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ALAN V. OPPENHEIM: OK.
00:00:54.450 --> 00:00:56.820
In the last several
lectures, we've
00:00:56.820 --> 00:01:01.940
been discussing the
Fourier and Z-transforms.
00:01:01.940 --> 00:01:03.880
And we've seen in
particular that the Fourier
00:01:03.880 --> 00:01:07.320
and Z-transforms
provide us with a set
00:01:07.320 --> 00:01:12.030
of important analytical tools
for representing discrete time
00:01:12.030 --> 00:01:17.980
signals, and also for dealing
with discrete time systems.
00:01:17.980 --> 00:01:21.630
We saw, for example, that
through the use of the Fourier
00:01:21.630 --> 00:01:26.410
transform or the Z-transform,
we could convert convolution
00:01:26.410 --> 00:01:29.260
in the time domain
to multiplication
00:01:29.260 --> 00:01:32.260
in either the frequency domain
in the Fourier transform
00:01:32.260 --> 00:01:35.170
case, or more
generally, in the Z
00:01:35.170 --> 00:01:39.740
domain in the Z-transform case.
00:01:39.740 --> 00:01:43.790
Now one of the things that
it's important to recognize
00:01:43.790 --> 00:01:47.420
is that for the most part,
the Fourier transform
00:01:47.420 --> 00:01:52.010
and the Z-transform are
primarily analytical tools.
00:01:52.010 --> 00:01:55.700
That is, it would
be hard to imagine
00:01:55.700 --> 00:01:58.580
implementing, for
example, a discrete time
00:01:58.580 --> 00:02:02.630
system by first
computing the Fourier
00:02:02.630 --> 00:02:05.810
transform of the
sequences, multiplying
00:02:05.810 --> 00:02:08.810
the Fourier transforms
together, and then computing
00:02:08.810 --> 00:02:10.789
the inverse transform.
00:02:10.789 --> 00:02:14.450
One of the reasons that that's
obviously a difficult thing
00:02:14.450 --> 00:02:18.230
to do computationally
is that we saw,
00:02:18.230 --> 00:02:23.240
as we discussed, for example,
the Fourier transform,
00:02:23.240 --> 00:02:25.790
that the Fourier
transform is a function
00:02:25.790 --> 00:02:27.980
of a continuous variable.
00:02:27.980 --> 00:02:31.340
That is, omega in
the Fourier transform
00:02:31.340 --> 00:02:33.472
is a continuous variable.
00:02:33.472 --> 00:02:35.180
So that, in fact, if
we wanted to compute
00:02:35.180 --> 00:02:38.030
the Fourier
transform explicitly,
00:02:38.030 --> 00:02:40.880
we would have to compute
it at an infinite number
00:02:40.880 --> 00:02:42.970
of frequencies.
00:02:42.970 --> 00:02:48.990
Similarly, we have a situation
like that for the Z-transform.
00:02:48.990 --> 00:02:51.060
Well, today I'd
like to introduce
00:02:51.060 --> 00:02:55.890
a third transform,
which I'll refer to as
00:02:55.890 --> 00:02:59.570
the discrete Fourier transform.
00:02:59.570 --> 00:03:03.930
The discrete Fourier
transform is similar in style
00:03:03.930 --> 00:03:06.600
to the Fourier transform
and the Z-transform,
00:03:06.600 --> 00:03:11.010
as we've been talking about,
in the sense that more or less,
00:03:11.010 --> 00:03:13.590
the discrete Fourier
transform maps
00:03:13.590 --> 00:03:17.640
convolution to multiplication.
00:03:17.640 --> 00:03:20.050
Also, there are
properties of Fourier
00:03:20.050 --> 00:03:22.620
transform-- the discrete
Fourier transform--
00:03:22.620 --> 00:03:25.380
that are similar to the
properties of the Fourier
00:03:25.380 --> 00:03:29.820
transform and Z-transform, as
we've been talking about them.
00:03:29.820 --> 00:03:33.330
But the discrete
Fourier transform
00:03:33.330 --> 00:03:36.600
is different than
the other transforms
00:03:36.600 --> 00:03:41.520
that we've been discussing in
a number of important respects.
00:03:41.520 --> 00:03:47.930
One of the respects, one of
the reasons that it's different
00:03:47.930 --> 00:03:54.380
is that it is a transform that
can be explicitly evaluated.
00:03:54.380 --> 00:03:57.890
And consequently,
it is important,
00:03:57.890 --> 00:04:01.580
not only in the analysis
of discrete time systems,
00:04:01.580 --> 00:04:06.200
but also, as we'll see, in the
implementation of discrete time
00:04:06.200 --> 00:04:07.130
systems.
00:04:07.130 --> 00:04:10.970
That is, many digital signal
processing algorithms,
00:04:10.970 --> 00:04:16.339
as we'll see, actually involve
the explicit computation
00:04:16.339 --> 00:04:19.399
of the discrete Fourier
transform, which
00:04:19.399 --> 00:04:23.720
is the Fourier transform that
we're about to introduce.
00:04:23.720 --> 00:04:27.290
Now let me try to explain
a little about what
00:04:27.290 --> 00:04:29.030
the discrete
Fourier transform is
00:04:29.030 --> 00:04:33.040
before we look at it in detail.
00:04:33.040 --> 00:04:37.060
Basically, the discrete
Fourier transform
00:04:37.060 --> 00:04:40.480
is related to the
Fourier transform
00:04:40.480 --> 00:04:43.000
that we've been
discussing in the sense
00:04:43.000 --> 00:04:47.590
that it corresponds to samples
of the Fourier transform,
00:04:47.590 --> 00:04:52.540
or more generally, samples
of the Z-transform.
00:04:52.540 --> 00:04:58.000
Now that requires that we
impose some restrictions
00:04:58.000 --> 00:05:00.820
on the sequences that
we're representing
00:05:00.820 --> 00:05:02.440
through that transform.
00:05:02.440 --> 00:05:07.420
And as we'll see, it turns out
that we can represent sequences
00:05:07.420 --> 00:05:09.730
that are of finite length.
00:05:09.730 --> 00:05:14.290
That is, only have a finite
number of non-zero samples.
00:05:14.290 --> 00:05:17.920
We can represent those sequences
by samples of their Fourier
00:05:17.920 --> 00:05:19.120
transform.
00:05:19.120 --> 00:05:24.960
Those samples then correspond to
the discrete Fourier transform.
00:05:24.960 --> 00:05:28.380
Well, there are some
issues that arise
00:05:28.380 --> 00:05:30.780
in looking at the discrete
Fourier transform,
00:05:30.780 --> 00:05:34.830
or DFT, as I'll refer to it.
00:05:34.830 --> 00:05:39.720
And a number of these
issues relate to the fact
00:05:39.720 --> 00:05:44.220
that the discrete
Fourier transform
00:05:44.220 --> 00:05:49.750
has properties that are somewhat
different, and also somewhat
00:05:49.750 --> 00:05:52.570
similar, to the Fourier
transform properties
00:05:52.570 --> 00:05:54.820
that we've been discussing.
