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To my understanding, pole-zero plots are used to analyze or visualize transfer functions. Suppose there is some very simple system, for example a simple low-pass filter (so it is linear and time-invariant). How does one calculate the pole-zero plot of such system?

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    $\begingroup$ You have a transfer function $H(s)$ in continuous time or $H(z)$ in discrete-time. The pole zero-plot shows the locations of the zeros and poles of $H(s)$ or $H(z)$ in the complex plane. If this doesn't answer your question, you should probably edit it to make it clear what it is that you don't understand. $\endgroup$ – Matt L. Oct 11 '15 at 14:13
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Suppose you are given a system with transfer function

$$H(z)=\frac{(1-3z^{-1})(1-7z^{-1})}{(1-4z^{-1})(1-6z^{-1})} $$

Poles

Poles are the values of $z$ for which the entire function will be infinity or undefined. So, they will be the roots of the denominators, right?

Look here, what values of $z$ will turn the transfer function tend to infinity? Obviously it's $z= 4$ and $z=6$, because if you let $z$ equal 4 or 6, the denominator will be zero, which means the transfer function will tend to infinity.

Zeros

Zeros are the values of z for which the transfer function will be zero. As you have guessed correctly, zeros come from numerator. In this case, zeros are $z= 3$ and $z=7$, cause if you put $z= 3$ or $z=7$, the numerator will be zero, that means the whole transfer function will be zero.

So here poles are $z=4$ and $z=6$, and zeros are $z=3$ and $z=7$.

Pole-Zero Plot

About finding the Pole zero plot, you draw a complex plane. Then you put the values of poles as 'X' marks and zeros as 'O' marks. Here I took the liberty of drawing the pole zero plot of the system:

Pole zero plot of the above system

So, for low pass filter, you find out the transfer function, then the poles and zeros.

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