Editing Z-transform (section)

==Linear constant-coefficient difference equation==
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the [[Autoregressive moving average model|autoregressive moving-average]] equation:

:<math>\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q} .</math>

Both sides of the above equation can be divided by <math>\alpha_0</math> if it is not zero. By normalizing with <math>\alpha_0{=}1,</math> the LCCD equation can be written

:<math>y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}.</math>

This form of the LCCD equation is favorable to make it more explicit that the "current" output <math>y[n]</math> is a function of past outputs <math>y[n-p],</math> current input <math>x[n],</math> and previous inputs <math>x[n-q].</math>

===Transfer function===
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields:

:<math>Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}</math>

where <math>X(z)</math> and <math>Y(z)</math> are the z-transform of <math>x[n]</math> and <math>y[n],</math> respectively. (Notation conventions typically use capitalized letters to refer to the z-transform of a signal denoted by a corresponding lower case letter, similar to the convention used for notating Laplace transforms.)

Rearranging results in the system's [[transfer function]]:

:<math>H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + \cdots + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + \cdots + z^{-N} \alpha_N}.</math>

===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.

===Output response===
If such a system <math>H(z)</math> is driven by a signal <math>X(z)</math> then the output is <math>Y(z) = H(z)X(z).</math> By performing [[partial fraction]] decomposition on <math>Y(z)</math> and then taking the inverse Z-transform the output <math>y[n]</math> can be found. In practice, it is often useful to fractionally decompose <math>\textstyle \frac{Y(z)}{z}</math> before multiplying that quantity by <math>z</math> to generate a form of <math>Y(z)</math> which has terms with easily computable inverse Z-transforms.