00:05:54.820 --> 00:05:56.890
There are lots of
ways of introducing
00:05:56.890 --> 00:05:59.830
the discrete Fourier transform.
00:05:59.830 --> 00:06:04.990
And the one that I guess I
find the most interesting,
00:06:04.990 --> 00:06:09.190
and the most satisfactory, is
to relate the discrete Fourier
00:06:09.190 --> 00:06:12.850
transform to the
discrete Fourier
00:06:12.850 --> 00:06:16.690
series for periodic sequences.
00:06:16.690 --> 00:06:19.960
And in particular,
relate the notion
00:06:19.960 --> 00:06:25.060
of finite length sequences
to periodic sequences.
00:06:25.060 --> 00:06:27.280
Well, let me explain in
a little more detail what
00:06:27.280 --> 00:06:30.280
I mean by that.
00:06:30.280 --> 00:06:33.720
Let's consider, first
of all, a sequence
00:06:33.720 --> 00:06:40.950
x of n, which I'll restrict to
be a finite length sequence.
00:06:40.950 --> 00:06:46.090
The sequence, for example,
one example indicated here,
00:06:46.090 --> 00:06:53.290
has a set of non-zero values
only over a finite range
00:06:53.290 --> 00:06:56.010
of the argument n.
00:06:56.010 --> 00:06:59.790
Here is a sequence that
is of finite length.
00:06:59.790 --> 00:07:03.370
And I've chosen to refer to it
as a sequence of finite length
00:07:03.370 --> 00:07:08.540
capital N. 0 outside
the range from little n
00:07:08.540 --> 00:07:12.260
equals 0 to little n
equals capital N minus 1.
00:07:12.260 --> 00:07:15.730
It's 0 outside that range.
00:07:15.730 --> 00:07:19.150
It's interesting,
just as an aside,
00:07:19.150 --> 00:07:22.180
to note that for this
particular sequence,
00:07:22.180 --> 00:07:25.690
it's also actually 0 outside
the range from little n
00:07:25.690 --> 00:07:29.410
equals 1 to capital N minus 1.
00:07:29.410 --> 00:07:33.970
And in general, obviously, if
I talk about a finite length
00:07:33.970 --> 00:07:37.780
sequence of length
capital N, I can also
00:07:37.780 --> 00:07:40.810
refer to it as a finite
length sequence of length
00:07:40.810 --> 00:07:44.530
greater than capital N. That
is, the important statement
00:07:44.530 --> 00:07:46.990
about the sequence
being finite length,
00:07:46.990 --> 00:07:51.010
of finite length capital N, is
that the sequence values are
00:07:51.010 --> 00:07:56.740
0 outside the range 0
to capital N minus 1.
00:07:56.740 --> 00:07:59.200
Although obviously,
the sequence could also
00:07:59.200 --> 00:08:02.065
be 0 inside that range
for some of the values.
00:08:04.630 --> 00:08:10.430
Now the basic notion that
leads to the discrete Fourier
00:08:10.430 --> 00:08:13.600
transform, or one way of
looking at the discrete Fourier
00:08:13.600 --> 00:08:18.040
transform, is to recognize
that if I have a finite length
00:08:18.040 --> 00:08:21.510
sequence, as I
have here, I could
00:08:21.510 --> 00:08:25.950
construct from that sequence
a periodic sequence.
00:08:25.950 --> 00:08:31.200
And let me denote the periodic
sequence by x tilde of n.
00:08:31.200 --> 00:08:34.169
In general, by the way,
when I refer to a sequence
00:08:34.169 --> 00:08:36.030
with a tilde on it,
that will always
00:08:36.030 --> 00:08:38.760
correspond to a
periodic sequence.
00:08:38.760 --> 00:08:42.030
And let me construct
this periodic sequence
00:08:42.030 --> 00:08:45.180
by simply taking the
finite length sequence
00:08:45.180 --> 00:08:49.770
and repeating it over and over
again with a period of capital
00:08:49.770 --> 00:08:52.930
N. In other words,
I can construct
00:08:52.930 --> 00:08:58.890
the periodic sequence, x
tilde of n equal to x of n
00:08:58.890 --> 00:09:02.370
plus x of n shifted
to the left by capital
00:09:02.370 --> 00:09:07.560
N and x of n shifted to the
right by capital N and 2n, 2
00:09:07.560 --> 00:09:10.620
capital N, and 3
capital N, et cetera.
00:09:10.620 --> 00:09:14.370
In other words, just
simply taking this sequence
00:09:14.370 --> 00:09:17.580
and repeating it
over and over again
00:09:17.580 --> 00:09:20.100
with a period of capital N.
00:09:20.100 --> 00:09:24.660
So obviously I can generate,
from a finite length
00:09:24.660 --> 00:09:29.050
sequence, a periodic sequence.
00:09:29.050 --> 00:09:32.160
And in fact, I could get
the finite length sequence
00:09:32.160 --> 00:09:38.010
back from the periodic sequence
by simply extracting one period
00:09:38.010 --> 00:09:40.140
of this periodic sequence.
00:09:40.140 --> 00:09:46.680
In other words, I can get x of
n back from x tilde of n simply
00:09:46.680 --> 00:09:53.100
by multiplying by unity for
n between 0 and capital N
00:09:53.100 --> 00:10:00.650
minus 1, and 0
outside that range.
00:10:00.650 --> 00:10:02.200
Now the important point here--
00:10:02.200 --> 00:10:04.270
there are a couple
of important points.
00:10:04.270 --> 00:10:08.260
One of them is that if I have
a finite length sequence,
00:10:08.260 --> 00:10:11.230
I can turn it into
a periodic sequence.
00:10:11.230 --> 00:10:13.540
If I have a periodic
sequence, I can
00:10:13.540 --> 00:10:15.460
get back to the
finite length sequence
00:10:15.460 --> 00:10:18.280
simply by extracting one period.
00:10:18.280 --> 00:10:22.450
Or what that essentially
says is that there really
00:10:22.450 --> 00:10:25.900
isn't much difference between
a finite length sequence
00:10:25.900 --> 00:10:28.030
and a periodic sequence.
00:10:28.030 --> 00:10:30.430
That is, a finite
length sequence
00:10:30.430 --> 00:10:34.440
is defined by capital N values.
00:10:34.440 --> 00:10:37.950
A periodic sequence is also
defined by capital N values
00:10:37.950 --> 00:10:41.490
because once I specify
a single period,
00:10:41.490 --> 00:10:44.460
then I don't have
any more degrees
00:10:44.460 --> 00:10:48.060
of freedom in specifying
the rest of the sequence.
00:10:48.060 --> 00:10:51.300
And that's an important point
to keep in mind, particularly
00:10:51.300 --> 00:10:53.440
as we go through the
next several lectures.
00:10:53.440 --> 00:10:58.610
A finite length sequence is very
similar to a periodic sequence
00:10:58.610 --> 00:11:03.690
in that both of them are simply
defined by capital N values.
