Stability analysis (pole-zero plot) in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

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

A pole is a point in the complex plane where the system's transfer function becomes infinite. It indicates frequencies where the system's output can grow very large.

beginner

What does a zero represent in a pole-zero plot?

A zero is a point in the complex plane where the system's transfer function becomes zero. It shows frequencies where the system output is completely canceled.

beginner

How can you tell if a system is stable by looking at its poles on the pole-zero plot?

If all poles lie inside the unit circle (for discrete systems) or in the left half of the complex plane (for continuous systems), the system is stable. Poles outside these regions mean instability.

intermediate

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

The unit circle defines the boundary for stability in discrete-time systems. Poles inside the unit circle mean the system's response will die out over time, ensuring stability.

intermediate

What is the significance of the distance of poles from the origin in a pole-zero plot?

The distance of poles from the origin relates to how fast the system's response decays or grows. Poles closer to the origin usually mean faster decay and more stability.

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

A Inside the unit circle

B On the unit circle

C Outside the unit circle

D Anywhere in the complex plane

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

What does a zero on the pole-zero plot 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 points where the system output becomes zero at certain frequencies.

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

A On the imaginary axis

B Right half of the complex plane

C Left half of the complex plane

D Anywhere on the complex plane

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

What happens if a pole lies exactly on the unit circle in a discrete system?

A System is stable

B System output is zero

C System is unstable

D System is marginally stable

Poles on the unit circle indicate marginal stability, meaning the system neither grows nor decays.

Why do poles closer to the origin usually mean faster decay in system response?

A Because they represent higher frequencies

B Because they have smaller magnitude leading to faster damping

C Because they are zeros

D Because they are outside the unit circle

Poles with smaller magnitude cause the system's response to decay faster, improving stability.

Explain how to determine system stability using a pole-zero plot for a discrete-time system.

Think about where poles must be located relative to the unit circle.

Describe the roles of poles and zeros in shaping the behavior of a system using the pole-zero plot.

Consider how poles and zeros affect the system output at different frequencies.