Causality and stability
The impulse response of a causal LTI system is zero for n<0. Therefore, the impulse response of a causal LTI system is determined from the transfer function by using right-sided inverse transforms. A pole that is inside the unit circle in the z-plane contributes an exponentially increasing term. A pole on the unit circle contributes a complex sinusoid.
Alternatively, if we know a system is stable, then the impulse response is absolutely summable, and the DTFT of the impulse response exists. It follows that the ROC must include the unit circle in the z-plane. Hence, the relationship between the location of a pole and the unit circle determines the component of the impulse response associated with that pole. A pole inside the unit circle contributes a right-sided decaying exponential term to the impulse response, while a pole outside the unit circle contributes a left-sided decaying exponential term to the impulse response. A stable response cannot contain any increasing exponential terms, since an increasing exponential is not absolutely integrable.
Another situation is what if the system is both causal and stable has all the poles inside the unit circle?
In this situation, a pole that is on the left half of the z-plane contributes a right-sided decaying exponential term to the impulse response. We cannot have a pole in the right half of the z-plane, however, because a pole in the right half of the z-plane, however, because a pole in the right half will contribute either a left-sided decaying exponential that is not causal or a right-sided increasing exponential that results in an unstable impulse response. That is, the inverse Laplace transform of a pole in the right half of the z-plane is either stable or causal, but it cannot be both. Systems that are stable and causal must have all their poles in the left half of the z-plane.
The relationship between the location of a pole and the unit circle determines the component of the impulse response associated with that pole. A pole inside the circle contributes a right-sided decaying exponential term to the impulse response, while a pole outside the unit circle contributes a left-sided decaying exponential term to the impulse response.
If H(z) is written in the pole-zero form, the zeros of H(z) are the poles of H ^inv* (z), and the poles of H(z) are the zeros of H ^inv* (z). Any system described by a rational transfer function has an inverse system of this form.
We are often interested in inverse systems that are both stable and causal, so we can implement a system H ^inv * (z) that reverses the distortion introduced by H(z) to a signal of interest. H^ inv* (z) is both stable and causal if all of its poles are inside the unit circle. Since the poles of H^inv* (z) are the zeros of H(z), we conclude that a stable and causal inverse of an LTI system H(z) exists if and only if all the zeros of are inside the unit circle. If H(z) has any zeros outside the unit circle, then a stable and causal inverse system does exist. As with a continuous-time minimum-phase system, there is a unique relationship between the magnitude and phase responses of a discrete-time minimum-phase system. That is, the phase response of a minimum-phase system is ly determined by the magnitude response. Alternatively, the magnitude response of a unique minimum-phase system is uniquely determined by the phase response.
Determining the frequency response from poles and zeroes.
As we know that the frequency response is obtained f the transfer function by substituting e^iΩ for z in H(z). That is, the frequency response responds to the transfer function evaluated on the unit circle in the z-plane. That is the ROC includes the unit circle. A pole that is very close to the unit circle will cause a large peak in |H(e^iΩ)| at the frequency corresponding to the phase angle of the pole. Hence, zeros tend to pull the frequency response magnitude down, while poles tend to push it up. The size of |H(e^iΩ)| depends on how far it is from the unit circle.
Computational structures for implementing discrete-time LTI systems
Discrete-time LTI systems lend themselves to implementation in computers. In order to write the computer program that determines the system output from the input, we must first specify the order in which each computation is to be performed. The z-transform is often used to develop such computational structures to implement discrete-time LTI systems. The freedom to choose between alternative implementations can be used to optimize some criteria associated with computation, such as the number of numerical operations or the sensitivity of the system to numerical rounding of computations.
Causality and stability