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[MUSIC PLAYING]
PROFESSOR: In the last several
lectures, we've talked about a
generalization of the
continuous-time Fourier
transform and a very similar
strategy also applies to
discrete-time, and that's what
we want to begin to deal with
in today's lecture.
So what we want to talk about
is generalizing the Fourier
transform, and what this will
lead to in discrete-time is a
notion referred to as
the z-transform.
Now, just as in continuous-time,
in
discrete-time the Fourier
transform corresponded to a
representation of a sequence
as a linear combination of
complex exponentials.
So this was the synthesis
equation.
And, of course, there is the
corresponding analysis equation.
And as you recall, and as is the
same for continuous-time,
the reason that we picked
complex exponentials was
because of the fact that they
are eigenfunctions of linear
time-invariant systems.
In other words, if you have a
complex exponential into a
linear time-invariant system,
the output is a complex
exponential.
And the change in complex
amplitude, which corresponds
to the frequency response, in
fact is what led to the
definition of the Fourier
transform.
In particular, it is the Fourier
transform of the
impulse response.
Well, that set of notions is,
more or less, identical to the
way we motivated the Laplace
transform in the
continuous-time case, in the
Fourier transform in the
continuous-time case.
And just as in continuous-time,
there are a
set of signals more general than
the complex exponentials,
which are also eigenfunctions
of linear
time-invariant systems.
In particular, in discrete-time,
if we had
instead of an exponential e to
the j omega n, we had a more
general complex number z, that
the signal z to the n is also
an eigenfunction of a linear
time-invariant system for any
particular z.
So we can see that by
substituting that into the
convolution sum and recognizing,
again, very
strongly paralleling the
continuous-time argument, that
we can rewrite this factor as
z to the n z to the minus k.
And because of the fact that
it's a sum on k and this term
doesn't depend on k, we can
take that term out.
And the conclusion is that if
we have z to the n as an
input, that the output is of the
same form times a factor
which depends on z.
But of course, doesn't depend
on k because that is summed
out as we form the summation.
So, in fact, this summation
corresponds to a complex
number, which we'll denote as H
of z, where z will represent
a more general complex number.
Namely, z is r e to the j omega,
r being the magnitude
of this complex number
and omega, of
course, being the angle.
So for a linear time-invariant
system, a more general complex
exponential sequence of this
form generates as an output a
complex exponential sequence of
the same form with a change
in amplitude which we're
representing as H of z,
recognizing the fact that it's
going to be a function of what
that complex number is.
And this amplitude factor is
given by this summation.
And it is this summation which
is defined as the z-transform
of the sequence H of n.
Now, let me stress that--
and I'll continue to stress this
as the lecture goes on.
That much of what we've said is
directly parallel to what
we said in the continuous-time
case.
And what we've simply done
is to expand from complex
exponentials with a purely
imaginary exponent, complex
exponential time functions or
sequences of that form, to
ones that have more general
complex exponential factors.
Now, we have a mapping here from
the impulse response to
the amplitude or the eigenvalue
associated with
that input, and this mapping is
what is referred to as the
z-transform.
So H of z is, in fact,
the z-transform
of the impulse response.
And if we consider applying
this mapping as we did in
continuous-time in a similar
argument, applying this
mapping to a sequence whether
or not it corresponds to the
impulse response or a linear
time-invariant system, that
leads then to the z-transform of
a general sequence x of n.
The z-transform being defined
by this relationship.
And again, notationally, we'll
often represent a time
function and a z-transform
through a shorthand notation,
just indicating that x of z is
the z-transform of x of n.
So we've kind of motivated the
development in a manner
exactly identical to
what we had done
with the Laplace transform.
Kind of the idea that if you
look at the eigenvalue
associated with a linear
time-invariant system, that
essentially generates a mapping
between the sequence--
system impulse response
and a function of z.
And that corresponds to the
z-transform here, it
corresponded to the Laplace
transform in the
continuous-time case.
That same argument is also the
kind of argument that we use
to lead us into the Fourier
transform originally.
And once again, what you would
expect is that the z-transform
has a very close and important
relationship
to the Fourier transform.
And indeed, that relationship
turns out to be, more or less,
identical to the relationship
between the Laplace transform
and the Fourier transform
in continuous-time.
Well, let's look at
the relationship.
