第 10 讲：离散时间傅里叶级数
本讲内容：离散时间线性时不变系统对复指数信号的响应；周期信号的表示：成谐波关系的复指数信号的线性组合；与连续时间情形的异同；分析式与综合式；周期信号的逼近与收敛性；非周期离散时间信号的傅里叶表示，即离散时间傅里叶变换；周期信号的傅里叶变换表示。
授课教师：艾伦·V. 奥本海姆教授
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[MUSIC PLAYING]
PROFESSOR: Over the last several lectures, we developed
the Fourier representation for continuous-time signals.
What I'd now like to do is develop a similar
representation for discrete-time.
And let me begin the discussion by reminding you of
what our basic motivation was.
The idea is that what we wanted to do was exploit the
properties of linearity and time invariance for linear
time-invariant systems.
So in the case of linear time-invariant systems, the
basic idea was to consider decomposing the input as a
linear combination of basic inputs.
And then, because of linearity, the output could be
expressed as a linear combination of corresponding
outputs where psi sub i is the output due to phi sub i.
So basically, what we attempted to do was decompose
the input, and then reconstruct the output through
a linear combination of the outputs to those basic inputs.
We then focused on the notion of choosing the basic inputs
with two criteria in mind.
One was to choose them so that a broad class of signals could
be constructed out of those basic inputs.
And the second was to choose the basic inputs, so that the
response to those was easy to compute.
And as you recall, one representation that we ended
up with, with those basic criteria in mind, was the
representation through convolution.
And then in beginning the discussion of the Fourier
representation of continuous-time signals, we
chose as another set of basic inputs complex exponentials.
So for continuous-time, we chose a set of basic inputs
which were complex exponentials.
The motivation there was the fact that the complex
exponentials have what we refer to as the
eigenfunction property.
Namely, if we put complex exponentials into our
continuous-time systems, then the output is a complex
exponential of the same form with only a
change in complex amplitude.
And that change in complex amplitude is what we referred
to as the frequency response of the system.
And also, by the way, as it developed later, that
frequency response, as you should now recognize from this
expression, is in fact the Fourier transform, the
continuous-time Fourier transform of the system
impulse response.
And the notion of decomposing a signal as a linear
combination of these complex exponentials is what, first
the Fourier series representation, and then later
the Fourier transform representation
corresponded to.
And finally, to remind you of one additional point, the fact
is that because of the eigenfunction
property, the response--
once we have decomposed the input as a linear combination
of complex exponentials, the response to that linear
combination is straightforward to compute once we know the
frequency response because of the eigenfunction property.
Now, basically the same strategy and many of the same
ideas work in discrete-time, paralleling almost exactly
what happened in continuous-time.
So the similarities between discrete-time and
continuous-time are very strong.
Although as we'll see, there are a number of differences.
And it's important as we go through the discussion to
illuminate not only the similarities, but obviously
also the differences.
Well, let's begin with the eigenfunction property, and
let me just state that just as in continuous-time, if we
consider a set of basic signals, which are complex
exponential rules, then discrete-time linear time-m
invariant systems have the eigenfunction property.
Namely, if we put a complex exponential into the system,
the response is a complex exponential at the same
complex frequency, and simply multiplied by an appropriate
complex factor, or constant.
And just as we did in continuous-time, we will be
referring to this complex constant, which is a function,
of course, the frequency of the complex exponential input.
We'll be referring to this as the frequency response.
And although it's not particularly evident at this
point, as the discussion develops through this lecture,
what in fact will happen is very much paralleling
continuous-time.
This particular expression, in fact, will correspond to what
we'll refer to as the Fourier transform, the discrete-time
Fourier transform of the system impulse response.
So there, of course, there's a very strong parallel between
continuous time and discrete time.
Now, just as we did in continuous-time, let's begin
the discussion by first concentrating on periodic--
the representation through complex exponentials of
periodic sequences, and then we'll generalize that
discussion to the representation
of aperiodic signals.
