The following content is
provided under a Creative
Commons license.
Your support will help MIT
OpenCourseWare continue to
offer high quality educational
resources for free.
To make a donation or view
additional materials from
hundreds of MIT courses, visit
MIT OpenCourseWare at
ocw.mit.edu.
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.
