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PROFESSOR: Last time, we began
to address the issue of
00:00:58.910 --> 00:01:02.610
building continuous time
signals out of a linear
00:01:02.610 --> 00:01:05.560
combination of complex
exponentials.
00:01:05.560 --> 00:01:10.110
And for the class of periodic
signals specifically, what
00:01:10.110 --> 00:01:13.350
this led to was the Fourier
series representation for
00:01:13.350 --> 00:01:15.020
periodic signals.
00:01:15.020 --> 00:01:17.270
Let me just summarize
the results that we
00:01:17.270 --> 00:01:19.110
developed last time.
00:01:19.110 --> 00:01:23.880
For periodic signals, we had
the continuous-time Fourier
00:01:23.880 --> 00:01:29.220
series, where we built the
periodic signal out of a
00:01:29.220 --> 00:01:33.250
linear combination of
harmonically related complex
00:01:33.250 --> 00:01:34.750
exponentials.
00:01:34.750 --> 00:01:39.030
And what that led to was what
we referred to as the
00:01:39.030 --> 00:01:41.050
synthesis equation.
00:01:41.050 --> 00:01:45.900
And we briefly addressed the
issue of when this in fact
00:01:45.900 --> 00:01:50.360
builds, when this in fact is a
complete representation of the
00:01:50.360 --> 00:01:55.970
periodic signal, and in essence,
what we presented was
00:01:55.970 --> 00:02:00.540
conditions either for x of t
being square integrable or x
00:02:00.540 --> 00:02:03.090
of t being absolutely
integrable.
00:02:03.090 --> 00:02:06.880
Then, the other side of the
Fourier series is what I
00:02:06.880 --> 00:02:10.380
referred to as the analysis
equation.
00:02:10.380 --> 00:02:14.120
And the analysis equation was
the equation that told us how
00:02:14.120 --> 00:02:19.110
we get the Fourier series
coefficients from x of t.
00:02:19.110 --> 00:02:23.430
And so this equation together
with the synthesis equation
00:02:23.430 --> 00:02:26.600
represent the Fourier
series description
00:02:26.600 --> 00:02:29.470
for periodic signals.
00:02:29.470 --> 00:02:34.580
Now what we'd like to do is
extend this idea to provide a
00:02:34.580 --> 00:02:39.460
mechanism for building
non-periodic signals also out
00:02:39.460 --> 00:02:42.500
of a linear combination of
complex exponentials.
00:02:42.500 --> 00:02:47.310
And the basic idea behind doing
this is very simple and
00:02:47.310 --> 00:02:50.680
also very clever as I
indicated last time.
00:02:50.680 --> 00:02:54.850
Essentially, the thought is the
following, if we have a
00:02:54.850 --> 00:02:59.220
non-periodic signal or aperiodic
signal, we can think
00:02:59.220 --> 00:03:04.720
of constructing a periodic
signal by simply periodically
00:03:04.720 --> 00:03:08.230
replicating that aperiodic
signal.
00:03:08.230 --> 00:03:12.520
So for example, if I have an
aperiodic signal as I've
00:03:12.520 --> 00:03:18.870
indicated here, I can consider
building a periodic signal,
00:03:18.870 --> 00:03:23.700
where I simply take this
original signal and repeat it
00:03:23.700 --> 00:03:27.910
at multiples of some
period t 0.
00:03:27.910 --> 00:03:30.380
Now, two things to recognize
about this.
00:03:30.380 --> 00:03:36.155
One is that the periodic
signal is equal to the
00:03:36.155 --> 00:03:39.150
aperiodic signal over
one period.
00:03:39.150 --> 00:03:45.020
And the second is that as the
period goes to infinity then,
00:03:45.020 --> 00:03:52.490
in fact, the periodic signal
goes to the aperiodic signal.
00:03:52.490 --> 00:03:59.360
So the basic idea then is to
use the Fourier series to
00:03:59.360 --> 00:04:03.920
represent the periodic signal,
and then examine the Fourier
00:04:03.920 --> 00:04:08.880
series expression as we let
the period go to infinity.
00:04:08.880 --> 00:04:14.150
Well, let's quickly see how this
develops in terms of the
00:04:14.150 --> 00:04:16.130
associated equations.
00:04:16.130 --> 00:04:22.079
Here, again, we have the
periodic signal.
00:04:22.079 --> 00:04:26.920
And what we want to inquire into
is what happens to the
00:04:26.920 --> 00:04:30.690
Fourier series expression for
this as we let the period go
00:04:30.690 --> 00:04:31.750
to infinity.
00:04:31.750 --> 00:04:35.140
As that happens, whatever
Fourier series representation
00:04:35.140 --> 00:04:40.260
we end up with will correspond
also to a representation for
00:04:40.260 --> 00:04:42.850
this aperiodic signal.
00:04:42.850 --> 00:04:44.220
Well, let's see.
00:04:44.220 --> 00:04:48.040
The Fourier series synthesis
expression for the periodic
00:04:48.040 --> 00:04:53.650
signal expresses x tilde of t,
the periodic signal as a
00:04:53.650 --> 00:04:57.230
linear combination of
harmonically related complex
00:04:57.230 --> 00:05:01.510
exponentials with the
fundamental frequency omega 0
00:05:01.510 --> 00:05:04.370
equaled to 2 pi divided
by the period.