00:11:03.690 --> 00:11:05.670
One way to think of
that, by the way,
00:11:05.670 --> 00:11:11.190
is to think of taking a finite
length sequence of capital N
00:11:11.190 --> 00:11:13.040
values.
00:11:13.040 --> 00:11:15.440
Instead of drawing it
along a straight line
00:11:15.440 --> 00:11:19.130
as I've done here, imagine
taking this finite length
00:11:19.130 --> 00:11:21.440
sequence and wrapping it
around the circumference
00:11:21.440 --> 00:11:23.380
of a cylinder.
00:11:23.380 --> 00:11:27.400
So we start with the finite
length sequence and just simply
00:11:27.400 --> 00:11:31.330
display it wrapped around the
circumference of a cylinder.
00:11:31.330 --> 00:11:34.060
As we run around the
cylinder, over and over
00:11:34.060 --> 00:11:40.780
again, what we see is the
periodic sequence x tilde of n.
00:11:40.780 --> 00:11:43.810
So in a sense, the
periodic sequence
00:11:43.810 --> 00:11:46.240
is just simply like the
finite length sequence,
00:11:46.240 --> 00:11:48.730
but wrapped on a cylinder
instead of laid out
00:11:48.730 --> 00:11:50.260
in a straight line.
00:11:50.260 --> 00:11:52.810
And that's also a
picture that will
00:11:52.810 --> 00:11:55.420
recur several times as we
go through the next several
00:11:55.420 --> 00:11:56.530
lectures.
00:11:56.530 --> 00:12:00.130
So we can generate the periodic
sequence from the finite length
00:12:00.130 --> 00:12:01.690
sequence.
00:12:01.690 --> 00:12:07.700
We can also recover
the sequence x of n
00:12:07.700 --> 00:12:12.440
from the periodic sequence by
extracting just one period,
00:12:12.440 --> 00:12:18.020
as I've indicated here, x of n
is x tilde of n in the range,
00:12:18.020 --> 00:12:21.530
little n between 0
and capital N minus 1.
00:12:21.530 --> 00:12:24.200
And it's equal to 0, otherwise.
00:12:24.200 --> 00:12:27.770
Or what we'll use as a
convenient form of notation
00:12:27.770 --> 00:12:33.560
for that is to express x of
n, the finite length sequence,
00:12:33.560 --> 00:12:38.480
as the periodic sequence,
x tilde of n multiplied
00:12:38.480 --> 00:12:44.750
by another sequence, which I'll
denote as R sub capital N of n,
00:12:44.750 --> 00:12:46.880
where R sub capital
N of n is just
00:12:46.880 --> 00:12:49.810
simply a rectangular sequence.
00:12:49.810 --> 00:12:54.530
So R sub capital N of n is
1 for little n between 0
00:12:54.530 --> 00:13:00.010
and capital N minus 1, and
it's equal to 0, otherwise.
00:13:00.010 --> 00:13:00.510
All right.
00:13:00.510 --> 00:13:03.490
So it's important to
begin, at this point,
00:13:03.490 --> 00:13:08.150
to think of finite length
and periodic sequences
00:13:08.150 --> 00:13:10.940
as more or less the same
type of thing in the sense
00:13:10.940 --> 00:13:15.010
that it's easy to go back and
forth from one to the other.
00:13:15.010 --> 00:13:18.240
Now, why is this point
of view important?
00:13:18.240 --> 00:13:22.140
Well, we know certainly in
the continuous time case
00:13:22.140 --> 00:13:26.250
that a periodic time function
can be represented by a Fourier
00:13:26.250 --> 00:13:27.730
series.
00:13:27.730 --> 00:13:30.760
In the discrete time
case, a periodic sequence,
00:13:30.760 --> 00:13:34.960
likewise, can be represented
by a Fourier series.
00:13:34.960 --> 00:13:38.440
And the idea, that
is, the key point
00:13:38.440 --> 00:13:41.410
behind the discrete
Fourier transform
00:13:41.410 --> 00:13:46.450
is that we can use the
Fourier series representation
00:13:46.450 --> 00:13:50.590
of the periodic sequence to
represent the finite length
00:13:50.590 --> 00:13:51.670
sequence.
00:13:51.670 --> 00:13:53.960
That is, that, in
essence, provides
00:13:53.960 --> 00:13:58.720
a Fourier kind of representation
for a finite length sequence.
00:13:58.720 --> 00:14:04.930
So we have the notion then
that the periodic sequence x
00:14:04.930 --> 00:14:10.940
tilde of n has a Fourier
series representation.
00:14:10.940 --> 00:14:15.660
We can compute the
discrete Fourier series
00:14:15.660 --> 00:14:18.780
of this periodic sequence.
00:14:18.780 --> 00:14:20.190
And as we'll see--
00:14:20.190 --> 00:14:21.840
not in this lecture,
but as we'll
00:14:21.840 --> 00:14:24.690
see in more detail
in the next lecture--
00:14:24.690 --> 00:14:29.160
it is discrete Fourier
series representation
00:14:29.160 --> 00:14:33.120
of this periodic
sequence that is
00:14:33.120 --> 00:14:38.820
what we'll call the discrete
Fourier transform of x of n.
00:14:38.820 --> 00:14:42.240
So let's begin then
with a discussion
00:14:42.240 --> 00:14:46.740
of the discrete Fourier
series of periodic sequences
00:14:46.740 --> 00:14:51.630
with the idea that we'll be
applying that representation
00:14:51.630 --> 00:14:54.810
to the representation of
finite length sequence.
00:14:54.810 --> 00:14:57.180
And that representation
is what will
00:14:57.180 --> 00:15:01.490
correspond to the discrete
Fourier transform.
00:15:01.490 --> 00:15:01.990
OK.
00:15:01.990 --> 00:15:06.930
So we want to talk about
the discrete Fourier series.
00:15:06.930 --> 00:15:13.430
We're considering a sequence x
tilde of n, which is periodic,
00:15:13.430 --> 00:15:19.160
and it's period
we'll call capital N.
00:15:19.160 --> 00:15:21.860
In the continuous
time case, we know
00:15:21.860 --> 00:15:24.470
that if we have a
periodic time function,
00:15:24.470 --> 00:15:28.910
we can represent it as a linear
combination of harmonically
00:15:28.910 --> 00:15:31.070
related complex exponentials.
00:15:31.070 --> 00:15:33.920
That's the Fourier
series representation
00:15:33.920 --> 00:15:36.150
in the continuous time case.
00:15:36.150 --> 00:15:39.500
And I'll just simply
state without proof
00:15:39.500 --> 00:15:42.500
that the same kind
of relationship
00:15:42.500 --> 00:15:44.750
holds in the discrete time case.
00:15:44.750 --> 00:15:50.900
That is, we can represent a
periodic sequence x tilde of n
00:15:50.900 --> 00:15:56.540
as a linear combination of
complex exponentials, which
00:15:56.540 --> 00:16:00.260
are harmonically related
to the frequency.
