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[MUSIC PLAYING]
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PROFESSOR: In the
last lecture, we
00:00:53.340 --> 00:00:56.760
considered the frequency
response of linear shift
00:00:56.760 --> 00:00:59.130
and variance systems.
00:00:59.130 --> 00:01:01.350
In this lecture, what
I would like to do
00:01:01.350 --> 00:01:06.990
is extend some of those ideas
to the notion of the Fourier
00:01:06.990 --> 00:01:12.390
transform, which I'll refer to
as the discrete time Fourier
00:01:12.390 --> 00:01:14.280
transform.
00:01:14.280 --> 00:01:19.020
So let me begin by
reminding you of one
00:01:19.020 --> 00:01:22.360
of the key results
from last time,
00:01:22.360 --> 00:01:24.720
namely, the notion
of the frequency
00:01:24.720 --> 00:01:28.590
response of a linear
shift invariant system.
00:01:28.590 --> 00:01:31.440
For a linear shift
invariant system,
00:01:31.440 --> 00:01:36.090
we saw last time that
one of the key properties
00:01:36.090 --> 00:01:40.290
was that for a complex
exponential input,
00:01:40.290 --> 00:01:44.460
the output is a complex
exponential sequence
00:01:44.460 --> 00:01:50.100
with the same complex frequency,
but with a change in amplitude.
00:01:50.100 --> 00:01:52.470
And the change in
amplitude, that
00:01:52.470 --> 00:01:55.890
is the function h
of e to the j omega,
00:01:55.890 --> 00:02:00.740
we refer to as the frequency
response of the system.
00:02:00.740 --> 00:02:04.070
The expression that we had
for the frequency response,
00:02:04.070 --> 00:02:06.890
in terms of the unit
sample response,
00:02:06.890 --> 00:02:11.570
was that the frequency
response h of e to the j omega,
00:02:11.570 --> 00:02:15.230
is given by the sum of
h of n, the unit sample
00:02:15.230 --> 00:02:20.355
response of the system,
multiplied by e to the minus j
00:02:20.355 --> 00:02:21.290
omega n.
00:02:21.290 --> 00:02:24.800
So this is a relationship
that tells us,
00:02:24.800 --> 00:02:28.040
in terms of the unit sample
response of the system,
00:02:28.040 --> 00:02:31.280
how to get the
frequency response.
00:02:31.280 --> 00:02:34.970
There were two properties
of the frequency response
00:02:34.970 --> 00:02:37.630
that I stressed last time.
00:02:37.630 --> 00:02:41.810
One of them was the fact
that the frequency response
00:02:41.810 --> 00:02:46.070
is a function of a
continuous variable, omega.
00:02:46.070 --> 00:02:50.300
That is, complex
exponential inputs
00:02:50.300 --> 00:02:55.790
can have the frequency variable
omega, vary continuously.
00:02:55.790 --> 00:02:59.090
Omega is a continuous
variable, whereas n
00:02:59.090 --> 00:03:02.160
is the discrete variable.
00:03:02.160 --> 00:03:05.840
So this was one
important property.
00:03:05.840 --> 00:03:10.220
The second important property is
that the frequency response is
00:03:10.220 --> 00:03:13.290
a periodic function of omega.
00:03:13.290 --> 00:03:17.510
And the period is equal 2 pi.
00:03:17.510 --> 00:03:21.730
H of e to the j omega
is equal to h of e
00:03:21.730 --> 00:03:28.810
to the j omega plus 2 pi
k, where k is any integer.
00:03:28.810 --> 00:03:34.540
Now one of the sort of heuristic
explanations or justifications
00:03:34.540 --> 00:03:37.780
for why the frequency
response is periodic,
00:03:37.780 --> 00:03:40.840
as I've tried to indicate
in several lectures,
00:03:40.840 --> 00:03:45.670
is basically tied to the fact
that complex exponentials
00:03:45.670 --> 00:03:53.170
or sinusoidal inputs, are very
periodically with frequency
00:03:53.170 --> 00:03:56.020
over an interval 0 to 2 pi.
00:03:56.020 --> 00:03:59.500
That is, once we've looked at
them for omega between 0 and 2
00:03:59.500 --> 00:04:02.395
pi, if we go further
than that in frequency,
00:04:02.395 --> 00:04:05.710
all we see are the same
complex exponentials
00:04:05.710 --> 00:04:09.100
or the same
sinusoids over again.
00:04:09.100 --> 00:04:11.650
So that, in essence,
is the reason
00:04:11.650 --> 00:04:16.690
why the frequency response is
a periodic function of omega.
00:04:16.690 --> 00:04:19.510
Well, one of the
things we'd like to do
00:04:19.510 --> 00:04:22.810
is develop an
inverse relationship,
00:04:22.810 --> 00:04:27.190
which permits us to get
the unit sample response
00:04:27.190 --> 00:04:29.950
from the frequency response.
00:04:29.950 --> 00:04:35.920
One heuristic notion
or style of developing
00:04:35.920 --> 00:04:38.380
such an inverse
relationship, we can
00:04:38.380 --> 00:04:44.980
get by observing that since
the frequency response is
00:04:44.980 --> 00:04:47.830
a function of a
continuous variable omega,
00:04:47.830 --> 00:04:51.890
and in fact, it's a
periodic function of omega,
00:04:51.890 --> 00:04:54.550
then it should have a Fourier
series representation.
00:04:54.550 --> 00:04:58.050
That is, it's a continuous
periodic function.
00:04:58.050 --> 00:05:00.670
A continuous periodic
function has a Fourier series
00:05:00.670 --> 00:05:02.810
representation.
00:05:02.810 --> 00:05:07.640
Well in fact, if we
look at this expression,
00:05:07.640 --> 00:05:11.620
we see that indeed what
this is is a Fourier series
00:05:11.620 --> 00:05:17.140
expansion of this
periodic function in terms
00:05:17.140 --> 00:05:19.630
of a complex Fourier series.
00:05:19.630 --> 00:05:24.460
The Fourier series expressed in
terms of complex exponentials.
00:05:24.460 --> 00:05:26.890
Well, what are the Fourier
series coefficients?
00:05:26.890 --> 00:05:31.120
The Fourier series
coefficients are the values
00:05:31.120 --> 00:05:35.830
of the unit sample response, in
other words, the values h of n.
00:05:35.830 --> 00:05:39.880
So in fact, this has the
form of a Fourier series.
00:05:39.880 --> 00:05:42.790
That means that these are
the Fourier coefficients.
00:05:42.790 --> 00:05:46.480
That means that we can get these
Fourier coefficients in terms
00:05:46.480 --> 00:05:49.630
of h of e to the j
omega, the way we always
00:05:49.630 --> 00:05:52.660
get Fourier series
coefficients in terms
00:05:52.660 --> 00:05:56.510
of a periodic
continuous function.