00:05:04.370 --> 00:05:11.530
And the analysis equation tells
us what the relationship
00:05:11.530 --> 00:05:16.180
is for the coefficients in terms
of the periodic signal.
00:05:16.180 --> 00:05:20.215
Now, I indicated that the
periodic signal and the
00:05:20.215 --> 00:05:23.370
aperiodic signal are equal
over one period.
00:05:23.370 --> 00:05:27.640
We recognize that this
integration, in fact, only
00:05:27.640 --> 00:05:29.460
occurs over one period.
00:05:29.460 --> 00:05:33.840
And so we can re-express this
in terms of our original
00:05:33.840 --> 00:05:35.510
aperiodic signal.
00:05:35.510 --> 00:05:38.730
So this tells us the Fourier
series coefficients in
00:05:38.730 --> 00:05:40.012
terms of x of t.
00:05:43.320 --> 00:05:49.360
Now, if we look at this
expression, which is the
00:05:49.360 --> 00:05:53.020
expression for the Fourier
coefficients of the aperiodic
00:05:53.020 --> 00:05:58.690
signal, one of the things to
recognize is that in effect
00:05:58.690 --> 00:06:06.420
what this represents are samples
of an integral, where
00:06:06.420 --> 00:06:12.350
we can think of the variable
omega taking on values that
00:06:12.350 --> 00:06:14.830
are integer multiples
of omega 0.
00:06:14.830 --> 00:06:18.600
Said another way, let's define
a function, as I've indicated
00:06:18.600 --> 00:06:24.540
here, which is this integral,
where we may think of omega as
00:06:24.540 --> 00:06:27.930
being a continuous variable and
then the Fourier series
00:06:27.930 --> 00:06:35.890
coefficients correspond
to substituting for
00:06:35.890 --> 00:06:38.660
omega k omega 0.
00:06:38.660 --> 00:06:42.710
Now, one reason for doing that,
as we'll see, is that in
00:06:42.710 --> 00:06:47.480
fact, this will turn out to
provide us with a mechanism
00:06:47.480 --> 00:06:51.250
for a Fourier representation
of x of t.
00:06:51.250 --> 00:06:56.360
And this, in fact, then, is an
envelope of the Fourier series
00:06:56.360 --> 00:06:57.340
coefficients.
00:06:57.340 --> 00:07:03.530
In other words, t 0 the period
times the coefficients is
00:07:03.530 --> 00:07:07.850
equal to this integral add
integer multiples of omega 0.
00:07:10.540 --> 00:07:14.360
So this, in effect, tells us how
to get the Fourier series
00:07:14.360 --> 00:07:19.490
coefficients of the periodic
signal in terms of samples of
00:07:19.490 --> 00:07:20.140
an envelope.
00:07:20.140 --> 00:07:23.770
And that will become a very
important notion shortly.
00:07:23.770 --> 00:07:28.370
And that, in effect, will
correspond to an analysis
00:07:28.370 --> 00:07:32.280
equation to represent the
aperiodic signal.
00:07:32.280 --> 00:07:35.770
Now, let's look at the
synthesis equation.
00:07:35.770 --> 00:07:40.200
Recall that in the synthesis
our strategy is to build a
00:07:40.200 --> 00:07:44.190
periodic signal and let the
period go to infinity.
00:07:44.190 --> 00:07:50.160
Well, here is the expression
for the synthesis of the
00:07:50.160 --> 00:07:56.510
periodic signal now expressed
in terms of samples of this
00:07:56.510 --> 00:08:01.840
envelope function, and where
I've simply used the fact or
00:08:01.840 --> 00:08:06.870
the substitution that t 0 is 2
pi over omega 0, and so I have
00:08:06.870 --> 00:08:10.830
an omega 0 here and
a 1 over 2 pi.
00:08:10.830 --> 00:08:14.280
And the reason for doing that,
as we'll see in a minute, is
00:08:14.280 --> 00:08:17.440
that this then turns
into an integral.
00:08:17.440 --> 00:08:20.880
Specifically, then, the
synthesis equation that we
00:08:20.880 --> 00:08:26.690
have is what I've
indicated here.
00:08:26.690 --> 00:08:30.300
We would now want to examine
this as the period goes to
00:08:30.300 --> 00:08:33.990
infinity, which means
that omega 0 becomes
00:08:33.990 --> 00:08:36.140
infinitesimally small.
00:08:36.140 --> 00:08:39.820
And without dwelling on the
details, and with my
00:08:39.820 --> 00:08:43.440
suggesting that you give this
a fair amount of reflection,
00:08:43.440 --> 00:08:47.830
in fact, what happens as the
period goes to infinity is
00:08:47.830 --> 00:08:53.910
that this summation approaches
an integral over omega, where
00:08:53.910 --> 00:08:57.650
omega 0 becomes the differential
in omega, and the
00:08:57.650 --> 00:09:00.020
periodic signal, of course,
approach is
00:09:00.020 --> 00:09:02.740
the aperiodic signal.
00:09:02.740 --> 00:09:08.240
So the resulting equation that
we get out of the original
00:09:08.240 --> 00:09:14.680
Fourier series synthesis
equation is the equation that
00:09:14.680 --> 00:09:21.730
I indicate down here, x of t
synthesized in terms of this
00:09:21.730 --> 00:09:25.720
integral, which is what the
Fourier series approaches as
00:09:25.720 --> 00:09:28.780
omega 0 goes to 0.
