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Introduction to DSP - IIR filters: poles and zeros

The IIR filter's transfer function is a ratio of terms.

  • if the numerator becomes zero, the transfer functionwill also become zero - this is called a zero of the function
  • if the denominator becomes zero, we have a division by zero - the function can become infinitely large - this is called a pole of the function
Filter frequency response equation

The positions of poles (very large values) affects the stability of the filter:

Digram showing positons of poles and zeros

The shape of the transfer function H(z) is determined by the positions of its poles and zeroes:

Visualisation of the transfer function

This can be visualised using the rubber sheet analogy:

  • imagine the Argand diagram laid out on the floor
  • place tall vertical poles at the poles of the function
  • stretch a rubber sheet over the poles
  • at zeroes, pin the rubber sheet to the floor
  • the rubber sheet will take up a shape which is determined by the position of the poles and zeroes

Thanks are due to Jim Richardson for the rubber sheet analogy, which came to mind while he was an instructor officer at the Royal Naval Engineering College, Devonport.

Now the frequency response is the transfer function H(z) evaluated around the unit circle on the Argand diagram of z:

Argand diagram of z

and since the shape of the transfer function can be determined from the positions of its poles and zeroes, so can the frequency response.

unit circle

The frequency response can be determined by tracing around the unit circle on the Argand diagram of the z plane:

  • project poles and zeroes radially to hit the unit circle
  • poles cause bumps
  • zeroes cause dips
  • the closer to the unit circle, the sharper the feature

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