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PROFESSOR: Last time, we began to address the issue of
building continuous time signals out of a linear
combination of complex exponentials.
And for the class of periodic signals specifically, what
this led to was the Fourier series representation for
periodic signals.
Let me just summarize the results that we
developed last time.
For periodic signals, we had the continuous-time Fourier
series, where we built the periodic signal out of a
linear combination of harmonically related complex
exponentials.
And what that led to was what we referred to as the
synthesis equation.
And we briefly addressed the issue of when this in fact
builds, when this in fact is a complete representation of the
periodic signal, and in essence, what we presented was
conditions either for x of t being square integrable or x
of t being absolutely integrable.
Then, the other side of the Fourier series is what I
referred to as the analysis equation.
And the analysis equation was the equation that told us how
we get the Fourier series coefficients from x of t.
And so this equation together with the synthesis equation
represent the Fourier series description
for periodic signals.
Now what we'd like to do is extend this idea to provide a
mechanism for building non-periodic signals also out
of a linear combination of complex exponentials.
And the basic idea behind doing this is very simple and
also very clever as I indicated last time.
Essentially, the thought is the following, if we have a
non-periodic signal or aperiodic signal, we can think
of constructing a periodic signal by simply periodically
replicating that aperiodic signal.
So for example, if I have an aperiodic signal as I've
indicated here, I can consider building a periodic signal,
where I simply take this original signal and repeat it
at multiples of some period t 0.
Now, two things to recognize about this.
One is that the periodic signal is equal to the
aperiodic signal over one period.
And the second is that as the period goes to infinity then,
in fact, the periodic signal goes to the aperiodic signal.
So the basic idea then is to use the Fourier series to
represent the periodic signal, and then examine the Fourier
series expression as we let the period go to infinity.
Well, let's quickly see how this develops in terms of the
associated equations.
Here, again, we have the periodic signal.
And what we want to inquire into is what happens to the
Fourier series expression for this as we let the period go
to infinity.
As that happens, whatever Fourier series representation
we end up with will correspond also to a representation for
this aperiodic signal.
Well, let's see.
The Fourier series synthesis expression for the periodic
signal expresses x tilde of t, the periodic signal as a
linear combination of harmonically related complex
exponentials with the fundamental frequency omega 0
equaled to 2 pi divided by the period.
And the analysis equation tells us what the relationship
is for the coefficients in terms of the periodic signal.
Now, I indicated that the periodic signal and the
aperiodic signal are equal over one period.
We recognize that this integration, in fact, only
occurs over one period.
And so we can re-express this in terms of our original
aperiodic signal.
So this tells us the Fourier series coefficients in
terms of x of t.
Now, if we look at this expression, which is the
expression for the Fourier coefficients of the aperiodic
signal, one of the things to recognize is that in effect
what this represents are samples of an integral, where
we can think of the variable omega taking on values that
are integer multiples of omega 0.
Said another way, let's define a function, as I've indicated
here, which is this integral, where we may think of omega as
being a continuous variable and then the Fourier series
coefficients correspond to substituting for
omega k omega 0.
Now, one reason for doing that, as we'll see, is that in
fact, this will turn out to provide us with a mechanism
for a Fourier representation of x of t.
And this, in fact, then, is an envelope of the Fourier series
coefficients.
In other words, t 0 the period times the coefficients is
equal to this integral add integer multiples of omega 0.
So this, in effect, tells us how to get the Fourier series
coefficients of the periodic signal in terms of samples of
an envelope.
And that will become a very important notion shortly.
And that, in effect, will correspond to an analysis
equation to represent the aperiodic signal.
Now, let's look at the synthesis equation.
Recall that in the synthesis our strategy is to build a
periodic signal and let the period go to infinity.
Well, here is the expression for the synthesis of the
periodic signal now expressed in terms of samples of this
envelope function, and where I've simply used the fact or
the substitution that t 0 is 2 pi over omega 0, and so I have
an omega 0 here and a 1 over 2 pi.
And the reason for doing that, as we'll see in a minute, is
that this then turns into an integral.
Specifically, then, the synthesis equation that we
have is what I've indicated here.
We would now want to examine this as the period goes to
infinity, which means that omega 0 becomes
infinitesimally small.
And without dwelling on the details, and with my
suggesting that you give this a fair amount of reflection,
in fact, what happens as the period goes to infinity is
that this summation approaches an integral over omega, where
omega 0 becomes the differential in omega, and the
periodic signal, of course, approach is
the aperiodic signal.
So the resulting equation that we get out of the original
Fourier series synthesis equation is the equation that
I indicate down here, x of t synthesized in terms of this
integral, which is what the Fourier series approaches as
omega 0 goes to 0.