First of all, what we recognize
is that if we
compare the Fourier transform
expression for a sequence and
the z-transform expression for
the same sequence that they
involve, essentially,
the same operations.
And in fact, since z is of the
form r e to the j omega, if we
want this sum to look like this
sum, then that would mean
that we would choose z equal
to e to the j omega.
Said another way, the
z-transform, when z is e to
the j omega, is going to reduce
to the Fourier transform.
So we have a relationship like
the one, again, that we had
between the Laplace transform
and the Fourier transform in
continuous-time.
Namely that for a certain set
of values of the complex
variable, the transform, the
z-transform, reduces to the
Fourier transform.
So if we have x of z, the
z-transform, and we look at
that for z equal to e to the j
omega, and z equal to e to the
j omega is similar to saying
that we're looking at that for
the magnitude of z equal to 1.
We're specifically choosing
r equal to 1, which is the
magnitude of z.
Then this is equal
to the Fourier
transform of the sequence.
So the z-transform for z equal
to e to the j omega is the
Fourier transform,
and so this then
corresponds to x of omega.
Namely, the Fourier transform.
Well, we now have ourselves in a
similar situation, again, to
what we had when we talked about
the Laplace transform.
Namely, a notational
awkwardness, or inconvenience,
which we can resolve by simply
redefining some of our notation.
In particular, the awkwardness
relates to the fact that
whereas we've been writing our
Fourier transforms this way,
as x of omega, if we were to
express x of z and look at it,
it's equal to e to
the j omega.
We end up with the independent
variable being e to the j
omega rather than omega.
Well, in fact, the Fourier
transform is
a function of omega.
It's also a function of
e to the j omega.
And now what we can see is that
given the fact that we
want to generalize the Fourier
transform to the z-transform,
it's convenient now to use as
notation for the Fourier
transform x of z with z equal
to e to the j omega.
Namely, our Fourier transforms
will now be written as I've
indicated here.
So just summarizing that, our
new notation is that the
independent variable on the
Fourier transform is now going
to be expressed as e to the j
omega rather than as omega.
It's a minor notational change,
but I recognize the
fact that it's somewhat
confusing initially, and takes
a few minutes to sit down and
just get it straightened out.
It's very similar
to what we did
with the Laplace transform.
But let me draw your attention
to the fact that in the
Laplace transform, the
independent variable that we
ended up with in talking about
the Fourier transform is
different than what we're
ending up with here.
In particular, before we
had j omega, now we
have e to the j omega.
And the reason for that is
simply that whereas in
continuous time we were talking
about functions of the
form e to the st, now we're
talking about sequences of the
form z to the n.
So we have one relationship
between the Fourier transform
and the z-transform.
Namely, the fact that for the
magnitude of z equal to 1, the
z-transform reduces to the
Fourier transform.
Now, in the Laplace transform,
we also had another important
relationship and observation,
which was the fact that the
Laplace transform
was the Fourier
transform of x of t modified.
And how was it modified?
It was modified by multiplying
by a decaying or growing
exponential, depending on what
the real part of s is.
Well, we have a very similar
situation with the
z-transform.
In particular, in addition to
the fact that the z-transform
for z equal to e to the j omega
reduces to the Fourier
transform, we'll see that the
z-transform for other values
of z is the Fourier transform
of the sequence with an
exponential weighting.
And let's see where
that comes from.
Here we have the general
expression for the
z-transform.
And recognizing that z is a
complex number which we're
expressing in polar form as r e
to the j omega, substituting
that in, this summation now
becomes x of n, r e to the j
omega to the minus n.
We can factor out these two
terms, r to the minus n and e
to the minus j omega n.
And combining the r to the minus
n with x of n and the e
to the minus j omega n being
treated separately, what we
end up with is the summation
that I have here.
Well, what this says is that
the z-transform, which is
this, at z equal to r e to the
j omega, is in fact the
Fourier transform of what?
It's the Fourier transform
of x of n multiplied by r
to the minus n.
So that is the expression
that we have here.
And in continuous-time, we had
the Laplace transform as the
Fourier transform of x of t
e to the minus sigma t.
Here we have the Fourier
transform of x of n r
to the minus n.
Now, something to just reflect
on for a minute is--
because it tends to cause a
little bit of problem with the
algebra later on if you're
attention isn't drawn to it,
is that we're talking about
multiplying x of n times r to
the minus n.