00:16:00.260 --> 00:16:04.100
Or equivalently, the
reciprocal of the period,
00:16:04.100 --> 00:16:08.360
so that this forms a
general relationship
00:16:08.360 --> 00:16:13.510
for a discrete Fourier series
of a periodic sequence x
00:16:13.510 --> 00:16:15.440
tilde of n.
00:16:15.440 --> 00:16:17.510
That is, these are the
harmonically related
00:16:17.510 --> 00:16:22.310
complex exponentials, just as
we form linear combinations
00:16:22.310 --> 00:16:26.120
of harmonically related complex
exponentials for the Fourier
00:16:26.120 --> 00:16:29.560
series in a
continuous time case.
00:16:29.560 --> 00:16:31.190
What are the Fourier
coefficients?
00:16:31.190 --> 00:16:34.840
Well, it's, of course,
these capital X tildes
00:16:34.840 --> 00:16:41.030
of k, which we'll have a little
more to say about in a minute.
00:16:41.030 --> 00:16:46.520
Well, notice that I haven't
specified as of yet any limits
00:16:46.520 --> 00:16:48.560
on this summation.
00:16:48.560 --> 00:16:52.910
And in particular,
what we need to examine
00:16:52.910 --> 00:16:57.140
is how many distinct
periodically or harmonically
00:16:57.140 --> 00:17:01.590
related complex
exponentials there are.
00:17:01.590 --> 00:17:06.819
Well, let's take a look at
the complex exponential,
00:17:06.819 --> 00:17:10.020
the set of complex
exponentials e to the j, 2
00:17:10.020 --> 00:17:13.869
pi over capital N, little nk.
00:17:13.869 --> 00:17:16.000
And the statement
that I want to make
00:17:16.000 --> 00:17:20.290
is that these complex
exponentials are periodic.
00:17:20.290 --> 00:17:23.060
Of course, we know that
they're periodic in n.
00:17:23.060 --> 00:17:25.429
But they're are
also periodic in k.
00:17:28.300 --> 00:17:34.366
As we vary k from 0
to capital N minus 1,
00:17:34.366 --> 00:17:38.240
we generate all of the
possible harmonically related
00:17:38.240 --> 00:17:41.900
complex exponentials with
this fundamental frequency, 2
00:17:41.900 --> 00:17:44.810
pi over capital N.
Well, we can see
00:17:44.810 --> 00:17:52.800
that very simply by substituting
in for k, k plus capital N.
00:17:52.800 --> 00:17:56.310
And then recognizing
that we can break
00:17:56.310 --> 00:17:59.430
this complex exponential
into the product of two
00:17:59.430 --> 00:18:06.420
complex exponentials e to the
j, 2 pi over capital N, nk,
00:18:06.420 --> 00:18:11.970
and e to the j, 2 pi
over capital N times
00:18:11.970 --> 00:18:15.600
little n times capital N.
00:18:15.600 --> 00:18:19.370
Well, these capital N's
cancel each other out.
00:18:19.370 --> 00:18:23.450
This factor is then e to
the j, 2 pi times little n.
00:18:23.450 --> 00:18:27.650
Well, any integer multiple
of 2 pi up here then just
00:18:27.650 --> 00:18:31.250
simply reduces this to unity.
00:18:31.250 --> 00:18:36.790
So that, in fact, e to the j,
2 pi over capital N, little n
00:18:36.790 --> 00:18:40.095
times k plus capital
N is the same as
00:18:40.095 --> 00:18:45.310
e to the j, 2 pi over
capital N, little n times k.
00:18:45.310 --> 00:18:47.620
Well, that shouldn't be
surprising, actually,
00:18:47.620 --> 00:18:50.740
because that's a point
that's come up time
00:18:50.740 --> 00:18:53.170
and again as we've been
going through these lectures.
00:18:53.170 --> 00:18:59.950
The point being that
sinusoids in the discrete time
00:18:59.950 --> 00:19:04.750
domain, as we vary
sinusoids in frequency--
00:19:04.750 --> 00:19:10.400
we've seen time and again,
in the range 0 to 2 pi--
00:19:10.400 --> 00:19:14.230
in fact, those are all the
sinusoids that we can generate.
00:19:14.230 --> 00:19:16.180
And if we keep
going in frequency,
00:19:16.180 --> 00:19:19.070
we just simply see the
same ones over again.
00:19:19.070 --> 00:19:22.210
So this is a
consequence of that.
00:19:22.210 --> 00:19:24.420
But for the discrete
Fourier series,
00:19:24.420 --> 00:19:25.900
it says an important thing.
00:19:25.900 --> 00:19:29.710
It says that in forming
the discrete Fourier
00:19:29.710 --> 00:19:36.410
series, once we've used the
complex exponentials for k
00:19:36.410 --> 00:19:39.890
between 0 and capital
N minus 1, we've
00:19:39.890 --> 00:19:42.440
used all the
complex exponentials
00:19:42.440 --> 00:19:45.680
with this fundamental
frequency that we have.
00:19:45.680 --> 00:19:48.080
And if we keep
going with k, we're
00:19:48.080 --> 00:19:51.800
just simply going to see the
same complex exponentials over
00:19:51.800 --> 00:19:53.900
and over again.
00:19:53.900 --> 00:19:56.480
What does that say about
the discrete Fourier series?
00:19:56.480 --> 00:19:59.900
It says that in
the representation
00:19:59.900 --> 00:20:06.590
of the discrete Fourier
series, the limits on this sum
00:20:06.590 --> 00:20:10.400
don't range from minus
infinity to plus infinity.
00:20:10.400 --> 00:20:16.265
They run simply from 0
to capital N minus 1.
00:20:16.265 --> 00:20:21.500
So once I've looked at
these linear combination
00:20:21.500 --> 00:20:24.500
of these complex
exponentials for k between 0
00:20:24.500 --> 00:20:28.880
and capital N minus 1, there
are no new complex exponentials
00:20:28.880 --> 00:20:32.790
with that fundamental frequency
that I'm able to find.
00:20:32.790 --> 00:20:37.980
So this then is the form of
the discrete Fourier series.
00:20:37.980 --> 00:20:42.150
There's one additional
insertion that I'd like to make.
00:20:42.150 --> 00:20:47.190
And this is just simply
a normalization factor.
00:20:47.190 --> 00:20:51.090
I want to multiply this by 1
over capital N. Obviously, that
00:20:51.090 --> 00:20:53.200
doesn't make any
essential difference.
00:20:53.200 --> 00:20:55.770
It's just a factor
for normalization.
00:20:55.770 --> 00:21:00.900
And it plays a role which is
similar in the continuous time
00:21:00.900 --> 00:21:07.540
case to the role that 2
pi or 1 over 2 pi plays.
00:21:07.540 --> 00:21:08.050
All right.
00:21:08.050 --> 00:21:12.470
So here we have the
discrete Fourier series.
00:21:12.470 --> 00:21:16.370
It's relatively
straightforward to show
00:21:16.370 --> 00:21:21.110
that you can obtain the
Fourier series coefficients x
00:21:21.110 --> 00:21:25.430
tilde of k from x
tilde of n through
00:21:25.430 --> 00:21:29.180
an inverse relationship,
which is the relationship
00:21:29.180 --> 00:21:31.920
that I've indicated here.
