Transfer Function from Poles and Zeroes

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Build a transfer function out of a collection of poles (in red) and zeroes (in blue) in the complex plane. When poles and zeroes are complex conjugates, the amplitude of the transfer function is symmetric and the phase is antisymmetric, which corresponds to a real impulse response. Otherwise, the impulse response is complex.

Contributed by: Philippe Thévenaz (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: A single zero in creates a high-pass filter, since the amplitude term (in blue, top-right graph) shows attenuation near the origin. The phase term (in purple, bottom-right graph) is linear but has a discontinuity at the origin.

Snapshot 2: This filter has lowpass characteristics. It has a real impulse response since the poles and zeroes are symmetric. The attenuation is good, but there are strong ripples in the pass-band.

Snapshot 3: This filter is characterized by symmetric zeroes but asymmetric poles; therefore, its impulse response is complex. This can also be observed from its phase, which does not satisfy Hermitian symmetry, nor does the amplitude.

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