Pole-zero analysis for stability in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a pole in the context of pole-zero analysis?

A pole is a value of the complex variable where the system's transfer function becomes infinite. It represents natural frequencies where the system's output can grow very large.

beginner

What does a zero represent in pole-zero analysis?

A zero is a value of the complex variable where the system's transfer function becomes zero. It indicates frequencies that the system blocks or cancels out.

intermediate

How do poles relate to system stability?

For a system to be stable, all poles must lie inside the unit circle in the z-plane (discrete systems) or in the left half of the s-plane (continuous systems). Poles outside these regions cause instability.

intermediate

Why is the unit circle important in discrete-time pole-zero analysis?

The unit circle defines the boundary for stability in discrete-time systems. Poles inside the unit circle mean the system's output will not grow unbounded over time.

advanced

What happens if a pole lies exactly on the unit circle in discrete-time systems?

If a pole lies exactly on the unit circle, the system is marginally stable. The output neither grows nor decays but can oscillate indefinitely.

Where must all poles lie for a discrete-time system to be stable?

A Anywhere in the complex plane

B Outside the unit circle

C On the unit circle

D Inside the unit circle

For discrete-time systems, stability requires all poles to be inside the unit circle in the complex plane.

What does a zero of a system's transfer function indicate?

A Frequency where output is zero

B Frequency where output is infinite

C Frequency where input is zero

D Frequency where system is unstable

Zeros are frequencies where the system's output becomes zero, effectively blocking those frequencies.

In continuous-time systems, where must poles lie for stability?

A Right half of the s-plane

B On the imaginary axis

C Left half of the s-plane

D Anywhere in the s-plane

Continuous-time systems are stable only if all poles lie in the left half of the s-plane.

What is the effect of a pole on the unit circle in a discrete system?

A System is marginally stable

B System is unstable

C System is stable

D System output is zero

A pole on the unit circle means the system is marginally stable, causing sustained oscillations.

Which of the following best describes a pole-zero plot?

A Plot showing input vs output

B Graph of poles and zeros in the complex plane

C Time-domain signal plot

D Frequency response magnitude plot

A pole-zero plot shows the locations of poles and zeros in the complex plane, helping analyze system behavior.

Explain how pole locations affect the stability of a discrete-time system.

Think about how the system's output behaves over time depending on pole positions.

Describe the difference between poles and zeros in a system's transfer function.

Consider what happens to the system output at poles and zeros.