00:21:31.920 --> 00:21:36.540
So this is then, in essence,
the inverse discrete Fourier
00:21:36.540 --> 00:21:40.140
series, or equivalently,
the relationship
00:21:40.140 --> 00:21:43.320
for obtaining the
discrete Fourier
00:21:43.320 --> 00:21:49.160
series coefficients from the
periodic sequence x tilde of n.
00:21:49.160 --> 00:21:53.750
Now notice that I've happened
to write the Fourier series
00:21:53.750 --> 00:21:57.710
coefficients with
a tilde over them,
00:21:57.710 --> 00:22:04.310
implying that those coefficients
are themselves periodic.
00:22:04.310 --> 00:22:08.490
Well, are they periodic,
or aren't they periodic?
00:22:08.490 --> 00:22:11.520
What I've been saying here and
what I've been saying here--
00:22:11.520 --> 00:22:14.250
and I've spent a long
time saying this--
00:22:14.250 --> 00:22:18.330
that there are only a finite
number of complex exponentials.
00:22:18.330 --> 00:22:21.915
Once I've looked at them in the
range 0 to capital N minus 1,
00:22:21.915 --> 00:22:24.930
I've seen all the
ones I can see.
00:22:24.930 --> 00:22:29.080
And in that sense, the
Fourier series coefficients,
00:22:29.080 --> 00:22:32.730
x tilde of k, are finite.
00:22:32.730 --> 00:22:35.731
That is, there are only
a finite number of them.
00:22:35.731 --> 00:22:37.730
Well, the important point
is that there are only
00:22:37.730 --> 00:22:42.530
a finite number of
distinct coefficients.
00:22:42.530 --> 00:22:45.470
Also in this
relationship, whether I
00:22:45.470 --> 00:22:48.995
consider these as
periodic or not periodic,
00:22:48.995 --> 00:22:51.530
I still only use a
finite number of them.
00:22:51.530 --> 00:22:55.970
That is, I only use them in
the range k equals 0 to capital
00:22:55.970 --> 00:22:57.620
N minus 1.
00:22:57.620 --> 00:23:02.390
And how I choose to interpret
capital X tilde of k
00:23:02.390 --> 00:23:05.720
outside the range 0
to capital N minus 1
00:23:05.720 --> 00:23:09.290
is going to have absolutely
no effect on the evaluation
00:23:09.290 --> 00:23:11.850
of this summation.
00:23:11.850 --> 00:23:15.970
Well, it turns out to be
convenient to interpret
00:23:15.970 --> 00:23:21.690
the Fourier series coefficients
as being periodic in k.
00:23:21.690 --> 00:23:25.380
Well, in fact, the relationship
as I've written it here
00:23:25.380 --> 00:23:28.920
makes it evident, in this
particular relationship,
00:23:28.920 --> 00:23:31.170
that the coefficients
are periodic.
00:23:31.170 --> 00:23:39.940
In other words, if I substitute
in for k, k plus capital N,
00:23:39.940 --> 00:23:45.130
then I'll get back exactly the
same relation that I have here.
00:23:45.130 --> 00:23:49.450
Because of the fact that e to
the minus j 2 pi over capital
00:23:49.450 --> 00:23:53.710
N times little n
times k plus capital
00:23:53.710 --> 00:24:00.740
N is equal to e the
minus j 2 pi over capital
00:24:00.740 --> 00:24:04.490
N times little n times k.
00:24:04.490 --> 00:24:06.430
So that if I simply examined--
00:24:06.430 --> 00:24:09.330
I asked from this relationship--
00:24:09.330 --> 00:24:14.000
what does capital X tilde of k
plus capital N come out to be?
00:24:14.000 --> 00:24:17.450
If I simply substitute that
in, then because of the fact
00:24:17.450 --> 00:24:20.620
that these two complex
exponentials are the same,
00:24:20.620 --> 00:24:22.370
I'll find that x--
00:24:22.370 --> 00:24:26.740
capital X tilde of k
is equal to capital X
00:24:26.740 --> 00:24:30.920
tilde of k plus capital N, k
plus 2 capital N, et cetera.
00:24:30.920 --> 00:24:35.000
That is, it's a periodic
sequence, although I only
00:24:35.000 --> 00:24:38.300
use one period of
it in reconstructing
00:24:38.300 --> 00:24:42.130
the periodic sequence
x tilde of n.
00:24:42.130 --> 00:24:45.880
I choose to do that
mainly because of duality.
00:24:45.880 --> 00:24:50.080
That is, mainly because
it is convenient to think
00:24:50.080 --> 00:24:54.400
of these as a periodic
sequence so that I
00:24:54.400 --> 00:24:57.490
have one periodic
sequence representing
00:24:57.490 --> 00:25:03.460
another periodic sequence, this
being a periodic sequence in k
00:25:03.460 --> 00:25:12.080
with a period of capital N, and
this being a periodic sequence
00:25:12.080 --> 00:25:16.340
in n with a period
also of capital N.
00:25:16.340 --> 00:25:19.040
So now with a discrete
Fourier series,
00:25:19.040 --> 00:25:22.160
there's a duality
in the, what we
00:25:22.160 --> 00:25:25.280
could call the time domain
and the frequency domain.
00:25:25.280 --> 00:25:27.410
And the duality is there
in part because we've
00:25:27.410 --> 00:25:32.240
chosen to represent, or think of
the Fourier series coefficients
00:25:32.240 --> 00:25:35.550
as periodic, as a
periodic sequence,
00:25:35.550 --> 00:25:38.930
although we only use a
finite number of those values
00:25:38.930 --> 00:25:43.640
in actually explicitly
evaluating the Fourier
00:25:43.640 --> 00:25:46.800
series for x tilde of n.
00:25:46.800 --> 00:25:50.040
Well, this then is
the Fourier series.
00:25:50.040 --> 00:25:53.400
Let me finally rewrite
it one other way,
00:25:53.400 --> 00:25:58.050
which just introduces some
notation that's convenient.
00:25:58.050 --> 00:26:03.130
Let me define w
sub capital N as e
00:26:03.130 --> 00:26:08.350
to the minus j 2 pi over
capital N. In that case,
00:26:08.350 --> 00:26:12.510
then just simply rewriting
the Fourier series,
00:26:12.510 --> 00:26:15.480
we have x tilde of
n is 1 over capital
00:26:15.480 --> 00:26:21.240
N, the sum of capital X
tilde of k, w sub capital
00:26:21.240 --> 00:26:24.180
N to the minus nk.
00:26:24.180 --> 00:26:27.240
And the Fourier
series coefficients
00:26:27.240 --> 00:26:31.240
expressed in terms
of capital W sub n
00:26:31.240 --> 00:26:39.110
is the sum of x tilde of n,
w sub capital N to the nk.
00:26:39.110 --> 00:26:42.740
Well, the discrete
Fourier series
00:26:42.740 --> 00:26:46.370
has properties,
just as the Fourier
00:26:46.370 --> 00:26:50.730
transform and the Z-transform
has had a number of properties.
