really mean.
00:01:29.610 --> 00:01:33.250
It turns out that if you put
just a little bit of measure
00:01:33.250 --> 00:01:38.000
theory into your thinking, it
makes all these problems very
00:01:38.000 --> 00:01:39.710
much simpler.
00:01:39.710 --> 00:01:42.540
So you remember we talked about
what a measurable set
00:01:42.540 --> 00:01:45.400
was last time.
00:01:45.400 --> 00:01:50.380
What a measurable set is, is a
set which you can essentially
00:01:50.380 --> 00:01:55.150
break up into a countable
union of intervals, and
00:01:55.150 --> 00:02:03.030
something which is bounded by
a countble set of intervals
00:02:03.030 --> 00:02:05.340
which have arbitrarily
small measure.
00:02:05.340 --> 00:02:09.310
So you can take all of these
sets of zero measure --
00:02:09.310 --> 00:02:12.970
countable sets, cantor sets, all
of these things, and you
00:02:12.970 --> 00:02:16.140
can just throw all of those
out from consideration.
00:02:16.140 --> 00:02:19.960
That's what makes measure theory
useful and interesting
00:02:19.960 --> 00:02:22.530
and what simplifies things.
00:02:22.530 --> 00:02:26.750
So we then said that a function
is measurable if each
00:02:26.750 --> 00:02:31.930
of these sets here -- if the
set of times for which a is
00:02:31.930 --> 00:02:34.860
less than or equal to u of t is
less than or equal to b --
00:02:34.860 --> 00:02:37.500
these things are these
level sets here.
00:02:37.500 --> 00:02:42.920
It's a set of values, t, along
this axis in which the
00:02:42.920 --> 00:02:47.470
function is nailed between any
two of these points here.
00:02:52.230 --> 00:02:55.370
For a function to be measurable,
all of these sets
00:02:55.370 --> 00:02:57.910
in here have to be measurable.
00:02:57.910 --> 00:03:01.390
We haven't given you any
examples of sets which are not
00:03:01.390 --> 00:03:04.990
measurable, and therefore, we
haven't given you any examples
00:03:04.990 --> 00:03:07.720
of functions which are
not measurable.
00:03:07.720 --> 00:03:15.360
It's very, very difficult to
construct such functions, and
00:03:15.360 --> 00:03:18.480
since it's so difficult to
construct them, it's easier to
00:03:18.480 --> 00:03:23.510
just say that all of the
functions we can think of are
00:03:23.510 --> 00:03:24.540
measurable.
00:03:24.540 --> 00:03:27.550
That includes an awful lot more
functions than you ever
00:03:27.550 --> 00:03:30.640
deal with in engineering.
00:03:30.640 --> 00:03:37.410
If you ever make a serious
mistake in your engineering
00:03:37.410 --> 00:03:41.670
work because of thinking that a
function is measurable when
00:03:41.670 --> 00:03:45.750
it's not measurable, I
will give you $1,000.
00:03:45.750 --> 00:03:48.610
I make that promise that you
because I don't think it's
00:03:48.610 --> 00:03:51.290
ever happened in history, I
don't think it ever will
00:03:51.290 --> 00:03:58.440
happen in history, and that's
just how sure I am of it.
00:03:58.440 --> 00:04:01.580
Unless you try to deliberately
make such a mistake in order
00:04:01.580 --> 00:04:03.080
to collect $1,000.
00:04:03.080 --> 00:04:08.090
Of course, that's not fair.
00:04:08.090 --> 00:04:13.900
Then we said the way we define
this approximation to the
00:04:13.900 --> 00:04:17.650
integral, namely, we always
defined it from underneath,
00:04:17.650 --> 00:04:21.810
and therefore, if we started to
take these intervals here
00:04:21.810 --> 00:04:25.960
with epsilon and split them in
half, the thing that happens
00:04:25.960 --> 00:04:29.210
is you always get extra
components put in.
00:04:29.210 --> 00:04:33.560
So that as you make the scaling
finer and finer, what
00:04:33.560 --> 00:04:36.970
happens is the approximation
into the integral gets larger
00:04:36.970 --> 00:04:38.080
and larger.
00:04:38.080 --> 00:04:40.970
Because of that, if you're
dealing with a non-negative
00:04:40.970 --> 00:04:45.800
function, the only two things
that can happen, as you go to
00:04:45.800 --> 00:04:50.350
the limit of finer and finer
scaling, is one, you come to a
00:04:50.350 --> 00:04:53.830
finite limit, and two, you come
to an infinite limit.
00:04:53.830 --> 00:04:55.900
Nothing else can happen.
00:04:55.900 --> 00:05:00.440
You see that's remarkably simple
as mathematics go.
00:05:00.440 --> 00:05:04.340
This is dealing with all of
these functions you can ever
00:05:04.340 --> 00:05:06.980
think of, and those are the
only two things that can
00:05:06.980 --> 00:05:08.230
happen here.
00:05:12.660 --> 00:05:17.370
So then we went on to say that
a function as L1, if it's
00:05:17.370 --> 00:05:21.820
measurable, and if the integral
of its magnitude is
00:05:21.820 --> 00:05:23.830
less than infinity, and
so far we're still
00:05:23.830 --> 00:05:27.710
talking about real functions.
00:05:27.710 --> 00:05:38.520
If u of t is L1, then you can
take the integral of u of t
00:05:38.520 --> 00:05:41.210
and you can split it
up into two things.
00:05:41.210 --> 00:05:45.790
You can split it up into the set
of times over which u of t
00:05:45.790 --> 00:05:48.630
is positive, and the
set of times over
00:05:48.630 --> 00:05:51.270
which u of t is negative.
00:05:51.270 --> 00:05:54.880
The set of times over which u of
t is equal to zero doesn't
00:05:54.880 --> 00:05:56.900
contribute to the integral
at all so you can
00:05:56.900 --> 00:05:59.410
forget about that.
00:05:59.410 --> 00:06:03.040
This is well-defined as the
function is measurable.
00:06:03.040 --> 00:06:06.540
What's now happening is that
if the integral of this
00:06:06.540 --> 00:06:11.060
magnitude is less than infinity,
then both this has
00:06:11.060 --> 00:06:14.890
to be less than infinity, and
this has to be less than
00:06:14.890 --> 00:06:18.860
infinity, which says that so
long as you're dealing with L1
00:06:18.860 --> 00:06:24.540
functions, you never have to
worry about this nasty
00:06:24.540 --> 00:06:28.320
situation where the negative
part of the function is
00:06:28.320 --> 00:06:32.060
infinite, the positive part of
the function is infinite, and
00:06:32.060 --> 00:06:35.610
therefore, the difference, plus
infinity, minus infinity
00:06:35.610 --> 00:06:38.090
doesn't make any sense at all.
00:06:38.090 --> 00:06:41.360
And instead of having an
integral which is infinite or
00:06:41.360 --> 00:06:45.030
finite, you have a function
which might just be undefined
00:06:45.030 --> 00:06:46.040
completely.
00:06:46.040 --> 00:06:50.090
Well this says that if the
function is L1, you can't ever
00:06:50.090 --> 00:06:54.990
have any problem with integrals
of this sort thing
00:06:54.990 --> 00:06:55.710
being undefined.
00:06:55.710 --> 00:06:59.800
So, and in fact, u of t is
always integral and has a
00:06:59.800 --> 00:07:02.560
finite value in this case.
00:07:02.560 --> 00:07:06.340
Now, we say that a complex
function is measurable if both
00:07:06.340 --> 00:07:09.210
the real part is measurable
and the imaginary part is
00:07:09.210 --> 00:07:10.350
measurable.
00:07:10.350 --> 00:07:13.470
Therefore, you don't have to
worry about complex functions
00:07:13.470 --> 00:07:18.000
as really being any more than
twice as complicated as real
00:07:18.000 --> 00:07:20.530
functions, and conceptually
they aren't any more
00:07:20.530 --> 00:07:24.340
complicated at all because you
just treat them as two
00:07:24.340 --> 00:07:27.100
separate functions, and
everything you know about real
00:07:27.100 --> 00:07:32.170
functions you now know about
complex functions.
00:07:32.170 --> 00:07:34.570
Now, as far as Fourier theory
is concerned --
00:07:34.570 --> 00:07:39.490
Fourier series, Fourier
integrals, discrete time,
00:07:39.490 --> 00:07:45.200
Fourier transforms -- all of
these things, if u of t is an
00:07:45.200 --> 00:07:51.880
L1 function, then u of t times e
to the 2 pi ft has to be L1,
00:07:51.880 --> 00:07:55.350
and this has to be true for
all possible values of f.
00:07:55.350 --> 00:07:56.860
Why is that?
00:07:56.860 --> 00:08:00.330
Well, it's just that the
absolute value of u of t for
00:08:00.330 --> 00:08:04.650
any given t, is exactly the same
as the absolute value of
00:08:04.650 --> 00:08:08.510
u of t times e to
the 2 pi i ft.
00:08:08.510 --> 00:08:13.170
Namely, this is a quantity whose
magnitude is always 1,
00:08:13.170 --> 00:08:18.870
and therefore, this quantity,
this magnitude is equal to
00:08:18.870 --> 00:08:20.850
this magnitude.
00:08:20.850 --> 00:08:24.150
Therefore, when you integrate
this magnitude, you get the
00:08:24.150 --> 00:08:28.460
same thing as when you integrate
this magnitude.
00:08:28.460 --> 00:08:30.830
So, there's no problem here.
00:08:33.370 --> 00:08:37.640
So what that says, using this
idea, too, of positive and
00:08:37.640 --> 00:08:41.420
negative parts, it says that the
integral of u of t times e
00:08:41.420 --> 00:08:47.710
to the 2 pi i ft dt has to
exist for all real f.
00:08:47.710 --> 00:08:51.300
That covers Fourier series
and Fourier integral.
00:08:51.300 --> 00:08:54.980
We haven't talked about the
Fourier integral yet, but it
00:08:54.980 --> 00:09:00.510
says that just by defining u of
t to be L1, you avoid all
00:09:00.510 --> 00:09:04.570
of these problems of when these
Fourier integrals exist
00:09:04.570 --> 00:09:05.700
and when they don't exist.
00:09:05.700 --> 00:09:06.950
They always exist.
00:09:11.580 --> 00:09:14.100
So let's talk about the Fourier
transform then since
00:09:14.100 --> 00:09:18.140
we're backing into this
slowly anyway.
00:09:18.140 --> 00:09:21.540
What you've learned in probably
two or three classes
00:09:21.540 --> 00:09:25.490
by now is that the Fourier
transform is
00:09:25.490 --> 00:09:27.630
defined in this way.
00:09:27.630 --> 00:09:31.740
Namely, the transform, a
function of frequency, u hat
00:09:31.740 --> 00:09:37.120
of f, we use hats here instead
of capital letters because we
00:09:37.120 --> 00:09:41.160
like to use capital letters
for random variables.
00:09:41.160 --> 00:09:44.500
So that this transform here
is the integral from minus
00:09:44.500 --> 00:09:49.140
infinity to infinity of the
original function, we start
00:09:49.140 --> 00:09:53.950
with u of t times e to the
minus 2 pi ift dt.
00:09:53.950 --> 00:09:55.950
Now, what's the nice
part about this?
00:09:55.950 --> 00:10:02.600
The nice part about this is that
if u of t is L1, then in
00:10:02.600 --> 00:10:08.590
fact this Fourier transform
has to exist everywhere.
00:10:08.590 --> 00:10:12.710
Namely, what you've learned
before in all of these classes
00:10:12.710 --> 00:10:17.600
says essentially that if these
functions are well-behaved
00:10:17.600 --> 00:10:20.140
then these transforms exist.
00:10:20.140 --> 00:10:23.750
What you've learned about
well-behaved is completely
00:10:23.750 --> 00:10:27.060
circular, namely, a function
is well-behaved if the
00:10:27.060 --> 00:10:31.040
transform exists, and the
transform exists if the
00:10:31.040 --> 00:10:32.690
function is well-behaved.
00:10:32.690 --> 00:10:38.670
You have no idea of what kinds
of functions the Fourier
00:10:38.670 --> 00:10:40.740
transform exists and
what kinds of
00:10:40.740 --> 00:10:42.270
functions it doesn't exist.
00:10:42.270 --> 00:10:45.650
There's another thing buried
in here, in this transform
00:10:45.650 --> 00:10:46.980
relationship.
00:10:46.980 --> 00:10:50.660
Namely, the first thing is we're
trying to define the
00:10:50.660 --> 00:10:54.390
transform function of frequency
in terms of the
00:10:54.390 --> 00:10:56.020
function of time.
00:10:56.020 --> 00:10:57.860
Well and good.