So let's consider first a periodic signal, or in
general, signals which are periodic.
Period denoted by capital N. And then, of course, the
fundamental frequency is 2 pi divided by capital N.
Now, we can consider exponentials which have this
as a fundamental frequency, or which are harmonics of that,
and that would correspond to the class of complex
exponentials of the form e to the jk omega 0 n.
So these complex exponentials then, as k varies, are complex
exponentials that are harmonically related, all of
which are periodic with the same period capital N.
Although the fundamental period is
different for each of these.
Each of them being related by an integer amount.
Now, again, just as we did in continuous time, we can
consider attempting to build our periodic signal out of a
linear combination of these.
And so we consider a periodic signal, which is a weighted
sum of these complex exponentials.
And, of course, this periodic signal--
this is a periodic signal.
This can be verified, more or less, in a straightforward way
by substitution.
And, of course, one of the things that we'll want to
address shortly is how broad a class of signals, again, can
be represented by this sum?
And another question obviously will be, how do we determine
the coefficients a sub k?
However, before we do that, let me focus on an important
distinction between continuous-time and
discrete-time in the context of these complex exponentials
and this representation.
When we talked about complex exponentials and sinusoids
early in the course, one of the differences that we saw
between continuous-time and discrete-time is that in
continuous-time, as we vary the frequency variable, we see
different complex exponentials as omega varies.
Whereas, in discrete-time, we saw, in fact, that there was a
periodicity.
Or said another way, it's straightforward to verify that
if we think of this class of complex exponentials.
That, in fact, if we consider varying k by adding to it
capital N, where capital N is the period of the fundamental
complex exponential.
Then in fact, if we replace k by k plus capital N, we'll see
exactly the same complex exponentials over again.
Now, what does that say?
What it says is that if I consider this class of complex
exponentials, as k varies from 0 through capital N minus 1,
we will see all of the ones that there are to see.
There aren't anymore.
And so, in fact, if we can build x of n out of this
linear combination, then we better be able to do it as k
varies from 0 up to N minus 1.
Because beyond that, we'll simply see the same complex
exponentials over again.
So, for example, if k takes on the value capital N, that will
be exactly the same complex exponential as if
k is equal to 0.
So in fact, this sum ranges only over capital N of the
distinct complex exponentials.
Let's say, for example, from 0 to capital N minus 1 .
Although, in fact, since these complex exponentials repeat in
k, I could actually consider instead of from 0 to N minus
1, I could consider from 1 to N, or from 2 to
N plus 1, or whatever.
Or said another way, in this representation, I could
alternatively choose k outside this range, thinking of these
coefficients simply as periodically repeating in k
because of the fact that these complex exponentials
periodically repeat in k.
So, in fact, in place of this expression, it will be common
in writing the Fourier series expression to write it as I've
indicated here, where the implication is that these
Fourier coefficients periodically repeat as k
continues to repeat outside the interval from
0 to N minus 1.
And so this notation, in fact, says that what we're going to
use is k ranging over one period of this periodic
sequence, which is the Fourier series coefficients.
So the expression that we have then for the Fourier series
I've repeated here.
And the implication now is that the
a sub k's are periodic.
They periodically repeat because, of course, these
exponentials periodically repeat.
This indicates that we only use them over one period.
And now we can inquire as to how we determine the
coefficients a sub k.
Well, we can formally go through this much as we did in
the continuous-time case.
And we do, in fact, do that in the text, which involves
substituting some sums and interchanging the orders of
summation, et cetera.
But let me draw your attention to the fact that this, in
fact, can be thought of as capital N equations and
capital N unknowns.
In other words, we know x of n over a period, and so we know
what the left-hand side of this is for capital N values.
And we'd like to determine these constants a sub k.
Well, it turns out that there is a nice convenient
closed-form expression for that.
And, in fact, if we evaluate the closed-form expression
through any of a variety of algebraic manipulations, we
end up then with the analysis equation.