00:26:50.730 --> 00:26:55.850
And again, as we've done
with the other transforms,
00:26:55.850 --> 00:26:58.940
I won't spend a lot
of time on the details
00:26:58.940 --> 00:27:03.440
of either enumerating the
properties or proving them.
00:27:03.440 --> 00:27:06.320
But let me just
illustrate one or two
00:27:06.320 --> 00:27:11.880
to give you some idea as to
what these properties involve.
00:27:11.880 --> 00:27:16.560
Well, first of all, as we've
talked about in the Fourier
00:27:16.560 --> 00:27:19.530
transform and
Z-transform cases, there
00:27:19.530 --> 00:27:25.620
is a shifting property that
tells us how the Fourier series
00:27:25.620 --> 00:27:28.440
coefficients of a
periodic sequence
00:27:28.440 --> 00:27:31.260
are related to the Fourier
series coefficients
00:27:31.260 --> 00:27:33.120
of that sequence shifted.
00:27:33.120 --> 00:27:39.240
And in particular, it turns out
that if capital X of k, capital
00:27:39.240 --> 00:27:41.730
X tilde of k are
the Fourier series
00:27:41.730 --> 00:27:45.420
coefficients for
little x tilde of n,
00:27:45.420 --> 00:27:47.970
then the Fourier
series coefficients
00:27:47.970 --> 00:27:53.590
for that sequence shifted, that
is, little x tilde of n plus m,
00:27:53.590 --> 00:27:56.260
corresponds to multiplying
the Fourier series
00:27:56.260 --> 00:28:04.940
coefficients by w sub
capital N to the minus km.
00:28:04.940 --> 00:28:10.430
Shifting the sequence involves
multiplying the Fourier series
00:28:10.430 --> 00:28:14.150
coefficients by a complex
exponential, which
00:28:14.150 --> 00:28:16.170
is what this is.
00:28:16.170 --> 00:28:20.930
And that is similar to what
we've seen with the Fourier
00:28:20.930 --> 00:28:25.490
transform, shifting a sequence,
multiply the Fourier transform
00:28:25.490 --> 00:28:28.060
by a complex exponential.
00:28:28.060 --> 00:28:32.520
And the Z-transform, we had
exactly the same situation.
00:28:32.520 --> 00:28:36.480
We have a dual
relationship, which
00:28:36.480 --> 00:28:40.440
expresses the result of
shifting the Fourier series
00:28:40.440 --> 00:28:42.130
coefficients.
00:28:42.130 --> 00:28:45.120
If we shift the Fourier
series coefficients,
00:28:45.120 --> 00:28:49.320
the result is multiplication
by a complex exponential
00:28:49.320 --> 00:28:54.310
of the original sequence,
little x tilde of n.
00:28:54.310 --> 00:28:56.020
In fact, one of
the things that's
00:28:56.020 --> 00:28:59.650
true with the discrete Fourier
series that hasn't been true
00:28:59.650 --> 00:29:02.910
with the Fourier transform
or the Z-transform
00:29:02.910 --> 00:29:07.990
is there is a strong
duality between the time
00:29:07.990 --> 00:29:12.610
domain, the discrete time
domain, and the Fourier
00:29:12.610 --> 00:29:15.370
or frequency domain.
00:29:15.370 --> 00:29:21.690
In particular, we begin with
a discrete periodic sequence,
00:29:21.690 --> 00:29:24.540
and we end up in the
Fourier domain with, again,
00:29:24.540 --> 00:29:27.310
a discrete periodic sequence.
00:29:27.310 --> 00:29:29.590
In fact, something
to think about
00:29:29.590 --> 00:29:34.670
is the fact that the Fourier
series coefficients we've said,
00:29:34.670 --> 00:29:38.920
or we've chosen to interpret
them, as being periodic.
00:29:38.920 --> 00:29:42.040
That implies that they
themselves have a Fourier
00:29:42.040 --> 00:29:43.850
series representation.
00:29:43.850 --> 00:29:46.120
And something to
think about is, what
00:29:46.120 --> 00:29:48.730
are the Fourier
series coefficients
00:29:48.730 --> 00:29:49.970
of that periodic sequence?
00:29:54.340 --> 00:29:54.910
All right.
00:29:54.910 --> 00:29:59.110
So one important point then
is that we have this duality
00:29:59.110 --> 00:30:00.820
between the two domains.
00:30:00.820 --> 00:30:05.110
And essentially, any property
that we have in the time domain
00:30:05.110 --> 00:30:07.420
will find the dual
property in the frequency
00:30:07.420 --> 00:30:10.150
domain for the discrete
Fourier series.
00:30:10.150 --> 00:30:12.910
And that will the
hold true when,
00:30:12.910 --> 00:30:16.340
later on in the next lecture, we
talk about the discrete Fourier
00:30:16.340 --> 00:30:18.470
transform.
00:30:18.470 --> 00:30:21.890
Another useful property,
which we've also
00:30:21.890 --> 00:30:26.780
talked about for the
Fourier transform,
00:30:26.780 --> 00:30:29.540
are the set of
symmetry properties.
00:30:29.540 --> 00:30:34.160
And in particular, if
we consider x tilde of n
00:30:34.160 --> 00:30:38.260
to be a real a
periodic sequence,
00:30:38.260 --> 00:30:40.960
then there are
symmetry relationships
00:30:40.960 --> 00:30:45.430
between the real part
of the Fourier series,
00:30:45.430 --> 00:30:48.940
and symmetry relations for the
imaginary part of the Fourier
00:30:48.940 --> 00:30:50.080
series.
00:30:50.080 --> 00:30:54.280
In particular, we can think of
expressing the Fourier series
00:30:54.280 --> 00:30:59.840
coefficients in terms of
their real part plus j
00:30:59.840 --> 00:31:03.010
times the imaginary part.
00:31:03.010 --> 00:31:06.940
And the symmetry
relationships that result
00:31:06.940 --> 00:31:10.660
are that if the periodic
sequence is real,
00:31:10.660 --> 00:31:14.530
then the real part of the
Fourier series coefficients
00:31:14.530 --> 00:31:21.490
are even, x sub R of k is
equal to x sub R of minus k.
00:31:21.490 --> 00:31:26.140
And that's what we
refer to as the property
00:31:26.140 --> 00:31:28.690
of, in the case of the
Fourier transform, the Fourier
00:31:28.690 --> 00:31:31.300
transform being even.
00:31:31.300 --> 00:31:34.270
We can also write this
in another way that
00:31:34.270 --> 00:31:36.190
will become important
when we talk about
00:31:36.190 --> 00:31:38.960
the discrete Fourier transform.
00:31:38.960 --> 00:31:42.490
In particular, since this
is a periodic sequence,
00:31:42.490 --> 00:31:44.440
since s sub R of--
00:31:44.440 --> 00:31:47.110
since x of k is
periodic, obviously
00:31:47.110 --> 00:31:49.370
it's real part is
also periodic--
00:31:49.370 --> 00:31:54.760
we can replace k by k
plus capital N, or k
00:31:54.760 --> 00:31:58.630
minus capital N. And
we can rewrite this
00:31:58.630 --> 00:32:04.750
as a statement that says
that x sub R tilde of k
00:32:04.750 --> 00:32:12.390
is equal to x sub R tilde
of capital N minus k.