00:09:28.780 --> 00:09:34.400
And we had previously that
x of omega was in fact an
00:09:34.400 --> 00:09:36.900
envelope function.
00:09:36.900 --> 00:09:39.660
And we have then the
corresponding Fourier
00:09:39.660 --> 00:09:45.240
transform analysis equation,
which tells us how we arrive
00:09:45.240 --> 00:09:49.930
at that envelope in
terms of x of t.
00:09:49.930 --> 00:09:54.410
So we now have an analysis
equation and a synthesis
00:09:54.410 --> 00:09:59.150
equation, which in effect
expresses for us how to build
00:09:59.150 --> 00:10:03.910
x of t in terms of
infinitesimally finely spaced
00:10:03.910 --> 00:10:06.750
complex exponentials.
00:10:06.750 --> 00:10:11.690
The strategy to review it, and
which I'd like to illustrate
00:10:11.690 --> 00:10:21.090
with a succession of overlays,
was to begin with our
00:10:21.090 --> 00:10:28.780
aperiodic signal, as I indicate
here, and then we
00:10:28.780 --> 00:10:34.160
constructed from that
a periodic signal.
00:10:34.160 --> 00:10:39.880
And this periodic signal has a
Fourier series, and we express
00:10:39.880 --> 00:10:43.550
the Fourier series coefficients
of this as
00:10:43.550 --> 00:10:46.630
samples of an envelope
function.
00:10:46.630 --> 00:10:51.520
The envelope function is what I
indicate on the curve below.
00:10:51.520 --> 00:10:55.390
So this is the envelope of the
Fourier series coefficients.
00:10:55.390 --> 00:11:02.640
For example, if the period t 0
was four times t1, then the
00:11:02.640 --> 00:11:06.270
Fourier series coefficients that
we would end up with is
00:11:06.270 --> 00:11:10.480
this set of samples
of the envelope.
00:11:10.480 --> 00:11:15.500
If instead we doubled that
period, then the Fourier
00:11:15.500 --> 00:11:18.470
series coefficients that
we end up with
00:11:18.470 --> 00:11:22.180
are more finely spaced.
00:11:22.180 --> 00:11:28.470
And as t 0 continues to
increase, we get more and more
00:11:28.470 --> 00:11:33.290
finely spaced samples of this
envelope function, and as t 0
00:11:33.290 --> 00:11:37.030
goes to infinity in fact, what
we get is every single point
00:11:37.030 --> 00:11:41.080
on the envelope, and that
provides us with the
00:11:41.080 --> 00:11:44.780
representation for the
aperiodic signal.
00:11:44.780 --> 00:11:50.530
Let me, just to really emphasize
the point, show this
00:11:50.530 --> 00:11:52.520
example once again.
00:11:52.520 --> 00:11:55.800
But now, let's look at
it dynamically on
00:11:55.800 --> 00:11:58.440
the computer display.
00:11:58.440 --> 00:12:01.220
So here we have the square
wave, and below it, the
00:12:01.220 --> 00:12:03.010
Fourier series coefficients.
00:12:03.010 --> 00:12:06.490
And we now want to look at the
Fourier series coefficients as
00:12:06.490 --> 00:12:09.550
the period of the square wave
starts to increase.
00:12:17.760 --> 00:12:21.260
And what we see is that
these look like
00:12:21.260 --> 00:12:24.100
samples of an envelope.
00:12:24.100 --> 00:12:26.970
And in fact, the envelope
of the Fourier series
00:12:26.970 --> 00:12:30.090
coefficients is shown in
the bottom [? trace, ?]
00:12:30.090 --> 00:12:33.300
and to emphasize in fact that
it is the envelope let's
00:12:33.300 --> 00:12:37.400
superimpose it on top of the
Fourier series coefficients
00:12:37.400 --> 00:12:38.840
that we've generated so far.
00:12:48.340 --> 00:12:48.710
OK.
00:12:48.710 --> 00:12:54.020
Now, let's increase the period
even further, and we'll see
00:12:54.020 --> 00:12:57.910
the Fourier series coefficients
fill in under
00:12:57.910 --> 00:13:02.780
that envelope function
even more.
00:13:02.780 --> 00:13:07.980
And in fact, as the period gets
large enough, what we
00:13:07.980 --> 00:13:13.570
begin to get a sense of is that
we're sampling more and
00:13:13.570 --> 00:13:15.940
more finely this envelope.
00:13:15.940 --> 00:13:19.060
And in fact, in the limit,
as the period goes off to
00:13:19.060 --> 00:13:23.160
infinity, the samples basically
will represent every
00:13:23.160 --> 00:13:25.050
single point on the envelope.
00:13:25.050 --> 00:13:28.030
Well, this is about as
far as we want to go.
00:13:28.030 --> 00:13:33.130
Let's once again, plot the
envelope function, and again,
00:13:33.130 --> 00:13:36.160
to emphasize that we've
generated samples of that,
00:13:36.160 --> 00:13:39.175
let's superimpose that on the
Fourier series coefficients.
00:13:46.300 --> 00:13:52.000
So what we have then is now
our Fourier transform
00:13:52.000 --> 00:13:57.530
representation, the continuous
time Fourier transform with
00:13:57.530 --> 00:14:01.960
the synthesis equation expressed
as an integral, as
00:14:01.960 --> 00:14:06.870
I've indicated here, and this
integral is what the Fourier
00:14:06.870 --> 00:14:12.820
series sum went to as we let the
period go to infinity or
00:14:12.820 --> 00:14:14.720
the frequency go to zero.
