Z Transform

Z-transform & DTFT

  • In DTFT's equation: X(jw) = Zn:-8~8x[n]e-jwn, 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. x[n] = u[n]; x[n] = 0.5a*u[-n]; x[n] = sin(nw0)
  • Z-transform is a extension of DTFT, and it is designed to deal with these unsummable signals.
  • Z-transform's equation: X(z) = Zn:-8~8x[n]z-n, where z = rejw. Therefore, the Z-transform of x[n] is actually the DTFT of r-nx[n]. By adjusting the value of "r", we can always make the DTFT of r-nx[n] (ie. Z-transform of x[n]) to be convergent. Therefore, the range of r is called range of convergent.
  • Since z = rejw = 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.
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