How to read the Zero/Pole Diagram in DSP?

zero pole diagram

Hello all,

I noticed that when we evaluate a digital filter, we always focus on the Zero/Pole Diagram, but how to read such a diagram from a view of engineering?

The problem is, if I have a lot of digital filters to choose, can I choose one with the "BEST" Zero/Pole Diagram, but how to determine it is the best from the diagram?

Is it possible? Thanks!

DAVY

pole zero diagram

choose the filter which its zero and poles are not too near with themselves.i men choose zeros and poles which their diffrence with each other is larger.why?because by quantization a zero may cancel a pole or vice versa. i hope i could help u.

pole diagram

Hello Davy,

The main characteristics of a digital filter can be seen from a wiew to its zero-pole location, as in an analog filter. In an analog filter, the frequency axis (location of complex exponential signals) is the immaginary axis of the s-plane, while in the digital filter it is the unit circle of the z-plane.

For visualize the transfer function:
Take a point of the complex plane (e.g., for DC it is s=0 fon an analog system or z=1 for a digital one). Trace a vector from the point of interest to each pole and each zero.

The magnitude of the transfer function is (with a multiplicative constant):
the product of the modulus of the vectors that go to the zeros...
divised by...
the product of the modulus of the vectors that go to the poles.

If a point is very close to a zero, there is low gain at that frequency (gain=0 at the frequency of the zero);
If a point is very close to a pole, there is high gain at that frequency (gain=infinite at the frequency of the pole);

The phase of the transfer function is:
the sum of the angles (wrt the horizontal axis) of the vectors that go to the zeros...
minus...
the sum of the angles (wrt the horizontal axis) of the vectors that go to the poles

For instance, a low-pass IIR filter has poles around Z=1 and zeros around the unit circle. A passband IIR filter haz poles aroun the central frequency ans zeros at lower and higher frequencies, etc. FIR filters have only zeros, distributed on the unit circle in the attenuation bands, and outside the unit circle in the zone of pass bands, in such a way that the transfer function is conformed. A high attenuation in the attenuation band is obtained with lots of zeros on the unit circle.

These are the general characteristics. Very “fine details” that allow a comparision of two filters of the same type (e.g. two low-pass filters), as the ripple of the transfer function, are more difficult to visualize. But, as stated before, a filter with more dense zeros in the attenuation band is expected to have a better rejection at that band.

Regards

Z