And the analysis equation, which tells us how to get the
coefficients a sub k from x of n is what I've indicated here.
And so this tells us how from x of n to get the a sub k's.
And, of course, the first equation tells us how x of n
is built up out of the a sub k.
Notice incidentally that there is a strong duality between
these two equations.
And that's a duality that we'll return to, actually
toward the end of the next lecture.
Now, there is a real difference between the way
those equations look and the way the continuous-time
Fourier series looked.
In the continuous-time case, let me remind you that it
required an infinite number of coefficients to build up this
continuous-time function.
And so this was not simply a matter of identifying how to
invert capital N or a finite number of equations and a
finite number of unknowns.
And the analysis equation was an integration as opposed to
the synthesis equation, which is a summation.
So there is a real difference there between the
continuous-time and discrete-time cases.
And the difference arises, to a large extent, because of
this notion that in discrete-time, the complex
exponentials are periodic in their frequency.
So we have then to summarize the synthesis equation and the
analysis equation for the discrete-time Fourier series.
Again, x of n, our original signal is periodic.
And, of course, the complex exponentials
involved are periodic.
They're periodic obviously in n.
But in contrast to continuous-time,
these repeat in k.
In other words, as k omega 0 goes outside a range that
covers a 2 pi interval.
And because of that, we're imposing, in a sense, the
interpretation that the a sub k's are
likewise a periodic sequence.
And in fact, if we look at the analysis equation, as we let k
vary outside the range from 0 to N minus 1, what you can
easily verify by substitution in here is that this sequence
will, in fact, periodically repeat.
So to underscore the difference between the
continuous-time and discrete-time cases, we have
this periodicity in the time domain, and that's a
periodicity that is, of course, true in discrete-time
and it's also true in continuous-time if we replace
the integer variable by the discrete-time time variable.
And we also, in discrete-time, have this periodicity in k, or
in k omega 0.
And correspondingly, a periodicity in the Fourier
coefficients.
And that is a set of properties that does not
happen in continuous-time.
And it is that that essentially leads to all of
the important differences between discrete-time Fourier
representations and continuous-time Fourier
representations.
Now, just quickly, let me draw your attention to the issue of
convergence and when a sequence can and can't be
represented, et cetera.
And recall that in the continuous-time case, we
focused on convergence in the context either of conditions,
which I referred to as square integrability, or another set
of conditions, which were the Dirichlet conditions.
And there was this issue about when the signal does and
doesn't converge at discontinuities, et cetera.
Let me just simply draw your attention to the fact that in
the discrete-time case, what we have is the representation
of the periodic signal as a sum of a
finite number of terms.
This represents capital N equations
and capital N unknowns.
If we consider earth the partial sum, namely taking a
smaller number of terms, then simply what happens is as the
number of terms increases to the finite number required to
represent x of n, we simply end up with the partial sum
representing the finite [? length ?] sequence.
What all that boils down to is the statement that in
discrete-time there really are no convergence issues as there
were in continuous-time.
OK, well let's look at an example of the Fourier series
representation for a particular signal.
And the one that I've picked here is a simple one.
Namely, a constant, a sine term, and a cosine term.
Now, for this particular example, we can expand this
out directly in terms of complex exponentials and
essentially recognize this as a sum of complex exponentials.
It's examined in more detail in Example 5.2 in the text.
And if we look at the Fourier series coefficients, we can
either look at it in terms of real and imaginary parts or
magnitude and angle.
On the left side here, I have the real part of the Fourier
coefficients.
And let me draw your attention to the fact that I've drawn
this to specifically illuminate the periodicity of
the Fourier series coefficients with a period of
capital N.
So here are the Fourier coefficients.
And, in fact, it's this line that represents the DC, or
constant term, and these two lines that represent the
cosine term.
And of course, these are the three terms that are required.
Or equivalently, this one, this one, and this one.
