# pole/zero plot & impulse response

in a mock exam of dsp was asked this question, and even in knowing how to compute DFT's of the plot, I wondered if there was a fast way of pairing these plots to their respective impulse response?

• One more hint: make sure you look a the Y-axis labelling – Hilmar Jun 17 '19 at 5:25

It's pretty simple if you know a few basic things:

• a single pole on the real axis corresponds to an exponential sequence. The elements of that sequence either have the same sign, or their signs alternate, depending on the angle of the pole ($$0$$ or $$\pi$$). This should allow you to pair figures $$(c)$$ and $$(e)$$ with figures $$(1)$$ and $$(6)$$.
• a complex conjugate pole pair corresponds to a sinusoid with exponentially decaying or increasing amplitude, depending on whether the poles are inside or outside the unit circle. This should help you pair figure $$(f)$$ with the corresponding sequence.
• if there are only zeros in the PZ-diagram (and poles at the origin or at infinity), then the corresponding sequence has finite length. Note that since all shown sequences are causal, there must be the same number of poles at the origin as there are zeros. Those poles are not shown in the figures, which could be considered bad practice.
• linear phase sequences have certain symmetry properties in the time domain as well as in their PZ-diagrams.
• the formula for the geometric series helps with figure $$(5)$$.

Also take a look at this related question and its answers.

a5,b2,c6,d4,e1,f3

• Awesome thank you very much! :) – james Jun 16 '19 at 15:42