pole zero stuff

Hi all.
I am confused about pole - zero diagrams and how they relate to filter 
response.
Also confused about unit circle and IIR filters.
Can anyone help?
yours
Richard 

richard pickworth wrote:

> Hi all.
> I am confused about pole - zero diagrams and how they relate to filter 
> response.
> Also confused about unit circle and IIR filters.
> Can anyone help?
> yours
> Richard 
> 
> 
The locations of a filter's poles and zeros dictate it's response.  In 
the z domain any system (including a filter) with a pole outside the 
unit circle is unstable.  IIR filters have poles at z != 0.

Try this, it may help: 
http://www.wescottdesign.com/articles/zTransform/z-transforms.html

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

This may help you or it may confuse you, but the way I think about it
in terms of a bode (frequency response) plot ....

as you go up in frequency....

a pole is a break point  downwards (low pass filter)  at 6 dB / ocatve

a zero is a breakpoint upwards (high pass filter)  at 6 dB / octave


a complex pole pair  (often incorrectly just called a pole) is a
resonator creates a peak that depends upon the Q

a complex zero pair is a notch filter or trap that also depends upon
the Q


These descriptions may not be 100% mathematically correct but they
serve me pretty well.


Mark

Mark wrote:
> This may help you or it may confuse you, but the way I think about it
> in terms of a bode (frequency response) plot ....
> 
> as you go up in frequency....
> 
> a pole is a break point  downwards (low pass filter)  at 6 dB / ocatve
> 
> a zero is a breakpoint upwards (high pass filter)  at 6 dB / octave
> 
> 
> a complex pole pair  (often incorrectly just called a pole) is a
> resonator creates a peak that depends upon the Q
> 
> a complex zero pair is a notch filter or trap that also depends upon
> the Q
> 
> 
> These descriptions may not be 100% mathematically correct but they
> serve me pretty well.
> 
> 
> Mark
> 
They're very useful approximations for a system modelled in the Laplace 
domain, and they're still useful for systems modelled in the z domain, 
but only as long as the frequency is no more than about 1/5th or 1/10th 
the sampling rate -- above that frequency the amplitude (and phase) 
response no longer follows the nice approximations.  I gave up on trying 
to make precise sketches of Bode plots in the z domain years ago -- now 
I just dump the thing into MathCad or SciLab and make a plot numerically.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

"richard pickworth" <richard.pickworth@btopenworld.com> wrote in message 
news:dcaolt$2bd$1@nwrdmz02.dmz.ncs.ea.ibs-infra.bt.com...
> Hi all.
> I am confused about pole - zero diagrams and how they relate to filter 
> response.
> Also confused about unit circle and IIR filters.
> Can anyone help?
> yours
> Richard

Richard,

If you're visually oriented then you may like this:

Plot the poles and zeros in the s-plane or the z-plane.
Imagine that the plane is covered flat with an elastic sheet.
Imagine that there is a thumb tack holding the sheet down at the location of 
each zero.
Imagine that there is a pencil holding the sheet up from the plane at the 
location of each pole.
Now, really, the poles would be of infinite height so you need to do a bit 
of visualization there.
The resulting surface approximates the shape of the magnitude over the 
plane.

Now, visualize how the surface of the sheet varies if you traverse the jw 
axis in the s-plane or the unit circle in the z-plane.
If there are zeros on the unit circle then the height of the sheet is zero.
If there are poles near the unit circle then the height of the sheet is 
affected as well.

Imagine one pole in the z-plane at the orgin - the entire sheet is lifted 
and the height above the unit circle is uniform.
Now, put one zero at fs/2 or pi on the unit circle.  The entire sheet 
remains above the plane over the unit circle except at pi where it's at zero 
height.

It's not as good as using more careful plots but it can be useful in 
thinking about how certain filters work.

Fred 

richard pickworth wrote:
> Hi all.
> I am confused about pole - zero diagrams and how they relate to filter
> response.
> Also confused about unit circle and IIR filters.
> Can anyone help?

Exercise for the student:

Draw a unit circle.  Label the circle from 0 to SampleRate/2 Hz.
Draw your poles and zeros.  For some point on the circle,
calculate the distances to all the poles and zeros.  Multiply
together the distance to all the zeros and the reciprocal of the
distance to all the poles.  How does the resulting product and
the frequency of your selected point on the unit circle relate
to your expected IIR filter response?

Long ago I remember writing an Apple II program which would allow
me to use the joystick to move around a pole or zero, then (slowly)
have plotted the resulting distance product for some (not large)
number of points around a circle or vertical line.  (Should run a
bit faster on a modern PC :)


IMHO. YMMV.
-- 
rhn A.T nicholson D.o.T c-O-m

This is a great thought experiment Fred! I wonder what the coefficient
of elasticity of the elastic sheet would have to be for the analogy to
be exactly correct? I reckon it might have to be a weird type of
elastic that doesn't snap when stretched to infinity.

porterboy76@yahoo.com wrote:
> This is a great thought experiment Fred! I wonder what the coefficient
> of elasticity of the elastic sheet would have to be for the analogy to
> be exactly correct? I reckon it might have to be a weird type of
> elastic that doesn't snap when stretched to infinity.

The properties of the necessary sheet are simple: it must sustain no 
shear, and its weight must be negligible compared to its surface 
tension. A soap film does nicely. The infinite heights of the poles can 
be simulated by raising the sheet to an appropriate height with a 
flat-ended dowel.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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porterboy76@yahoo.com wrote:

> This is a great thought experiment Fred! I wonder what the coefficient
> of elasticity of the elastic sheet would have to be for the analogy to
> be exactly correct? I reckon it might have to be a weird type of
> elastic that doesn't snap when stretched to infinity.
> 
As long as the coefficient of elasticity (I suspect that the correct 
term is modulus of elasticity) is constant and the elastic can be 
stretched a good long way then you're fine.

This may or may not be a cool thing to do on a computer so you can 
rotate the thing with a mouse and visualize things -- someone with 
MatLab should give it a try.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott wrote:
> porterboy76@yahoo.com wrote:
> 
>> This is a great thought experiment Fred! I wonder what the coefficient
>> of elasticity of the elastic sheet would have to be for the analogy to
>> be exactly correct? I reckon it might have to be a weird type of
>> elastic that doesn't snap when stretched to infinity.
>>
> As long as the coefficient of elasticity (I suspect that the correct 
> term is modulus of elasticity) is constant and the elastic can be 
> stretched a good long way then you're fine.
> 
> This may or may not be a cool thing to do on a computer so you can 
> rotate the thing with a mouse and visualize things -- someone with 
> MatLab should give it a try.

The property of supporting no shear implies that the tension is 
everywhere the same and independent of elongation. That is exactly the 
property of a soap film. The forces on charges in an electric field in 
space can be visualized the same way. When the deformation is not large, 
the approximation achieved with a sheet of rubber is quite good. Before 
digital computers were available to do simulations, electron tubed were 
designed with the help of rubber-sheet models. A steel ball rolling on 
the sheet would follow the trajectory of an electron in the tube.

Soap-film models are also useful for determining the polar moment of 
inertia of irregular shapes.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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