And then because of the periodicity of the Fourier
series coefficients, this simply periodically repeats.
So here is the real part and below it I show
the imaginary part.
And in the imaginary part, incidentally let me draw your
attention to the fact that it's this term and this term
in the imaginary part that represent the sinusoid.
Whereas it's the symmetric terms in the real part the
represent the cosine.
OK, let's look at another example.
This is another example from the text, and one that we'll
be making frequent reference to in this particular lecture.
And what it is, is a square wave.
And I've expressed the Fourier series coefficients, which are
algebraically developed in the text.
I've expressed the Fourier series coefficients as samples
of an envelope function.
And so I've expressed it as samples of this particular
function, which is referred to as a sin
nx over sin x function.
And let me just compare it to a continuous-time example,
which is the continuous-time square wave, where with the
continuous-times square wave the form of the Fourier series
coefficients was as samples of what we refer to as a sin x
over x function.
Now, the sin nx over sin x function, which is the
envelope of the Fourier series coefficients for the
discrete-time periodic square wave plays the role--
and we'll see it very often in discrete-time--
that sin x over x does in continuous-time.
And, in fact, we should understand right from the
beginning that the sin x over x envelope couldn't possibly
be the envelope of the discrete-time Fourier series
coefficients.
And one obvious reason is that it is not periodic.
What we require, of course, from the discussion that I've
just gone through is periodicity of the
coefficients.
And then consequently, also periodicity of the envelope in
the discrete-time case.
So once again, if we look back at the algebraic expression
that I have, it's samples of the sin nx over sine x
function that represent the Fourier series coefficients of
this periodic square wave.
Now, in the representation in the continuous-time case, we
essentially had used the concept of an envelope to
represent the Fourier series coefficients, and the notion
that the Fourier series coefficients were samples of
an envelope.
And that is the same notion that we'll be using in
discrete-time.
So again for this square wave example, then what we have is
an envelope function, the sin nx over sin x envelope
function for a particular value of the period.
Here indicated with a period of 10 samples.
These samples of this envelope function would then represent
the Fourier series coefficients.
If we increased the period, then we would simply have a
finer spacing on the samples of the envelope function to
get the Fourier series coefficients.
And likewise, if we increase the period still further, what
we would have is an even finer spacing.
So actually, as the period increases, and recall we used
this in continuous-time also.
As the period increases, we can view the Fourier series
coefficients as samples of an envelope.
And as the period increases, the sample spacing
gets finer and finer.
And in fact, as the period goes off essentially to
infinity, the samples of the envelope, in effect, become
the envelope.
And recall also that this was essentially the trick that we
used in continuous-time to allow us to develop or utilize
the Fourier series to provide a representation of aperiodic
signals as a linear combination of complex
exponentials.
In particular, what we did in the continuous-time case when
we had an aperiodic signal was to consider constructing a
periodic signal for which the aperiodic
signal was one period.
And then we developed the notion that since the periodic
signal has a Fourier series, and since as the period of the
periodic signal increases and goes to infinity, the periodic
signal represents the aperiodic signal.
Then, essentially, the Fourier series provides us with a
representation.
Now, we can do exactly the same thing in the
discrete-time case.
The statement is exactly the same, except that in the
discrete-time case, instead of t as the independent variable,
we simply make exactly the same statement, but with our
discrete-time variable n.
So the basic notion then in representing a discrete-time
aperiodic signal is to first construct a periodic signal.
Here we have the aperiodic signal.
We construct a periodic signal by simply periodically
replicating the aperiodic signal.
The periodic signal and the aperiodic signal are identical
for one period.
And as the period goes off to infinity, it's the Fourier
series representation of the periodic signal that provides
a representation of the aperiodic signal.
Again, to return to the example that we have been kind
of working through this lecture.
Namely, the periodic square wave.
If we have an aperiodic signal, which is a rectangle,
and we construct a periodic signal.
And now we consider letting this
period increase to infinity.