00:10:57.860 --> 00:11:01.810
Then we try to define a function
of time in terms of a
00:11:01.810 --> 00:11:04.700
function of frequency.
00:11:04.700 --> 00:11:10.130
What's hidden here is if you
start with u of t, you then
00:11:10.130 --> 00:11:13.770
get a function of frequency,
and this sort of implicitly
00:11:13.770 --> 00:11:16.040
says that you get back
again the same
00:11:16.040 --> 00:11:18.680
thing you started with.
00:11:18.680 --> 00:11:21.730
That, in fact, is the most
ticklish part of all of this.
00:11:21.730 --> 00:11:27.270
So that if we start with an L1
function u of t, we wind up
00:11:27.270 --> 00:11:30.590
with a Fourier transform,
u hat of f.
00:11:30.590 --> 00:11:32.670
Then u hat of f goes in here.
00:11:32.670 --> 00:11:37.620
We hope we might get back
the sum transform here.
00:11:37.620 --> 00:11:40.420
The nasty thing, and one of the
reasons we're going to get
00:11:40.420 --> 00:11:47.080
away from L1 in just a minute,
is that if a function is L1,
00:11:47.080 --> 00:11:51.200
it's Fourier transform is
not necessarily L1.
00:11:51.200 --> 00:11:54.720
What that means is that you
have to learn all of this
00:11:54.720 --> 00:11:58.130
stuff about L1 functions, and
then as soon as you take the
00:11:58.130 --> 00:12:03.810
Fourier transform, bingo, it's
all gone up in a lot of smoke,
00:12:03.810 --> 00:12:07.730
and you have to start all over
again saying something about
00:12:07.730 --> 00:12:10.820
what properties the transform
might have.
00:12:10.820 --> 00:12:14.790
But anyway, it's nice to start
with because when u of t is
00:12:14.790 --> 00:12:18.745
L1, we know that this function
actually exists.
00:12:18.745 --> 00:12:21.120
It actually exists as
a complex number.
00:12:21.120 --> 00:12:23.750
It exists as a complex
number for every
00:12:23.750 --> 00:12:26.160
possible real f here.
00:12:26.160 --> 00:12:29.840
Namely, there aren't any if,
and's, but's or maybe's here,
00:12:29.840 --> 00:12:32.290
there's nothing like L2
convergence that we were
00:12:32.290 --> 00:12:35.260
talking about a little
bit before.
00:12:35.260 --> 00:12:37.160
This just exists, period.
00:12:40.340 --> 00:12:45.070
There's something more here,
that if this is L1, this not
00:12:45.070 --> 00:12:48.945
only exists but it's also a
continuous function, and those
00:12:48.945 --> 00:12:51.030
don't prove that.
00:12:51.030 --> 00:12:54.860
If you've taken a course in
analysis and you know what a
00:12:54.860 --> 00:12:58.850
complex function is and you're
quite patient, you can sit
00:12:58.850 --> 00:13:02.090
down and actually show
this yourselves, but
00:13:02.090 --> 00:13:03.340
it's a bit of a pain.
00:13:05.750 --> 00:13:07.890
So for a well-behaved function,
the first integral
00:13:07.890 --> 00:13:11.780
exists for all f, the second
exists for all t, and results
00:13:11.780 --> 00:13:13.670
in the original u of t.
00:13:13.670 --> 00:13:17.570
But then more specifically,
if u of t is L1, the first
00:13:17.570 --> 00:13:21.740
integral exists for all f and
it's a continuous function.
00:13:21.740 --> 00:13:24.950
It's also a bounded function, as
we'll see in a little bit.
00:13:24.950 --> 00:13:28.510
If u hat of f is L1, the second
integral exists for all
00:13:28.510 --> 00:13:31.410
t, and u of t is continuous.
00:13:31.410 --> 00:13:35.900
Therefore, if you assume at
the output at the onset of
00:13:35.900 --> 00:13:40.180
things that both this is
L1 and this is L1, then
00:13:40.180 --> 00:13:43.400
everything is also continuous
and you have a very nice
00:13:43.400 --> 00:13:46.840
theory, which doesn't apply to
too many of the things that
00:13:46.840 --> 00:13:48.730
you're interested in.
00:13:48.730 --> 00:13:50.720
And I'll explain why
in a little bit.
00:13:54.230 --> 00:13:58.510
Anyway, for these well-behaved
functions, we have all of
00:13:58.510 --> 00:14:02.640
these relationships that I'm
sure you've learned in
00:14:02.640 --> 00:14:04.820
whatever linear systems
course you've taken.
00:14:07.400 --> 00:14:09.980
Since you should know all of
these things, I just want to
00:14:09.980 --> 00:14:14.640
briefly talk about them I mean
this linearity idea is
00:14:14.640 --> 00:14:16.380
something you would
just use without
00:14:16.380 --> 00:14:19.210
thinking about it, I think.
00:14:19.210 --> 00:14:21.730
In other words, if you'd never
learned this and you were
00:14:21.730 --> 00:14:25.830
trying to work out a problem you
would just use it anyway,
00:14:25.830 --> 00:14:30.380
because anything respectable has
to have -- well, anything
00:14:30.380 --> 00:14:32.820
respectable, again,
means anything
00:14:32.820 --> 00:14:34.070
which has this property.
00:14:36.620 --> 00:14:39.590
This conjugate property you can
derive that easily from
00:14:39.590 --> 00:14:43.050
the Fourier transform
relationships also, if you
00:14:43.050 --> 00:14:45.010
have a well-behaved function.
00:14:45.010 --> 00:14:49.760
This quantity here, this
duality, is particularly
00:14:49.760 --> 00:14:57.890
interesting because it isn't
really duality, it's something
00:14:57.890 --> 00:15:00.820
called hermitian duality.
00:15:00.820 --> 00:15:04.650
You start out with this formula
here to go to there
00:15:04.650 --> 00:15:08.650
and you use almost the same
formula to get back again.
00:15:08.650 --> 00:15:13.710
The only difference is instead
of a minus 2 pi ift, you have
00:15:13.710 --> 00:15:15.960
a plus 2 pi ift.
00:15:15.960 --> 00:15:19.400
In other words, this is the
conjugate of this, which is
00:15:19.400 --> 00:15:22.480
why this is called hermitian
duality.
00:15:22.480 --> 00:15:27.100
But aside from that, everything
you learn about the
00:15:27.100 --> 00:15:31.480
Fourier transform, you also
automatically know about the
00:15:31.480 --> 00:15:35.740
inverse Fourier transform for
these well-behaved functions.
00:15:38.360 --> 00:15:40.980
Otherwise you don't know whether
the other one exists
00:15:40.980 --> 00:15:45.170
or not, and we'll certainly
get into that.
00:15:45.170 --> 00:15:49.290
So, the duality is expressed
this way, the Fourier
00:15:49.290 --> 00:15:54.520
transform of, bleah.
00:15:54.520 --> 00:15:58.900
If you take a function, u of
t, and regard that as a
00:15:58.900 --> 00:16:04.410
Fourier transform, then --
00:16:04.410 --> 00:16:07.800
I always have trouble
saying this.
00:16:07.800 --> 00:16:11.660
If you take a function, u of t,
and then you regard that as
00:16:11.660 --> 00:16:13.480
a function of frequency --
00:16:13.480 --> 00:16:16.840
OK, that's this -- and then you
regard it as a function of
00:16:16.840 --> 00:16:21.170
minus frequency, namely, you
substitute minus f for t in
00:16:21.170 --> 00:16:23.330
whatever time function
you start with.
00:16:23.330 --> 00:16:27.170
You start with a time function u
of t, you substitute minus f
00:16:27.170 --> 00:16:31.300
for t, which gives you a
function of frequency.
00:16:31.300 --> 00:16:35.780
The inverse Fourier transform
of that is what you get by
00:16:35.780 --> 00:16:40.660
taking the Fourier transform
of u of t, and then
00:16:40.660 --> 00:16:44.360
substituting t for f in it.
00:16:44.360 --> 00:16:47.990
It's much harder to
say it than to do.
00:16:47.990 --> 00:16:50.270
This time shift, you've
all seen that.
00:16:50.270 --> 00:16:53.880
If you shift a function in
time, the only thing that
00:16:53.880 --> 00:16:58.150
happens is you get this
rotating term in it.
00:16:58.150 --> 00:17:02.410
Same thing for a frequency
shift.
00:17:02.410 --> 00:17:05.200
You want to have an interesting
exercise, take
00:17:05.200 --> 00:17:09.320
time shift plus duality and
derive the scaling and
00:17:09.320 --> 00:17:13.380
frequency from it -- it has to
work, and of course, it does.
00:17:13.380 --> 00:17:14.640
Scaling --
00:17:14.640 --> 00:17:17.360
there's this relationship
here.
00:17:17.360 --> 00:17:20.440
This one is always
a funny one.
00:17:20.440 --> 00:17:25.480
It's a little strange because
when you scale here, it's not
00:17:25.480 --> 00:17:29.250
too surprising that when you
take a function and you squash
00:17:29.250 --> 00:17:32.120
it down that the Fourier
transform
00:17:32.120 --> 00:17:35.160
gets squashed upwards.
00:17:35.160 --> 00:17:38.840
Because in a sense what you're
doing when you squash a
00:17:38.840 --> 00:17:42.040
function down you're making
everything happen faster than
00:17:42.040 --> 00:17:45.080
it did before, which means that
all the frequencies in it
00:17:45.080 --> 00:17:47.420
are going to get higher
than they are before.
00:17:47.420 --> 00:17:50.780
But also when you squash it
down, the amount of energy in
00:17:50.780 --> 00:17:53.570
the function is going
to go down.
00:17:53.570 --> 00:17:56.600
One of the most important
properties that we're going to
00:17:56.600 --> 00:18:00.470
find and what you ought to know
already is that when you
00:18:00.470 --> 00:18:05.790
take the energy in a function,
you get the same answer as you
00:18:05.790 --> 00:18:09.190
get when you take the energy
in the Fourier transform.
00:18:09.190 --> 00:18:13.940
Namely, you integrate u of f
squared over frequency and you
00:18:13.940 --> 00:18:16.620
get the same as if you
integrate u of t
00:18:16.620 --> 00:18:18.450
squared over time.
00:18:18.450 --> 00:18:20.710
That's an important check
that you use on
00:18:20.710 --> 00:18:22.560
all sorts of things.
00:18:22.560 --> 00:18:25.490
The thing that happens now then
when you're scaling is
00:18:25.490 --> 00:18:28.550
when you scale a function, u
of t, you bring it down and
00:18:28.550 --> 00:18:30.950
you spread it out when
you bring it down.
00:18:30.950 --> 00:18:36.070
The frequency function goes up
so the energy in the time
00:18:36.070 --> 00:18:39.590
function goes down, the energy
in the frequency function goes
00:18:39.590 --> 00:18:44.400
up, and you need something in
order to keep the energy
00:18:44.400 --> 00:18:47.090
relationship working properly.
00:18:47.090 --> 00:18:48.480
This is what you need.
00:18:48.480 --> 00:18:50.260
Actually, if you derive
this, the t
00:18:50.260 --> 00:18:53.360
just falls out naturally.
00:18:53.360 --> 00:19:00.210
So we get the same thing if we
do scaling and frequency.
00:19:00.210 --> 00:19:03.010
I don't think I put that
down but it's the same
00:19:03.010 --> 00:19:04.090
relationship.
00:19:04.090 --> 00:19:06.110
There's differentiation.
00:19:06.110 --> 00:19:10.230
Differentiation we won't talk
about or use it a whole lot.
00:19:10.230 --> 00:19:14.120
All of these things turn out
to be remarkably robust.
00:19:14.120 --> 00:19:18.820
When you're dealing with L1
functions or L2 functions and
00:19:18.820 --> 00:19:21.300
you scale them or you shift
them or do any of those
00:19:21.300 --> 00:19:25.980
things, if they're L1, they're
still L1 after you're through,
00:19:25.980 --> 00:19:28.910
if they're L2, they're L2
after you're through.
00:19:28.910 --> 00:19:31.760
If you differentiate
a function, all
00:19:31.760 --> 00:19:33.000
those properties change.
00:19:33.000 --> 00:19:35.950
You can't be sure of
anything anymore.
00:19:35.950 --> 00:19:39.380
There's convolution, which I'm
sure you've derived many
00:19:39.380 --> 00:19:44.290
times, and which one of the
exercises derives again.
00:19:44.290 --> 00:19:52.370
There's correlation, and what
this sort of relationship says
00:19:52.370 --> 00:19:55.790
is taking products at the
frequency domain is the same
00:19:55.790 --> 00:20:00.220
as going through this
convolution relationship and
00:20:00.220 --> 00:20:01.470
the time domain.