The question is, for r greater
than 1, does r to the minus n
increase exponentially as n
increases or does it decrease?
We're talking about
r to the minus n.
If r is greater than 1, if
the magnitude of r is
greater than 1.
For example, if it's equal
to 2, r to the minus n
is 1/2 to the n.
And so, in fact, that decreases
exponentially.
Or more generally, the larger
r is, the faster r to the
minus n decays with
increasing n.
Well, let's just look at some
examples of the z-transform.
And examples that I've picked,
again, are examples directly
out of the text.
And so the details of the
algebra you can look at more
leisurely as you sit
with the textbook.
Let's consider, first of all, an
exponential sequence x of n
equals a to the n times
the unit step.
So 0 for negative time and an
exponential for positive time.
And the Fourier transform,
as we've seen in earlier
lectures, is 1 over 1 minus
a e to the minus j omega.
But this doesn't always
converge.
In particular, for convergence
of the Fourier transform, we
would require absolute
summability of
the original sequence.
And that, in turn, requires
that the magnitude of a be
less than 1.
So the Fourier transform is
this, provided that the
magnitude of a is less than 1.
And what is the Fourier
transform if the magnitude of
a is not less than 1?
Well, the answer is
that, in that
case, it doesn't converge.
Now, let's look at
the z-transform.
The z-transform is the sum from
minus infinity to plus
infinity of a to the
n z to the minus n
times the unit step.
The unit step will change
the lower limit to 0.
So it's the sum from
0 to infinity.
And this is of the form
a times z to the
minus 1 to the n.
So we're summing from 0 to
infinity a times z to the
minus 1 to the n.
That sum is 1 over 1 minus
a z to the minus 1.
But in order for that sum to
converge, we require that the
magnitude of a times z to the
minus 1 be less than 1.
Now, the z-transform is the
Fourier transform of the
sequence a to the n times
r to the minus n.
And this statement about the
z-transform converging is
exactly identical to the
statement that what we're
requiring is that the magnitude
of a times r to the
minus 1 be less than 1, where
this represents the
exponential factor that we have
that in effect is applied
to the sequence, so that the
Fourier transform becomes the
z-transform.
And so, if we put this
condition, we can interpret
this condition in exactly the
same way that we interpret the
condition on convergence of
the Fourier transform.
So from what we've worked out
here then, what we have is the
z-transform of a to the n times
u of n is 1 over 1 minus
a z to the minus 1.
That works for any value of a
provided that we pick the
value of z correctly.
In particular, we have to
pick the set of values
of z, so that what?
So that the magnitude of a
times z to the minus 1
is less than 1.
Or equivalently, so that the
magnitude of z is greater than
the magnitude of a.
So associated with the
z-transform of this sequence
is this algebraic expression,
and this set of values on z
for which that algebraic
expression is valid.
And just as with the Laplace
transform, this range of
values is referred to as the
region of convergence of the
z-transform.
Now, again, as we saw with the
Laplace transform, it's
important to recognize that in
specifying or having worked
out the z-transform of a
sequence, it's not just the
algebraic expression, but also
the region of convergence
that's required to uniquely
specify it.
To emphasize that further,
here is Example
10.2 from the text.
And if you work that one
through, what you find is
that, algebraically, the
z-transform of this sequence
is 1 over 1 minus a
z to the minus 1.
Identical algebraically to
what we had up here.
But now with a region of
convergence, which is the
magnitude of z less than
the magnitude of a.
In contrast to this example,
where the region of
convergence was the magnitude
of z greater than the
magnitude of a.
So again, it requires not just
the algebraic expression, but
also requires a specification of
the region of convergence.
And also, as with the Laplace
transform, it's convenient in
looking at the z-transform
to represent it
in the complex plane.
In this case, the complex
plane referred to as the
z-plane, whereas in
continuous-time when we talked
about the Laplace transform,
it was the s-plane.
z, of course, because z is the
complex variable in terms of
which we're representing
the z-transform.
So we will be representing the
z-transform in terms of
representations in the complex
plane, real part
and imaginary part.
But I've also identified
a circle here.
And you could wonder,
well, what's the
significance of the circle?