00:05:56.510 --> 00:05:59.440
Well, the resulting
expression then
00:05:59.440 --> 00:06:03.700
is that the Fourier series
coefficients, or equivalently
00:06:03.700 --> 00:06:08.080
the unit sample response,
is equal to 1 over 2 pi
00:06:08.080 --> 00:06:10.570
times the integral over
a period, which I've
00:06:10.570 --> 00:06:15.850
taken as minus pi to plus
pi, of h of e to the j omega,
00:06:15.850 --> 00:06:19.600
e to the omega n, d omega.
00:06:19.600 --> 00:06:21.860
This is just simply
the expression
00:06:21.860 --> 00:06:25.270
for Fourier series
coefficients in terms
00:06:25.270 --> 00:06:28.510
of the continuous
periodic function
00:06:28.510 --> 00:06:30.250
that we're dealing with.
00:06:30.250 --> 00:06:34.990
Well that's, in a sense,
a heuristic derivation.
00:06:34.990 --> 00:06:40.720
We can in fact verify that
this expression is valid simply
00:06:40.720 --> 00:06:46.450
by substituting in to the
integral, the expression for h
00:06:46.450 --> 00:06:48.950
of e to the j omega.
00:06:48.950 --> 00:06:51.910
And if we do that,
we end up with,
00:06:51.910 --> 00:06:55.060
for the expression for
h of e to the j omega,
00:06:55.060 --> 00:07:01.420
we have the sum of h of k,
e to the minus j omega k.
00:07:01.420 --> 00:07:07.810
Substituting in, the
result is this integral.
00:07:07.810 --> 00:07:13.300
If we interchange the order
of summation and integration,
00:07:13.300 --> 00:07:17.480
then the resulting
expression is shown here.
00:07:17.480 --> 00:07:22.480
And we can observe
that this integral--
00:07:22.480 --> 00:07:28.540
First of all, let's consider
it for n not equal to k.
00:07:28.540 --> 00:07:33.130
If n is not equal to k, this
is just simply the integral
00:07:33.130 --> 00:07:37.150
over a period of 2 pi
of a complex exponential
00:07:37.150 --> 00:07:40.155
whose period is 2 pi.
00:07:40.155 --> 00:07:42.030
The integral of the real
part is the integral
00:07:42.030 --> 00:07:45.780
of a cosine over an
integral number of periods.
00:07:45.780 --> 00:07:47.700
And the integral of
the imaginary part
00:07:47.700 --> 00:07:52.210
is the integral of the sine over
an integral number of periods.
00:07:52.210 --> 00:07:56.550
Obviously then, this integral,
if n is not equal to k
00:07:56.550 --> 00:07:57.825
is going to be 0.
00:07:57.825 --> 00:08:02.640
So for n not equal to k,
the value of this integral
00:08:02.640 --> 00:08:12.140
is 0, whereas for n equal
to k, this exponent is 0,
00:08:12.140 --> 00:08:17.360
the exponential is unity, unity
integrated from minus pi to pi
00:08:17.360 --> 00:08:18.920
is equal to 2 pi.
00:08:18.920 --> 00:08:22.190
So for n equal to k, the
value of the integral
00:08:22.190 --> 00:08:24.710
is equal to 2 pi.
00:08:24.710 --> 00:08:28.850
Well that means that the
only term in this sum that
00:08:28.850 --> 00:08:32.990
is going to contribute to
the answer is the term for k
00:08:32.990 --> 00:08:36.514
equal to n, because for all
the others the integral is 0.
00:08:36.514 --> 00:08:41.179
So for k equal to n, that's
the only non-zero term, for k
00:08:41.179 --> 00:08:43.280
equal to n, the value
of this integral
00:08:43.280 --> 00:08:47.300
is 2 pi, which will cancel
out this 1 over 2 pi
00:08:47.300 --> 00:08:50.976
and consequently we get h of n.
00:08:50.976 --> 00:08:57.200
All right, so this is, in a
sense, a formal justification
00:08:57.200 --> 00:09:02.330
for the fact that, indeed this
is the inverse relationship
00:09:02.330 --> 00:09:07.280
to express h of n terms
of the frequency response.
00:09:07.280 --> 00:09:11.300
Although, in fact it's more
useful in terms of insight
00:09:11.300 --> 00:09:14.090
to think of it as
the Fourier series
00:09:14.090 --> 00:09:18.080
coefficients of a function,
a periodic function,
00:09:18.080 --> 00:09:22.280
of the continuous
variable omega.
00:09:22.280 --> 00:09:30.410
Now this in essence, is a
Fourier domain representation
00:09:30.410 --> 00:09:33.020
for the unit sample
response of a system.
00:09:33.020 --> 00:09:37.310
And in fact, we can think of
this as a Fourier transform
00:09:37.310 --> 00:09:43.760
or a time domain, Fourier
domain transform pair,
00:09:43.760 --> 00:09:47.660
relating the unit sample
response of a system
00:09:47.660 --> 00:09:49.490
and the frequency
response of the system.
00:09:49.490 --> 00:09:51.840
And we can go back and forth.
00:09:51.840 --> 00:09:59.240
Now one of the obvious
facts is that any sequence,
00:09:59.240 --> 00:10:03.320
if we wanted to, we could
think of as the unit sample
00:10:03.320 --> 00:10:05.950
response of a system.
00:10:05.950 --> 00:10:11.170
Well, that suggests then
that this transform pair
00:10:11.170 --> 00:10:13.810
isn't restricted
to just sequences
00:10:13.810 --> 00:10:17.200
that we explicitly
identify as the unit sample
00:10:17.200 --> 00:10:18.490
response of a system.
00:10:18.490 --> 00:10:22.930
It in fact permits the
representation of an arbitrary,
00:10:22.930 --> 00:10:25.120
not quite arbitrary as
we'll see in a while,
00:10:25.120 --> 00:10:31.000
but the representation of a
more general class of sequences
00:10:31.000 --> 00:10:33.160
than just the ones
that we explicitly
00:10:33.160 --> 00:10:36.490
identify as the unit sample
response of a system.
00:10:36.490 --> 00:10:40.030
That is, we can generalize
this set of ideas
00:10:40.030 --> 00:10:46.480
to the Fourier transform,
which is a frequency domain
00:10:46.480 --> 00:10:51.760
representation of an arbitrary,
again not quite arbitrary,
00:10:51.760 --> 00:10:55.270
but for the time being we'll
consider arbitrary sequence,
00:10:55.270 --> 00:10:57.160
x of n.
00:10:57.160 --> 00:11:02.860
Well, the Fourier transform
then is defined as x of e
00:11:02.860 --> 00:11:06.910
to the j omega, which
is the sum of x of n, e
00:11:06.910 --> 00:11:12.110
to the minus j omega n, the
Fourier transform x of e
00:11:12.110 --> 00:11:17.760
to the j omega, of
a sequence, x of n,
00:11:17.760 --> 00:11:21.840
is essentially the frequency
response of a system whose
00:11:21.840 --> 00:11:24.960
unit sample response
would be x of n.
