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Introduction to DSP - IIR filters: the meaning of z

z is a complex number:

z equation

When drawn on the Argand diagram, z has the curious property that it can only have a magnitude of 1:z has magnitude of 1

So z, which is the variable used in our frequency response, traces a unit circle on the Argand diagram.

At first sight, z can have no value off the unit circle.

But if we use a mathematical fiction for a moment, we can imagine that the frequency f could itself be a complex number:

frequency as a complex number

In which case, the j from imaginary frequency component can cancel the j in the z term:

cancelling j

and the imaginary component of frequency introduces a straightforward exponential decay on top of the complex oscillation: showing that z can be off the unit circle, and that if it is this relates to transient response. The imaginary frequency has to do with transient response, while the real frequency (both real as in actual, and real as in the real part of a complex number) has to do with steady state oscillation.

For real frequencies z lies on the unit circle. Values of the transfer function H(z) for z off the unit circle relate to transient terms:

Positions of z on the unit circle

The position of z, inside or outside the unit circle, determines the stability of transient terms:

  • if z is inside the unit circle, the transient terms will die away
  • if z is on the unit circle, oscillations will be in a steady state
  • if z is outside the unit circle, the transient terms will increase

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