Recall that in the discussion
that we just came from, when
we talked about the relationship
between the
z-transform and the Fourier
transform, the z-transform
reduces to the Fourier transform
when the magnitude
of z is equal to 1.
The magnitude of z
equal to 1 in the
complex plane is a circle.
And that circle, in fact,
is a circle of radius 1.
And so it's on this contour
in the z-plane that the
z-transform reduces to the
Fourier transform.
And we'll see some additional
significance of
that as we go along.
Just again to emphasize the
relationships and differences
with continuous-time, with the
Laplace transform it's the
behavior in the s-plane on the
j omega axis that corresponds
to the Fourier transform.
Here it's the behavior on the
unit circle where the
z-transform corresponds to
the Fourier transform.
Now, we'll be talking--
as we did with the Laplace
transform, we'll be talking
very often about transforms
which are rational, and
rational transforms as we'll
see, represent systems which
are characterized by linear
constant coefficient
difference equations.
And so for the rational
z-transforms, we'll again find
it convenient to use a
representation in terms of
poles and zeroes
in the z-plane.
So let's look at our example
as we've worked it out
previously, Example 10.1.
And with this sequence, the
z-transform is 1 divided by 1
minus a z to the minus 1.
And we happen to have written
it as a function of z
to the minus 1.
Clearly, we can rewrite this by
multiplying numerator and
denominator by z, and this would
equivalently then be z
divided by z minus a.
And so if we were to represent
this through a pole-zero plot,
we would have a 0 at the origin
corresponding to this
factor and a pole at z equals
a corresponding to the
denominator factor.
And so the pole-zero pattern for
this is then a pole at z
equal to a and a 0
at the origin.
Now, let me just comment quickly
about the fact that we
had written this as 1 over 1
minus a z to the minus 1, and
that seems kind of strange
because perhaps we should have
multiplied through by z.
Let me just indicate that as
you'll see as you work
examples, it's very typical for
the z-transform to come
out as a function of
z to the minus 1.
And so very typically, you'll
get to recognize that things
will be expressed in terms of
factors involving terms like 1
minus a z to the minus 1, rather
than factors of the
form z minus a.
Well, here is the one example
that we had referred to.
And if we consider another
example, the other example,
which was example 10.2 consists
of an algebraic
expression as I indicate here.
But its region of validity is
the magnitude of z less than
the magnitude of a.
And that corresponds to the
same pole-zero plot, but a
region of convergence which
is inside this circle.
Whereas, in the previous case,
with the pole-zero plot, we
had a region of convergence
which was for the magnitude of
z greater than the
magnitude of a.
So these two examples, this one
and the other one, have
exactly the same pole-zero
pattern and they're
distinguished by their region
of convergence.
Now, notice incidentally that
in this particular case, the
region of convergence includes
the unit circle provided that
the magnitude of a
is less than 1.
And so, in fact, that would say
that the sequence has a
Fourier transform
that converges.
Namely, with the magnitude
of z equal to 1.
Whereas, in this example, the
region of convergence does not
include the unit circle.
And so, in fact, we cannot
look at x of z for the
magnitude of z equal to 1.
And so this example, with the
magnitude of a less than 1,
does not have a Fourier
transform that converges.
Well, assuming that the
magnitude of a is less than 1
and the Fourier transform
converges, we can, in fact,
look at the Fourier transform
by observing what happens as
we go around the unit circle.
We had seen this with the
Laplace transform in terms of
observing what happened as we
move along the j omega axis.
And here again, we can use the
vectors as we trace out the
unit circle.
And in particular, what we would
be looking at in this
case is the ratio of the zero
vector to the pole vector.
For example, if we were looking
at the magnitude of
the z-transform, the magnitude
of the z-transform would be
the ratio of the length of this
vector to the length of
this vector.
And to observe the Fourier
transform, we would observe
how those vectors change in
length as we move around the
unit circle.
And as we move around the unit
circle, what we would trace
out in terms of the ratio of the
lengths of those vectors
is the Fourier transform.
Well, let's focus on that also
in the context of a slightly
different z-transform.
In the z-transform here as we'll
see in a later lecture,
is the z-transform associated
with a second
order difference equation.
It has a denominator factor
which has two poles
associated with it.
And so here, if we assumed that
the Fourier transform of
the associated sequence
converged, then again we would
look at the behavior
of this as we moved
around the unit circle.
