Editing Z-transform (section)

===Zeros and poles===
From the [[fundamental theorem of algebra]] the [[numerator]] has <math>M</math> [[root of a function|roots]] (corresponding to zeros of ''<math>H</math>'') and the [[denominator]] has <math>N</math> roots (corresponding to poles).  Rewriting the [[transfer function]] in terms of [[zeros and poles]]
:<math>H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})\cdots(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})\cdots(1 - p_N z^{-1})} ,</math>
where <math>q_k</math> is the <math>k^\text{th}</math> zero and <math>p_k</math> is the <math>k^\text{th}</math> pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the [[pole–zero plot]].

In addition, there may also exist zeros and poles at <math>z{=}0</math> and <math>z{=}\infty.</math> If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.

By factoring the denominator, [[partial fraction]] decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the [[impulse response]] and the linear constant coefficient difference equation of the system.