Next: LTI Systems Characterized by Up: Z_Transform Previous: Z-Transform of Typical Signals

# Analysis of LTI Systems by Z-Transform

Due to its convolution property, the z-transform is a powerful tool to analyze LTI systems

As discussed before, when the input is the eigenfunction of all LTI system, i.e., , the operation on this input by the system can be found by multiplying the system's eigenvalue to the input:

• Causal LTI systems

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 impulse at is at rest for , i.e., . Its response to a general input is:

Due to the properties of the ROC, we know that

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 .

• Stable LTI systems

An LTI system is stable if its response to any bounded input is also bounded for all :

As the output and input of an LTI is related by convolution, we have:

and

which obviously requires:

In other words, if the impulse response function of an LTI system is absolutely integrable, then the system is stable. We can show that this condition is also necessary, i.e., all stable LTI systems' impulse response functions are absolutely integrable. Now we have:

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 .

• Causal and stable LTI systems

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

As shown before, without specifying the ROC, this could be the z-transform of one of the two possible time signals .
• If ROC is , the system is causal.
• If , i.e., unit circle can be included in ROC, the system is stable;
• If , i.e., unit circle cannot be included in ROC, the system is unstable;
• If ROC is , the system is anti-causal.
• If , i.e., unit circle cannot be included in ROC, the system is unstable;
• If , i.e., unit circle can be included in ROC, the system is stable;

 inside ROC outside ROC, causal, stable causal, unstable outside ROC inside ROC, anti-causal, unstable anti-causal, stable

Next: LTI Systems Characterized by Up: Z_Transform Previous: Z-Transform of Typical Signals
Ruye Wang 2014-10-28