00:11:24.960 --> 00:11:30.120
So then obviously from what
we just finished discussing,
00:11:30.120 --> 00:11:34.170
we have an inverse relationship
that tells us what x of n
00:11:34.170 --> 00:11:37.260
is in terms of x of
e to the j omega.
00:11:37.260 --> 00:11:39.660
And that then is
this relationship.
00:11:42.420 --> 00:11:51.310
Well this then provides a
transform pair, or a frequency
00:11:51.310 --> 00:11:54.850
domain, time domain
relationship between sequences
00:11:54.850 --> 00:11:57.430
and frequency domain functions.
00:11:57.430 --> 00:12:03.550
And it's useful, in fact,
to interpret this expression
00:12:03.550 --> 00:12:08.920
somewhat heuristically again
as basically corresponding
00:12:08.920 --> 00:12:13.630
to a decomposition of
a sequence, x of n,
00:12:13.630 --> 00:12:20.590
in terms of complex exponentials
with incremental amplitude.
00:12:20.590 --> 00:12:25.660
This is basically an expression
that says that x of n is a sum,
00:12:25.660 --> 00:12:29.200
except that it's sum sort of
in the limit which corresponds
00:12:29.200 --> 00:12:34.060
to an integral, of a set
of complex exponentials
00:12:34.060 --> 00:12:36.340
with amplitudes
that are essentially
00:12:36.340 --> 00:12:40.720
given by the Fourier transform
x of e to the j omega.
00:12:40.720 --> 00:12:43.900
Well in fact, you can see
this a little more explicitly
00:12:43.900 --> 00:12:47.860
if we consider this integral
as the limiting form
00:12:47.860 --> 00:12:51.700
of a sum, the limit,
as delta omega goes
00:12:51.700 --> 00:12:57.940
to 0, of the sum of x of
e to the j k delta omega,
00:12:57.940 --> 00:13:04.630
times delta omega over 2 pi,
e to the j k delta omega n.
00:13:04.630 --> 00:13:10.090
This limit is by definition,
what this integral is.
00:13:10.090 --> 00:13:13.930
So an important
point to keep in mind
00:13:13.930 --> 00:13:18.670
is that basically the
Fourier transform corresponds
00:13:18.670 --> 00:13:22.750
to a decomposition of
a sequence in terms
00:13:22.750 --> 00:13:27.220
of a linear combination
of complex exponentials
00:13:27.220 --> 00:13:30.730
with incremental amplitudes.
00:13:30.730 --> 00:13:34.120
There are a number
of reasons why that's
00:13:34.120 --> 00:13:36.130
an important point of view.
00:13:36.130 --> 00:13:39.340
One of the reasons
is that it leads
00:13:39.340 --> 00:13:43.710
to a very important property
of linear shift invariance
00:13:43.710 --> 00:13:51.100
systems, which I refer to
as the convolution property,
00:13:51.100 --> 00:14:00.430
and which states that if I
have the convolution of two
00:14:00.430 --> 00:14:07.630
sequences, x of n and y
of n, then the Fourier
00:14:07.630 --> 00:14:12.340
transform of those is the
product of their Fourier
00:14:12.340 --> 00:14:13.780
transforms.
00:14:13.780 --> 00:14:19.940
And this of course, shouldn't
be an h, it should be a y.
00:14:19.940 --> 00:14:23.270
Or this shouldn't be a
y, it should be an h.
00:14:23.270 --> 00:14:26.960
The important property is
that the convolution of two
00:14:26.960 --> 00:14:30.620
sequences has, as its
Fourier transform,
00:14:30.620 --> 00:14:35.000
the product of the
Fourier transforms.
00:14:35.000 --> 00:14:39.800
Well let's look at again,
somewhat heuristically,
00:14:39.800 --> 00:14:41.840
an argument that
at least justifies
00:14:41.840 --> 00:14:45.920
this, keeping in mind that
we could go through this more
00:14:45.920 --> 00:14:48.650
formally, plugging sums
into integrals and integrals
00:14:48.650 --> 00:14:51.570
into sums, but
let's not do that.
00:14:51.570 --> 00:14:56.870
Let's look at this from a
somewhat heuristic point
00:14:56.870 --> 00:14:58.860
of view.
00:14:58.860 --> 00:15:03.350
First of all, we
have that again,
00:15:03.350 --> 00:15:06.320
a property that we
began the lecture with,
00:15:06.320 --> 00:15:10.450
that for a linear
shift invariant system,
00:15:10.450 --> 00:15:13.760
a complex exponential
input gives us
00:15:13.760 --> 00:15:18.140
at the output, the same
complex exponential, that
00:15:18.140 --> 00:15:20.480
is the same frequency,
and only one
00:15:20.480 --> 00:15:26.150
of them, multiplied by
h of e to the j omega 0.
00:15:26.150 --> 00:15:30.500
That's a consequence of
linearity and shift invariance.
00:15:30.500 --> 00:15:35.330
We know also, that because of
the fact that the system is
00:15:35.330 --> 00:15:40.190
linear, if we have
a linear combination
00:15:40.190 --> 00:15:45.530
of complex exponentials, then
the output of this system
00:15:45.530 --> 00:15:49.790
is going to be the
same linear combination
00:15:49.790 --> 00:15:54.260
of complex exponentials, with
the amplitudes multiplied
00:15:54.260 --> 00:15:57.410
by h of e to the j omega k.
00:15:57.410 --> 00:16:00.440
In other words, each
of these exponentials
00:16:00.440 --> 00:16:05.510
has as an output ase of k times
h of e to the j omegus of k, e
00:16:05.510 --> 00:16:08.900
to the j omegus of k n.
00:16:08.900 --> 00:16:11.090
A sum of those at
the input gives us
00:16:11.090 --> 00:16:13.340
a sum at the output
because of the fact
00:16:13.340 --> 00:16:16.010
that the system is linear.
00:16:16.010 --> 00:16:18.410
All right, so let's
take an input, which
00:16:18.410 --> 00:16:23.570
is an arbitrary input, more
or less arbitrary, an input
00:16:23.570 --> 00:16:32.590
x of n, which I can express in
terms of its Fourier transform,
00:16:32.590 --> 00:16:38.620
in terms of this inverse
Fourier transform relationship.
00:16:38.620 --> 00:16:44.500
All right, now m as I
emphasized just a minute ago,
00:16:44.500 --> 00:16:51.760
is a decomposition of x of
n as a linear combination
00:16:51.760 --> 00:16:54.190
of complex exponentials.
00:16:54.190 --> 00:16:57.430
This is a linear combination
of complex exponentials
00:16:57.430 --> 00:17:00.140
at the input, so
what's the output?