And the ratio of the lengths
of the appropriate vectors
would describe for us the
frequency response.
I'm sorry, the Fourier
transform.
So the Fourier transform
magnitude would consist of the
ratio of the lengths of the zero
vectors divided by the
lengths of the pole vectors.
And one thing that we observe
is that as we move in
frequency omega in the vicinity
of this pole, this
pole vector, in fact, reaches
a minimum length.
That would mean that it's
reciprocal would be maximum.
And then, as we sweep past,
the lengths of these two
vectors would increase.
The zero vectors, of course,
would retain the same length
no matter where we were
on the unit circle.
So, in fact, if we looked
at the Fourier transform
associated with this pole-zero
pattern, if this was, for
example, represented the
z-transform of the impulse
response or a linear
time-invariant system, the
corresponding frequency response
would be what I
plotted out below.
And so it would peak.
And in fact, where it would peak
is in the vicinity of the
frequency location of the pole
as I indicate up here.
So as we sweep past this pole
then, in fact, this Fourier
transform .
Peaks.
Well, this notion of looking at
the frequency response as
we move around the unit circle
is a very important notion.
And it's important to recognize
it's the unit circle
we're talking about here,
whereas before we were talking
about the j omega axis.
And to emphasize this further,
let me just show this example.
And in fact, the previous
example with the computer
displays, so that we can see the
frequency response as it
sweeps out as we go around
the unit circle.
So here we have the pole-zero
pattern for the
second order example.
And to generate the Fourier
transform, we want to look at
the behavior of the pole and
zero vectors as we move around
the unit circle.
So first, let's display
the vectors.
And here we have them displayed
to the point
corresponding to
zero frequency.
And the magnitude of the Fourier
transform will be, as
we discussed, the magnitude of
the length of the zero vector
is divided by the
magnitude of the
length of the pole vectors.
Shown below will be the
Fourier transform.
And we have the Fourier
transform displayed here from
0 to 2 pi, rather than from
minus pi to pi as it was
displayed in the transparency.
Because of the periodicity of
the Fourier transform, both of
those are equivalent.
Now we're sweeping away from
omega equals 0 and the lengths
of the pole vectors
have changed.
And that, of course,
generates a change
in the Fourier transform.
And as we continued the
process further, if we
increase frequency, as we sweep
closer to the location
of the pole, the pole vector
decreases in length
dramatically.
And that generates a residence
in the Fourier transform, very
similar to what we saw
in continuous-time.
Now as we continue to sweep
further, what will happen is
that that pole vector
will begin to
increase in length again.
And so, in fact, the magnitude
of the Fourier
transform will decrease.
And we see that here
as we sweep toward
omega equal to pi.
Now, notice in this process that
the length of the zero
vectors has stayed the same
because of the fact that the
zeroes are at the origin, and no
matter where we are in the
unit circle, the length of
those vectors is unity.
So they don't influence in this
example the magnitude,
but they would, of course,
influence the phase.
Now we want to continue sweeping
from omega equal to
pi around to 2 pi.
And because of the symmetry in
the Fourier transform, what we
will see in the magnitude is
identical to what we would see
if we swept from omega equal
to pi back clockwise to
omega equals 0.
In particular now, as we're
increasing frequency, notice
that the length of the pole
vector associated with the
lower half plane pole
is decreasing.
And so, in fact, that
corresponds to generating a
resonance as we sweep
past that pole
location as we are here.
And then finally, that pole
vector increases in length as
we begin to approach omega
equal to 2 pi.
Or equivalently, as we approach
omega equal to 0.
Now finally, let's also look
at the Fourier transform
associated with the first
order example that we
discussed earlier
in the lecture.
And so what we'll want to look
at is the Fourier transform as
the pole and zero
vectors change.
Once again, the Fourier
transform will be displayed on
a scale from 0 to 2 pi, a
frequency scale from 0 to 2
pi, rather than minus
pi to pi.
And we want to observe the pole
and zero vectors as we
sweep around the unit circle.
We display first the pole
and zero vectors at
omega equal to 0.
And as the frequency increases,
the pole vector
increases in length.
The zero vector, since the zero
is at the origin, will
have constant length
no matter where we
are on the unit circle.
Although it would affect the
phase, which we are not
displaying here.
