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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.