00:14:14.720 --> 00:14:19.080
The corresponding analysis
equation, which we have here,
00:14:19.080 --> 00:14:23.670
the analysis equation being
the expression for the
00:14:23.670 --> 00:14:26.970
envelope of the Fourier series
coefficients for the
00:14:26.970 --> 00:14:30.620
periodically replicated
signal.
00:14:30.620 --> 00:14:33.690
And in shorthand notation,
we would think
00:14:33.690 --> 00:14:35.370
of x of t and [? its ?]
00:14:35.370 --> 00:14:40.380
Fourier transform as a pair,
as I've indicated here.
00:14:40.380 --> 00:14:45.830
And the Fourier transform, as
we'll emphasize in several
00:14:45.830 --> 00:14:49.800
examples, and certainly as is
consistent with the Fourier
00:14:49.800 --> 00:14:54.590
Series, is a complex valued
function even
00:14:54.590 --> 00:14:55.730
when x of t is real.
00:14:55.730 --> 00:14:59.690
So with x of t real, we end up
with a Fourier transform,
00:14:59.690 --> 00:15:01.770
which is a complex function.
00:15:01.770 --> 00:15:06.060
Just as the Fourier series
coefficients were complex for
00:15:06.060 --> 00:15:08.250
a real value time function.
00:15:08.250 --> 00:15:12.420
So we could alternatively, as
with the Fourier series,
00:15:12.420 --> 00:15:18.150
express the Fourier transform in
terms of it's real part and
00:15:18.150 --> 00:15:23.530
imaginary part, or
alternatively, in terms of its
00:15:23.530 --> 00:15:25.520
magnitude and its angle.
00:15:28.270 --> 00:15:33.560
All right, now let's look at an
example of a time function
00:15:33.560 --> 00:15:35.750
in its Fourier transform.
00:15:35.750 --> 00:15:39.680
And so let's consider an
example, which in fact is an
00:15:39.680 --> 00:15:41.350
example worked out
in the text.
00:15:41.350 --> 00:15:44.500
It's example 4.7 in the text.
00:15:44.500 --> 00:15:49.520
And this is our old familiar
friend the exponential.
00:15:49.520 --> 00:15:53.230
It's Fourier transform is the
integral from minus infinity
00:15:53.230 --> 00:15:57.950
to plus infinity, x of t, e
to the minus j omega t dt.
00:15:57.950 --> 00:16:03.960
And so, if we substitute in x
of t and combine these two
00:16:03.960 --> 00:16:09.200
exponentials together, these two
exponentials combined are
00:16:09.200 --> 00:16:15.130
e to the minus t times
a plus j omega.
00:16:15.130 --> 00:16:21.070
And if we carry out the
integration of this, we end up
00:16:21.070 --> 00:16:25.790
with the expression indicated
here and provided now, and
00:16:25.790 --> 00:16:32.830
this is important, provided that
a is greater than 0, then
00:16:32.830 --> 00:16:36.310
at the upper limit, this
exponential becomes 0.
00:16:36.310 --> 00:16:38.870
At the lower limit, of
course, it's one.
00:16:38.870 --> 00:16:44.460
And so what we have finally is
for the Fourier transform
00:16:44.460 --> 00:16:48.780
expression 1 over
a plus j omega.
00:16:52.700 --> 00:16:56.970
Now, this Fourier transform as
I indicated is a complex
00:16:56.970 --> 00:16:58.380
valued function.
00:16:58.380 --> 00:17:04.210
Let's just take a look at what
it looks like graphically.
00:17:04.210 --> 00:17:10.640
We have the expression for the
Fourier transform pair, e to
00:17:10.640 --> 00:17:11.910
the minus a t times [? a ?]
00:17:11.910 --> 00:17:12.839
[? step ?].
00:17:12.839 --> 00:17:16.980
And its Fourier transform is
1 over a plus j omega.
00:17:16.980 --> 00:17:22.609
And I indicated that that's
true for a greater than 0.
00:17:22.609 --> 00:17:26.710
Now, in the expression that we
just worked out, if a is less
00:17:26.710 --> 00:17:33.360
than 0, in fact, the expression
doesn't converge e
00:17:33.360 --> 00:17:38.800
to the minus a t for a
negative as t goes to
00:17:38.800 --> 00:17:44.170
infinity, blows up, and so in
fact the Fourier transform
00:17:44.170 --> 00:17:48.990
doesn't converge except for the
case where a is greater
00:17:48.990 --> 00:17:52.930
than 0 And in fact, there is a
more detailed discussion of
00:17:52.930 --> 00:17:54.990
convergence issues
in the text.
00:17:54.990 --> 00:17:58.000
The convergence issues are very
much the same for the
00:17:58.000 --> 00:18:00.860
Fourier transform as they are
for the Fourier series.
00:18:00.860 --> 00:18:03.510
And in fact, that's not
surprising, because we
00:18:03.510 --> 00:18:05.360
developed the Fourier
transform out of a
00:18:05.360 --> 00:18:07.410
consideration of the
Fourier series.
00:18:07.410 --> 00:18:10.970
So the convergence conditions as
you'll see as you refer in
00:18:10.970 --> 00:18:15.310
detail to the text relate to
whether the time function is
00:18:15.310 --> 00:18:18.400
absolutely integrable under
one set of conditions and
00:18:18.400 --> 00:18:22.290
square integrable under another
set of conditions.