00:17:00.140 --> 00:17:04.329
Well, the output is then going
to be a linear combination
00:17:04.329 --> 00:17:08.560
of complex exponentials
with the amplitude
00:17:08.560 --> 00:17:11.560
of each one of the
input exponentials,
00:17:11.560 --> 00:17:15.670
multiplied by the frequency
response of the system
00:17:15.670 --> 00:17:18.050
at that frequency.
00:17:18.050 --> 00:17:23.020
So the result is that with
this as the input, what
00:17:23.020 --> 00:17:30.780
we have to get at
the output is this,
00:17:30.780 --> 00:17:34.260
the same linear
combination, the only change
00:17:34.260 --> 00:17:38.070
being that the complex
amplitudes of the input
00:17:38.070 --> 00:17:43.050
are multiplied by h
of e to the j omega.
00:17:43.050 --> 00:17:48.360
Well, this of course, is the
expression for the output.
00:17:48.360 --> 00:17:52.785
We put in x of n, we know
that we're getting out y of n.
00:17:55.430 --> 00:18:01.340
So obviously this then must
be the Fourier transform
00:18:01.340 --> 00:18:04.140
of the output of the system.
00:18:04.140 --> 00:18:07.460
So what this says then
is that the Fourier
00:18:07.460 --> 00:18:13.110
transform of the output y of e
to the j omega, has to be this,
00:18:13.110 --> 00:18:17.270
it has to be x of e to
the j omega times h of e
00:18:17.270 --> 00:18:20.150
to the j omega.
00:18:20.150 --> 00:18:26.180
And basically-- well, and
we know also that y of n
00:18:26.180 --> 00:18:30.650
is going to be equal to x
of n convolved with h of n.
00:18:30.650 --> 00:18:36.470
So basically, this justifies
the convolution property, that
00:18:36.470 --> 00:18:39.500
is the convolution
of two sequences
00:18:39.500 --> 00:18:43.430
has as its Fourier transform,
the product of the Fourier
00:18:43.430 --> 00:18:46.700
transform of each of the
individual sequences.
00:18:46.700 --> 00:18:48.230
Very important property.
00:18:48.230 --> 00:18:52.790
And although we can go through
a formal derivation of this,
00:18:52.790 --> 00:18:58.380
in fact the basic reason for
it is tied to the arguments
00:18:58.380 --> 00:18:59.670
that I've outlined here.
00:18:59.670 --> 00:19:02.430
And in terms of insight,
I feel that it's
00:19:02.430 --> 00:19:05.760
more important to understand
this way of looking at it
00:19:05.760 --> 00:19:10.380
than to understand
a formal derivation.
00:19:10.380 --> 00:19:13.410
Well, this is interesting,
also important,
00:19:13.410 --> 00:19:18.270
and also should be familiar
to you in terms of things
00:19:18.270 --> 00:19:19.930
that you're used
to thinking about
00:19:19.930 --> 00:19:22.740
for continuous time systems.
00:19:22.740 --> 00:19:25.290
Obviously, in
continuous time systems
00:19:25.290 --> 00:19:27.930
the same type of property holds.
00:19:27.930 --> 00:19:32.670
That is, that the
Fourier transform changes
00:19:32.670 --> 00:19:35.400
convolution to multiplication.
00:19:35.400 --> 00:19:39.690
And in fact, it permits the
description of a linear shift
00:19:39.690 --> 00:19:44.640
invariant system to be in
terms of multiplication
00:19:44.640 --> 00:19:47.062
rather than in terms
of convolution.
00:19:47.062 --> 00:19:48.770
And as we'll see in
a number of lectures,
00:19:48.770 --> 00:19:55.080
that basically the basis
for the notions of filtering
00:19:55.080 --> 00:19:57.130
and some other very
important notions,
00:19:57.130 --> 00:19:59.950
and notions of modulation etc.
00:19:59.950 --> 00:20:03.150
OK, key property,
this is a key property
00:20:03.150 --> 00:20:06.540
of linear shift invariant
systems and the Fourier
00:20:06.540 --> 00:20:08.130
transform.
00:20:08.130 --> 00:20:11.910
There are, of course,
lots of other properties
00:20:11.910 --> 00:20:14.910
of linear shift
invariant systems--
00:20:14.910 --> 00:20:17.040
sorry, other
properties, there are
00:20:17.040 --> 00:20:20.280
other properties, obviously, of
linear shift invariant systems.
00:20:20.280 --> 00:20:24.780
There are other properties
of Fourier transforms that
00:20:24.780 --> 00:20:30.000
are important, both in terms of
interpreting Fourier transforms
00:20:30.000 --> 00:20:34.500
and in terms of computing
Fourier transforms
00:20:34.500 --> 00:20:37.590
of a variety of sequences.
00:20:37.590 --> 00:20:42.210
A lot of these properties
will be developed in the text
00:20:42.210 --> 00:20:45.000
and also in the study
guide, so you'll
00:20:45.000 --> 00:20:47.340
have to do the work
rather than me.
00:20:47.340 --> 00:20:51.720
But let me just indicate one
class of properties that,
00:20:51.720 --> 00:20:55.020
again, should be
very familiar to you
00:20:55.020 --> 00:20:59.130
if you relate your thinking
back to continuous time
00:20:59.130 --> 00:21:00.990
Fourier transforms.
00:21:00.990 --> 00:21:04.890
And that is the class
of symmetry properties
00:21:04.890 --> 00:21:08.610
for the special case in which
the sequences that we're
00:21:08.610 --> 00:21:13.320
talking about are
real sequences.
00:21:13.320 --> 00:21:19.020
Well the basic symmetry
property is that for x of n
00:21:19.020 --> 00:21:25.680
real the Fourier transform is
a conjugate symmetric function,
00:21:25.680 --> 00:21:30.930
x of e to the j omega is
equal to x conjugate of e
00:21:30.930 --> 00:21:34.410
to the minus j omega.
00:21:34.410 --> 00:21:39.030
And we can see that,
essentially in a straightforward
00:21:39.030 --> 00:21:44.000
way, that is the derivation is
effectively straightforward.
00:21:44.000 --> 00:21:47.700
Here's x of e to the j
omega, the sum of x of n, e
00:21:47.700 --> 00:21:50.730
to the minus j omega n.
00:21:50.730 --> 00:21:54.150
Here is x of e to
the minus j omega.
00:21:54.150 --> 00:21:56.670
Well, the only difference
between that and that
00:21:56.670 --> 00:22:00.000
is that we replace
omega by minus omega,
00:22:00.000 --> 00:22:02.400
so this becomes a plus sign.
00:22:02.400 --> 00:22:07.110
Of course, it's not this that
we want from this expression,
00:22:07.110 --> 00:22:09.450
it's the conjugate of that.
00:22:09.450 --> 00:22:15.120
So we want to complex
conjugate this.
00:22:15.120 --> 00:22:18.940
Well, we will do the same
on the left--hand side--
00:22:18.940 --> 00:22:21.040
on the right-hand side.