00:32:12.390 --> 00:32:14.190
It shouldn't be
particularly evident
00:32:14.190 --> 00:32:20.060
why we want to do that here,
although it becomes important
00:32:20.060 --> 00:32:22.340
when we apply some
of these notions
00:32:22.340 --> 00:32:24.890
to the representation of
finite length sequence
00:32:24.890 --> 00:32:28.880
and sequences in the
discrete Fourier transform.
00:32:28.880 --> 00:32:31.910
Likewise, for the imaginary
part of the Fourier series
00:32:31.910 --> 00:32:36.040
coefficients, we end up
with a symmetry property
00:32:36.040 --> 00:32:39.670
that says that the imaginary
part of the Fourier series
00:32:39.670 --> 00:32:41.275
coefficients are odd.
00:32:44.010 --> 00:32:47.940
And again, as we did with the
real part of the Fourier series
00:32:47.940 --> 00:32:51.000
coefficients, we
can rewrite this
00:32:51.000 --> 00:32:58.690
to say that x sub i tilde of
k is equal to minus x sub i
00:32:58.690 --> 00:33:03.300
tilde of N minus k.
00:33:06.800 --> 00:33:10.460
Finally, we can, in
a similar manner,
00:33:10.460 --> 00:33:14.270
look at the magnitude of the
Fourier series coefficients
00:33:14.270 --> 00:33:17.630
and the angle of the
Fourier series coefficients.
00:33:17.630 --> 00:33:21.620
And just as we've seen
with the Fourier transform,
00:33:21.620 --> 00:33:24.710
the result that we'll
find is that the magnitude
00:33:24.710 --> 00:33:27.350
of the Fourier
series coefficients
00:33:27.350 --> 00:33:31.610
are even, an even function of k.
00:33:31.610 --> 00:33:36.080
And the angle of the
Fourier series coefficients
00:33:36.080 --> 00:33:39.880
are an odd function of k.
00:33:39.880 --> 00:33:42.370
So we have symmetry
properties like we
00:33:42.370 --> 00:33:44.890
had with the Fourier transform.
00:33:44.890 --> 00:33:49.990
An important thing to keep track
of at this point, because we'll
00:33:49.990 --> 00:33:53.410
want to refer back
to this when we talk
00:33:53.410 --> 00:33:55.930
about the discrete
Fourier transform,
00:33:55.930 --> 00:33:59.590
is that in talking about
the property of even or odd,
00:33:59.590 --> 00:34:03.190
we can either talk
about it as we do here,
00:34:03.190 --> 00:34:05.890
or in terms of a shift--
00:34:05.890 --> 00:34:09.070
that is, in terms of a
statement that x sub bar of k
00:34:09.070 --> 00:34:13.510
is x sub R of capital N minus
k, which essentially relates
00:34:13.510 --> 00:34:18.580
the evenness of this
periodic sequence
00:34:18.580 --> 00:34:24.770
to the relationship between
values within one period.
00:34:24.770 --> 00:34:30.120
Finally, we have another very
important property, which
00:34:30.120 --> 00:34:32.569
is the convolution property.
00:34:32.569 --> 00:34:34.110
This is a property
that, again, we've
00:34:34.110 --> 00:34:38.020
had for the Fourier transform
and for the Z-transform.
00:34:38.020 --> 00:34:40.530
It's a property that
states, essentially,
00:34:40.530 --> 00:34:45.810
that the convolution of
two periodic sequences
00:34:45.810 --> 00:34:50.550
results in the multiplication
of the discrete Fourier series
00:34:50.550 --> 00:34:54.570
coefficients, with just one
minor twist, which again will
00:34:54.570 --> 00:34:58.140
become important when we relate
this to the discrete Fourier
00:34:58.140 --> 00:34:59.490
transform.
00:34:59.490 --> 00:35:03.630
In particular, we have
a periodic sequence,
00:35:03.630 --> 00:35:05.640
x1 tilde of n.
00:35:05.640 --> 00:35:09.120
And here its Fourier
series coefficients.
00:35:09.120 --> 00:35:13.110
A second periodic
sequence, x2 tilde of n,
00:35:13.110 --> 00:35:16.200
with its Fourier
series coefficients.
00:35:16.200 --> 00:35:20.220
And what we'd like
to ask is, what
00:35:20.220 --> 00:35:26.010
is the periodic
sequence, x3 tilde of n,
00:35:26.010 --> 00:35:28.740
whose Fourier series
coefficients are
00:35:28.740 --> 00:35:34.290
the product of x1 tilde
of k and x2 tilde of k?
00:35:34.290 --> 00:35:37.680
In other words, if we multiply
the Fourier series coefficients
00:35:37.680 --> 00:35:41.910
together, what is the sequence
that that corresponds to?
00:35:41.910 --> 00:35:45.390
The answer, which involves just
simply a little bit of algebra
00:35:45.390 --> 00:35:50.280
to verify, is that
the resulting sequence
00:35:50.280 --> 00:35:57.690
is a sum of x1 tilde of
m, x2 tilde of n minus m.
00:35:57.690 --> 00:36:03.000
Well, that looks exactly
like a convolution
00:36:03.000 --> 00:36:05.700
with one important difference.
00:36:05.700 --> 00:36:09.150
And that is that the
limits on the summation
00:36:09.150 --> 00:36:12.240
don't go from minus
infinity to plus infinity
00:36:12.240 --> 00:36:14.370
as they did in the
case of the Fourier
00:36:14.370 --> 00:36:16.770
transform and Z-transform.
00:36:16.770 --> 00:36:19.020
Ordinary linear
convolution, as we've always
00:36:19.020 --> 00:36:21.300
been talking about
it, involved the sum
00:36:21.300 --> 00:36:23.460
from minus infinity
to plus infinity,
00:36:23.460 --> 00:36:27.780
of x1 of m, x2 of n minus m.
00:36:27.780 --> 00:36:31.780
Here we have, as the
limits on the sum, 0
00:36:31.780 --> 00:36:33.960
to capital N minus 1.
00:36:33.960 --> 00:36:38.670
The summation is carried
out only over one period.
00:36:38.670 --> 00:36:45.610
We have also a dual property, in
other words, the property that
00:36:45.610 --> 00:36:50.710
relates the Fourier series
coefficients of the product
00:36:50.710 --> 00:36:54.310
of two sequences to the
Fourier series coefficients
00:36:54.310 --> 00:36:56.620
of the individual sequences.
00:36:56.620 --> 00:37:00.460
And because of the duality now
that we have between the time
00:37:00.460 --> 00:37:04.280
domain and the frequency
domain for the Fourier series,
00:37:04.280 --> 00:37:06.850
these Fourier
series coefficients
00:37:06.850 --> 00:37:11.890
are, again, the convolution
of the two Fourier series
00:37:11.890 --> 00:37:15.470
coefficients for x1
of n and x2 of n.