00:18:22.290 --> 00:18:28.110
OK, now, if we plot the Fourier
transform, let's first
00:18:28.110 --> 00:18:30.870
consider the shape of
the time function.
00:18:30.870 --> 00:18:34.580
And as I indicated, we're
restricting the time function
00:18:34.580 --> 00:18:37.630
so that the exponential
factor a is positive.
00:18:37.630 --> 00:18:40.730
In other words, e to the
minus a t decays
00:18:40.730 --> 00:18:42.920
as t goes to infinity.
00:18:42.920 --> 00:18:48.830
The magnitude of the Fourier
transform is as I indicate
00:18:48.830 --> 00:18:53.970
here and the phase below it.
00:18:53.970 --> 00:18:59.350
And there are a number of things
we can see about the
00:18:59.350 --> 00:19:02.630
magnitude and phase of the
Fourier transform for this
00:19:02.630 --> 00:19:07.990
example, which in fact we'll
see in the next lecture are
00:19:07.990 --> 00:19:10.210
properties that apply
more generally.
00:19:10.210 --> 00:19:13.810
For example, the fact that the
Fourier transform magnitude is
00:19:13.810 --> 00:19:17.440
an even function of frequency,
and the phase is an odd
00:19:17.440 --> 00:19:19.870
function of frequency.
00:19:19.870 --> 00:19:26.810
Now, let me also draw your
attention to the fact that on
00:19:26.810 --> 00:19:31.220
this curve we have both positive
frequencies and
00:19:31.220 --> 00:19:32.830
negative frequencies.
00:19:32.830 --> 00:19:35.950
In other words, in our
expression for the Fourier
00:19:35.950 --> 00:19:40.220
transform, it requires
both omega
00:19:40.220 --> 00:19:43.920
positive and omega negative.
00:19:43.920 --> 00:19:46.700
This, of course, was exactly
the same in the case of the
00:19:46.700 --> 00:19:48.160
Fourier series.
00:19:48.160 --> 00:19:53.530
And the reason you should recall
and keep in mind is
00:19:53.530 --> 00:19:56.390
related to the fact that we're
building our signals out of
00:19:56.390 --> 00:20:01.010
complex exponentials, which
require both positive values
00:20:01.010 --> 00:20:03.900
of omega and negative
values of omega.
00:20:03.900 --> 00:20:07.000
Alternatively, if we chosen
other representation, which
00:20:07.000 --> 00:20:10.060
turns out notationally to be
much more difficult, namely
00:20:10.060 --> 00:20:13.300
sines and cosines, then we would
in fact only consider
00:20:13.300 --> 00:20:14.700
positive frequencies.
00:20:14.700 --> 00:20:19.100
So it's important to keep in
mind that, in our case, both
00:20:19.100 --> 00:20:23.050
with the Fourier series and the
Fourier transform, we deal
00:20:23.050 --> 00:20:26.400
and require both positive and
negative frequencies in order
00:20:26.400 --> 00:20:27.650
to build our signals.
00:20:29.910 --> 00:20:34.220
Now, in the graphical
representation that I've shown
00:20:34.220 --> 00:20:39.210
here, I've chosen a linear
amplitude scale and a linear
00:20:39.210 --> 00:20:40.360
frequency scale.
00:20:40.360 --> 00:20:44.070
And that's one graphical
representation for the Fourier
00:20:44.070 --> 00:20:47.550
transform that we'll
typically use.
00:20:47.550 --> 00:20:54.110
There's another one that very
commonly arises, which I'll
00:20:54.110 --> 00:20:57.450
just briefly indicate
for this example.
00:20:57.450 --> 00:21:04.240
And that is what's referred to
as a bode plot in which the
00:21:04.240 --> 00:21:07.640
magnitude is displayed on
a log amplitude and log
00:21:07.640 --> 00:21:09.160
frequency scale.
00:21:09.160 --> 00:21:11.940
And the phase is displayed
on a log frequency scale.
00:21:11.940 --> 00:21:14.140
Let me show you what I mean.
00:21:14.140 --> 00:21:17.560
Here is the general expression
for the bode plot.
00:21:17.560 --> 00:21:23.080
The bode plot expresses for us
the amplitude in terms of the
00:21:23.080 --> 00:21:26.430
logarithm to the base
10 of the magnitude.
00:21:26.430 --> 00:21:33.020
And it also expresses the angle
in both cases expressed
00:21:33.020 --> 00:21:37.810
as a function of a logarithmic
frequency axis.
00:21:37.810 --> 00:21:45.240
So here is the amplitude
as I've displayed it.
00:21:45.240 --> 00:21:50.110
And this is a log magnitude
scale, a logarithmic frequency
00:21:50.110 --> 00:21:54.280
scale as indicated by the fact
that as we move in equal
00:21:54.280 --> 00:21:58.190
increments along this axis,
we change frequency
00:21:58.190 --> 00:22:00.050
by a factor of 10.
00:22:00.050 --> 00:22:07.440
And similarly, what we have is
a display for the phase again
00:22:07.440 --> 00:22:10.190
on a log frequency scale.