00:22:21.040 --> 00:22:26.100
And so we would conjugate
that and conjugate
00:22:26.100 --> 00:22:33.910
this, which replaces this
plus sign with a minus sign.
00:22:33.910 --> 00:22:39.180
But we're talking about
a real sequence, x of n,
00:22:39.180 --> 00:22:41.920
so x conjugate of
n is just x of n.
00:22:41.920 --> 00:22:46.870
In other words, this is just
the x of an all over again.
00:22:46.870 --> 00:22:51.300
And so we have that x conjugate
of e to the minus j omega
00:22:51.300 --> 00:22:54.180
is the sum of x of
n, e to the minus j
00:22:54.180 --> 00:22:57.960
omega n, which is just
what we have up here.
00:22:57.960 --> 00:23:01.860
So obviously then,
these two are equal.
00:23:01.860 --> 00:23:08.070
So for a real sequence,
the Fourier transform
00:23:08.070 --> 00:23:10.660
is a conjugate
symmetric function.
00:23:10.660 --> 00:23:14.430
This is what we'll call
conjugate symmetric.
00:23:14.430 --> 00:23:18.390
Well let's press that
a little further.
00:23:18.390 --> 00:23:22.290
We have x of e to
the j omega, which
00:23:22.290 --> 00:23:26.430
I can represent in
terms of its real part
00:23:26.430 --> 00:23:31.810
and its imaginary part as
obviously x real, x sub r of e
00:23:31.810 --> 00:23:38.270
to the j omega, plus j times
x sub i of e to the j omega.
00:23:38.270 --> 00:23:42.870
And the conjugate
symmetric counterpart
00:23:42.870 --> 00:23:50.420
is this with omega replaced by
minus omega and the expression
00:23:50.420 --> 00:23:51.770
conjugated.
00:23:51.770 --> 00:23:56.390
So we have x sub r of
e to the minus j omega,
00:23:56.390 --> 00:24:02.090
minus j times x sub i of
e to the minus j omega.
00:24:02.090 --> 00:24:04.910
And we know that
these two are equal.
00:24:04.910 --> 00:24:07.460
Well, if these two are
equal, then these two
00:24:07.460 --> 00:24:11.430
are equal, and so are these.
00:24:11.430 --> 00:24:15.510
The real part of x
of e to the j omega
00:24:15.510 --> 00:24:21.300
must be equal to the real part
of x of e to the minus j omega.
00:24:21.300 --> 00:24:24.150
In other words,
the real part has
00:24:24.150 --> 00:24:30.480
to be the same if omega is
replaced by minus omega.
00:24:30.480 --> 00:24:34.380
That means then that the real
part of the Fourier transform
00:24:34.380 --> 00:24:38.790
is an even function of omega.
00:24:38.790 --> 00:24:43.230
Meaning that if we replace
omega by minus omega
00:24:43.230 --> 00:24:47.760
then the real part
doesn't change.
00:24:47.760 --> 00:24:50.310
The imaginary part, on
the other hand, does.
00:24:50.310 --> 00:24:54.960
In particular, on the basis
of what we're saying here
00:24:54.960 --> 00:24:59.970
and the equality of these, the
imaginary part x sub i of e
00:24:59.970 --> 00:25:03.900
to the j omega, must
be equal to minus,
00:25:03.900 --> 00:25:07.530
don't forget the minus
sign, minus x sub i of e
00:25:07.530 --> 00:25:09.630
to the minus j omega.
00:25:09.630 --> 00:25:15.810
And that says then, that if we
replace omega by minus omega,
00:25:15.810 --> 00:25:19.350
then the sign of the
imaginary part changes.
00:25:19.350 --> 00:25:24.720
So the imaginary part, in fact,
is an odd function of omega.
00:25:24.720 --> 00:25:28.590
The real part is an
even function of omega.
00:25:28.590 --> 00:25:32.550
Well, from this,
or from this, we
00:25:32.550 --> 00:25:37.650
can also show that the magnitude
of the Fourier transform
00:25:37.650 --> 00:25:42.520
is an even function of omega.
00:25:42.520 --> 00:25:47.220
And the angle of the
Fourier transform
00:25:47.220 --> 00:25:51.610
is an odd function of omega.
00:25:51.610 --> 00:25:56.460
And those, of course,
are identical to what
00:25:56.460 --> 00:25:59.640
we know is true for the
continuous time Fourier
00:25:59.640 --> 00:26:01.230
transform.
00:26:01.230 --> 00:26:04.530
Remember however, again, that
these are periodic functions,
00:26:04.530 --> 00:26:09.140
whereas in the continuous
time case, they're not.
00:26:09.140 --> 00:26:12.480
All right well, this
is an introduction
00:26:12.480 --> 00:26:17.820
to the Fourier transform.
00:26:17.820 --> 00:26:22.230
One of the things
that we've refrained
00:26:22.230 --> 00:26:25.020
from doing in all
of these lectures
00:26:25.020 --> 00:26:30.000
is tying our
development too closely
00:26:30.000 --> 00:26:35.940
to the notion of continuous
time signals, and in particular,
00:26:35.940 --> 00:26:38.640
avoiding to some
extent, the notion
00:26:38.640 --> 00:26:43.800
of interpreting discrete time
signals as just simply sampled
00:26:43.800 --> 00:26:47.100
replicas of continuous
time signals.
00:26:47.100 --> 00:26:49.650
And we'll continue to do
that throughout this set
00:26:49.650 --> 00:26:51.160
of lectures.
00:26:51.160 --> 00:26:54.960
But in particular
Fourier transform,
00:26:54.960 --> 00:27:01.980
I think that it's
instructive to tie together,
00:27:01.980 --> 00:27:04.170
at least in terms
of some insight
00:27:04.170 --> 00:27:07.980
into the relationship, the
continuous time Fourier
00:27:07.980 --> 00:27:12.090
transform of obviously
continuous time signal,
00:27:12.090 --> 00:27:17.820
and the discrete time Fourier
transform for a sequence that's
00:27:17.820 --> 00:27:20.970
obtained by periodic sampling.
00:27:20.970 --> 00:27:25.150
That is, equally spaced sampling
of the continuous time signal.
00:27:25.150 --> 00:27:29.460
So what I'd like to do now is
focus on that relationship,
00:27:29.460 --> 00:27:33.660
emphasizing again, that
not all sequences arise
00:27:33.660 --> 00:27:37.560
by periodic sampling of
continuous time signals.
00:27:37.560 --> 00:27:40.200
But for the cases
in which they do,
00:27:40.200 --> 00:27:43.440
the relationship between
the continuous time
00:27:43.440 --> 00:27:46.690
and discrete time Fourier
transform is instructive.