We would first have this set of samples of the envelope.
As the period increases, we would decrease the sample
spacing to this set of samples.
As the period increases further, it would be this set
of samples.
And as the period goes off to infinity, it's every point on
the envelope.
In fact, what the representation of the
aperiodic signal is, is the envelope.
OK, well, so that's the basic notion.
It's no different than what we did in the
continuous-time case.
And mathematically, it develops in very much the same
way as in the continuous-time case.
Specifically, here is our representation through the
Fourier series of the--
here is a representation through the envelope function.
And this is the Fourier series synthesis equation where the
equation below tells us how we get these Fourier coefficients
or the envelope from x of n.
Now, x tilde of n is the periodic signal.
And we know that over one period, which is the only
interval over which we use it, in fact, this is identical to
the aperiodic signal.
And so, in fact, we can rewrite this equation simply
by substituting in instead of x tilde, the original
aperiodic signal.
And now we can use infinite limits on this sum.
And what we would want to examine, mathematically, is
what happens to the top equation as we let the period
go off to infinity?
And what happens is exactly identical, mathematically, to
continuous-time.
I won't belabor the details.
Essentially it's this sum that goes to an integral.
Omega 0, which is the fundamental frequency, is
going towards 0.
In fact, becomes the differential in the integral.
And in the second equation, of course, this then
becomes x of omega.
And as N goes to infinity then, what the Fourier series
becomes is the Fourier transform as summarized by the
bottom two equations.
So although there is a little bit of mathematical trickery.
Or let's not call it trickery, but subtlety, to be tracked
through in detail.
The important conceptual thing to think about is this notion
that we take the aperiodic signal, form a periodic
signal, let the period go off to infinity.
In which case, the Fourier series coefficients become
these envelopes functions.
And incidentally, mathematically, one of the
sums ends up going to an integral.
So what we have then is the discrete-time Fourier
transform, which is a representation of
an aperiodic signal.
And we have the synthesis equation, which I show as the
top equation on this transparency.
And this is the integral that the Fourier series synthesis
equation went to as the period went off to infinity.
And we have the corresponding analysis equation, which is
shown below, where this tells us the Fourier transform.
In effect, the envelope or the Fourier series coefficients of
that periodic signal.
And here represented in terms of the aperiodic signal.
So we have the analysis equation
and synthesis equation.
There are a number of things to focus on
as you look at this.
And we'll talk about some of its properties actually in the
next lecture.
But some of the points that I'd like you to think about
and focus on is the fact that now there is somewhat of an
imbalance or lack of duality between the time domain and
frequency domain.
x of n, which is our aperiodic signal,
is of course, discrete.
It's Fourier transform, x of omega, is a function of a
continuous variable.
Omega is a continuous variable.
That is essentially what represents the envelope.
Also, in the time domain x of n is aperiodic.
It's not a periodic function.
However, in the frequency domain, remember that the
Fourier series coefficients were always periodic.
Well, this envelope function then is also periodic with a
period in omega of 2 pi.
Once again, the reason for the periodicity, it all stems back
to the fact that when we talk about complex exponentials--
and recall back to the early lectures.
In discrete-time, as the frequency variable covers a
range of 2 pi, when you proceed past that range, you
simply see the same complex exponentials
over and over again.
And so obviously, anything that we do with them would
have to be periodic in that frequency variable.
All right, notationally, we'll, again, represent the
discrete-time Fourier transform pair
as I indicated here.
And since it's a complex function of frequency may, on
occasion, want to either represent it in rectangular
form as I indicate in this equation, or in polar form as
I indicate in this equation.
Let's look at an example.
And, of course, one example that we can look at is the one
that has kind of been tracking us through this lecture, which
is the example of a rectangle.
Now, the rectangle, if we refer back to our argument of
how we get a Fourier representation for an
aperiodic signal, we would form a periodic signal where
this is repeated.
And that's our square wave example.