And we had previously that x of omega was in fact an
envelope function.
And we have then the corresponding Fourier
transform analysis equation, which tells us how we arrive
at that envelope in terms of x of t.
So we now have an analysis equation and a synthesis
equation, which in effect expresses for us how to build
x of t in terms of infinitesimally finely spaced
complex exponentials.
The strategy to review it, and which I'd like to illustrate
with a succession of overlays, was to begin with our
aperiodic signal, as I indicate here, and then we
constructed from that a periodic signal.
And this periodic signal has a Fourier series, and we express
the Fourier series coefficients of this as
samples of an envelope function.
The envelope function is what I indicate on the curve below.
So this is the envelope of the Fourier series coefficients.
For example, if the period t 0 was four times t1, then the
Fourier series coefficients that we would end up with is
this set of samples of the envelope.
If instead we doubled that period, then the Fourier
series coefficients that we end up with
are more finely spaced.
And as t 0 continues to increase, we get more and more
finely spaced samples of this envelope function, and as t 0
goes to infinity in fact, what we get is every single point
on the envelope, and that provides us with the
representation for the aperiodic signal.
Let me, just to really emphasize the point, show this
example once again.
But now, let's look at it dynamically on
the computer display.
So here we have the square wave, and below it, the
Fourier series coefficients.
And we now want to look at the Fourier series coefficients as
the period of the square wave starts to increase.
And what we see is that these look like
samples of an envelope.
And in fact, the envelope of the Fourier series
coefficients is shown in the bottom [? trace, ?]
and to emphasize in fact that it is the envelope let's
superimpose it on top of the Fourier series coefficients
that we've generated so far.
OK.
Now, let's increase the period even further, and we'll see
the Fourier series coefficients fill in under
that envelope function even more.
And in fact, as the period gets large enough, what we
begin to get a sense of is that we're sampling more and
more finely this envelope.
And in fact, in the limit, as the period goes off to
infinity, the samples basically will represent every
single point on the envelope.
Well, this is about as far as we want to go.
Let's once again, plot the envelope function, and again,
to emphasize that we've generated samples of that,
let's superimpose that on the Fourier series coefficients.
So what we have then is now our Fourier transform
representation, the continuous time Fourier transform with
the synthesis equation expressed as an integral, as
I've indicated here, and this integral is what the Fourier
series sum went to as we let the period go to infinity or
the frequency go to zero.
The corresponding analysis equation, which we have here,
the analysis equation being the expression for the
envelope of the Fourier series coefficients for the
periodically replicated signal.
And in shorthand notation, we would think
of x of t and [? its ?]
Fourier transform as a pair, as I've indicated here.
And the Fourier transform, as we'll emphasize in several
examples, and certainly as is consistent with the Fourier
Series, is a complex valued function even
when x of t is real.
So with x of t real, we end up with a Fourier transform,
which is a complex function.
Just as the Fourier series coefficients were complex for
a real value time function.
So we could alternatively, as with the Fourier series,
express the Fourier transform in terms of it's real part and
imaginary part, or alternatively, in terms of its
magnitude and its angle.
All right, now let's look at an example of a time function
in its Fourier transform.
And so let's consider an example, which in fact is an
example worked out in the text.
It's example 4.7 in the text.
And this is our old familiar friend the exponential.
It's Fourier transform is the integral from minus infinity
to plus infinity, x of t, e to the minus j omega t dt.
And so, if we substitute in x of t and combine these two
exponentials together, these two exponentials combined are
e to the minus t times a plus j omega.
And if we carry out the integration of this, we end up
with the expression indicated here and provided now, and
this is important, provided that a is greater than 0, then
at the upper limit, this exponential becomes 0.
At the lower limit, of course, it's one.
And so what we have finally is for the Fourier transform
expression 1 over a plus j omega.
Now, this Fourier transform as I indicated is a complex
valued function.
Let's just take a look at what it looks like graphically.
We have the expression for the Fourier transform pair, e to
the minus a t times [? a ?]
[? step ?].
And its Fourier transform is 1 over a plus j omega.
And I indicated that that's true for a greater than 0.
Now, in the expression that we just worked out, if a is less
than 0, in fact, the expression doesn't converge e
to the minus a t for a negative as t goes to
infinity, blows up, and so in fact the Fourier transform
doesn't converge except for the case where a is greater
than 0 And in fact, there is a more detailed discussion of
convergence issues in the text.
The convergence issues are very much the same for the
Fourier transform as they are for the Fourier series.
And in fact, that's not surprising, because we
developed the Fourier transform out of a
consideration of the Fourier series.