00:20:01.470 --> 00:20:04.930
Of course, there's a dual
relation to that, which you
00:20:04.930 --> 00:20:07.680
don't use very often but
it still exists.
00:20:07.680 --> 00:20:16.280
Correlation is you actually get
correlation by using one
00:20:16.280 --> 00:20:20.420
of the conjugate properties on
the convolution and I'm sure
00:20:20.420 --> 00:20:22.060
you've all seen that.
00:20:25.640 --> 00:20:29.700
That's something you should
all have been using and
00:20:29.700 --> 00:20:31.910
familiar with for a long time.
00:20:31.910 --> 00:20:38.100
Two special cases of the Fourier
transform is that u of
00:20:38.100 --> 00:20:42.990
zero is what happens when you
take the Fourier transform and
00:20:42.990 --> 00:20:45.440
you evaluate it at
t equals zero.
00:20:45.440 --> 00:20:50.570
You get u of zero is just the
integral of u hat of f.
00:20:50.570 --> 00:20:55.090
u hat of zero is just the
integral of u of t.
00:20:55.090 --> 00:20:57.560
What do you use these for?
00:20:57.560 --> 00:21:01.350
Well the thing I use them for
is this half the time when
00:21:01.350 --> 00:21:06.630
you're working out a problem
it's obvious by inspection
00:21:06.630 --> 00:21:10.240
what this integral is or it's
obvious by inspection what
00:21:10.240 --> 00:21:16.210
this integral is, and by doing
that you can check whether you
00:21:16.210 --> 00:21:18.500
have all the constants
right in the
00:21:18.500 --> 00:21:20.880
transform that you've taken.
00:21:20.880 --> 00:21:24.270
I don't know anybody who can
take Fourier transforms
00:21:24.270 --> 00:21:27.480
without getting at least one
constant wrong at least half
00:21:27.480 --> 00:21:29.440
the time they do it.
00:21:29.440 --> 00:21:32.210
I probably get one constant
wrong about three-quarters of
00:21:32.210 --> 00:21:35.500
the time that I do it, and I'm
sure I'll do it here in class
00:21:35.500 --> 00:21:36.330
a number of times.
00:21:36.330 --> 00:21:43.720
I hope you find it, but it's one
of these things we all do.
00:21:43.720 --> 00:21:46.610
This is one of the best ways
of finding out what you've
00:21:46.610 --> 00:21:49.700
done and going back
and checking it.
00:21:49.700 --> 00:21:55.460
Parseval's Theorem, Parseval's
Theorem is really just this
00:21:55.460 --> 00:21:59.730
convolution equation which
we're applying
00:21:59.730 --> 00:22:02.520
at tau equals zero.
00:22:02.520 --> 00:22:05.490
You take the convolution, you
apply it at tau equals zero
00:22:05.490 --> 00:22:06.350
and what do you get?
00:22:06.350 --> 00:22:12.620
You get the integral of u of t
times some other conjugate --
00:22:12.620 --> 00:22:15.610
the conjugate of the other
function is equal to the
00:22:15.610 --> 00:22:21.560
integral of u hat of
f times the complex
00:22:21.560 --> 00:22:24.190
conjugate of v of f.
00:22:26.840 --> 00:22:31.050
Much more important than this
is what happens if v happens
00:22:31.050 --> 00:22:34.980
to be the same as u, and that
gives you the energy equation
00:22:34.980 --> 00:22:36.640
here, which is what I
was talking about.
00:22:36.640 --> 00:22:42.720
It says the integral in a
function you can find it two
00:22:42.720 --> 00:22:45.580
ways, either by looking
at it in time or by
00:22:45.580 --> 00:22:47.810
looking at it in frequency.
00:22:47.810 --> 00:22:50.450
I urge you to always think
about doing this whenever
00:22:50.450 --> 00:22:55.000
you're working problems, because
often the Fourier
00:22:55.000 --> 00:22:59.520
integral it's very easy to find
the integral and the and
00:22:59.520 --> 00:23:02.300
the time function is
very difficult.
00:23:02.300 --> 00:23:05.570
A good example of this
is sinc functions
00:23:05.570 --> 00:23:08.220
and rectangular functions.
00:23:08.220 --> 00:23:12.800
Anybody can take a rectangular
function, square it and
00:23:12.800 --> 00:23:14.470
integrate it.
00:23:14.470 --> 00:23:17.770
It takes a good deal of skill
if you don't use this
00:23:17.770 --> 00:23:21.890
relationship to take a sinc
function, to square it and to
00:23:21.890 --> 00:23:24.760
integrate it.
00:23:24.760 --> 00:23:28.630
You can do it if you're skillful
at integration, you
00:23:28.630 --> 00:23:31.170
might regard it as a challenge,
but after you get
00:23:31.170 --> 00:23:34.550
done you realize that you've
really wasted a lot of time
00:23:34.550 --> 00:23:37.170
because this is the right
way of doing it here.
00:23:43.980 --> 00:23:48.480
Now, as I mentioned before,
it's starting to look like
00:23:48.480 --> 00:23:52.750
Fourier series and Fourier
integrals are much nicer when
00:23:52.750 --> 00:23:58.450
you have L1 functions, and they
are, but L1 functions are
00:23:58.450 --> 00:24:01.880
not terribly useful
as far as most
00:24:01.880 --> 00:24:04.710
communication functions go.
00:24:04.710 --> 00:24:08.690
In other words, not enough
functions are L1 to provide
00:24:08.690 --> 00:24:11.860
suitable models for the
communication systems that we
00:24:11.860 --> 00:24:13.110
want to look at.
00:24:16.710 --> 00:24:19.820
In fact, most of the models that
we're going to look at,
00:24:19.820 --> 00:24:23.810
the functions that we're dealing
with are L2 functions.
00:24:23.810 --> 00:24:28.330
One of the reasons for this is
this sinc function, sine x
00:24:28.330 --> 00:24:31.120
over x function is not L1.
00:24:34.500 --> 00:24:38.660
A sinc function just goes down
as 1 over t, and since it goes
00:24:38.660 --> 00:24:42.350
down as 1 over t, you take the
absolute value of it and you
00:24:42.350 --> 00:24:45.810
integrate 1 over t and what do
you get when you integrate it
00:24:45.810 --> 00:24:48.450
from minus infinity
to infinity.
00:24:48.450 --> 00:24:53.340
A function that's 1 over t,
well, if you really want to go
00:24:53.340 --> 00:24:56.650
through the trouble of
integrating it, you can
00:24:56.650 --> 00:25:00.650
integrate it over limits and you
get limits where you have
00:25:00.650 --> 00:25:04.190
to evaluate the limits on
a logarithmic function.
00:25:04.190 --> 00:25:05.840
When you get all done
with that you
00:25:05.840 --> 00:25:06.950
get an infinite value.
00:25:06.950 --> 00:25:07.830
And you can see this.
00:25:07.830 --> 00:25:11.610
You could take 1 over t and you
just look at it, as you go
00:25:11.610 --> 00:25:14.050
further and further out it just
gets bigger and bigger
00:25:14.050 --> 00:25:15.570
without limit.
00:25:15.570 --> 00:25:18.220
So, sinc t is not L1.
00:25:18.220 --> 00:25:21.230
Sinc function is a function
we'd like to use.
00:25:21.230 --> 00:25:26.000
Any function with a
discontinuity can't be the
00:25:26.000 --> 00:25:30.390
Fourier transform of
any L1 function.
00:25:30.390 --> 00:25:32.620
In other words, we said that
if you take the Fourier
00:25:32.620 --> 00:25:36.920
transform of an L1 function, one
of the nice things about
00:25:36.920 --> 00:25:40.830
it is you get a continuous
function.
00:25:40.830 --> 00:25:43.520
One of the nasty things
about it is you get
00:25:43.520 --> 00:25:45.900
a continuous function.
00:25:45.900 --> 00:25:50.080
Since you get a continuous
function it says that any time
00:25:50.080 --> 00:25:52.970
you want to deal with transforms
which are not
00:25:52.970 --> 00:26:02.050
continuous, you can't be talking
about time functions
00:26:02.050 --> 00:26:03.420
which are L1.
00:26:03.420 --> 00:26:06.440
One of the frequency functions
we want to look at a great
00:26:06.440 --> 00:26:10.030
deal is a frequency function
corresponding to a
00:26:10.030 --> 00:26:12.020
band-limited function.
00:26:12.020 --> 00:26:15.370
When you take a band-limited
function you just chop it off
00:26:15.370 --> 00:26:19.000
at some frequency, and usually
when you chop it off, you chop
00:26:19.000 --> 00:26:22.000
it off and get a
discontinuity.
00:26:22.000 --> 00:26:26.300
When you chop it off and get a
discountinuity, bingo, the
00:26:26.300 --> 00:26:30.090
time function you're dealing
with cannot be L1.
00:26:30.090 --> 00:26:34.340
It has to dribble away
much too slowly as
00:26:34.340 --> 00:26:35.700
time goes to infinity.
00:26:35.700 --> 00:26:39.760
That's an extraordinarily
important thing to remember.
00:26:39.760 --> 00:26:44.210
Any time you get a function
which is discontinuous in the
00:26:44.210 --> 00:26:49.840
frequency domain, the function
cannot go to zero any faster
00:26:49.840 --> 00:26:53.660
in a time domain than 1 over
t and vice versa in the
00:26:53.660 --> 00:26:55.860
frequency domain.
00:26:55.860 --> 00:26:59.460
L1 functions sometimes
have infinite energy.
00:26:59.460 --> 00:27:03.080
In other words, sinc
t is not L1 --
00:27:03.080 --> 00:27:07.120
well, that's not a good example
because that's not L1,
00:27:07.120 --> 00:27:11.370
and it also has infinite energy,
but you can just as
00:27:11.370 --> 00:27:14.720
easily find functions which drop
off a little more slowly
00:27:14.720 --> 00:27:18.170
than sinc t, and which
have infinite
00:27:18.170 --> 00:27:21.300
energy because they--.
00:27:21.300 --> 00:27:22.980
Excuse me.
00:27:22.980 --> 00:27:27.720
Sometimes you have functions
which go off to infinity too
00:27:27.720 --> 00:27:32.760
fast as you approach time equals
zero, things which are
00:27:32.760 --> 00:27:34.960
a little bit like impulses
but not really.
00:27:34.960 --> 00:27:38.740
Impulses are awful and we'll
talk about them in a minute,
00:27:38.740 --> 00:27:42.490
because they don't have finite
energy as we said before.
00:27:42.490 --> 00:27:46.590
We have functions which just
slowly go off to infinity and
00:27:46.590 --> 00:27:50.540
they are L2 but they
aren't L1.
00:27:50.540 --> 00:27:52.800
Why do we care about those
weird functions?
00:27:52.800 --> 00:27:56.020
We care about them, as I said
before, because we would like
00:27:56.020 --> 00:27:59.680
to be able to make statements
which are simple which we can
00:27:59.680 --> 00:28:02.260
believe in.
00:28:02.260 --> 00:28:05.800
In other words, you don't want
to go through a course like
00:28:05.800 --> 00:28:10.550
this with a whole bunch of
things that you have to leave.
00:28:10.550 --> 00:28:12.930
It's very nice to have some
things that you really
00:28:12.930 --> 00:28:17.670
believe, and whether you believe
them or not, it's nice
00:28:17.670 --> 00:28:19.490
to have theorems about them.
00:28:19.490 --> 00:28:22.210
Even if you don't believe the
theorems, at least you have
00:28:22.210 --> 00:28:25.590
theorems so you can fool
other people about
00:28:25.590 --> 00:28:27.920
it, if nothing else.
00:28:27.920 --> 00:28:31.350
Well, it turns out that L2
functions are really the right
00:28:31.350 --> 00:28:33.530
class to look at here.
00:28:45.440 --> 00:28:47.480
Oh, I think I left out
one of the most
00:28:47.480 --> 00:28:48.540
important things here.
00:28:48.540 --> 00:28:51.360
Maybe I put it down here.
00:28:51.360 --> 00:28:53.130
No, probably not.
00:28:53.130 --> 00:28:57.480
One of the reasons we want to
deal with L2 functions is if
00:28:57.480 --> 00:29:04.460
you're dealing with compression,
for example, and
00:29:04.460 --> 00:29:09.220
you take a function, if the
function has infinite energy
00:29:09.220 --> 00:29:15.300
in it, then one of the things
that happens is that any time
00:29:15.300 --> 00:29:19.060
you expand it into any kind of
orthonormal expansion or
00:29:19.060 --> 00:29:22.260
orthongonal expansion, which
we'll talk about later, you
00:29:22.260 --> 00:29:26.000
have coefficients, which
have infinite energy.
