Due to its convolution property, the z-transform is a powerful tool to analyze LTI
An LTI system is causal if its output depends only on the current
and past input (but not the future). Assuming the system is initially at
rest with zero output
, then its response to an
at is at rest for , i.e., .
Its response to a general input is:
If an LTI system is causal (with a right sided impulse response function for ), then the ROC of its transfer function is the exterior of a circle including infinity. In particular, when is rational, then the system is causal if and only if its ROC is the exterior of a circle outside the out-most pole, and the order of numerator is no greater than the order of the denominator.
Note the requirement for the orders of the numerator and denominator guarantees the existence of even when .
An LTI system is stable if its response to any bounded input is also
bounded for all :
An LTI system is stable if and only if its impulse response is absolutely
summable, i.e., the frequency response function
exits, i.e. the
ROC of its transfer function includes the unit circle .
From the two properties above, we also see that
A causal LTI system with a rational transfer function is stable
if and only if all poles of are inside the unit circle of the z-plane,
i.e., the magnitudes of all poles are smaller than 1.
Example: The transfer function of an LTI is
|inside ROC||outside ROC,|
|causal, stable||causal, unstable|
|outside ROC||inside ROC,|
|anti-causal, unstable||anti-causal, stable|