So the convergence conditions as you'll see as you refer in
detail to the text relate to whether the time function is
absolutely integrable under one set of conditions and
square integrable under another set of conditions.
OK, now, if we plot the Fourier transform, let's first
consider the shape of the time function.
And as I indicated, we're restricting the time function
so that the exponential factor a is positive.
In other words, e to the minus a t decays
as t goes to infinity.
The magnitude of the Fourier transform is as I indicate
here and the phase below it.
And there are a number of things we can see about the
magnitude and phase of the Fourier transform for this
example, which in fact we'll see in the next lecture are
properties that apply more generally.
For example, the fact that the Fourier transform magnitude is
an even function of frequency, and the phase is an odd
function of frequency.
Now, let me also draw your attention to the fact that on
this curve we have both positive frequencies and
negative frequencies.
In other words, in our expression for the Fourier
transform, it requires both omega
positive and omega negative.
This, of course, was exactly the same in the case of the
Fourier series.
And the reason you should recall and keep in mind is
related to the fact that we're building our signals out of
complex exponentials, which require both positive values
of omega and negative values of omega.
Alternatively, if we chosen other representation, which
turns out notationally to be much more difficult, namely
sines and cosines, then we would in fact only consider
positive frequencies.
So it's important to keep in mind that, in our case, both
with the Fourier series and the Fourier transform, we deal
and require both positive and negative frequencies in order
to build our signals.
Now, in the graphical representation that I've shown
here, I've chosen a linear amplitude scale and a linear
frequency scale.
And that's one graphical representation for the Fourier
transform that we'll typically use.
There's another one that very commonly arises, which I'll
just briefly indicate for this example.
And that is what's referred to as a bode plot in which the
magnitude is displayed on a log amplitude and log
frequency scale.
And the phase is displayed on a log frequency scale.
Let me show you what I mean.
Here is the general expression for the bode plot.
The bode plot expresses for us the amplitude in terms of the
logarithm to the base 10 of the magnitude.
And it also expresses the angle in both cases expressed
as a function of a logarithmic frequency axis.
So here is the amplitude as I've displayed it.
And this is a log magnitude scale, a logarithmic frequency
scale as indicated by the fact that as we move in equal
increments along this axis, we change frequency
by a factor of 10.
And similarly, what we have is a display for the phase again
on a log frequency scale.
And I indicated that there is a symmetry to the Fourier
transform, and so in fact, we can infer from this particular
picture what it looks like for the negative frequencies as
well as for the positive frequencies.
Now, what we've done so far is to develop the Fourier
transform on the basis, the Fourier transform of an
aperiodic signal on the basis of periodically repeating it
and recognizing that the Fourier series coefficients
are samples of an envelope and that these become more finely
spaced as frequency increases.
And in fact, we can go back to our original equation in which
we developed an envelope function, and what we had
indicated is that the Fourier series coefficients were
samples of this envelope.
We then defined this envelope as the Fourier transform of
this aperiodic signal.
So that provided us with a way-- and it was a mechanism--
for getting a representation for an aperiodic signal.
Now, suppose that we have instead a periodic signal, are
there, in fact, some statements that we can make
about how the Fourier series coefficients of that are
related to the Fourier transform of something.
Well, in fact, this statement tells us
exactly how to do that.
What this statement says is that, in fact, the Fourier
series coefficients are samples of the Fourier
transform of one period?
So if we now consider a periodic signal, we can in
fact get the Fourier series coefficients of that periodic
signal by considering the Fourier
transform of one period.
Said another way, the Fourier series coefficients are
proportional to samples of the Fourier
transform of one period.
So if we consider this a periodic signal, computed as
Fourier transform, and selected these samples that I
indicate here, namely samples equally spaced in omega by
integer multiples of omega 0, then in fact, those would be
the Fourier series coefficients.
So we can go back to our example previously that
involved the square wave.
And now, in this case, we could argue that if in fact it
was the periodic signal that we started with, we could get
the Fourier series coefficients of that by
thinking about the Fourier transform of one period, which
I indicate here.
And then the Fourier series coefficients of the periodic
signal, in fact, are the appropriate set of samples of
this envelope.
All right, now, we have a way of getting the Fourier series
coefficients from the Fourier transform of one period.
We originally derived the Fourier transform of one
period from the Fourier series.
What would, in fact, be nice is if we could incorporate the
Fourier series and the Fourier transform
within a common framework.
And in fact, it turns out that there is a very convenient way
of doing that almost by definition.
Essentially, if we consider what the equation for the
synthesis looks like in both cases, we can in effect define
a Fourier transform for the periodic signal, which we know
is represented by its Fourier series coefficients.