00:29:26.000 --> 00:29:30.860
In other words, they have
infinite values, or the sum
00:29:30.860 --> 00:29:34.370
squared of the coefficients
is equal to infinity.
00:29:34.370 --> 00:29:37.850
When we try to compress those
we're going to find that no
00:29:37.850 --> 00:29:42.840
matter how we do it our
mean square error
00:29:42.840 --> 00:29:45.100
is going to be infinite.
00:29:45.100 --> 00:29:50.020
Yes, we will talk about that
later when we get to talking
00:29:50.020 --> 00:29:52.130
about expansions.
00:29:52.130 --> 00:29:56.220
So for all those reasons L1
isn't the right thing, L2 is
00:29:56.220 --> 00:29:58.800
the right thing.
00:29:58.800 --> 00:30:03.510
A function going from the reals
into the complexes, in
00:30:03.510 --> 00:30:07.350
other words, a complex valued
function is L2 if it's
00:30:07.350 --> 00:30:12.530
measurable, and if this
integral is less than
00:30:12.530 --> 00:30:15.950
infinity, in other words, if
that has finite energy.
00:30:15.950 --> 00:30:18.620
Primarily it means it has finite
energy because all the
00:30:18.620 --> 00:30:21.390
functions you can think
of are measurable.
00:30:21.390 --> 00:30:23.890
So it really says you're dealing
with functions that
00:30:23.890 --> 00:30:25.620
have finite energy here.
00:30:31.510 --> 00:30:34.040
So, let's go on to Fourier
transforms then.
00:30:37.670 --> 00:30:39.950
Interesting simple theorem.
00:30:39.950 --> 00:30:42.850
I think I stated this
last time, also.
00:30:42.850 --> 00:30:50.980
If a function is L2 and its time
limited, it's also L1.
00:30:50.980 --> 00:30:55.130
So we've already found that if
functions are L1, they have
00:30:55.130 --> 00:30:58.360
Fourier transforms that exist.
00:30:58.360 --> 00:31:03.070
The reason for this is if you
square u of t, take the
00:31:03.070 --> 00:31:07.320
magnitude squared of u of t for
any given t, it has to be
00:31:07.320 --> 00:31:11.550
less than or equal to the
sum of u of t plus 1.
00:31:11.550 --> 00:31:13.080
In fact, it has to be
less than that.
00:31:13.080 --> 00:31:14.330
Why?
00:31:19.530 --> 00:31:22.520
How would you prove this
if you had to prove it?
00:31:25.980 --> 00:31:29.770
Well, you say gee, this is
two separate terms here.
00:31:29.770 --> 00:31:32.730
Why don't I look at two
separate cases.
00:31:32.730 --> 00:31:38.320
The two separate cases are
first, suppose u of t itself
00:31:38.320 --> 00:31:40.930
as a magnitude less than 1.
00:31:40.930 --> 00:31:45.530
If u of t has and magnitude less
than 1 and you square it,
00:31:45.530 --> 00:31:47.630
you get something
even smaller.
00:31:47.630 --> 00:31:51.770
So, any time u of t has a
magnitude less than 1, u
00:31:51.770 --> 00:31:57.190
squared of t is less than
or equal to u of t.
00:31:57.190 --> 00:31:59.530
Blah blah blah blah blah
blah blah blah.
00:31:59.530 --> 00:32:00.930
If u of t --
00:32:10.020 --> 00:32:11.270
what did I do here?
00:32:19.320 --> 00:32:22.620
No wonder I couldn't explain
this to you.
00:32:27.330 --> 00:32:30.620
Let's try it that way
and see if it works.
00:32:30.620 --> 00:32:33.118
If you can prove something, turn
it around and see if you
00:32:33.118 --> 00:32:36.350
can prove it then.
00:32:36.350 --> 00:32:38.160
Now, two cases.
00:32:38.160 --> 00:32:41.940
First one, let's suppose
that u of t is less
00:32:41.940 --> 00:32:43.440
than or equal to 1.
00:32:43.440 --> 00:32:46.430
Well then, u of t is less
than or equal to 1.
00:32:46.430 --> 00:32:50.160
And this is positive so this is
less than or equal to that.
00:32:50.160 --> 00:32:53.420
Let's look at the other case.
u of t is greater than or
00:32:53.420 --> 00:32:55.560
equal to 1, magnitude.
00:32:55.560 --> 00:32:57.840
Well then, it's less
than or equal to u
00:32:57.840 --> 00:32:59.540
of t magnitude squared.
00:32:59.540 --> 00:33:02.530
So either way this is less
than or equal to that.
00:33:02.530 --> 00:33:07.390
What that says is the integral
over any finite limits of u of
00:33:07.390 --> 00:33:12.700
t dt is less than or equal
to the integral of this.
00:33:12.700 --> 00:33:15.870
Well, the integral of this
splits up into the integral
00:33:15.870 --> 00:33:22.940
magnitude u of t squared dt
plus the integral of 1.
00:33:22.940 --> 00:33:25.900
Now, the integral of 1 over
any finite limits is
00:33:25.900 --> 00:33:28.300
just b minus a.
00:33:28.300 --> 00:33:30.860
That's where the finite
limits come in.
00:33:30.860 --> 00:33:34.120
Finite limits say you don't have
to worry about this term
00:33:34.120 --> 00:33:36.050
because it's finite.
00:33:36.050 --> 00:33:42.650
So that says at any time you
have an L2 function over a
00:33:42.650 --> 00:33:46.710
finite range, that function
is also L1
00:33:46.710 --> 00:33:48.960
over that finite range.
00:33:48.960 --> 00:33:52.830
Which says that any time you
take a Fourier transform of an
00:33:52.830 --> 00:33:57.610
L2 function, which is only
non-zero over a finite range,
00:33:57.610 --> 00:34:00.910
bingo, it's L1 and you get all
these nice properties.
00:34:00.910 --> 00:34:03.910
It has to exist, it has to be
continuous, it has to be
00:34:03.910 --> 00:34:06.120
bounded, and all of
that neat stuff.
00:34:09.040 --> 00:34:14.060
So for any L2 function u of t,
what I'm going to try to do
00:34:14.060 --> 00:34:17.850
now, and I'm just copying what
a guy by the name of
00:34:17.850 --> 00:34:20.810
Plancherel did a
long time ago.
00:34:20.810 --> 00:34:24.530
The thing that Plancherel did
was he said how do I know when
00:34:24.530 --> 00:34:28.260
a Fourier transform
exists or not.
00:34:28.260 --> 00:34:31.040
I would like to make
it exist for L2
00:34:31.040 --> 00:34:33.900
functions, how do I do it?
00:34:33.900 --> 00:34:37.000
Well, he said OK, the thing I'm
going to do is to take the
00:34:37.000 --> 00:34:42.290
function u of t and I'm first
going to truncate it.
00:34:42.290 --> 00:34:45.290
In fact, if you think in terms
of Reimann integration and
00:34:45.290 --> 00:34:48.940
things like that, any time you
take an integral from minus
00:34:48.940 --> 00:34:53.140
infinity to plus infinity,
what do you mean by it?
00:34:53.140 --> 00:34:56.250
You mean the limit as you
integrate the function over
00:34:56.250 --> 00:35:00.510
finite limits and then you let
the limits go to infinity.
00:35:00.510 --> 00:35:04.230
So all we're doing is
the same trick here.
00:35:04.230 --> 00:35:09.180
So we're going to take u of t,
we're going to truncate it to
00:35:09.180 --> 00:35:13.900
some minus a to plus a over some
finite range, no matter
00:35:13.900 --> 00:35:15.720
how big a happens to be.
00:35:15.720 --> 00:35:20.570
We're going to call the function
b sub a of t, u of t
00:35:20.570 --> 00:35:22.580
truncated to these limits.
00:35:22.580 --> 00:35:27.690
In other words, u of t times a
rectangular function evaluated
00:35:27.690 --> 00:35:30.400
at t over 2a.
00:35:30.400 --> 00:35:34.650
Now, can all of you look at this
function and see that it
00:35:34.650 --> 00:35:38.560
just means truncate from
minus a to plus a?
00:35:38.560 --> 00:35:39.430
No.
00:35:39.430 --> 00:35:44.260
Well, you should learn
to do this.
00:35:44.260 --> 00:35:47.170
One of the ways to do it is
to say OK, the rectangular
00:35:47.170 --> 00:35:52.330
function is defined as having
the value 1 between minus 1/2
00:35:52.330 --> 00:35:55.330
and plus 1/2 and it's zero
everywhere else.
00:35:57.860 --> 00:36:00.570
I think I said that before
in class, didn't I?
00:36:00.570 --> 00:36:03.650
Certainly it's in the notes.
00:36:03.650 --> 00:36:16.490
Rectangle ft equals 1 for
t less than or equal
00:36:16.490 --> 00:36:19.960
to 1/2, zero else.
00:36:23.980 --> 00:36:27.360
So with this definition you
just evaluate what happens
00:36:27.360 --> 00:36:32.870
when t is equal to minus a, you
get rectangle of minus a
00:36:32.870 --> 00:36:36.150
over 2a, which is minus 1/2.
00:36:36.150 --> 00:36:39.910
When t is equal to a, you're up
to the other limit, so this
00:36:39.910 --> 00:36:44.650
function is 1 for t between
minus a and plus a and zero
00:36:44.650 --> 00:36:46.230
everywhere else.
00:36:46.230 --> 00:36:50.940
Please get used to using this
and become a little facile at
00:36:50.940 --> 00:36:55.800
sorting out what it means
because it's a very handy way
00:36:55.800 --> 00:36:59.300
to avoid writing awkward
things like this.
00:36:59.300 --> 00:37:05.870
So va of t by what we've said
is both L2 and its L1.
00:37:05.870 --> 00:37:10.770
We started out with a function
which is L2 and we truncated.
00:37:10.770 --> 00:37:16.360
Then according to this theorem
here, this function va of t is
00:37:16.360 --> 00:37:18.670
now time limited.
00:37:18.670 --> 00:37:24.300
It's also L2, and therefore, by
the theorem it's also L1.
00:37:24.300 --> 00:37:29.660
Therefore, it's continue and
you can take the Fourier
00:37:29.660 --> 00:37:35.700
transform of it -- v hat a of
f exists for all f and it's
00:37:35.700 --> 00:37:36.590
continuous.
00:37:36.590 --> 00:37:42.990
So this function is just the
Fourier transform that you get
00:37:42.990 --> 00:37:46.400
when you truncate the function,
which is what you
00:37:46.400 --> 00:37:48.780
would think of as a
way to find the
00:37:48.780 --> 00:37:51.840
Fourier transform anyway.
00:37:51.840 --> 00:37:54.810
I mean if this is not a
reasonable approximation to
00:37:54.810 --> 00:37:58.650
the Fourier transform a
function, you haven't modeled
00:37:58.650 --> 00:38:01.530
the function very well.
00:38:01.530 --> 00:38:05.470
Because when a gets
extraordinarily large, if
00:38:05.470 --> 00:38:08.320
there's anything of significance
that happens
00:38:08.320 --> 00:38:13.300
before year ten to the minus 6
or which happens after year
00:38:13.300 --> 00:38:17.120
ten to the plus 6, and you're
dealing with electronic
00:38:17.120 --> 00:38:21.530
speeds, your models don't
make any sense.
00:38:21.530 --> 00:38:26.050
So for anything of any interest,
these functions here
00:38:26.050 --> 00:38:29.550
are going to start approximating
u of t, and
00:38:29.550 --> 00:38:32.590
therefore we hope this will
start approximating the
00:38:32.590 --> 00:38:35.520
Fourier transform of u of t.
00:38:35.520 --> 00:38:37.500
Who can make that more
precise for me?
00:38:40.440 --> 00:38:44.500
What happens when u of
t has finite energy?
00:38:47.210 --> 00:38:51.900
If it has finite energy it means
that the integral of u
00:38:51.900 --> 00:38:54.410
of t magnitude squared over the
00:38:54.410 --> 00:38:58.460
infinite interval is finite.
00:38:58.460 --> 00:39:01.650
So you start integrating it
over bigger and bigger
00:39:01.650 --> 00:39:04.580
minus a to plus a.
00:39:04.580 --> 00:39:08.630
What happens is as the minus
a to plus a gets bigger and
00:39:08.630 --> 00:39:11.180
bigger and bigger, you're
including more and more of the
00:39:11.180 --> 00:39:16.760
function, so that the integral
of u of t squared over that
00:39:16.760 --> 00:39:19.460
bigger and bigger interval has
to be getting closer and
00:39:19.460 --> 00:39:23.780
closer to the overall
integral of u of t.
00:39:23.780 --> 00:39:29.800
Which says that the energy in u
of t minus the a of t has to
00:39:29.800 --> 00:39:33.760
get very, very small
as a gets large.