00:27:46.690 --> 00:27:55.170
So let's take a look
at the relationship
00:27:55.170 --> 00:28:02.280
between some continuous time
and discrete time Fourier
00:28:02.280 --> 00:28:07.860
transforms when we obtain
the discrete time signal
00:28:07.860 --> 00:28:11.740
by sampling a
continuous time signal.
00:28:11.740 --> 00:28:17.790
Well, we begin of course,
with a continuous time,
00:28:17.790 --> 00:28:22.470
time function which I denote by
x sub a of t, a sort of meaning
00:28:22.470 --> 00:28:24.030
analog.
00:28:24.030 --> 00:28:27.030
And the steps that
we would go through
00:28:27.030 --> 00:28:31.590
to convert that to a
sequence are first of all,
00:28:31.590 --> 00:28:36.550
to go through a sampler,
which I've indicated here.
00:28:36.550 --> 00:28:42.510
And the output of the sampler is
then a sampled continuous time
00:28:42.510 --> 00:28:46.500
version of this signal.
00:28:46.500 --> 00:28:49.740
This essentially
is the input signal
00:28:49.740 --> 00:28:53.810
multiplied by an impulse train.
00:28:53.810 --> 00:28:56.540
So I've indicated
that here, here's
00:28:56.540 --> 00:28:58.970
the continuous time input.
00:28:58.970 --> 00:29:02.630
Here is the
continuous time output
00:29:02.630 --> 00:29:05.030
of the sampler,
which is an impulse
00:29:05.030 --> 00:29:09.020
train with the envelope
of the impulse train
00:29:09.020 --> 00:29:14.430
being the continuous time
function x sub a of t.
00:29:14.430 --> 00:29:16.460
All right, well this
isn't the sequence.
00:29:16.460 --> 00:29:19.430
This is just simply
an impulse train.
00:29:19.430 --> 00:29:21.980
To turn this into
a sequence we need
00:29:21.980 --> 00:29:26.350
to go into a box,
which I've labeled
00:29:26.350 --> 00:29:30.470
c slash d, meaning continuous
time to discrete time
00:29:30.470 --> 00:29:32.180
converter.
00:29:32.180 --> 00:29:37.250
And the output then
is a sequence x of n.
00:29:37.250 --> 00:29:42.290
The sequence values being
samples of x sub a of t
00:29:42.290 --> 00:29:46.640
at the sampling
instances n times capital
00:29:46.640 --> 00:29:50.180
T. In other words,
it's converting
00:29:50.180 --> 00:29:55.500
the areas of these impulses
into sequence values.
00:29:55.500 --> 00:30:01.520
Now I've illustrated it here
for one choice for the sampling
00:30:01.520 --> 00:30:06.140
interval capital T.
Let's, down here,
00:30:06.140 --> 00:30:10.250
illustrate it for a sampling
interval that's twice as long.
00:30:10.250 --> 00:30:14.840
Well, of course we have the
same continuous time input.
00:30:14.840 --> 00:30:19.610
The sampled output,
x sub a, tilde of t,
00:30:19.610 --> 00:30:24.410
has the same envelope, but
the spacing of the impulses
00:30:24.410 --> 00:30:28.970
is twice what it is here,
and that I've indicated.
00:30:28.970 --> 00:30:32.150
The envelope, of
course, is the same.
00:30:32.150 --> 00:30:34.690
But at the output of
the continuous time
00:30:34.690 --> 00:30:38.190
to discrete time
converter, what do we have?
00:30:38.190 --> 00:30:43.970
Well, we have the areas
of these impulses lined up
00:30:43.970 --> 00:30:50.090
along this axis, again,
at integer values of n.
00:30:50.090 --> 00:30:54.020
That is, the spacing
of the lines,
00:30:54.020 --> 00:30:56.210
when we look at
the sequence here,
00:30:56.210 --> 00:31:00.290
must be exactly the same as
the spacing of the lines here,
00:31:00.290 --> 00:31:03.650
it's just that the values are
different because we picked out
00:31:03.650 --> 00:31:06.150
samples at different instance.
00:31:06.150 --> 00:31:11.420
So in fact, the
envelope here is indeed
00:31:11.420 --> 00:31:15.560
a compressed version of the
envelope that we had here.
00:31:15.560 --> 00:31:17.670
Very important point.
00:31:17.670 --> 00:31:22.760
The point being that no matter
what the sampling rate is
00:31:22.760 --> 00:31:26.600
the sequence values, when we
line them up as a sequence,
00:31:26.600 --> 00:31:31.760
are going to fall at integer
values along this argument,
00:31:31.760 --> 00:31:36.450
n, always at intervals
that correspond to 1.
00:31:36.450 --> 00:31:39.320
Whereas the output
of the sampler
00:31:39.320 --> 00:31:43.700
had impulses occurring at a
spacing of capital T, which
00:31:43.700 --> 00:31:46.610
in this case, was capital
T 1, and in this case
00:31:46.610 --> 00:31:49.520
is capital T 2.
00:31:49.520 --> 00:31:51.380
All right well,
this is essentially
00:31:51.380 --> 00:31:56.090
the sampling process plus the
conversion to a discrete time
00:31:56.090 --> 00:31:57.540
signal.
00:31:57.540 --> 00:32:00.050
And now let's take
a look at what this
00:32:00.050 --> 00:32:07.430
means in terms of the Fourier
transform of the discrete time
00:32:07.430 --> 00:32:11.210
signal as compared with
the Fourier transform
00:32:11.210 --> 00:32:15.090
of the continuous time signal.
00:32:15.090 --> 00:32:17.820
Let's do this in two steps.
00:32:17.820 --> 00:32:20.820
First of all, we
have the sampler.
00:32:20.820 --> 00:32:25.650
Here is the input x sub a of
t, continuous time function.
00:32:25.650 --> 00:32:29.390
Here is the output, x sub a
tilde of t, a continuous time
00:32:29.390 --> 00:32:30.800
function.
00:32:30.800 --> 00:32:34.520
And the output of the
sampler is the input
00:32:34.520 --> 00:32:40.220
to the sampler, multiplied
by an impulse train.
00:32:40.220 --> 00:32:44.990
Or equivalently,
it's an impulse train
00:32:44.990 --> 00:32:47.660
with the areas of
the impulses given
00:32:47.660 --> 00:32:52.190
by the values of x sub
a of t at the times
00:32:52.190 --> 00:32:54.560
that the impulses occur.
00:32:54.560 --> 00:32:57.230
Naught, by the way, is what
I'm using as the notation
00:32:57.230 --> 00:33:02.640
to designate a unit impulse,
unit continuous time impulse.
00:33:02.640 --> 00:33:05.810
Now in the Fourier
transformed domain then,
00:33:05.810 --> 00:33:09.500
the continuous time
Fourier transform,
00:33:09.500 --> 00:33:13.970
is the convolution of
the continuous time
00:33:13.970 --> 00:33:17.720
Fourier transform
of x sub a of t,
00:33:17.720 --> 00:33:21.770
convolved with the Fourier
transform of the impulse train,
00:33:21.770 --> 00:33:25.184
which is an impulse train
in the frequency domain.