00:22:10.190 --> 00:22:15.000
And I indicated that there is
a symmetry to the Fourier
00:22:15.000 --> 00:22:19.570
transform, and so in fact, we
can infer from this particular
00:22:19.570 --> 00:22:23.360
picture what it looks like for
the negative frequencies as
00:22:23.360 --> 00:22:24.695
well as for the positive
frequencies.
00:22:29.560 --> 00:22:39.450
Now, what we've done so far
is to develop the Fourier
00:22:39.450 --> 00:22:44.285
transform on the basis, the
Fourier transform of an
00:22:44.285 --> 00:22:48.170
aperiodic signal on the basis
of periodically repeating it
00:22:48.170 --> 00:22:52.420
and recognizing that the Fourier
series coefficients
00:22:52.420 --> 00:22:57.200
are samples of an envelope and
that these become more finely
00:22:57.200 --> 00:22:59.990
spaced as frequency increases.
00:22:59.990 --> 00:23:08.080
And in fact, we can go back to
our original equation in which
00:23:08.080 --> 00:23:13.810
we developed an envelope
function, and what we had
00:23:13.810 --> 00:23:19.200
indicated is that the Fourier
series coefficients were
00:23:19.200 --> 00:23:22.920
samples of this envelope.
00:23:22.920 --> 00:23:31.290
We then defined this envelope
as the Fourier transform of
00:23:31.290 --> 00:23:33.760
this aperiodic signal.
00:23:33.760 --> 00:23:38.670
So that provided us with a way--
and it was a mechanism--
00:23:38.670 --> 00:23:44.810
for getting a representation
for an aperiodic signal.
00:23:44.810 --> 00:23:50.820
Now, suppose that we have
instead a periodic signal, are
00:23:50.820 --> 00:23:54.660
there, in fact, some statements
that we can make
00:23:54.660 --> 00:23:58.050
about how the Fourier series
coefficients of that are
00:23:58.050 --> 00:24:02.240
related to the Fourier transform
of something.
00:24:02.240 --> 00:24:05.780
Well, in fact, this
statement tells us
00:24:05.780 --> 00:24:07.390
exactly how to do that.
00:24:07.390 --> 00:24:11.000
What this statement says is
that, in fact, the Fourier
00:24:11.000 --> 00:24:16.140
series coefficients are
samples of the Fourier
00:24:16.140 --> 00:24:19.180
transform of one period?
00:24:19.180 --> 00:24:28.450
So if we now consider a periodic
signal, we can in
00:24:28.450 --> 00:24:33.080
fact get the Fourier series
coefficients of that periodic
00:24:33.080 --> 00:24:35.980
signal by considering
the Fourier
00:24:35.980 --> 00:24:38.630
transform of one period.
00:24:38.630 --> 00:24:42.400
Said another way, the Fourier
series coefficients are
00:24:42.400 --> 00:24:45.510
proportional to samples
of the Fourier
00:24:45.510 --> 00:24:47.670
transform of one period.
00:24:47.670 --> 00:24:53.030
So if we consider this a
periodic signal, computed as
00:24:53.030 --> 00:25:00.380
Fourier transform, and selected
these samples that I
00:25:00.380 --> 00:25:06.780
indicate here, namely samples
equally spaced in omega by
00:25:06.780 --> 00:25:12.260
integer multiples of omega 0,
then in fact, those would be
00:25:12.260 --> 00:25:14.680
the Fourier series
coefficients.
00:25:14.680 --> 00:25:22.890
So we can go back to our
example previously that
00:25:22.890 --> 00:25:27.190
involved the square wave.
00:25:27.190 --> 00:25:31.910
And now, in this case, we could
argue that if in fact it
00:25:31.910 --> 00:25:36.490
was the periodic signal that we
started with, we could get
00:25:36.490 --> 00:25:40.610
the Fourier series coefficients
of that by
00:25:40.610 --> 00:25:46.950
thinking about the Fourier
transform of one period, which
00:25:46.950 --> 00:25:49.590
I indicate here.
00:25:49.590 --> 00:25:53.920
And then the Fourier series
coefficients of the periodic
00:25:53.920 --> 00:26:00.070
signal, in fact, are the
appropriate set of samples of
00:26:00.070 --> 00:26:01.320
this envelope.
00:26:04.710 --> 00:26:11.400
All right, now, we have a way of
getting the Fourier series
00:26:11.400 --> 00:26:14.930
coefficients from the Fourier
transform of one period.
00:26:14.930 --> 00:26:18.100
We originally derived the
Fourier transform of one
00:26:18.100 --> 00:26:21.420
period from the Fourier
series.
00:26:21.420 --> 00:26:25.820
What would, in fact, be nice is
if we could incorporate the
00:26:25.820 --> 00:26:28.790
Fourier series and the
Fourier transform
00:26:28.790 --> 00:26:30.530
within a common framework.
00:26:30.530 --> 00:26:34.390
And in fact, it turns out that
there is a very convenient way
00:26:34.390 --> 00:26:38.040
of doing that almost
by definition.
00:26:38.040 --> 00:26:44.380
Essentially, if we consider
what the equation for the
00:26:44.380 --> 00:26:51.640
synthesis looks like in both
cases, we can in effect define
00:26:51.640 --> 00:26:56.820
a Fourier transform for the
periodic signal, which we know
00:26:56.820 --> 00:26:59.360
is represented by its Fourier
series coefficients.