00:39:33.760 --> 00:39:36.560
That's one of the reasons why
we like to deal with finite
00:39:36.560 --> 00:39:38.200
energy functions.
00:39:38.200 --> 00:39:42.650
By definition they cannot have
a appreciable energy outside
00:39:42.650 --> 00:39:43.990
of very large limits.
00:39:43.990 --> 00:39:46.950
How large those large limits
have to be depends on the
00:39:46.950 --> 00:39:50.600
function, but if you make them
large enough you will always
00:39:50.600 --> 00:39:55.550
get negligible energy outside
of those limits.
00:39:55.550 --> 00:39:59.580
So then we can take the Fourier
transform of this
00:39:59.580 --> 00:40:03.750
function within those limits and
we get something which we
00:40:03.750 --> 00:40:07.500
hope is going to be a reasonable
approximation of
00:40:07.500 --> 00:40:10.300
the Fourier transform
of u of t.
00:40:16.400 --> 00:40:19.820
That's what Plancherel said.
00:40:19.820 --> 00:40:26.610
Plancherel said if we have an
L2 function, u of t, then
00:40:26.610 --> 00:40:30.900
there is an L2 function u hat
of f, which is really the
00:40:30.900 --> 00:40:33.270
Fourier transform of u of t.
00:40:33.270 --> 00:40:39.060
Some people call this the
Plancherel transform of u of t
00:40:39.060 --> 00:40:42.990
and say that indeed Plancherel
was the one that invented
00:40:42.990 --> 00:40:44.790
Fourier transforms
or Plancherel
00:40:44.790 --> 00:40:47.110
transforms for L2 functions.
00:40:47.110 --> 00:40:50.190
That's probably giving him a
little too much credit, and
00:40:50.190 --> 00:40:54.360
Fourier somewhat less
than due credit.
00:40:54.360 --> 00:40:57.590
But it was a neat theorem.
00:40:57.590 --> 00:41:03.140
What he said is that there is
a function u hat of f, which
00:41:03.140 --> 00:41:06.240
we'll call the Fourier
transform, which has the
00:41:06.240 --> 00:41:13.660
property that when you take the
integral of the difference
00:41:13.660 --> 00:41:20.830
between u hat of f and the
transform of b sub a of t,
00:41:20.830 --> 00:41:22.800
when you take the integral
of this dt --
00:41:22.800 --> 00:41:26.080
in other words, when you
evaluate the energy in the
00:41:26.080 --> 00:41:34.880
difference between u hat
of f and v sub a of f,
00:41:34.880 --> 00:41:37.570
that goes to zero.
00:41:37.570 --> 00:41:39.960
Well this isn't a big deal.
00:41:39.960 --> 00:41:43.090
In other words, this is
plausible, since this integral
00:41:43.090 --> 00:41:46.610
has to go to zero for an L2
function, that's what we just
00:41:46.610 --> 00:41:52.760
said, then therefore, using the
energy relation, this also
00:41:52.760 --> 00:41:55.570
has to go to zero.
00:41:55.570 --> 00:41:58.830
So is this another example where
Plancherel just came
00:41:58.830 --> 00:42:03.510
along at the right time and he
said something totally trivial
00:42:03.510 --> 00:42:05.960
and became famous
because of it?
00:42:05.960 --> 00:42:09.320
I mean as I've urged all of
you, work on problems that
00:42:09.320 --> 00:42:10.680
other people haven't
worked on.
00:42:10.680 --> 00:42:12.940
If you're lucky, you
will do something
00:42:12.940 --> 00:42:15.700
trivial and become famous.
00:42:15.700 --> 00:42:18.130
As another piece of philosophy,
you become far
00:42:18.130 --> 00:42:20.670
more famous for doing something
trivial than for
00:42:20.670 --> 00:42:23.990
doing something difficult,
because everybody remembers
00:42:23.990 --> 00:42:25.870
something trivial.
00:42:25.870 --> 00:42:28.390
And if you do something
difficult nobody even
00:42:28.390 --> 00:42:32.030
understands it.
00:42:32.030 --> 00:42:38.150
But no, it wasn't that because
there's something hidden here.
00:42:38.150 --> 00:42:42.500
He says a function
like this exists.
00:42:42.500 --> 00:42:47.740
In other words, the problem is
these functions get closer and
00:42:47.740 --> 00:42:50.690
closer to something.
00:42:50.690 --> 00:42:53.440
They get closer and closer to
each other as a gets bigger
00:42:53.440 --> 00:42:55.490
and bigger.
00:42:55.490 --> 00:42:57.030
You can show that because
you have a
00:42:57.030 --> 00:42:59.920
handle on these functions.
00:42:59.920 --> 00:43:03.000
Whether they get closer and
closer to a real bonafide
00:43:03.000 --> 00:43:05.900
function or not is
another question.
00:43:09.950 --> 00:43:15.320
Back when you studied
arithmetic, if you were in any
00:43:15.320 --> 00:43:18.730
kind of advanced class studying
arithmetic, you
00:43:18.730 --> 00:43:20.980
studied the rational
numbers and the
00:43:20.980 --> 00:43:23.440
real numbers you remember.
00:43:23.440 --> 00:43:28.500
And you remember the problem of
what happens if you take a
00:43:28.500 --> 00:43:34.760
sequence of rational numbers
which is approaching a limit.
00:43:34.760 --> 00:43:37.700
There's a big problem there
because when you take a
00:43:37.700 --> 00:43:40.900
sequence of rational numbers
that approaches a limit, the
00:43:40.900 --> 00:43:43.680
limit might not be rational.
00:43:43.680 --> 00:43:47.250
In other words, when you take
sequences of things you can
00:43:47.250 --> 00:43:50.790
get out of the domain of the
things you're working with.
00:43:50.790 --> 00:43:54.030
Now, we can't get out of the
domain of being L2, but we
00:43:54.030 --> 00:43:56.770
might get out of domain of
measurable functions, we might
00:43:56.770 --> 00:43:59.820
get out of the domain
of functions at all.
00:43:59.820 --> 00:44:03.480
We can have all sorts of
strange things happen.
00:44:03.480 --> 00:44:07.640
The nice thing here, which
was really a theorem by
00:44:07.640 --> 00:44:08.490
[? Resenage ?]
00:44:08.490 --> 00:44:10.470
a long time ago.
00:44:10.470 --> 00:44:13.740
It says that when you take
cosine sequences of L2
00:44:13.740 --> 00:44:17.950
functions, they converge
to an L2 function.
00:44:17.950 --> 00:44:21.200
So that's really what's
involved in here.
00:44:21.200 --> 00:44:24.420
So maybe this should be called
the [? Resenage ?]
00:44:24.420 --> 00:44:26.880
transform, I don't know.
00:44:26.880 --> 00:44:31.220
But anyway, whatever this says,
the theorem says, the
00:44:31.220 --> 00:44:34.080
first part of Plancherel's
theorem says that this
00:44:34.080 --> 00:44:39.050
function exists and you get a
handle on it by taking this
00:44:39.050 --> 00:44:43.260
transform, making a bigger and
bigger, and it says it will
00:44:43.260 --> 00:44:47.010
converge to something in
this energy sense.
00:44:47.010 --> 00:44:50.480
Bingo, when you're all done
this goes to zero.
00:44:50.480 --> 00:44:54.750
We're going to denote this
function as a limit and a mean
00:44:54.750 --> 00:44:57.230
of the Fourier transform.
00:44:57.230 --> 00:45:00.970
In other words, we do have a
Fourier transform in the same
00:45:00.970 --> 00:45:03.630
sense that we had a Fourier
series before.
00:45:03.630 --> 00:45:06.275
We didn't know weather the
Fourier series would converge
00:45:06.275 --> 00:45:09.600
at every point, but we knew that
it converged at enough
00:45:09.600 --> 00:45:13.190
points, namely, almost
everywhere, everywhere but on
00:45:13.190 --> 00:45:14.710
a set of measure zero.
00:45:14.710 --> 00:45:18.910
It converges so that, in fact,
you get this kind of
00:45:18.910 --> 00:45:21.400
relationship.
00:45:21.400 --> 00:45:23.890
Now, do you have to
worry about that?
00:45:23.890 --> 00:45:24.270
No.
00:45:24.270 --> 00:45:28.340
Again, this is one of these
very nice things that says
00:45:28.340 --> 00:45:32.700
there is a Fourier transform,
you don't have to worry about
00:45:32.700 --> 00:45:36.200
what goes on at these oddball
sets where the function has
00:45:36.200 --> 00:45:38.400
discountinuities and
things like that.
00:45:38.400 --> 00:45:40.010
You can forget all of that.
00:45:40.010 --> 00:45:43.560
You can be as careless as you've
ever been, and now you
00:45:43.560 --> 00:45:46.640
know that it all works out
mathematically, so long as you
00:45:46.640 --> 00:45:48.600
stick to L2 functions.
00:45:48.600 --> 00:45:53.010
So, sticking to L2 functions
says you can be a careless
00:45:53.010 --> 00:45:56.850
engineer, you can use lousy
mathematics and you'll always
00:45:56.850 --> 00:45:58.510
get the right answer.
00:45:58.510 --> 00:46:02.600
So, it's nice for engineers,
it's nice for me.
00:46:02.600 --> 00:46:05.730
I don't like to be careful
all the time.
00:46:05.730 --> 00:46:11.130
I like to be careful once and
then solve that and go on.
00:46:11.130 --> 00:46:13.850
Well, because of time frequency
duality, you can do
00:46:13.850 --> 00:46:17.860
exactly the same thing in
the frequency domain.
00:46:17.860 --> 00:46:22.170
So, you start out defining some
b, which is bigger than
00:46:22.170 --> 00:46:26.470
zero which is arbitrarily large,
you define a finite
00:46:26.470 --> 00:46:34.850
bandwidth approximation as w hat
sub b of f is u hat of f.
00:46:34.850 --> 00:46:36.820
We now know that u
hat of f exists
00:46:36.820 --> 00:46:38.900
and it's an L2 function.
00:46:38.900 --> 00:46:42.220
u hat of f times this
rectangular function,
00:46:42.220 --> 00:46:43.240
that's f over 2b.
00:46:43.240 --> 00:46:48.510
In other words, it's u hat of f
truncated to a big bandwidth
00:46:48.510 --> 00:46:49.810
minus b to plus b.
00:46:52.860 --> 00:46:56.510
Since w sub b of f
is L1, as well as
00:46:56.510 --> 00:47:00.140
L2, this always exists.
00:47:00.140 --> 00:47:02.800
So long as you deal with a
finite bandwidth, this
00:47:02.800 --> 00:47:04.520
quantity exists.
00:47:04.520 --> 00:47:06.950
It exists for all t and r.
00:47:06.950 --> 00:47:08.980
It's continuous.
00:47:08.980 --> 00:47:11.980
The second part of Plancherel's
theorem then says
00:47:11.980 --> 00:47:16.550
that the limit as b goes to
infinity of the integral of u
00:47:16.550 --> 00:47:21.130
of t minus this truncated
function, magnitude squared
00:47:21.130 --> 00:47:25.420
the energy in that,
goes to zero.
00:47:25.420 --> 00:47:29.060
This now is a little different
than what we did before.
00:47:29.060 --> 00:47:31.720
It's easier in the sense that
we don't have to worry about
00:47:31.720 --> 00:47:35.120
the existence of this function
because we started out with
00:47:35.120 --> 00:47:37.530
this to start with.
00:47:37.530 --> 00:47:40.710
It's a little harder because
we know that a function
00:47:40.710 --> 00:47:44.020
exists, but we don't know
that it's u of t.
00:47:44.020 --> 00:47:47.150
So, in fact, poor old Plancherel
had to do something
00:47:47.150 --> 00:47:52.040
other than just this very simple
argument that says all
00:47:52.040 --> 00:47:53.430
the energy works out right.
00:47:53.430 --> 00:47:56.720
He had to also show that you
really wind up with the right
00:47:56.720 --> 00:47:58.660
function when you get
all through with it.
00:47:58.660 --> 00:48:01.650
But again, this is the same
kind of energy convergence
00:48:01.650 --> 00:48:02.470
that we had before.
00:48:02.470 --> 00:48:02.840
Yeah?
00:48:02.840 --> 00:48:05.900
AUDIENCE: Could you discuss
non-uniqueness?
00:48:05.900 --> 00:48:07.940
Clearly, [INAUDIBLE]
00:48:07.940 --> 00:48:12.260
u hat f to satisfy Plancherel
1 and Plancherel 2.
00:48:12.260 --> 00:48:16.710
PROFESSOR: Yeah, in fact, any
two functions which are L2
00:48:16.710 --> 00:48:18.980
equivalent.