As the period goes to infinity, the Fourier
transform of this is represented by the envelope of
those Fourier series coefficients, and that was our
sin nx over sin x function, which in this particular case,
for these particular numbers, is sin 5 omega over 2 divided
by sin omega over 2.
And notice, of course, as we would expect--
notice that this is a periodic function of the frequency
variable omega repeating, of course, with a period of 2 pi.
Whereas, in the time domain, the function was not a
periodic function, it's aperiodic.
Now, let's look at another example.
Let's look at an example which is another signal that has
kind of popped its head up from time to time as the
lectures have gone along.
A signal which is another aperiodic signal, which is a
decaying exponential of this form with the factor a chosen
between 0 and 1.
And you can work out the algebra at your leisure.
Basically, if we substitute into the Fourier transform
analysis equation, it's this sum that we evaluate.
Because we have a unit step here which shuts this off for
n less than 0, we can change the limits on the sum.
This then corresponds to the sum over an infinite number of
terms of a geometric series.
And that, as we've seen before, is 1 divided by 1
minus a e to the minus j omega.
So let's look at what that looks like.
Here then we have, again, the expression in the time domain
and the expression in the frequency domain.
And let's, in particular, focus on what the magnitude of
the Fourier transform looks like.
It's as we show here.
And for the particular values of a that I pick, namely
between 0 and 1, it's larger at the origin
than it is at pi.
And then, of course, it is periodic.
And the periodicity is inherent in the Fourier
transform in discrete-time, so we really might only need to
look at this either from minus pi to pi, or from 0 to 2 pi.
The periodicity, of course, would imply what the rest of
this is for other values of omega.
Let me also draw your attention while we're on it to
the fact that--
observe that if a were, in fact, negative, then this
value would be less than this value.
And in fact, for a negative, the magnitude of the frequency
response would look like this except shifted by an amount in
omega equal to pi.
And this example will come up and play an important role in
our discussion next time, so try to keep it in mind.
And in fact, work it out more carefully between
now and next time.
And also, if you have a chance, focus on this issue of
how it looks with a positive as compared with a negative.
Now, we developed the Fourier transform by beginning with
the Fourier series.
We did that in continuous-time also.
What I'd like to do now, just as we did in continuous-time,
is now absorb the Fourier series within the broader
framework of the Fourier transform.
And there are two relationships between the
Fourier series and the Fourier transform, which are identical
to relationships that we had in the continuous-time case.
Let me remind you that in continuous-time we had the
statement that if we have a periodic signal, that in fact
the Fourier series coefficients of that periodic
signal is proportional to samples of the Fourier
transform of one period.
Well, in fact, let me remind you flows easily from all the
things that we built up so far, because of the fact that
the Fourier transform essentially, by definition, of
the way we developed it, is what we get as the Fourier
series coefficients, as we focus on one period, and then
let the period go off to infinity.
Well, looking at one period, the Fourier transform of that
then is the envelope of the Fourier series coefficients.
And so in continuous-time, we have this relationship.
And in discrete-time, we have precisely the same
relationship, except that here we're talking about an integer
variable as opposed to the continuous variable, and a
period of capital N as opposed to a period of t0.
OK, so once again, if we return to our example, or if
we return to a periodic signal.
If we have a periodic signal and we consider the Fourier
transform of one period, the Fourier series coefficients of
this periodic signal are, in fact, samples--
as stated mathematically in the bottom equation, samples
of the Fourier transform of one period.
So x of omega is the Fourier transform of one period.
a sub k's are the Fourier series coefficients of the
periodic signal.
And this relationship simply says they're related except
for scale factor through samples along
the frequency axis.
And, of course, we saw this in the context of
our square wave example.
In the square wave example, we have a periodic signal, which
is a periodic square wave.
And the Fourier transform of one period, in fact,
represents the envelope.
And here we have the envelope function.
Represents the envelope of the Fourier series coefficients.
And the Fourier series coefficients are samples.
