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.

Practice

1. What does it mean for a system to be stable in terms of its poles on the pole-zero plot?

easy

A. All poles must lie in the left half of the complex plane (negative real parts).

B. All poles must lie on the imaginary axis.

C. All zeros must lie in the right half of the complex plane.

D. Poles and zeros must be equal in number.

Solution

1. Step 1: Understand pole location and stability

   Stability depends on poles, not zeros. Poles with negative real parts mean the system's response decays over time.

2. Step 2: Identify correct pole location for stability

   Poles in the left half-plane (negative real parts) ensure the system is stable.

3. Final Answer:

   All poles must lie in the left half of the complex plane (negative real parts). -> Option A

4. Quick Check:

   Stable system = poles with negative real parts [OK]

Hint: Stable system means poles left of imaginary axis [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking poles on imaginary axis are stable
• Assuming zeros affect stability directly

2. Which of the following is the correct Python syntax to plot poles and zeros using the scipy.signal library?

easy

A. signal.pzmap(system, plot=True)

B. signal.pzplot(system)

C. signal.pzplot(system, plot=True)

D. signal.pzmap(system, plot=False)

Solution

1. Step 1: Recall scipy.signal function for pole-zero plot

   The function to plot poles and zeros is pzmap, not pzplot.

2. Step 2: Check correct usage with plotting enabled

   Using pzmap with plot=True plots the pole-zero map correctly.

3. Final Answer:

   signal.pzmap(system, plot=True) -> Option A

4. Quick Check:

   Use pzmap with plot=True for pole-zero plot [OK]

Hint: Use pzmap with plot=True to plot poles and zeros [OK]

Common Mistakes:

• Using non-existent function pzplot
• Forgetting plot=True argument
• Confusing function names

3. Given the poles of a system as poles = [-1+2j, -1-2j, 0.5], what can you say about the system's stability?

medium

A. The system is stable because poles are complex conjugates.

B. The system is stable because all poles have negative real parts.

C. The system is marginally stable because poles are on the imaginary axis.

D. The system is unstable because one pole has a positive real part.

Solution

1. Step 1: Analyze the real parts of each pole

   Poles are -1+2j (real part -1), -1-2j (real part -1), and 0.5 (real part 0.5).

2. Step 2: Determine stability from pole real parts

   Since one pole has positive real part (0.5), the system is unstable.

3. Final Answer:

   The system is unstable because one pole has a positive real part. -> Option D

4. Quick Check:

   Positive real pole means unstable system [OK]

Hint: Check if any pole has positive real part for instability [OK]

Common Mistakes:

• Ignoring positive real part pole
• Assuming complex conjugates guarantee stability
• Confusing marginal stability with instability

4. The following Python code attempts to plot poles and zeros but gives an error:

from scipy import signal
system = signal.TransferFunction([1, 3], [1, 2, 1])
signal.pzplot(system)

What is the error and how to fix it?

medium

A. TransferFunction requires discrete=True for pzplot to work.

B. TransferFunction arguments are reversed; swap numerator and denominator.

C. Function pzplot does not exist; use pzmap(system, plot=True) instead.

D. Import statement is wrong; use import scipy.signal as signal.

Solution

1. Step 1: Identify the function causing error

   The function pzplot does not exist in scipy.signal, causing the error.

2. Step 2: Use correct function to plot poles and zeros

   Use pzmap(system, plot=True) to plot poles and zeros correctly.

3. Final Answer:

   Function pzplot does not exist; use pzmap(system, plot=True) instead. -> Option C

4. Quick Check:

   Use pzmap, not pzplot, for pole-zero plots [OK]

Hint: Use pzmap with plot=True, not pzplot [OK]

Common Mistakes:

• Using non-existent pzplot function
• Swapping numerator and denominator unnecessarily
• Wrong import statement causing confusion

5. You have a system with poles at -0.5, -1+1j, -1-1j and zeros at 0, -2. Which statement best describes the system's stability and response?

hard

A. The system is marginally stable because of zeros at zero.

B. The system is stable with oscillatory decay due to complex poles.

C. The system is unstable because zeros are on the right half-plane.

D. The system is unstable due to poles with positive real parts.

Solution

1. Step 1: Check pole locations for stability

   All poles have negative real parts (-0.5, -1), so system is stable.

2. Step 2: Analyze pole types for response behavior

   Complex conjugate poles (-1±1j) cause oscillations that decay over time.

3. Step 3: Consider zeros' effect on stability

   Zeros do not affect stability directly; zeros at 0 and -2 are not in right half-plane.

4. Final Answer:

   The system is stable with oscillatory decay due to complex poles. -> Option B

5. Quick Check:

   Negative real poles + complex = stable oscillatory decay [OK]

Hint: Stable if poles negative; complex poles cause oscillations [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking zeros at zero cause instability
• Ignoring oscillatory nature of complex poles