00:48:18.980 --> 00:48:23.270
But you see the nice thing is
when you take this finite
00:48:23.270 --> 00:48:26.370
bandwidth approximation
there's only one.
00:48:26.370 --> 00:48:28.370
It's only when you get
to the limit that all
00:48:28.370 --> 00:48:31.720
of this mess occurs.
00:48:31.720 --> 00:48:34.390
If you take these different
possible functions, u hat of
00:48:34.390 --> 00:48:40.880
f, which just differ in these
negligible sets of measure
00:48:40.880 --> 00:48:44.720
zero, those don't affect
this integral.
00:48:44.720 --> 00:48:48.190
Sets of measure zero don't
affect integrals at all.
00:48:48.190 --> 00:48:51.640
So the mathematicians deal
with L2 theory by talking
00:48:51.640 --> 00:48:55.280
about equivalence classes
of functions.
00:48:55.280 --> 00:48:57.920
I find it hard to think about
equivalence classes of
00:48:57.920 --> 00:49:01.560
functions and partitioning the
set of all functions into a
00:49:01.560 --> 00:49:04.120
bunch of equivalence classes.
00:49:04.120 --> 00:49:07.375
So I just sort of remember in
the back of my mind that there
00:49:07.375 --> 00:49:11.630
are all these functions which
differ in a strange way.
00:49:11.630 --> 00:49:13.990
We'll talk about that more when
we get to the sampling
00:49:13.990 --> 00:49:16.580
theorem later today,
because there it
00:49:16.580 --> 00:49:18.300
happens to be important.
00:49:18.300 --> 00:49:20.470
Here it's not really important,
here we don't have
00:49:20.470 --> 00:49:23.410
to worry about it.
00:49:23.410 --> 00:49:26.570
Anyway, we can always get back
to the u of t that we started
00:49:26.570 --> 00:49:29.670
with in this way.
00:49:29.670 --> 00:49:33.920
Now, this says that all L2
functions have Fourier
00:49:33.920 --> 00:49:37.470
transforms in this
very nice sense.
00:49:37.470 --> 00:49:40.720
In other words, at this point
you don't have to worry about
00:49:40.720 --> 00:49:43.750
continuity, you don't have to
worry about how fast things
00:49:43.750 --> 00:49:47.070
drop off, you don't have to
worry about anything.
00:49:47.070 --> 00:49:50.660
So long as you have finite
energy functions, this
00:49:50.660 --> 00:49:53.840
beautiful result always
holds true.
00:49:53.840 --> 00:49:58.390
There always is a Fourier
transform, it always has this
00:49:58.390 --> 00:50:02.350
nice property that it has the
same energy as the function
00:50:02.350 --> 00:50:04.240
you started with.
00:50:04.240 --> 00:50:09.820
The only nasty thing, as Dave
pointed out, is that, in fact,
00:50:09.820 --> 00:50:12.270
it might not be a unique
function, but it's close
00:50:12.270 --> 00:50:13.490
enough to unique.
00:50:13.490 --> 00:50:15.790
It's unique in an engineering
sense.
00:50:25.020 --> 00:50:29.570
The other thing is that L2 wave
forms don't include some
00:50:29.570 --> 00:50:32.880
of your favorite wave forms.
00:50:32.880 --> 00:50:37.510
They don't include constants,
they don't include sine waves,
00:50:37.510 --> 00:50:40.610
and they don't include Dirac
impulse functions.
00:50:40.610 --> 00:50:42.640
All of them have infinite
energy.
00:50:42.640 --> 00:50:45.790
I pointed out in class, spent
quite a bit of time explaining
00:50:45.790 --> 00:50:49.580
why an impulse function had
infinite energy by looking at
00:50:49.580 --> 00:50:54.040
it as a very narrow pulse of
width epsilon and a height 1
00:50:54.040 --> 00:50:59.330
over epsilon, and showing that
the energy in that is 1 over
00:50:59.330 --> 00:51:02.710
epsilon, and as epsilon goes
to zero and the pulse gets
00:51:02.710 --> 00:51:07.270
narrower and narrower, bingo,
the energy goes to infinity.
00:51:07.270 --> 00:51:11.060
Constants are the same way, they
extend on and on forever.
00:51:11.060 --> 00:51:14.870
Therefore, they have infinite
energy, except if the constant
00:51:14.870 --> 00:51:17.220
happens to be zero.
00:51:17.220 --> 00:51:20.780
Sine waves are the same way,
they dribble on forever.
00:51:23.400 --> 00:51:28.560
So the question is are these
good models of reality?
00:51:28.560 --> 00:51:31.450
The answer is they're good for
some things and they're very
00:51:31.450 --> 00:51:33.810
bad for other things.
00:51:33.810 --> 00:51:37.420
The point in this course is
that if you're looking for
00:51:37.420 --> 00:51:43.140
wave forms that are good models
for either the kinds of
00:51:43.140 --> 00:51:46.930
functions that we're going to
quantize, namely, source wave
00:51:46.930 --> 00:51:50.320
forms, or if you're looking for
the kinds of things that
00:51:50.320 --> 00:51:54.170
we're going to transmit
on channels, these
00:51:54.170 --> 00:51:56.720
are very lousy functions.
00:51:56.720 --> 00:51:59.860
They don't make any sense in
a communication context.
00:52:02.660 --> 00:52:06.370
But anyway, where did these
things come from?
00:52:06.370 --> 00:52:09.660
Constants and sine waves
result from refusing to
00:52:09.660 --> 00:52:11.720
explicitly model when very
00:52:11.720 --> 00:52:14.920
long-lasting functions terminate.
00:52:14.920 --> 00:52:17.260
In other words, if you're
looking at a carrier function
00:52:17.260 --> 00:52:26.640
in a communication's system,
sine of 2 pi, f carrier times
00:52:26.640 --> 00:52:31.620
t, it just keeps on wiggling
around forever.
00:52:31.620 --> 00:52:35.030
Since you want to talk about
that over the complete time of
00:52:35.030 --> 00:52:39.680
interest, you don't want to say
what the time of interest
00:52:39.680 --> 00:52:44.190
is, you don't want to admit to
your employer that this thing
00:52:44.190 --> 00:52:47.560
is going to stop working after
one month because you want to
00:52:47.560 --> 00:52:50.320
let him think that he's going
to make a profit off of this
00:52:50.320 --> 00:52:55.300
forever, and you don't want to
commit to putting it into use
00:52:55.300 --> 00:52:57.740
in one month when you know it's
going to get delayed for
00:52:57.740 --> 00:52:58.670
a whole year.
00:52:58.670 --> 00:52:59.940
So you want to think of this as
00:52:59.940 --> 00:53:01.400
something which is permanent.
00:53:01.400 --> 00:53:04.770
You don't want to answer the
question at what time does it
00:53:04.770 --> 00:53:08.450
start and at what time does it
end, because for many of the
00:53:08.450 --> 00:53:11.120
questions you ask, you
can just regard
00:53:11.120 --> 00:53:15.110
it as going on forever.
00:53:15.110 --> 00:53:17.440
You have the same thing
with impulses.
00:53:17.440 --> 00:53:25.420
Impulses are always models
of short pulses.
00:53:25.420 --> 00:53:28.300
If you put these short pulses
through a filter, the only
00:53:28.300 --> 00:53:30.680
thing which is of interest
in them is what
00:53:30.680 --> 00:53:32.670
their integral is.
00:53:32.670 --> 00:53:35.040
And since the only thing of
interest is their integral,
00:53:35.040 --> 00:53:39.120
you call it an impulse and you
don't worry about just how
00:53:39.120 --> 00:53:43.800
narrow it is, except that it
has infinite energy and,
00:53:43.800 --> 00:53:46.980
therefore, whenever you start
to deal with a situation in
00:53:46.980 --> 00:53:51.870
which energy is important, these
becomes lousy models.
00:53:51.870 --> 00:53:54.320
So we can't use these
when we're
00:53:54.320 --> 00:53:56.070
talking about L2 functions.
00:53:56.070 --> 00:53:59.540
That's the price we pay for
dealing with L2 functions.
00:53:59.540 --> 00:54:03.320
But it's a small price because
for almost all the things
00:54:03.320 --> 00:54:06.570
we'll be dealing with, it's the
energy of the functions
00:54:06.570 --> 00:54:09.810
that are really important.
00:54:09.810 --> 00:54:12.790
So as communication wave forms,
infinite energy wave
00:54:12.790 --> 00:54:16.090
forms make mean square error
quantization results
00:54:16.090 --> 00:54:17.860
meaningless.
00:54:17.860 --> 00:54:20.980
In other words, when you sample
these infinite energy
00:54:20.980 --> 00:54:24.985
wave forms you get results that
don't make any sense, and
00:54:24.985 --> 00:54:28.250
they make those channel
results meaningless.
00:54:28.250 --> 00:54:30.940
Therefore, from now on, whether
I remember to say it
00:54:30.940 --> 00:54:34.620
or not, everything we deal with,
unless we're looking at
00:54:34.620 --> 00:54:38.740
counter examples to something,
is going to be an L2 function.
00:54:45.630 --> 00:54:48.310
Let's go on.
00:54:48.310 --> 00:54:52.430
I'm starting to feel like I'm
back in our signals and
00:54:52.430 --> 00:54:56.570
systems course, because at this
point I'm defining my
00:54:56.570 --> 00:55:00.100
third different kind
of transform.
00:55:00.100 --> 00:55:02.990
Fortunately, this is the last
transform we will have to talk
00:55:02.990 --> 00:55:06.050
about, so we're all
done with this.
00:55:06.050 --> 00:55:09.860
The other nice thing is that
the discrete time Fourier
00:55:09.860 --> 00:55:15.020
transform happens to be just the
time frequency dual of the
00:55:15.020 --> 00:55:17.110
Fourier series.
00:55:17.110 --> 00:55:24.480
So that whether you've ever
studied the dtft or not, you
00:55:24.480 --> 00:55:27.460
already know everything there
is to know about it, because
00:55:27.460 --> 00:55:30.550
the only things there are to
know about it are the results
00:55:30.550 --> 00:55:33.050
about Fourier series.
00:55:33.050 --> 00:55:36.280
So the theorem is really the
same theorem that we had for
00:55:36.280 --> 00:55:38.570
Fourier series.
00:55:38.570 --> 00:55:43.360
Assume that you have a function
of frequency, u hat
00:55:43.360 --> 00:55:47.020
of f -- before we had a function
of time, now we have
00:55:47.020 --> 00:55:49.590
a function of frequency.
00:55:49.590 --> 00:55:52.680
Suppose it's defined over
the interval minus w
00:55:52.680 --> 00:55:56.260
to plus w into c.
00:55:56.260 --> 00:55:59.260
In other words, a way we often
say that a function is
00:55:59.260 --> 00:56:03.410
truncated is to say it goes
from some interval into c.
00:56:03.410 --> 00:56:08.510
This is a complex function which
is non-zero only for f
00:56:08.510 --> 00:56:12.290
in this finite bandwidth
range.
00:56:12.290 --> 00:56:16.240
We want to assume that this
is L2 and thus, we
00:56:16.240 --> 00:56:19.510
know it's also L1.
00:56:19.510 --> 00:56:23.380
Then we're going to take the
Fourier coefficients.
00:56:23.380 --> 00:56:26.710
Before we thought of the
Fourier coefficients as
00:56:26.710 --> 00:56:30.020
corresponding to what goes on at
different frequencies, now
00:56:30.020 --> 00:56:33.180
we're going to regard them as
time quantities, and we'll see
00:56:33.180 --> 00:56:35.360
exactly why later.
00:56:35.360 --> 00:56:37.240
So we'll define these
as the Fourier
00:56:37.240 --> 00:56:39.690
coefficients of this function.
00:56:39.690 --> 00:56:43.280
So they're 1 over 2w times
this integral here.
00:56:43.280 --> 00:56:46.650
You remember before when we
dealt with the Fourier series,
00:56:46.650 --> 00:56:50.220
we went from minus t over
2 to plus t over 2.
00:56:50.220 --> 00:56:53.250
Now we're going from
minus w to plus w.
00:56:53.250 --> 00:56:53.850
Why?
00:56:53.850 --> 00:56:57.150
It's just convention, there's
no real reason.
00:56:57.150 --> 00:57:02.790
So that what's happening here
is that this is a Fourier
00:57:02.790 --> 00:57:06.310
series formula for a coefficient
where we're
00:57:06.310 --> 00:57:11.710
substituting w for t over 2.
00:57:11.710 --> 00:57:15.130
We're putting a plus sign in
the exponent instead of a
00:57:15.130 --> 00:57:15.990
minus sign.
00:57:15.990 --> 00:57:21.500
In other words, we're doing this
hermitian duality bit.