So what we have then is a relationship back to the
Fourier series coefficients from the Fourier transform
that tells us that for a periodic signal now, the
periodic signal--
the Fourier series coefficients are related, are
samples of the Fourier transform of one period.
Now, finally, to kind of bring things back in a circle and
exactly identical to what we did in the continuous-time
case, we can finally absorb the Fourier series in
discrete-time.
We can absorb it into the framework
of the Fourier transform.
Now, remember or recall how we did that when we tried to do a
similar sort of thing in continuous-time.
In continuous-time, what we essentially did is to develop
that, more or less, by definition.
We have a periodic signal.
The periodic signal is represented through a Fourier
series and Fourier series coefficients.
Essentially what I pointed out at that time
was that if we define--
take it as a definition, the Fourier transform of the
periodic signal as an impulse train where the amplitudes of
the impulses are proportional to the Fourier series
coefficients.
If we take that impulse train representation and simply plug
it into the Fourier transform synthesis equation, what we
end up with is the Fourier series synthesis equation.
So in continuous-time, we had used this definition of the
continuous-time Fourier transform
of a periodic signal.
And again, in discrete-time, it's simply a matter of using
exactly the same expression.
And using, instead, the appropriate variables related
to discrete-time rather than the variables related to
continuous-time.
So in discrete-time, if we have a periodic signal, the
Fourier transform of that periodic signal is defined as
an impulse train where the amplitudes of the impulses are
proportional to the Fourier series coefficients.
If this expression is substituted into the synthesis
equation for the Fourier transform, that will simply
then reduce to the synthesis equation
for the Fourier series.
So once more returning to our example, which is the square
wave example that we've carried through these
lectures, or through this lecture, we can see that
really what we're talking about really is
a notational change.
Here is the periodic signal and below it are the Fourier
series coefficients, where I've removed the envelope
function and just indicate the amplitudes of the coefficients
indexed, of course, on the coefficient number.
And so this represents a bar graph.
And if instead of talking about the Fourier series
coefficients, what I want to talk about is the Fourier
transform, the Fourier transform, in essence,
corresponds to simply redrawing that using impulses
and using an axis that is essentially indexed on the
fundamental frequency omega 0, rather than on the Fourier
series coefficient number k.
OK, so to summarize, what we've done is
to pretty much parallel--
somewhat more quickly, the kind of development that we
went through for continuous-time representation
through complex exponentials, paralleled that for the
discrete-time case.
And pretty much the conceptual underpinnings of the
development are identical in discrete-time and in
continuous-time.
We saw that there are some major differences, or
important differences between continuous-time and
discrete-time.
And the difference, essentially
relates to two aspects.
One aspect is the fact that in discrete-time, we have a
discrete representation in the time domain, whereas the
independent variable in the frequency domain is a
continuous variable.
Whereas in continuous-time for the Fourier transform, we had
a duality between the time domain and frequency domain.
The other very important difference tied back to the
difference between complex exponentials, continuous-time
and discrete-time.
In continuous-time, complex exponentials, as you vary the
frequency, generate distinct time functions.
In discrete-time, as you vary the frequency, once you've
covered a frequency interval of 2 pi, then you've seen all
the ones there are to see.
There are no more.
And this, in effect, imposes a periodicity on the Fourier
domain representation of discrete-time signals.
And some of those differences and, of course, lots of the
similarities will surface, both as we use this
representation and as we develop further properties.
In the next lecture, what we'll do is to focus in,
again, on the Fourier transform, the discrete-time
Fourier transform, develop or illuminate some of the
properties of the Fourier transform, and then see how
these properties can be used for a number of things.
For example, how the properties as they were in
continuous-time can be used to efficiently generate the
solution and analyze linear constant coefficient
difference equations.
And then beyond that, the concepts of filtering and
modulation.
And both the properties and interpretation, which will
very strongly parallel the kinds of developments along
those lines that we did in the last lecture.
Thank you.