00:37:15.470 --> 00:37:18.880
There's a normalization
factor, 1 over capital N.
00:37:18.880 --> 00:37:25.878
But again, the limits on the sum
involve 0 to capital N minus 1.
00:37:25.878 --> 00:37:29.870
So this is slightly different
than the convolution
00:37:29.870 --> 00:37:31.860
as we've been talking about.
00:37:31.860 --> 00:37:34.820
It is referred to
generally, and we'll
00:37:34.820 --> 00:37:38.420
be referring to it as
a periodic convolution.
00:37:38.420 --> 00:37:43.850
The important distinction being
that for this convolution,
00:37:43.850 --> 00:37:46.670
it involves a summation,
not for minus infinity
00:37:46.670 --> 00:37:51.500
to plus infinity, but simply a
summation over only one period
00:37:51.500 --> 00:37:54.270
of the periodic sequences.
00:37:54.270 --> 00:37:58.050
Well, to drive this
home, let me just
00:37:58.050 --> 00:38:05.580
show you with a simple example
what these sequences look like,
00:38:05.580 --> 00:38:08.610
and in particular, what
the shifting involves,
00:38:08.610 --> 00:38:11.490
and apply an
interpretation to that.
00:38:11.490 --> 00:38:13.972
And so let's return
to the view graph.
00:38:20.860 --> 00:38:27.350
What I'm indicating here
is a sequence x2 tilde
00:38:27.350 --> 00:38:36.390
of m, a sequence x1 tilde of
m, and to form the convolution
00:38:36.390 --> 00:38:38.790
of these two sequences.
00:38:38.790 --> 00:38:45.530
What I would like to do is
generate x2 tilde, in general,
00:38:45.530 --> 00:38:54.740
of n minus m, multiply that
by this, by x1 tilde of m,
00:38:54.740 --> 00:38:58.860
and then sum up the
result over one period.
00:38:58.860 --> 00:39:02.200
Well, I'm indicating,
by the way,
00:39:02.200 --> 00:39:06.320
in blue, just one period
of this periodic sequence.
00:39:06.320 --> 00:39:08.660
And similarly in
blue, one period
00:39:08.660 --> 00:39:10.890
of this periodic sequence.
00:39:10.890 --> 00:39:14.720
So let's examine, first
of all, the sequence
00:39:14.720 --> 00:39:22.370
which corresponds to replacing
this by x2 tilde of n minus m.
00:39:22.370 --> 00:39:25.610
And let's do it for
the specific case
00:39:25.610 --> 00:39:27.710
where little n is equal to 2.
00:39:27.710 --> 00:39:35.120
So I've illustrated here the
sequence x2 tilde of 2 minus m.
00:39:35.120 --> 00:39:38.870
And to generate this
sequence from this one
00:39:38.870 --> 00:39:42.300
involves essentially two steps.
00:39:42.300 --> 00:39:46.550
The first step is to
flip this sequence over,
00:39:46.550 --> 00:39:50.720
that's replacing m by minus m.
00:39:50.720 --> 00:39:55.790
And then the second step is to
shift the sequence by an amount
00:39:55.790 --> 00:39:58.610
little n, depending on
the argument that we
00:39:58.610 --> 00:40:00.440
want to stick in here.
00:40:00.440 --> 00:40:04.490
Now this is the
result of doing that.
00:40:04.490 --> 00:40:09.230
We can think of this as having
flipped this sequence over,
00:40:09.230 --> 00:40:17.020
and then shifted it by
two points to the right.
00:40:17.020 --> 00:40:22.040
And the result is then
this periodic sequence.
00:40:22.040 --> 00:40:26.860
An important thing
to keep in mind,
00:40:26.860 --> 00:40:29.920
or to look at-- and again,
this is a point that we'll be
00:40:29.920 --> 00:40:34.090
emphasizing in much more
detail in the next lecture--
00:40:34.090 --> 00:40:38.080
is that as we examine
these blue points,
00:40:38.080 --> 00:40:40.510
we could think of
having gotten those
00:40:40.510 --> 00:40:44.740
by simply flipping
one period of this
00:40:44.740 --> 00:40:48.430
and circularly shifting
that, circularly shifting
00:40:48.430 --> 00:40:53.820
those points, simply in the
range 0 to capital N minus 1.
00:40:53.820 --> 00:40:56.320
That's an interpretation
that will become much more
00:40:56.320 --> 00:41:00.190
evident in the next lecture.
00:41:00.190 --> 00:41:03.580
But then to form the
convolution of this sequence
00:41:03.580 --> 00:41:09.420
with this sequence, we then,
after having constructed
00:41:09.420 --> 00:41:14.350
x2 tilde of n minus m,
carry out the multiplication
00:41:14.350 --> 00:41:16.810
of these two sequences.
00:41:16.810 --> 00:41:18.720
The result of that
multiplication,
00:41:18.720 --> 00:41:23.440
I've indicated here, so that
for this particular example,
00:41:23.440 --> 00:41:27.650
these three points get
multiplied by these three.
00:41:27.650 --> 00:41:31.340
These four points,
rather, get multiplied
00:41:31.340 --> 00:41:32.990
by these four values.
00:41:32.990 --> 00:41:37.610
The rest of the points in that
one period get multiplied by 0.
00:41:37.610 --> 00:41:40.250
And then finally, to
evaluate the result
00:41:40.250 --> 00:41:44.530
of the periodic convolution
for little n equal to 2,
00:41:44.530 --> 00:41:50.391
we sum up those values in the
range 0 to capital N minus 1.
00:41:50.391 --> 00:41:55.480
So it operates very much like a
linear or ordinary convolution,
00:41:55.480 --> 00:41:59.020
as we've been talking about
over the last several lectures.
00:41:59.020 --> 00:42:03.640
The important difference is that
the summation is carried out
00:42:03.640 --> 00:42:07.241
just simply over one period.
00:42:07.241 --> 00:42:07.740
All right.
00:42:07.740 --> 00:42:12.780
Well, that completes
the discussion
00:42:12.780 --> 00:42:18.540
as we want to present it, of
the discrete Fourier series.
00:42:18.540 --> 00:42:20.850
As I indicated at the
beginning of the lecture,
00:42:20.850 --> 00:42:27.510
our objective was eventually
to develop a Fourier
00:42:27.510 --> 00:42:30.390
representation for finite
length sequences, that is,
00:42:30.390 --> 00:42:32.940
the discrete Fourier transform.
00:42:32.940 --> 00:42:37.530
And in the next lecture,
what we'll want to do
00:42:37.530 --> 00:42:40.770
is take the ideas of
the discrete Fourier
00:42:40.770 --> 00:42:44.070
series, as we've
talked about them here,
00:42:44.070 --> 00:42:48.510
and apply them to the
representation of finite length
00:42:48.510 --> 00:42:51.630
sequences, resulting
in what we'll call
00:42:51.630 --> 00:42:54.320
the discrete Fourier transform.