Z Transform

## Z-transform & DTFT

- In DTFT's equation:
**X(jw) = Z**, we know that for a convergent DTFT, the time series x[n] must be absolutely summable. However, some discrete signal is not absolutely summable. Ex._{n:-8~8}x[n]e^{-jwn}**x[n] = u[n]; x[n] = 0.5**^{a}*u[-n]; x[n] = sin(nw_{0}) - Z-transform is
**a extension of DTFT**, and it is designed to deal with these unsummable signals. - Z-transform's equation:
**X(z) = Z**, where_{n:-8~8}x[n]z^{-n}**z = re**. Therefore, the Z-transform of x[n] is actually the DTFT of r^{jw}^{-n}x[n]. By adjusting the value of "r", we can always make the**DTFT of r**(ie. Z-transform of x[n]) to be convergent. Therefore,^{-n}x[n]**the range of r is called range of convergent**. - Since z = re
^{jw}= r(cos(w) + jsin(w)), its best to use complex plane to represent it, that is z-plane. We can say that z-plane is specifically designed to represent the variable "z", where "r" is the radius of the z-plane, "w" is angle of z-plane.

## ROC

- ROC is the range of r that makes the Z-transform convergent (absolutely summable).
- When
**r = 1, z has value on the unit circle of the z-plane**, the Z-transform using this z is equal to DTFT. - Therefore, if a signal's
**DTFT is convergent**, its Z-transform's**ROC must include unit circle**.

page revision: 12, last edited: 22 May 2010 03:46