- Advanced Z-transform
In
mathematics andsignal processing , the advanced Z-transform is an extension of theZ-transform , to incorporate ideal delays that are not multiples of the sampling time. It takes the form:F(z, m) = sum_{k=0}^{infty} f(k T + m)z^{-k}
where
* "T" is the sampling period
* "m" (the "delay parameter") is a fraction of the sampling period 0, T).It is also known as the modified Z-transform.
The advanced Z-transform is widely applied, for example to model accurately 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
:Z left [ sum_{k=1}^{m} c_k f_k(t) ight] = sum_{k=1}^{m} c_k F(z, m).
Time shift
:Z left [ u(t - n T)f(t - n T) ight] = z^{-n} F(z, m).
Damping
:Z left [ f(t) e^{-a, t} ight] = e^{-a, m} F(e^{a, T} z, m).
Time multiplication
:Z left [ t^y f(t) ight] = left(-T z frac{d}{dz} + m ight)^y F(z, m).
Final value theorem
:lim_{k = infty} f(k T + m) = lim_{z = 1} (1-z^{-1})F(z, m).
Example
Consider the following example where f(t) = cos(omega t)
:F(z, m) = Z left [cos left(omega left(k T + m ight) ight) ight] :F(z, m) = Z left [cos (omega k T) cos (omega m) - sin (omega k T) sin (omega m) ight] :F(z, m) = cos(omega m) Z left [ cos (omega k T) ight] - sin (omega m) Z left [ sin (omega k T) ight] :F(z, m) = cos(omega m) frac{z left(z - cos (omega T) ight)}{z^2 - 2z cos(omega T) + 1} - sin(omega m) frac{z sin(omega T)}{z^2 - 2z cos(omega T) + 1}:F(z, m) = frac{z^2 cos(omega m) - z cos(omega(T - m))}{z^2 - 2z cos(omega T) + 1}.
If m=0 then F(z, m) reduces to the
Z-transform :F(z, m) = frac{z^2 - z cos(omega T)}{z^2 - 2z cos(omega T) + 1}
which is clearly just the Z-transform of f(t).
ee also
*
Z-transform Bibliography
*
Eliahu Ibraham Jury , "Theory and Application of the Z-Transform Method", Krieger Pub Co, 1973. ISBN 0-88275-122-0.
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