|
|
| Line 1: |
Line 1: |
| In [[mathematics]] and [[signal processing]], the '''advanced Z-transform''' is an extension of the [[Z-transform]], to incorporate ideal delays that are not multiples of the [[sampling rate|sampling time]]. It takes the form
| | Hello and welcome. My name is Irwin and I completely dig that name. Supervising is my occupation. To play baseball is the pastime he will never stop doing. Minnesota has usually been his house but his wife desires them to move.<br><br>Here is my page std testing at home ([http://3bbc.com/index.php?do=/profile-362966/info/ linked webpage]) |
| | |
| :<math>F(z, m) = \sum_{k=0}^{\infty} f(k T + m)z^{-k}</math>
| |
| | |
| where
| |
| * ''T'' is the sampling period
| |
| * ''m'' (the "delay parameter") is a fraction of the sampling period <math>[0, T).</math>
| |
| | |
| It is also known as the '''modified Z-transform'''.
| |
| | |
| The advanced Z-transform is widely applied, for example to accurately model processing delays in [[digital control]].
| |
| | |
| ==Properties==
| |
| If the delay parameter, ''m'', is considered fixed then all the properties of the Z-transform hold for the advanced Z-transform.
| |
| | |
| ===Linearity===
| |
| :<math>\mathcal{Z} \left\{ \sum_{k=1}^{n} c_k f_k(t) \right\} = \sum_{k=1}^{n} c_k F(z, m).</math>
| |
| | |
| ===Time shift===
| |
| :<math>\mathcal{Z} \left\{ u(t - n T)f(t - n T) \right\} = z^{-n} F(z, m).</math>
| |
| | |
| ===Damping===
| |
| :<math>\mathcal{Z} \left\{ f(t) e^{-a\, t} \right\} = e^{-a\, m} F(e^{a\, T} z, m).</math>
| |
| | |
| ===Time multiplication===
| |
| :<math>\mathcal{Z} \left\{ t^y f(t) \right\} = \left(-T z \frac{d}{dz} + m \right)^y F(z, m).</math>
| |
| | |
| ===Final value theorem===
| |
| :<math>\lim_{k \to \infty} f(k T + m) = \lim_{z \to 1} (1-z^{-1})F(z, m).</math>
| |
| | |
| ==Example==
| |
| Consider the following example where <math>f(t) = \cos(\omega t)</math>
| |
| | |
| :<math>\begin{align}
| |
| F(z, m) =& \mathcal{Z} \left\{ \cos \left(\omega \left(k T + m \right) \right) \right\} \\
| |
| =& \mathcal{Z} \left\{ \cos (\omega k T) \cos (\omega m) - \sin (\omega k T) \sin (\omega m) \right\} \\
| |
| =& \cos(\omega m) \mathcal{Z} \left\{ \cos (\omega k T) \right\} - \sin (\omega m) \mathcal{Z} \left\{ \sin (\omega k T) \right\} \\
| |
| =& \cos(\omega m) \frac{z \left(z - \cos (\omega T) \right)}{z^2 - 2z \cos(\omega T) + 1} - \sin(\omega m) \frac{z \sin(\omega T)}{z^2 - 2z \cos(\omega T) + 1} \\
| |
| =& \frac{z^2 \cos(\omega m) - z \cos(\omega(T - m))}{z^2 - 2z \cos(\omega T) + 1}
| |
| \end{align}</math>.
| |
| | |
| If <math>m=0</math> then <math>F(z, m)</math> reduces to the [[Z-transform]]
| |
| | |
| :<math>F(z, 0) = \frac{z^2 - z \cos(\omega T)}{z^2 - 2z \cos(\omega T) + 1}</math>
| |
| | |
| which is clearly just the Z-transform of <math>f(t).</math>
| |
| | |
| ==See also==
| |
| *[[Z-transform]]
| |
| | |
| ==Bibliography==
| |
| * [[Eliahu Ibraham Jury]], ''Theory and Application of the Z-Transform Method'', Krieger Pub Co, 1973. ISBN 0-88275-122-0.
| |
| | |
| {{DSP}}
| |
| | |
| [[Category:Transforms]]
| |
Hello and welcome. My name is Irwin and I completely dig that name. Supervising is my occupation. To play baseball is the pastime he will never stop doing. Minnesota has usually been his house but his wife desires them to move.
Here is my page std testing at home (linked webpage)