91Ó°ÊÓ

In signal processing, poles and zeros are fundamental concepts used to understand and manipulate the behavior of systems, particularly in the context of z-transforms. A _pole_ is a value of \( z \) that makes the denominator of a transfer function zero. Conversely, a _zero_ is a value of \( z \) that makes the numerator zero.

• Poles determine the system's stability and response characteristics.
• Zeros can be thought of as frequencies that are attenuated or nullified in the system's output.

In this exercise, we consider a pole at \( z = \frac{1}{2} \). The location of the pole—in this case, inside the unit circle—tells us about the potential stability for right-sided signals. Poles outside the unit circle would indicate potential stability for left-sided signals. For finite-duration signals, the concept of poles does not exist, as their z-transform is simply a polynomial with no denominator.

_Signal stability_ refers to whether the output of a system remains bounded for any bounded input. This stability is determined through the locations of poles.

• For a right-sided signal, stability is assured if all poles are inside the unit circle \(|z| < 1\).
• For left-sided signals, the poles must lie outside the unit circle \(|z| > 1\) to ensure stability.
• A two-sided signal, which extends indefinitely in both directions, requires complex analysis to determine stability, as some poles can be inside and some outside the unit circle.

In our exercise, the given pole at \( z = \frac{1}{2} \) suggests that the signal could be potentially stable if right-sided since the pole is inside the unit circle. However, if the signal were two-sided or left-sided, it would not meet the stability criteria due to pole locations.

A _right-sided signal_ is one where the non-zero values exist, or are primarily significant, for non-negative time indices (\( n \geq 0 \)). This type of signal's characteristics include:

• They tend to be stable if their poles are located inside the unit circle.
• They are often associated with causal systems, which only respond after input.
• Right-sided signals can converge and provide meaningful transforms easily when poles are within the unit circle.

Given the pole at \( z = \frac{1}{2} \), which is within the unit circle, the signal in question could be a right-sided signal. It meets the stability criterion because such a signal would not diverge, aligning well with usual expectations for discrete-time, right-sided signals.

_Discrete-time signals_ are sequences of values defined at discrete intervals, which is to say time progresses in separate steps. These are fundamental to digital signal processing (DSP) since most practical systems are digital and thus naturally deal with discrete-time signals.

• A valuable feature of these signals is their representation in the z-domain for analysis.
• They allow us to study behavior across frequency components through z-transforms.
• In evaluating such signals, the location of poles and zeros in the z-plane is crucial for understanding their duration and stability.

In considering a discrete-time signal's characteristics, it's essential to assess the pole location critically, as in this exercise. The pole at \( z = \frac{1}{2} \) offers insight into the signal's duration and whether it's likely stable as a right-sided signal. Such assessments help in designing filters and analyzing systems effectively.