And so the principle effect, the
only effect really on the
magnitude, is due to
the pole vector.
As the frequency continues to
increase, the pole vector
increases in length,
monotonically in fact.
And so that means that the
magnitude of the Fourier
transform will decrease
monotonically until we get
past omega equal to pi.
Here we are now at omega
equal to pi.
And when we continue sweeping
past this frequency around to
2 pi, then we will see basically
the same curve swept
out in reverse.
Since because of the symmetry,
again, of the Fourier
transform magnitude, sweeping
from pi to 2 pi is going to be
equivalent with regard to the
magnitude to sweeping
from pi back to 0.
And so now the pole vector
begins to decrease in length
and correspondingly, the
magnitude of the Fourier
transform will increase.
And that will continue until we
get around to omega equal
to 2 pi, which is equivalent,
of course, to
omega equal to 0.
And obviously, if we continue
to sweep around again, we
would simply trace out other
periods associated with the
Fourier transform.
Well, that hopefully gives you
kind of some feel for the
notion of sweeping around
the unit circle.
And of course, you can see that
because the circle is
periodic as we go around and
around, of course, what we'll
get is a periodic Fourier
transform, which is the way
Fourier transforms are
supposed to be.
Now, just as with the Laplace
transform, the region of
convergence of the z-transform,
as we've seen in
this example, is a very
important part of the
specification of the
z-transform.
And we can, in talking about
sequences and their
transforms, either specify
the region of convergence
implicitly, or we can specify
it explicitly.
We can, for example, say what
it is, as let's say the
magnitude of z being greater
than the magnitude of a.
Or we can recognize that the
region of convergence has
certain constraints associated
with certain properties of the
time function.
And in particular, there are
some important properties of
the region of convergence which
allow us, given that we
know certain characteristics of
the time function, to then
identify the region of
convergence by looking at the
pole-zero pattern.
For example, we recognize that
the region of convergence does
not contain any poles because of
the fact that at poles, the
z-transform, in fact, blows up
and, of course, can't converge
at that point.
Furthermore, the region of
convergence consists of a ring
in the z-plane centered
about the origin.
Recall that with the Laplace
transform, the region of
convergence consisted of
strips in the s-plane.
With the z-transform, the region
of convergence consists
of a ring, basically because of
the fact that the region of
convergence is dependent
on the magnitude of z.
Whereas, with the Laplace
transform, the region of
convergence was dependent
on the real part of s.
The fact that it's the magnitude
of z says, in
effect, that all values of z
that have the same magnitude
lie on a circle.
And so the region of convergence
you would expect
to be a concentric ring
in the z-plane.
Furthermore, as we've already
talked about and exploited
actually, convergence of the
Fourier transform is
equivalent to the statement that
the region of convergence
includes the unit circle
in the z-plane.
Now, we can also associate the
region of convergence with
issues about whether the
sequence is of finite duration
or right-sided or left-sided.
And let me sort of quickly
indicate again what the style
of the argument is.
If we have a finite duration
sequence, so that the sequence
is absolutely summable, and
therefore has a Fourier
transform that converges, then
because of the fact that it's
0 outside some interval,
I can multiply it by an
exponentially decaying
sequence or by an
exponentially growing
sequence.
And since I'm only doing this
over a finite interval, no
matter how I choose that
exponential, we'll end up with
an absolutely summable
product.
So if x of n is a finite
duration, then in fact the
region of convergence is the
entire z-plane, possibly with
the exception of the
origin or infinity.
On the other hand, if the
sequence is a right-sided
sequence, then we have to be
careful that we don't multiply
by an exponential that grows
too fast for positive time.
Or equivalently, we might have
to choose the exponential so
that it decays sufficiently
fast for positive time.
As a consequence of that, for a
right-sided sequence, if we
have a value which is in the
region of convergence, as long
as I multiply by exponentials
that decay faster than that
for positive time, then I'll
also have convergence.
In other words, all finite
values of z for which the
magnitude of z is greater than
this, so that the exponentials
die off even faster will also
be in the region of
convergence.
If we combine that statement
with the fact that there are
no poles in the region of
convergence, then we end up
with a statement similar
to what we had
with the Laplace transform.
Here, the statement is that if
the sequence is right-sided,
then the region of convergence
has to be outside the
outermost pole.