00:33:25.184 --> 00:33:26.600
That's a result
that you should be
00:33:26.600 --> 00:33:30.830
familiar with for
continuous time signals.
00:33:30.830 --> 00:33:34.250
Or equivalently, it's
given by 1 over t,
00:33:34.250 --> 00:33:37.160
times the sum of x
sub a of j omega,
00:33:37.160 --> 00:33:42.650
plus j 2 pi r over capital
T. Basically, what that means
00:33:42.650 --> 00:33:48.430
is that the Fourier transform
of this continuous time signal
00:33:48.430 --> 00:33:50.590
is equal to the
Fourier transform
00:33:50.590 --> 00:33:55.000
of this continuous time signal,
but repeated over and over
00:33:55.000 --> 00:34:00.610
again in frequency at intervals
of 2 pi over capital T.
00:34:00.610 --> 00:34:06.100
So this is sort of a standard
sampling theorem kind of result
00:34:06.100 --> 00:34:08.650
in the continuous time
case, and a result
00:34:08.650 --> 00:34:11.770
that you should be more
or less familiar with.
00:34:11.770 --> 00:34:16.929
Let's look at this now from
a different point of view.
00:34:16.929 --> 00:34:21.310
Again looking at x sub
a tilde of j omega.
00:34:21.310 --> 00:34:25.600
Well x sub a tilde of
j omega is the integral
00:34:25.600 --> 00:34:32.050
of x sub a tilde of t, e
to the minus j omega t d t.
00:34:32.050 --> 00:34:35.889
Substitute in for
x sub a tilde of t,
00:34:35.889 --> 00:34:38.949
the relationship in terms
of an impulse train,
00:34:38.949 --> 00:34:42.010
and interchange summation
and integration,
00:34:42.010 --> 00:34:46.659
and I think you could verify
in a very straightforward way
00:34:46.659 --> 00:34:51.820
that what you end up with is
an expression for the Fourier
00:34:51.820 --> 00:34:55.600
transform at the
output of the sampler
00:34:55.600 --> 00:35:01.270
as given by the sampling
values times e to the minus j n
00:35:01.270 --> 00:35:05.470
capital omega T. Capital omega
is a continuous frequency
00:35:05.470 --> 00:35:07.490
variable.
00:35:07.490 --> 00:35:10.580
Now we want to look
at the relationship
00:35:10.580 --> 00:35:15.890
between the Fourier transform,
continuous time of x sub a of t
00:35:15.890 --> 00:35:20.150
and the discrete time
Fourier transform of x of n.
00:35:20.150 --> 00:35:23.420
So we have the
next step, which is
00:35:23.420 --> 00:35:28.280
to put this impulse train into
the continuous to discrete time
00:35:28.280 --> 00:35:30.240
converter.
00:35:30.240 --> 00:35:33.510
And I remind you that we
just developed two results.
00:35:33.510 --> 00:35:37.020
One result was that x
sub a tilde of j omega
00:35:37.020 --> 00:35:39.450
was given by this expression.
00:35:39.450 --> 00:35:44.410
Also we developed that it
was given by this expression.
00:35:44.410 --> 00:35:46.800
Well what's the
Fourier transform
00:35:46.800 --> 00:35:50.820
of the output of the continuous
to discrete time converter?
00:35:50.820 --> 00:35:55.850
Well, it's just simply
x of e to the j omega,
00:35:55.850 --> 00:35:59.380
our discrete time
Fourier transform,
00:35:59.380 --> 00:36:05.880
which is equal to the
sum on n of x of n.
00:36:05.880 --> 00:36:11.770
But x of n is x sub a of n
capital T. So we have x sub a
00:36:11.770 --> 00:36:20.850
of n capital T, times e
to the minus j omega n.
00:36:20.850 --> 00:36:24.810
Well let's compare
this with this.
00:36:24.810 --> 00:36:29.520
We see that they're exactly
the same except that for omega,
00:36:29.520 --> 00:36:33.270
little omega here we
have capital omega times
00:36:33.270 --> 00:36:35.700
capital T there.
00:36:35.700 --> 00:36:40.080
Consequently, we can say that
the discrete time Fourier
00:36:40.080 --> 00:36:43.920
transform is equal
to the Fourier
00:36:43.920 --> 00:36:47.910
transform of the impulse train,
the continuous time Fourier
00:36:47.910 --> 00:36:52.590
transform of the impulse train,
with omega times capital T
00:36:52.590 --> 00:36:53.790
equal to little omega.
00:36:56.380 --> 00:37:01.540
And we had another expression
for the Fourier transform
00:37:01.540 --> 00:37:05.380
of the impulse train,
which we derived here.
00:37:05.380 --> 00:37:08.770
Consequently, the final
result that we end up with
00:37:08.770 --> 00:37:14.980
is that the Fourier transform
of the discrete time sequence
00:37:14.980 --> 00:37:21.490
is equal to 1 over t times
the sum of x sub a of j omega,
00:37:21.490 --> 00:37:28.360
plus j 2 pi r over capital
T, with omega replaced
00:37:28.360 --> 00:37:34.550
by little omega, divided by
capital T. In other words,
00:37:34.550 --> 00:37:38.110
we had the expression that
capital omega times capital T
00:37:38.110 --> 00:37:40.180
is equal to little omega.
00:37:40.180 --> 00:37:44.470
And this then tells us
what the Fourier transform
00:37:44.470 --> 00:37:47.890
of the sequence is in
terms of the Fourier
00:37:47.890 --> 00:37:51.080
transform of the output.
00:37:51.080 --> 00:37:53.510
Well, this is just
the equations.
00:37:53.510 --> 00:37:57.690
Let's take a look at what
this looks like graphically.
00:38:06.760 --> 00:38:13.960
I've depicted here
the continuous time
00:38:13.960 --> 00:38:20.380
Fourier transform of
some time function,
00:38:20.380 --> 00:38:22.480
and I picked the
Fourier transform
00:38:22.480 --> 00:38:26.500
that looks like a triangle.
00:38:26.500 --> 00:38:29.440
Well, first of all
we derived the fact
00:38:29.440 --> 00:38:33.040
that the impulse
train that results
00:38:33.040 --> 00:38:35.980
from sampling
little x sub a of t
00:38:35.980 --> 00:38:41.110
has a Fourier transform, which
is this, periodically repeated
00:38:41.110 --> 00:38:47.830
in frequency with a period in
frequency equal to 2 pi divided
00:38:47.830 --> 00:38:53.990
by capital 1, where capital
1 is the sampling rate.