00:57:21.500 --> 00:57:23.600
What's the other difference?
00:57:23.600 --> 00:57:26.090
We're interchanging time
and frequency.
00:57:26.090 --> 00:57:29.470
Aside from that it's exactly
the same theorem that we
00:57:29.470 --> 00:57:32.180
established -- well, that
we stated before.
00:57:32.180 --> 00:57:37.880
So we know that this quantity
here, since u hat of f is L1,
00:57:37.880 --> 00:57:41.690
this is finite and it exists
-- that's just a finite
00:57:41.690 --> 00:57:48.320
complex number and nothing more
-- for all integer k.
00:57:48.320 --> 00:57:54.580
Also, the convergence result
when we go back, this is the
00:57:54.580 --> 00:57:57.790
formula we try to use to go
back to the function we
00:57:57.790 --> 00:57:59.670
started with.
00:57:59.670 --> 00:58:02.020
It's just a finite
approximation
00:58:02.020 --> 00:58:04.300
to the Fourier series.
00:58:04.300 --> 00:58:08.350
We're saying that as k zero gets
larger and larger, this
00:58:08.350 --> 00:58:12.190
finite approximation approaches
the function in
00:58:12.190 --> 00:58:14.450
energy sense.
00:58:14.450 --> 00:58:19.420
So this is exactly what you
should mean by a Fourier
00:58:19.420 --> 00:58:20.430
series anyway.
00:58:20.430 --> 00:58:22.175
It's exactly what you
should mean by a
00:58:22.175 --> 00:58:24.110
Fourier transform anyway.
00:58:24.110 --> 00:58:28.610
As you go to the limit with more
and more terms you get
00:58:28.610 --> 00:58:34.010
something which is equal to what
you started with, except
00:58:34.010 --> 00:58:37.070
on the set of measure zero.
00:58:37.070 --> 00:58:40.240
In other words, it converges
everywhere where it matters.
00:58:40.240 --> 00:58:43.270
It converges to something
in the sense that the
00:58:43.270 --> 00:58:47.160
energy is the same.
00:58:47.160 --> 00:58:50.860
I said that in such a way that
makes it a little simpler than
00:58:50.860 --> 00:58:53.660
it really is.
00:58:53.660 --> 00:59:01.940
You can't always say that this
converges in any nice way.
00:59:01.940 --> 00:59:06.280
Next time I'm going to show
you a truly awful function
00:59:06.280 --> 00:59:10.910
which we'll use in the Fourier
series instead of dtft, which
00:59:10.910 --> 00:59:16.360
is time limited and which is
just incredibly messy and
00:59:16.360 --> 00:59:19.460
it'll show you why you have to
be a little bit careful about
00:59:19.460 --> 00:59:22.250
stating these results.
00:59:22.250 --> 00:59:26.250
But you don't have to worry
about it most of the time,
00:59:26.250 --> 00:59:29.560
because this theorem
is still true.
00:59:29.560 --> 00:59:31.960
It's just that you have to be a
little careful about how to
00:59:31.960 --> 00:59:35.230
interpret it because you don't
know whether this is going to
00:59:35.230 --> 00:59:37.340
reach a limit or not.
00:59:37.340 --> 00:59:39.720
All you know is that this will
be true, this energy
00:59:39.720 --> 00:59:42.960
difference goes to zero.
00:59:42.960 --> 00:59:48.060
Also, the energy in the function
is equal to the
00:59:48.060 --> 00:59:49.800
energy in the coefficients.
00:59:49.800 --> 00:59:52.610
This was the thing that we
found so useful with the
00:59:52.610 --> 00:59:54.810
Fourier series.
00:59:54.810 --> 01:00:00.420
It's why we can play this game
that we play with mean square
01:00:00.420 --> 01:00:06.550
quantization error of taking a
function and then turning it
01:00:06.550 --> 01:00:11.690
into a sequence of samples,
trying to quantize the samples
01:00:11.690 --> 01:00:16.050
for minimum mean square error
and associate the mean square
01:00:16.050 --> 01:00:18.590
error in the samples with
the mean square
01:00:18.590 --> 01:00:20.120
error on the function.
01:00:20.120 --> 01:00:24.280
You can't do that with anything
that I know of other
01:00:24.280 --> 01:00:25.830
than mean square error.
01:00:25.830 --> 01:00:28.790
If you want to deal with other
kinds of quantization errors,
01:00:28.790 --> 01:00:31.320
you have a real problem
going from
01:00:31.320 --> 01:00:34.170
coefficients to functions.
01:00:34.170 --> 01:00:43.310
And finally, for any set of
numbers u sub k, if the sum is
01:00:43.310 --> 01:00:46.470
less than infinity, in other
words, if you're dealing with
01:00:46.470 --> 01:00:52.770
a sequence of finite energy,
there always is such a
01:00:52.770 --> 01:00:54.540
frequency function.
01:00:54.540 --> 01:00:57.340
Many people when they use the
discrete time Fourier
01:00:57.340 --> 01:01:01.860
transform think of starting with
the sequence and taking a
01:01:01.860 --> 01:01:04.920
sequence and saying well it's
nice to think of this sequence
01:01:04.920 --> 01:01:09.270
in the frequency domain, and
then they say a function f
01:01:09.270 --> 01:01:13.100
exists such that this is true,
or they just say that this is
01:01:13.100 --> 01:01:16.300
equal to that without worrying
about the convergence at all,
01:01:16.300 --> 01:01:18.150
which is more common.
01:01:18.150 --> 01:01:20.620
But since we've already gone
through all of this for the
01:01:20.620 --> 01:01:25.760
Fourier series, we might as well
say it right here also.
01:01:25.760 --> 01:01:27.970
So there's really nothing
different here.
01:01:27.970 --> 01:01:34.090
But now the question is why
do these u of k's --
01:01:34.090 --> 01:01:38.060
I mean why do we think of those
as time coefficients?
01:01:38.060 --> 01:01:40.420
I mean what's really going
on in this discrete
01:01:40.420 --> 01:01:42.660
time Fourier transform.
01:01:42.660 --> 01:01:47.050
At this point it just looks like
a lot of mathematics and
01:01:47.050 --> 01:01:50.930
it's hard to interpret what
any of these things mean.
01:01:50.930 --> 01:01:53.040
Well, the next thing I want
to do is to go into
01:01:53.040 --> 01:01:55.760
the sampling theorem.
01:01:55.760 --> 01:01:59.480
The sampling theorem, in fact,
is going to interpret for you
01:01:59.480 --> 01:02:02.920
what this discrete time Fourier
transform is, because
01:02:02.920 --> 01:02:05.760
the sampling theorem and the
discrete time Fourier
01:02:05.760 --> 01:02:10.270
transform are just intimately
related, they're hand and
01:02:10.270 --> 01:02:21.280
glove with each other,
and that's the next
01:02:21.280 --> 01:02:22.800
thing we want to do.
01:02:22.800 --> 01:02:27.230
But first we have to re-write
this a little bit.
01:02:27.230 --> 01:02:32.180
We're going to say that this
frequency function is the
01:02:32.180 --> 01:02:35.810
limit in the mean of this --
01:02:35.810 --> 01:02:40.630
this rectangular function is
what we use just to make sure
01:02:40.630 --> 01:02:43.190
we're only talking about
frequency between
01:02:43.190 --> 01:02:47.250
minus w and plus w.
01:02:47.250 --> 01:02:50.355
The limit in the mean, there's a
little notational trick that
01:02:50.355 --> 01:02:55.510
we use so that we can think of
this as just a limit instead
01:02:55.510 --> 01:02:59.310
of thinking of it as this
crazy thing that we just
01:02:59.310 --> 01:03:02.120
derive, which is really
not so crazy.
01:03:04.900 --> 01:03:08.540
That means we can talk about
this transform without always
01:03:08.540 --> 01:03:14.060
rubbing our noses in all
of this mess here.
01:03:14.060 --> 01:03:16.900
It just means that once in
awhile we go back and think
01:03:16.900 --> 01:03:18.630
what does this really mean.
01:03:18.630 --> 01:03:22.110
It means convergence in energy
rather than convergence
01:03:22.110 --> 01:03:26.640
point-wise, because we might
not have convergence
01:03:26.640 --> 01:03:27.890
point-wise.
01:03:29.650 --> 01:03:34.660
So we're going to write this
also as the limit in the mean
01:03:34.660 --> 01:03:37.580
of the sum over k of u of k.
01:03:37.580 --> 01:03:40.480
We're going to glop all
of this together.
01:03:40.480 --> 01:03:45.940
This is just some function
of k and a frequency.
01:03:45.940 --> 01:03:50.920
So we're going to call that phi
sub k of f at some wave
01:03:50.920 --> 01:03:56.150
form, and the wave
form is this.
01:03:56.150 --> 01:04:01.940
What happens if you look at the
relationship between to
01:04:01.940 --> 01:04:06.030
phi k of f and phi
k prime of f?
01:04:06.030 --> 01:04:08.200
Namely, if you look at two
different functions.
01:04:11.220 --> 01:04:14.490
These two functions are
orthongonal to each other for
01:04:14.490 --> 01:04:21.570
the same reason that the
functions that the sinusoid
01:04:21.570 --> 01:04:27.040
you used in the Fourier series
are orthongonal.
01:04:27.040 --> 01:04:31.790
Namely, you take this function,
you multiply it by e
01:04:31.790 --> 01:04:35.450
to the minus 2 pi ik prime
f over 2w times
01:04:35.450 --> 01:04:37.070
this rectangular function.
01:04:37.070 --> 01:04:40.880
You integrate from minus w
to w and what do you get?
01:04:40.880 --> 01:04:43.490
You're just integrating
a sinusoid --
01:04:43.490 --> 01:04:45.880
the whole thing is one
big sinusoid --
01:04:45.880 --> 01:04:49.520
over one period of that sinusoid
or multiple periods
01:04:49.520 --> 01:04:50.630
of the sinusoid.
01:04:50.630 --> 01:04:54.650
Actually, k minus k prime
periods of the sinusoid.
01:04:54.650 --> 01:04:57.360
And when you integrate a
sinusoid over a period, what
01:04:57.360 --> 01:04:58.140
do you get?
01:04:58.140 --> 01:05:00.200
You get zero.
01:05:00.200 --> 01:05:03.330
So it says that these functions
are all orthongonal
01:05:03.330 --> 01:05:04.460
to each other.
01:05:04.460 --> 01:05:08.460
So, presto, we have another
orthongonal expansion just
01:05:08.460 --> 01:05:11.850
like the Fourier series gave us
an orthongonal expansion.
01:05:11.850 --> 01:05:14.020
And in fact, it's the same
orthongonal expansion.
01:05:31.050 --> 01:05:36.400
Now the next thing to observe
is that we have done the
01:05:36.400 --> 01:05:41.220
Fourier transform and we've also
done the discrete time
01:05:41.220 --> 01:05:43.070
Fourier transform.
01:05:43.070 --> 01:05:48.010
In both of them we're dealing
with some frequency function.
01:05:48.010 --> 01:05:50.720
Now we're dealing with some
frequency function which is
01:05:50.720 --> 01:05:54.560
limited to minus w to plus
w, but we have two
01:05:54.560 --> 01:05:57.570
expansions for it.
01:05:57.570 --> 01:06:00.740
We have the Fourier transform,
so we can go to a function u
01:06:00.740 --> 01:06:03.710
of t, and we also have
this discrete
01:06:03.710 --> 01:06:07.070
time Fourier transform.
01:06:07.070 --> 01:06:13.070
So u of t is equal to this
Fourier transform here.
01:06:13.070 --> 01:06:17.580
Again, I should write limit in
the mean here, but then I
01:06:17.580 --> 01:06:19.390
think about and I say
do I have to write
01:06:19.390 --> 01:06:21.220
limit in the mean?
01:06:21.220 --> 01:06:22.090
No.
01:06:22.090 --> 01:06:24.520
I don't need a limit in the
mean here because I'm
01:06:24.520 --> 01:06:26.970
integrating this over
finite limits.
01:06:26.970 --> 01:06:31.050
Since I'm taking a function over
finite limits, u hat of f
01:06:31.050 --> 01:06:36.150
is over these limits, is an L1
function, therefore, this
01:06:36.150 --> 01:06:37.010
integral exists.
01:06:37.010 --> 01:06:41.510
This function is a continuous
function.
01:06:41.510 --> 01:06:45.070
There aren't any sets of measure
zero involved here.
01:06:45.070 --> 01:06:49.740
This is one specific function
which is always the same.
01:06:49.740 --> 01:06:52.460
You know what it is exactly
at every point.
01:06:52.460 --> 01:06:56.370
At every point t,
this converges.
