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4.7.b Region of convergence and stability
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来自 École Polytechnique Fédérale de Lausanne 的课程
数字信号处理
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从本节课中
Module 4: Part 2 Filter Design
与讲师见面
Paolo PrandoniLecturer
School of Computer and Communication Science
Martin VetterliProfessor
School of Computer and Communication Sciences
One remaining question that we have to answer before we start using
the z-transform with reckless abandon is when does the z-transform exist?
For the z-transform,
existence is equivalent to convergence of the power series that defines it.
And so, we will define for each case a region of convergence.
That determines the points on the complex plane for which the z-transform exists.
One property to remember is that because the z-transform is a power series,
when it converges, it converges absolutely.
So that means that the convergence of z-transform depends only
on the magnitude of z and not of it's phase.
Let's look at three different cases.
The first case is the simplest.
When we have a finite support sequence,
the Z transform is the sum of a finite number of terms.
So convergence is not a problem.
In other cases the region of convergence of the z transform has circular symmetry.
In the sense that, because of the absolute convergence,
if the z transform converges in one point, it will converge for
all points in the plane with the same magnitude.
Finally, the region of convergence for
cause of sequences extends from the circle to infinity.
So, suppose that we have determined
that the region of convergence includes this circle.
Well if the sequence originates in the z transform is casual.
We know that the region of convergence will extend outwards from this circle.
And we will see later ways to determine the smallest circle
that defines the region of convergence.
Well this is just a better picture of what I just drew.
Given an LTI system, the transfer function is a ratio of polynomials,
as we have seen, and that's why it's also called a rational transfer function.
So, the rational transfer function has a polynomial of degree M-
1 at the numerator, and a polynomial of degree N- 1 at the denominator.
So we can factor it as the product at the numerator of capital M- 1 first order
terms, and at the denominator the product of capital N- 1 first order terms as well.
The roots of the first order terms in the numerator
are called the zeros of the transfer function.
Because they send the transfer function to zero.
On the other hand, the roots of the denominator, they're called the poles of
the transfer function, and those are really the trouble spots for
the region of convergence, because they send the denominator to zero.
So to resume the situation about the region of convergence of a rational
transfer function, we know that the region of convergence extends outwards for
a causal system.
And we know that the region of convergence cannot include any poles.
So if you put this together, we can say that the region of a convergence extends
outwards from a circle that touches the pole with the largest magnitude.
So here is an example of a pole zero plot where we represent with x's, the poles.
And with dots, the zeros of the function.
And if we try to draw the region of convergence.
Well, of course, the region of convergence cannot include this guy.
And cannot include these guys either.
The magnitude of these two poles is the same and
so these aren't really the largest magnitude poles on the plane and
the region of convergence will extend outwards starting from these poles.
The 0s do not affect the region of convergence of transfer function.
Now here is a very handy stability criterion that we can apply
once we know the transfer function of the system.
Remember from the stability theorem we said that necessary insufficient
condition for BIBO stability is that the impulse response is absolutely summable.
Now remember, the transfer function is the z transform of the impulse response and
because of the absolute convergence property of the z-transform when
the z-transform exists for z equal to 1 in magnitude,
then it means that the z-transform converges absolutely.
Which means that the impulse response is absolutely summable and vice versa.
So in conclusion, we can say that a system is stable If,
and only if, the region of convergence includes the units circle.
So, for instance here, we can see that all the poles of the transfer function
are inside the unit circle.
Therefore, the region of convergence will extend outwards including the unit circle
and the system is stable.
Vice versa, here we have a pole that is outside of the unit circle here.
And therefore the original convergence will not include the unit circle and
this will be an unstable system.
Finally, lets look at a cute little trick that will allow us to estimate
the frequency response of a system from its pole zero plot.
This is called the circus tent method, and you will understand why in a second.
The idea is to think of the magnitude of the z transform
like a rubber sheet that is spread over the complex plane.
This sheet is glued to the ground
where the zeroes of the transferred function are.
And the poles are like poles that push this rubber sheet up and the frequency
respond in magnitude is the profile of the sheet around the unit circle.
So let's have a look.
Imagine a simple pole zero plot like so while you have two zeros and two poles,
the complex poles are complex conjugate, as is the case for real valid systems.
Let's plot the complex plane in a 3D perspective
were on the vertical access we will put the magnitude of the G transform.
So, we put the zeros and the pulse down and
we can indicate the uplifting action of the poles with deltas like so.
Next, we drape our rubber sheet on top of the plane and
we put it at height of 1 to start with.
And then we start gluing it to the complex plane where the zeroes are.
So the first one is here and so it sticks the rubber sheet down there.
And the next one is on the unit circle, and
it will put the rubber sheets down there as well.
Now we can see the poles one at time and so
the first pole will start pushing up the rubber sheet.
Actually this thing in theory goes to infinity but
we stopped there just to get the idea and the second pole will do exactly the same.
And this is really the shape of the rubber sheet.
Once we've conceded all four components on the complex planner.
We can turn it around, we'll have a better look on how it unfolds.
So now the idea is that before you transform
is the value of the z transform around the unit circle which means,
it's the level curve computed around the unit circle on the rubber sheet.
And if we plot that, it will look like this.
In other words, the level curve will, intuitively,
dip down when it passes near a zero.
And will be pushed up when it passes near a pole.
This zero is actually on the unit circle, so
the level curve will touch the unit circle there.
We can turn it around a little bit to see it better.
As a final step, we're going to follow the angle from minus pi
all the way up to pi and plot the corresponding value of the level curve
to obtain an estimate of the frequency response.
Well clearly, this is something you can do roughly in your head
without heaving to draw the 3D picture.
But it gives you an idea how to interpret a pole zero plot.
In order to obtain the information on the frequency response of the system.