We can define a Fourier transform, and the definition
of the Fourier transform is as an impulse train, where the
coefficients in the impulse train are proportional, with a
proportionality factor of 2 pi for a more or less a
bookkeeping reason, proportional to the Fourier
series coefficients.
And the validity of this is, more or less, can be seen
essentially by substitution.
Specifically, here is then the synthesis equation for the
Fourier transform if we substitute this definition for
the Fourier transform of the periodic signal into this
expression then when we do the appropriate bookkeeping and
interchange the order of summation and integration the
impulse integrates out to the exponential
factor that we want.
So we have the exponential factor.
We have the Fourier series coefficients.
The 2 pis take care of each other, and what we're left
with is the synthesis equation for aperiodic signal in terms
of the Fourier transform, or in terms of its Fourier series
coefficients.
Now, we can just see this in terms of a simple example.
If we consider the example of a symmetric square wave, then
in effect what we're saying is that for this symmetric square
wave, this has a set of Fourier series coefficients,
which we worked out previously and which I indicate on this
figure with a bar graph.
And really all that we're saying is that, whereas these
Fourier series coefficients are indexed on an integer
variable k, and [? they're ?] bars not impulses.
If we simply redefine or define the Fourier transform
of the periodic signal as an impulse train, where the
weights of the impulses are 2 pi times the corresponding
Fourier series coefficients, then this, in fact, is what we
would use as the Fourier transform of
the periodic signal.
Now, we've kind of gone back and forth, and maybe even it
might seem like we've gone around in circles.
So let me just try to summarize the various
relationships and steps that we've gone through, keeping in
mind that one of our objectives was first to
develop a representation for aperiodic signals and then
attempt to incorporate within one framework both periodic
and aperiodic signals.
We began with an aperiodic signal.
And the strategy was to develop a Fourier
representation by constructing a periodic signal for which
that was one period.
And then we let the period go to infinity,
as I indicate here.
So we have an aperiodic signal.
We construct a periodic signal, x tilde of t for which
one period is the aperiodic signal.
X tilde of t, the periodic signal, has a Fourier series,
and as its period increases that approaches the aperiodic
signal, and the Fourier series of that approaches the Fourier
transform of the original aperiodic signal.
So that was the first step we took.
Now, the second thing that we recognize is that once we have
the concept of the Fourier transform, we can, in fact,
relate the Fourier series coefficients to the Fourier
transform of one period.
So the second statement that we made was that if in fact
we're trying to represent a periodic signal, we can get
the Fourier series coefficients of that by
computing the Fourier transform of one period and
then samples of that Fourier transform are, in fact, the
Fourier series coefficients for the periodic signal.
Then, the third step that we took was to inquire as to
whether there is a Fourier transform that can
appropriately be defined for the periodic signal, and the
mechanism for doing that was to recognize that if we simply
defined the Fourier transform of the periodic signal as an
impulse train, where the impulse heights or areas were
proportional to the Fourier series coefficients, then, in
fact, the Fourier transform synthesis equation reduced to
the Fourier series synthesis equation.
So the third step, then, was with a periodic signal.
The Fourier transform of that periodic signal, defined as an
impulse train, where the heights or areas of the
impulses are proportional to the Fourier series
coefficients, provides us with a mechanism for combining it
together the concepts or notation of the Fourier series
and Fourier transform.
So if we just took a very simple example, here is an
example in which we have an aperiodic signal, which is
just an impulse, and its Fourier
transform is just a constant.
We can think of a periodic signal associated with this,
which is this signal periodically replicated with a
spacing t 0.
The Fourier transform of this is a constant.
And this, of course, has a Fourier series representation.
So the Fourier transform of the original
impulse is just a constant.
The Fourier transform of the periodic signal is an impulse
train, where the heights of the impulses are proportional
to the Fourier series coefficients.
And, of course, we could previously have computed the
Fourier series coefficients for that impulse train, and
those Fourier series coefficients are
as I've shown here.
So in both of these cases, these in effect represent just
a change in notation, where here we have a bar graph, and
here we have an impulse train.
And both of these simply represent samples of what we
have above, which is the Fourier transform of the
original aperiodic signal.
Once again, I suspect that kind of moving back and forth
and trying to straighten out when we're talking about
periodic and aperiodic signals may require a little mental
gymnastics initially.
Basically, what we've tried to do is incorporate within one
framework a representation for both aperiodic and periodic
signals, and the Fourier transform provides us with a
mechanism to do that.
In the next lecture, I'll continue with the discussion
of the continuous-time Fourier transform in particular
focusing on a number of its properties, some of which
we've already seen, namely the symmetry properties.
We'll see lots of other properties that relate, of
course, both to the Fourier transform and
to the Fourier series.
Thank you.