01:06:56.370 --> 01:06:58.890
So then what we're going to do,
what the sampling theorem
01:06:58.890 --> 01:07:05.030
does is it relates this to
what you get with a dtft.
01:07:05.030 --> 01:07:09.090
So the sampling theorem says
let u hat of f be an L2
01:07:09.090 --> 01:07:20.050
function which goes from minus
ww to c, and let u of t be
01:07:20.050 --> 01:07:23.730
this, namely, that, which we
now know exists and is
01:07:23.730 --> 01:07:25.110
continuous.
01:07:25.110 --> 01:07:28.750
Define capital T as 1 over 2w.
01:07:28.750 --> 01:07:32.270
You don't have to do that if you
don't want to, but it's a
01:07:32.270 --> 01:07:35.250
little easier to think in terms
of some increment of
01:07:35.250 --> 01:07:37.880
time, T, here.
01:07:37.880 --> 01:07:42.120
Then u of t is continuous,
L2 and bounded.
01:07:42.120 --> 01:07:44.570
It's bounded by u of t less
than or equal to this.
01:07:44.570 --> 01:07:46.030
Why is that?
01:07:53.295 --> 01:07:57.670
It doesn't make any sense
as I stated it.
01:07:57.670 --> 01:07:58.940
Now it makes sense.
01:07:58.940 --> 01:08:01.720
OK, its magnitude is bounded.
01:08:01.720 --> 01:08:04.880
Its magnitude is bounded because
if you take u of t
01:08:04.880 --> 01:08:09.310
magnitude, it's equal to the
magnitude of this which is
01:08:09.310 --> 01:08:12.740
less than or equal to the
integral of the magnitude of
01:08:12.740 --> 01:08:18.130
this, which is equal to the
integral of the magnitude of
01:08:18.130 --> 01:08:22.370
just u hat of f, which
is what we have here.
01:08:22.370 --> 01:08:25.470
So all of that works nicely.
01:08:25.470 --> 01:08:29.520
So, u of t is a nice,
well-defined function.
01:08:29.520 --> 01:08:33.150
Then the other part of it is
that u of t is equal to the
01:08:33.150 --> 01:08:38.800
sum if its values at these
sample points times sinc of t
01:08:38.800 --> 01:08:40.810
minus kt over t.
01:08:40.810 --> 01:08:43.490
Now you've probably seen this
sampling theorem before.
01:08:43.490 --> 01:08:46.430
How many people haven't
seen this before?
01:08:46.430 --> 01:08:50.130
I mean aside from the question
of trying to do it --
01:08:50.130 --> 01:08:52.870
do it in a way that
makes sense.
01:08:52.870 --> 01:08:53.890
OK, you've all seen it.
01:08:53.890 --> 01:08:55.080
So good.
01:08:55.080 --> 01:09:00.160
What it's saying is you can
represent a function in terms
01:09:00.160 --> 01:09:03.310
of just knowing what
its samples are.
01:09:03.310 --> 01:09:06.890
Or you can take the function,
you can sample it, and when
01:09:06.890 --> 01:09:14.340
you sample it if you put these
little sinc hats around all
01:09:14.340 --> 01:09:17.180
the samples, you get back
to the function again.
01:09:17.180 --> 01:09:20.490
So you take all the samples,
you then put these sincs
01:09:20.490 --> 01:09:22.630
around them, add them
all up, and
01:09:22.630 --> 01:09:27.680
bingo, you got the function.
01:09:27.680 --> 01:09:29.880
Let's see why that's true.
01:09:35.820 --> 01:09:38.290
Here's the sinc function here.
01:09:38.290 --> 01:09:43.290
The important thing about sinc
t, sine pi t over pi t, which
01:09:43.290 --> 01:09:47.410
you can see by just looking at
the sine function, is it has
01:09:47.410 --> 01:09:51.040
the value 1 when t
is equal to zero.
01:09:51.040 --> 01:09:53.780
I mean to get that value 1, you
really have to go through
01:09:53.780 --> 01:09:57.830
a limiting operation here to
think of sine pi t when t is
01:09:57.830 --> 01:10:01.580
very small as being
approximately equal to pi t.
01:10:01.580 --> 01:10:05.580
When you divide pi t by pi t you
get 1, so that's its value
01:10:05.580 --> 01:10:08.490
there and value around there.
01:10:08.490 --> 01:10:13.580
At every other sample point,
namely, at t equals 1, sine of
01:10:13.580 --> 01:10:15.220
pi t is zero.
01:10:15.220 --> 01:10:19.200
At t equals 2, sine of pi
t is equal to zero.
01:10:19.200 --> 01:10:25.030
So the sinc function is 1 at
zero and is zero at every
01:10:25.030 --> 01:10:27.660
other integer point.
01:10:27.660 --> 01:10:32.240
Now to see why it's true and to
understand what the dtft is
01:10:32.240 --> 01:10:39.270
all about, note that we have
said that a frequency
01:10:39.270 --> 01:10:44.150
function, u hat of f, can be
expressed as the sum over k --
01:10:44.150 --> 01:10:47.530
and I should use a limit in the
mean here but I'm not --
01:10:47.530 --> 01:10:51.970
of uk times this transform
relationship here.
01:10:54.560 --> 01:10:59.450
Well, these are these functions
that we talked about
01:10:59.450 --> 01:11:00.700
awhile ago.
01:11:09.010 --> 01:11:09.980
They're these functions.
01:11:09.980 --> 01:11:15.840
They're the sinusoids, periodic
sinusoids in k
01:11:15.840 --> 01:11:18.510
truncated in frequency.
01:11:24.430 --> 01:11:31.880
So we know that u hat of f is
equal to that dtft expansion.
01:11:31.880 --> 01:11:35.360
If I take the inverse Fourier
transform of that, I can take
01:11:35.360 --> 01:11:38.490
the inverse Fourier transform
of all these functions and
01:11:38.490 --> 01:11:43.590
I'll get u of t is equal to the
sum over k, of uk pk of t.
01:11:43.590 --> 01:11:45.710
I'm being careless about
the mathematics here.
01:11:45.710 --> 01:11:48.510
I've been careful about
it all along.
01:11:48.510 --> 01:11:52.450
The notes does it carefully,
particularly in the appendix.
01:11:52.450 --> 01:11:54.750
I'm not going to worry
about that here.
01:11:54.750 --> 01:11:58.520
If I take the function pk of
f, which is this truncated
01:11:58.520 --> 01:12:03.340
sinusoid, and I take the inverse
transform of that --
01:12:03.340 --> 01:12:07.350
take the transform of
this, I get this.
01:12:07.350 --> 01:12:10.870
Can you see that just
by inspection?
01:12:10.870 --> 01:12:13.090
If you were really hot on these
things, if you just
01:12:13.090 --> 01:12:16.830
finished taking 6.003, you could
probably see that by
01:12:16.830 --> 01:12:18.630
inspection.
01:12:18.630 --> 01:12:23.490
If you remember all of those
relationships that we went
01:12:23.490 --> 01:12:26.650
through before, you can
see it by inspection.
01:12:26.650 --> 01:12:31.260
The Fourier transform of erect
function is a sinc function.
01:12:31.260 --> 01:12:35.520
This exponential here when you
go into the time domain
01:12:35.520 --> 01:12:39.010
corresponds to a time shift,
so that gives rise to this
01:12:39.010 --> 01:12:41.350
time shift here.
01:12:41.350 --> 01:12:45.150
The 1 over t is just one of
these constants you have to
01:12:45.150 --> 01:12:48.250
keep straight, and which I would
do just by integrating
01:12:48.250 --> 01:12:51.110
the things to see what I get.
01:12:51.110 --> 01:13:00.970
Finally, u of kt, if I look at
this, is just 1 over t times u
01:13:00.970 --> 01:13:05.220
of k, because these functions
here are these sinc functions,
01:13:05.220 --> 01:13:08.350
which are zero everywhere
but on their own point.
01:13:08.350 --> 01:13:17.240
So if I look at u of kt, it's a
sum over k of ck of kt, and
01:13:17.240 --> 01:13:26.160
ck of kt is only 1 when little
t is equal to k times capital
01:13:26.160 --> 01:13:30.690
T, and therefore, I get
that point there.
01:13:30.690 --> 01:13:35.740
Therefore, u of kt is just
1 over t times u of k.
01:13:35.740 --> 01:13:39.950
That finishes the sampling
theorem except for really
01:13:39.950 --> 01:13:47.110
tracing through all of these
things about convergence, but
01:13:47.110 --> 01:13:50.350
it also tells you what the
discrete time Fourier
01:13:50.350 --> 01:13:52.380
transform is.
01:13:52.380 --> 01:13:56.420
Because the discrete time
Fourier transform is just
01:13:56.420 --> 01:13:59.560
scaled samples of u of t.
01:13:59.560 --> 01:14:02.380
In other words, you start out
with this frequency function,
01:14:02.380 --> 01:14:05.110
you take the inverse Fourier
transform of it, you get a
01:14:05.110 --> 01:14:06.280
time function.
01:14:06.280 --> 01:14:10.050
You take the samples of that,
you scale them, and those are
01:14:10.050 --> 01:14:12.800
the coefficients in the
discrete time Fourier
01:14:12.800 --> 01:14:15.410
transform, which is what you
use discrete time Fourier
01:14:15.410 --> 01:14:17.100
transforms for.
01:14:17.100 --> 01:14:20.520
You think of sampling of
function, then you represent
01:14:20.520 --> 01:14:23.450
the function in terms of those
samples and then you want to
01:14:23.450 --> 01:14:26.820
go into the frequency domain as
a way of dealing with the
01:14:26.820 --> 01:14:30.110
properties of those samples, and
all you're doing is just
01:14:30.110 --> 01:14:34.690
going into the Fourier transform
of u of t, and all
01:14:34.690 --> 01:14:37.550
of that works out.
01:14:42.780 --> 01:14:46.370
There's one bizarre thing here,
and I'm going to talk
01:14:46.370 --> 01:14:50.170
about that more next time.
01:14:50.170 --> 01:14:54.340
That is that when you look at
this time frequency limited
01:14:54.340 --> 01:15:03.080
function, u hat of f, u hat of
f can be very badly behaved.
01:15:03.080 --> 01:15:06.340
You can have a frequency limited
function which does
01:15:06.340 --> 01:15:08.330
all sorts of crazy things.
01:15:08.330 --> 01:15:11.770
Since it's frequency limited
as inverse transform, it's
01:15:11.770 --> 01:15:12.820
beautifully behaved.
01:15:12.820 --> 01:15:15.460
It's just the sum of sinc
functions -- it's bounded,
01:15:15.460 --> 01:15:18.580
it's continuous and
everything else.
01:15:18.580 --> 01:15:22.180
When we go back into the
frequency domain it is just as
01:15:22.180 --> 01:15:23.980
ugly as can be.
01:15:23.980 --> 01:15:29.050
So what we have in the sampling
theorem, it comes out
01:15:29.050 --> 01:15:31.570
particularly clearly.
01:15:31.570 --> 01:15:36.240
Is that L1 functions in one
domain are nice, continuous,
01:15:36.240 --> 01:15:39.910
beautiful functions which are
not L1 in the other domain.
01:15:39.910 --> 01:15:44.260
So you sort of go from
L1 to continuous.
01:15:44.260 --> 01:15:47.360
Now when we're dealing with
L2 functions, they're not
01:15:47.360 --> 01:15:50.630
continuous, they're not anything
else, but you always
01:15:50.630 --> 01:15:52.380
go from L2 to L2.
01:15:52.380 --> 01:15:56.240
In other words, you can't get
out of the L2 domain, and
01:15:56.240 --> 01:16:01.140
therefore, when you're dealing
with L2 functions, all you
01:16:01.140 --> 01:16:04.700
have to worry about is L2
functions, because you always
01:16:04.700 --> 01:16:06.890
stay there no matter what.
01:16:06.890 --> 01:16:10.120
When we start talking about
stochastic processes, we'll
01:16:10.120 --> 01:16:12.510
find out you always
stay there also.
01:16:12.510 --> 01:16:15.450
In other words, you only have
to know about one thing.
01:16:15.450 --> 01:16:18.380
We've seen here that to
interpret what these Fourier
01:16:18.380 --> 01:16:23.230
transforms mean, it's nice to
have a little idea about L1
01:16:23.230 --> 01:16:28.400
functions also because when we
think of going to the limit,
01:16:28.400 --> 01:16:33.450
when we go to the limit we get
something which is badly
01:16:33.450 --> 01:16:37.390
behaved, and for these finite
time and finite frequency
01:16:37.390 --> 01:16:41.000
approximations, we have things
which are beautifully behaved.
01:16:41.000 --> 01:16:43.280
I'm going to stop now so we
can pass out the quizzes.