00:26:59.360 --> 00:27:04.150
We can define a Fourier
transform, and the definition
00:27:04.150 --> 00:27:09.960
of the Fourier transform is as
an impulse train, where the
00:27:09.960 --> 00:27:14.220
coefficients in the impulse
train are proportional, with a
00:27:14.220 --> 00:27:17.200
proportionality factor of 2
pi for a more or less a
00:27:17.200 --> 00:27:20.140
bookkeeping reason, proportional
to the Fourier
00:27:20.140 --> 00:27:22.220
series coefficients.
00:27:22.220 --> 00:27:28.430
And the validity of this is,
more or less, can be seen
00:27:28.430 --> 00:27:30.860
essentially by substitution.
00:27:30.860 --> 00:27:37.210
Specifically, here is then the
synthesis equation for the
00:27:37.210 --> 00:27:42.030
Fourier transform if we
substitute this definition for
00:27:42.030 --> 00:27:45.290
the Fourier transform of the
periodic signal into this
00:27:45.290 --> 00:27:51.310
expression then when we do the
appropriate bookkeeping and
00:27:51.310 --> 00:27:54.850
interchange the order of
summation and integration the
00:27:54.850 --> 00:27:58.690
impulse integrates out
to the exponential
00:27:58.690 --> 00:28:02.530
factor that we want.
00:28:02.530 --> 00:28:05.820
So we have the exponential
factor.
00:28:05.820 --> 00:28:07.990
We have the Fourier series
coefficients.
00:28:07.990 --> 00:28:11.450
The 2 pis take care of each
other, and what we're left
00:28:11.450 --> 00:28:16.620
with is the synthesis equation
for aperiodic signal in terms
00:28:16.620 --> 00:28:21.160
of the Fourier transform, or in
terms of its Fourier series
00:28:21.160 --> 00:28:23.600
coefficients.
00:28:23.600 --> 00:28:30.230
Now, we can just see this in
terms of a simple example.
00:28:30.230 --> 00:28:36.570
If we consider the example of a
symmetric square wave, then
00:28:36.570 --> 00:28:40.320
in effect what we're saying is
that for this symmetric square
00:28:40.320 --> 00:28:45.240
wave, this has a set of Fourier
series coefficients,
00:28:45.240 --> 00:28:50.020
which we worked out previously
and which I indicate on this
00:28:50.020 --> 00:28:52.440
figure with a bar graph.
00:28:52.440 --> 00:28:55.930
And really all that we're saying
is that, whereas these
00:28:55.930 --> 00:28:59.840
Fourier series coefficients
are indexed on an integer
00:28:59.840 --> 00:29:04.790
variable k, and [? they're ?]
bars not impulses.
00:29:04.790 --> 00:29:10.020
If we simply redefine or define
the Fourier transform
00:29:10.020 --> 00:29:15.360
of the periodic signal as an
impulse train, where the
00:29:15.360 --> 00:29:20.840
weights of the impulses are 2
pi times the corresponding
00:29:20.840 --> 00:29:25.750
Fourier series coefficients,
then this, in fact, is what we
00:29:25.750 --> 00:29:29.690
would use as the Fourier
transform of
00:29:29.690 --> 00:29:30.940
the periodic signal.
00:29:34.010 --> 00:29:39.080
Now, we've kind of gone back and
forth, and maybe even it
00:29:39.080 --> 00:29:42.180
might seem like we've gone
around in circles.
00:29:42.180 --> 00:29:45.880
So let me just try to summarize
the various
00:29:45.880 --> 00:29:49.490
relationships and steps that
we've gone through, keeping in
00:29:49.490 --> 00:29:52.670
mind that one of our objectives
was first to
00:29:52.670 --> 00:29:56.690
develop a representation for
aperiodic signals and then
00:29:56.690 --> 00:30:01.910
attempt to incorporate within
one framework both periodic
00:30:01.910 --> 00:30:05.530
and aperiodic signals.
00:30:05.530 --> 00:30:11.250
We began with an aperiodic
signal.
00:30:11.250 --> 00:30:15.290
And the strategy was to
develop a Fourier
00:30:15.290 --> 00:30:21.150
representation by constructing
a periodic signal for which
00:30:21.150 --> 00:30:22.860
that was one period.
00:30:22.860 --> 00:30:25.770
And then we let the period
go to infinity,
00:30:25.770 --> 00:30:27.580
as I indicate here.
00:30:27.580 --> 00:30:32.770
So we have an aperiodic
signal.
00:30:32.770 --> 00:30:37.760
We construct a periodic signal,
x tilde of t for which
00:30:37.760 --> 00:30:41.500
one period is the aperiodic
signal.
00:30:41.500 --> 00:30:45.550
X tilde of t, the periodic
signal, has a Fourier series,
00:30:45.550 --> 00:30:50.700
and as its period increases that
approaches the aperiodic
00:30:50.700 --> 00:30:57.050
signal, and the Fourier series
of that approaches the Fourier
00:30:57.050 --> 00:31:00.720
transform of the original
aperiodic signal.
00:31:00.720 --> 00:31:05.700
So that was the first
step we took.
00:31:05.700 --> 00:31:10.940
Now, the second thing that we
recognize is that once we have
00:31:10.940 --> 00:31:15.280
the concept of the Fourier
transform, we can, in fact,
00:31:15.280 --> 00:31:19.570
relate the Fourier series
coefficients to the Fourier
00:31:19.570 --> 00:31:22.210
transform of one period.