00:38:53.990 --> 00:38:57.110
And on the basis
of the expression
00:38:57.110 --> 00:39:02.060
that we derived relating the
discrete time Fourier transform
00:39:02.060 --> 00:39:05.430
and the continuous
time Fourier transform,
00:39:05.430 --> 00:39:07.970
the discrete time
Fourier transform
00:39:07.970 --> 00:39:12.830
looks exactly like this,
but with a re-normalization
00:39:12.830 --> 00:39:18.650
of the frequency axis, because
capital omega times capital T
00:39:18.650 --> 00:39:21.540
is equal to little omega.
00:39:21.540 --> 00:39:25.730
Yes, capital omega times capital
T is equal to little omega.
00:39:25.730 --> 00:39:33.170
So where capital omega is
equal to 2 pi over capital T,
00:39:33.170 --> 00:39:35.870
little omega is equal to 2 pi.
00:39:35.870 --> 00:39:40.760
So this point, which
was at pi over capital T
00:39:40.760 --> 00:39:44.930
ends up on the omega,
little omega axis at pi.
00:39:44.930 --> 00:39:50.180
So this picture just simply
gets scaled according
00:39:50.180 --> 00:39:53.690
to the relationship that
capital omega times capital T
00:39:53.690 --> 00:39:56.630
is equal to little omega.
00:39:56.630 --> 00:40:02.310
Well, this is, for one choice
of the sampling period.
00:40:02.310 --> 00:40:06.210
Obviously, if
capital T was large
00:40:06.210 --> 00:40:10.420
enough so that 2 pi over
t 1 was small enough,
00:40:10.420 --> 00:40:13.530
then each of these individual
replicas of the frequency
00:40:13.530 --> 00:40:16.200
response would interact.
00:40:16.200 --> 00:40:19.860
And we wouldn't have just
the simple separation
00:40:19.860 --> 00:40:22.380
of the spectra as we have here.
00:40:22.380 --> 00:40:26.940
We'd have an interaction,
which of course, is
00:40:26.940 --> 00:40:30.960
the interaction and the
relationship between when
00:40:30.960 --> 00:40:33.690
that interaction
occurs and capital T,
00:40:33.690 --> 00:40:37.740
is the basis for the well-known,
and hopefully, theorem
00:40:37.740 --> 00:40:41.550
that you're familiar with
namely, the sampling theorem.
00:40:41.550 --> 00:40:46.830
Well, to illustrate that if we
had a different sampling rate,
00:40:46.830 --> 00:40:50.580
say twice the sampling rate
that we have over here,
00:40:50.580 --> 00:40:53.700
so that capital T is
equal to 2 times capital
00:40:53.700 --> 00:41:01.060
T 1, then starting with the same
continuous time spectrum, what
00:41:01.060 --> 00:41:07.030
we have now is the spectra,
again periodically repeated,
00:41:07.030 --> 00:41:10.270
with a period again, which
is 2 pi over capital T,
00:41:10.270 --> 00:41:15.310
which in this case is
2 pi over capital T 2,
00:41:15.310 --> 00:41:17.740
but now they interact
and of course, we
00:41:17.740 --> 00:41:18.850
have to add these up.
00:41:18.850 --> 00:41:21.730
That was the expression
that we just derived.
00:41:21.730 --> 00:41:26.110
So in that case, we get some
interaction or aliasing,
00:41:26.110 --> 00:41:28.160
as it's referred to.
00:41:28.160 --> 00:41:32.320
And due to this
aliasing, the periodic,
00:41:32.320 --> 00:41:35.410
one period of this
periodic spectrum
00:41:35.410 --> 00:41:39.780
no longer resembles
the original spectrum.
00:41:39.780 --> 00:41:46.320
This is the Fourier transform
for the sampled continuous time
00:41:46.320 --> 00:41:46.980
function.
00:41:46.980 --> 00:41:49.230
That is, this is the
Fourier transform
00:41:49.230 --> 00:41:50.970
for the impulse train.
00:41:50.970 --> 00:41:54.330
And now if we renormalizing
the frequency axis
00:41:54.330 --> 00:41:58.050
so that we express this in terms
of the discrete time frequency
00:41:58.050 --> 00:42:04.560
variable, little omega, then
we simply scale this picture
00:42:04.560 --> 00:42:09.390
so that we end up with
2 pi over capital T,
00:42:09.390 --> 00:42:14.760
corresponding to 2
pi in little omega.
00:42:14.760 --> 00:42:19.080
Pi over capital T
corresponding to pi.
00:42:19.080 --> 00:42:20.910
And we see in fact, as--
00:42:20.910 --> 00:42:24.240
we better see, that is, it
better turn out this way,
00:42:24.240 --> 00:42:28.080
that the spectrum in
the discrete time case
00:42:28.080 --> 00:42:31.110
is a periodic function of omega.
00:42:31.110 --> 00:42:32.940
In other words, it's periodic.
00:42:32.940 --> 00:42:37.500
Furthermore, the period
is given by 2 pi,
00:42:37.500 --> 00:42:43.440
whereas here the period was
2 pi divided by capital T.
00:42:43.440 --> 00:42:46.290
All right, this perhaps
takes a little digesting.
00:42:46.290 --> 00:42:47.880
And you'll have
some chance to do
00:42:47.880 --> 00:42:53.040
that as we work some problems.
00:42:53.040 --> 00:42:55.980
We, meaning you, as
you work some problems
00:42:55.980 --> 00:42:59.580
in the study guide and
digest this a little
00:42:59.580 --> 00:43:01.540
while you're reading the text.
00:43:01.540 --> 00:43:04.310
But it's an important
relationship
00:43:04.310 --> 00:43:07.940
and it's important
to understand it.
00:43:07.940 --> 00:43:13.440
Now, this is basically
the Fourier transform,
00:43:13.440 --> 00:43:15.990
the relationship between
the discrete time
00:43:15.990 --> 00:43:18.990
and continuous time
Fourier transform.
00:43:18.990 --> 00:43:23.370
One of the difficulties
with the Fourier transform,
00:43:23.370 --> 00:43:28.440
which I've avoided illustrating
explicitly in this lecture,
00:43:28.440 --> 00:43:30.030
is that the Fourier
transform doesn't
00:43:30.030 --> 00:43:32.040
exist for all sequences.
00:43:32.040 --> 00:43:35.950
In particular, it doesn't
converge for all sequences.
00:43:35.950 --> 00:43:40.920
And this is a problem
which we can get around
00:43:40.920 --> 00:43:44.520
by generalizing the
notion of the Fourier
00:43:44.520 --> 00:43:49.260
transform to what we'll
call the z transform.
00:43:49.260 --> 00:43:53.010
And the z transform,
as you'll observe,
00:43:53.010 --> 00:43:57.660
is like, in the continuous time
case, the Laplace transform.
00:43:57.660 --> 00:44:01.000
And this is what we'll go
on to in the next lecture.
00:44:01.000 --> 00:44:02.500
Thanks
00:44:02.500 --> 00:44:04.650
[MUSIC PLAYING]