Essentially, because it has to
be outside someplace and can't
include any poles.
Finally, if we have a left-sided
sequence, then if
we have a value which is in the
region of convergence, all
values for which the magnitude
of z is less than that will
also be in the region
of convergence.
Or if x of z is rational, then
the region of convergence must
be inside the innermost pole.
And finally, if we have a
two-sided sequence, then
there's kind of a balance
between the exponential factor
that we use.
And so in that case, then the
region of convergence will be
a ring in the z-plane, and
essentially will extend
outward to a pole and
inward to a pole.
So if we had an algebraic
expression, let's say as we
have here, then we could
associate with that a region
of convergence outside
this pole.
And that would correspond to
a right-sided sequence.
Or we can associate with it a
region of convergence, which
is inside the innermost pole.
And that would correspond to
a left-sided sequence.
And the third and only other
possibility is a region of
convergence which lies between
these two poles.
And that would then correspond
to a two-sided sequence.
And notice incidentally because
of where I've placed
these poles, that this is the
only one for which the region
of convergence includes
the unit circle.
In other words, it's the only
one for which the Fourier
transform converges.
OK, now we've moved through
that fairly quickly.
And I've emphasized the fact
that it parallels very closely
what we did with the
Laplace transform.
What I'd like to do is just
conclude with a discussion of
how we get the time function
back again when we have the
z-transform including its
region of convergence.
Well, we can, first of all,
develop a more or less formal
expression.
And the algebra for this is gone
through in the text, and
you went through something
similar to this with the
Laplace transform in the
video course manual.
So I won't carry through
the details.
But basically, what we can use
to develop a formal expression
is the fact that the z-transform
is the Fourier
transform of the sequence
exponentially weighted.
So we can apply the inverse
transform to that, and that
gives us not x of n, but x of
n exponentially weighted.
And if we track that through,
then what we'll end up with is
an expression.
After we've taken care of a
few of the epsilons and
deltas, we'll end up with an
expression that expresses
formally the sequence x of n in
terms of the z-transform,
where this, in fact, is a
contour integral in the
complex plane.
And so there's a formal
expression, just as there's a
formal expression for the
Laplace transform.
But in fact, the more typical
procedure is to use
essentially transformed pairs
that we know together with the
idea of using a partial
fraction expansion.
So if we had a z-transform as
I indicate here, and if we
expand it out in a partial
fraction expansion, then we
can recognize, as we did in a
similar style with the Laplace
transform--
we can recognize that each
term, together with the
identified region of convergence
corresponds to an
exponential factor.
And so this term, together with
the fact we know that the
magnitude of z must be greater
than 2, allows us to recognize
this as similar to
the Example 10.1.
And in particular then, the
sequence associated with that
is what I indicate here.
And for the second term,
the sequence is
what I indicate here.
So what we're simply doing is
using the fact that we've
worked out the example going one
direction before, and now
we use that together with the
partial fraction expansion to
get the individual sequences
back again, and
then add them together.
There's one other method which
I'll just point to, which is
also elaborated on a little
more in the text.
But it's kind of the idea of
developing the inverse
z-transform by recognizing that
this z-transform formula,
in fact, is a power series.
So if we take x of z and expand
it in a power series,
then we can pick off the
values of x of n by
identifying the individual
coefficients in this expansion.
And so by simply doing long
division, for example, we can
also get the inverse
transform.
And that, by the way,
is very useful.
Particularly if we want to get
the inverse z-transform for a
z-transform expression,
which is not rational.
Now we've moved through
this fairly quickly.
On the other hand, I've stressed
that it's very
similar to what we went through
for the Laplace
transform, except for a very
important difference.
The principal difference really
being that with the
Laplace transform, it was the
j omega axis in the s-plane
that we focused attention on
when we were thinking about
the Fourier transform.
Here, the unit circle
in the z-plane plays
an important role.
What we'll see when we continue
this in the next
lecture is that there are
properties of the z-transform,
just as there were properties
of the Laplace transform.
And those properties allow us
to develop and exploit the
z-transform in the context of
systems describable by linear
constant coefficient difference
equations.
So in the next lecture, we'll
focus on some properties of
the z-transform, and then we'll
see how to use those
properties to help us in getting
further insight and
working with systems
describable
by difference equations.
Thank you.