00:31:22.210 --> 00:31:27.330
So the second statement that
we made was that if in fact
00:31:27.330 --> 00:31:33.070
we're trying to represent a
periodic signal, we can get
00:31:33.070 --> 00:31:35.900
the Fourier series coefficients
of that by
00:31:35.900 --> 00:31:42.340
computing the Fourier transform
of one period and
00:31:42.340 --> 00:31:47.380
then samples of that Fourier
transform are, in fact, the
00:31:47.380 --> 00:31:50.110
Fourier series coefficients
for the periodic signal.
00:31:53.210 --> 00:31:58.320
Then, the third step that we
took was to inquire as to
00:31:58.320 --> 00:32:02.340
whether there is a Fourier
transform that can
00:32:02.340 --> 00:32:06.420
appropriately be defined for the
periodic signal, and the
00:32:06.420 --> 00:32:11.260
mechanism for doing that was to
recognize that if we simply
00:32:11.260 --> 00:32:14.840
defined the Fourier transform
of the periodic signal as an
00:32:14.840 --> 00:32:19.420
impulse train, where the impulse
heights or areas were
00:32:19.420 --> 00:32:22.580
proportional to the Fourier
series coefficients, then, in
00:32:22.580 --> 00:32:29.630
fact, the Fourier transform
synthesis equation reduced to
00:32:29.630 --> 00:32:32.140
the Fourier series synthesis
equation.
00:32:32.140 --> 00:32:37.960
So the third step, then, was
with a periodic signal.
00:32:37.960 --> 00:32:42.280
The Fourier transform of that
periodic signal, defined as an
00:32:42.280 --> 00:32:44.610
impulse train, where the
heights or areas of the
00:32:44.610 --> 00:32:47.810
impulses are proportional
to the Fourier series
00:32:47.810 --> 00:32:53.040
coefficients, provides us with
a mechanism for combining it
00:32:53.040 --> 00:32:57.825
together the concepts or
notation of the Fourier series
00:32:57.825 --> 00:33:00.900
and Fourier transform.
00:33:00.900 --> 00:33:07.340
So if we just took a very simple
example, here is an
00:33:07.340 --> 00:33:12.950
example in which we have an
aperiodic signal, which is
00:33:12.950 --> 00:33:16.620
just an impulse,
and its Fourier
00:33:16.620 --> 00:33:20.670
transform is just a constant.
00:33:20.670 --> 00:33:24.730
We can think of a periodic
signal associated with this,
00:33:24.730 --> 00:33:29.820
which is this signal
periodically replicated with a
00:33:29.820 --> 00:33:32.300
spacing t 0.
00:33:32.300 --> 00:33:35.720
The Fourier transform of
this is a constant.
00:33:35.720 --> 00:33:38.970
And this, of course, has a
Fourier series representation.
00:33:38.970 --> 00:33:42.800
So the Fourier transform
of the original
00:33:42.800 --> 00:33:46.170
impulse is just a constant.
00:33:46.170 --> 00:33:51.830
The Fourier transform of the
periodic signal is an impulse
00:33:51.830 --> 00:33:56.170
train, where the heights of the
impulses are proportional
00:33:56.170 --> 00:33:58.830
to the Fourier series
coefficients.
00:33:58.830 --> 00:34:02.700
And, of course, we could
previously have computed the
00:34:02.700 --> 00:34:06.680
Fourier series coefficients for
that impulse train, and
00:34:06.680 --> 00:34:09.040
those Fourier series
coefficients are
00:34:09.040 --> 00:34:10.370
as I've shown here.
00:34:10.370 --> 00:34:14.290
So in both of these cases, these
in effect represent just
00:34:14.290 --> 00:34:16.909
a change in notation, where here
we have a bar graph, and
00:34:16.909 --> 00:34:19.199
here we have an impulse train.
00:34:19.199 --> 00:34:23.980
And both of these simply
represent samples of what we
00:34:23.980 --> 00:34:30.690
have above, which is the Fourier
transform of the
00:34:30.690 --> 00:34:31.940
original aperiodic signal.
00:34:35.179 --> 00:34:39.239
Once again, I suspect that kind
of moving back and forth
00:34:39.239 --> 00:34:42.239
and trying to straighten out
when we're talking about
00:34:42.239 --> 00:34:46.989
periodic and aperiodic signals
may require a little mental
00:34:46.989 --> 00:34:48.810
gymnastics initially.
00:34:48.810 --> 00:34:53.310
Basically, what we've tried to
do is incorporate within one
00:34:53.310 --> 00:34:58.200
framework a representation for
both aperiodic and periodic
00:34:58.200 --> 00:35:01.450
signals, and the Fourier
transform provides us with a
00:35:01.450 --> 00:35:04.410
mechanism to do that.
00:35:04.410 --> 00:35:07.640
In the next lecture, I'll
continue with the discussion
00:35:07.640 --> 00:35:11.310
of the continuous-time Fourier
transform in particular
00:35:11.310 --> 00:35:15.260
focusing on a number of its
properties, some of which
00:35:15.260 --> 00:35:18.000
we've already seen, namely
the symmetry properties.
00:35:18.000 --> 00:35:21.360
We'll see lots of other
properties that relate, of
00:35:21.360 --> 00:35:23.780
course, both to the Fourier
transform and
00:35:23.780 --> 00:35:24.970
to the Fourier series.
00:35:24.970 --> 00:35:26.220
Thank you.