Region of convergence in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the Region of Convergence (ROC) for Z-Transforms

Which statement correctly describes the Region of Convergence (ROC) for a causal discrete-time signal's Z-transform?

A The ROC is the exterior of a circle in the z-plane, including infinity.

B The ROC is the interior of a circle in the z-plane, excluding infinity.

C The ROC is a ring between two circles in the z-plane, excluding infinity.

D The ROC always includes the unit circle regardless of signal causality.

💡 Hint

Think about where the Z-transform converges for causal signals, which are zero for negative time indices.

For causal signals, the ROC is outside the outermost pole and extends to infinity, forming the exterior of a circle in the z-plane including infinity.

❓ Predict Output

intermediate

Determine the ROC from Poles of a Z-Transform

Given the poles of a Z-transform at 0.5 and 2, what is the ROC for a two-sided sequence?

Signal Processing

poles = [0.5, 2]
# ROC for two-sided sequence lies between poles
roc = (0.5, 2)
print(roc)

A (-2, -0.5)

B (0, 0.5)

C (2, infinity)

D (0.5, 2)

💡 Hint

Recall that for two-sided sequences, the ROC is a ring between poles.

Two-sided sequences have an ROC that lies between the poles, so here it is between 0.5 and 2.

❓ data_output

advanced

ROC Visualization Data for a Given Z-Transform

Given poles at 0.8 and 1.5, and zeros at 0.3 and 2, which data array correctly represents the ROC boundaries for a causal signal?

Signal Processing

poles = [0.8, 1.5]
zeros = [0.3, 2]
roc_boundary = [max(poles), float('inf')]
print(roc_boundary)

A [0.8, 2]

B [1.5, inf]

C [0.3, 1.5]

D [0.3, 0.8]

💡 Hint

For causal signals, ROC extends outside the outermost pole.

The ROC for a causal signal starts from the largest pole radius and extends to infinity.

🔧 Debug

advanced

Identify the Error in ROC Calculation Code

What error does the following code produce when calculating the ROC for a causal signal?

poles = [0.7, 1.2]
roc = (min(poles), float('inf'))
print(roc)

A The ROC should start from the maximum pole, not minimum, so output is incorrect.

B No error; output is correct.

C TypeError because float('inf') is not comparable.

D SyntaxError due to incorrect tuple definition.

💡 Hint

Consider which pole radius defines the start of the ROC for causal signals.

The ROC for causal signals starts from the largest pole radius, so using min(poles) is incorrect and leads to wrong ROC.

🚀 Application

expert

Determine Stability from ROC and Poles

A system has poles at 0.9 and 1.1. Which ROC indicates the system is stable?

A ROC: |z| > 1.1; System is unstable because ROC does not include unit circle.

B ROC: |z| < 0.9; System is stable because ROC includes unit circle.

C ROC: 0.9 < |z| < 1.1; System is stable because ROC includes unit circle.

D ROC: |z| > 0.9; System is unstable because ROC excludes unit circle.

💡 Hint

Recall that system stability requires the ROC to include the unit circle (|z|=1).

The ROC between 0.9 and 1.1 includes the unit circle, so the system is stable in this case.

Practice

1.

What does the Region of Convergence (ROC) represent in signal processing?

easy

A. The frequency range of the signal

B. The set of values where the signal's transform converges

C. The time duration of the signal

D. The amplitude range of the signal

Solution

1. Step 1: Understand the definition of ROC

   The ROC is the set of complex values for which the signal's transform (like Z-transform or Laplace transform) converges.

2. Step 2: Differentiate ROC from other signal properties

   Frequency range, time duration, and amplitude range describe different aspects, not convergence.

3. Final Answer:

   The set of values where the signal's transform converges -> Option B

4. Quick Check:

   ROC = convergence region [OK]

Hint: ROC means where transform sums up properly [OK]

Common Mistakes:

• Confusing ROC with frequency range
• Thinking ROC is about signal duration
• Mixing amplitude with convergence

2.

Which of the following is the correct syntax to denote the ROC of a Z-transform X(z)?

easy

A. ROC = \{z : |z| > r\}

B. ROC = \{z : z = 0\}

C. ROC = \{z : |z| = 0\}

D. ROC = \{z : |z| < 0\}

Solution

1. Step 1: Recall ROC form for causal signals

   For causal signals, ROC is outside a circle of radius r, so |z| > r.

2. Step 2: Check other options for correctness

   Options B, C, and D are invalid because |z| cannot be zero or negative for ROC.

3. Final Answer:

   ROC = \{z : |z| > r\} -> Option A

4. Quick Check:

   ROC syntax uses magnitude inequalities [OK]

Hint: ROC usually involves |z| > or < some radius [OK]

Common Mistakes:

• Using equality instead of inequality for ROC
• Setting ROC radius to zero
• Confusing ROC with single points

3.

Given the Z-transform X(z) = \frac{1}{1 - 0.5z^{-1}}, what is the ROC?

medium

A. |z| < 0.5

B. |z| >= 0.5

C. |z| = 0.5

D. |z| > 0.5

Solution

1. Step 1: Identify the pole location

   The denominator is zero when 1 - 0.5z^{-1} = 0, so z = 0.5.

2. Step 2: Determine ROC for causal sequence

   For a causal sequence, ROC is outside the outermost pole, so |z| > 0.5.

3. Final Answer:

   |z| > 0.5 -> Option D

4. Quick Check:

   ROC outside pole radius [OK]

Hint: ROC is outside the pole radius for causal signals [OK]

Common Mistakes:

• Choosing ROC inside the pole radius
• Confusing equality with inequality
• Ignoring pole location

4.

Find the error in this statement about ROC: "The ROC for a two-sided Z-transform always includes the unit circle |z| = 1."

medium

A. ROC never includes the unit circle

B. ROC always excludes the unit circle

C. ROC may or may not include the unit circle depending on poles

D. ROC is unrelated to the unit circle

Solution

1. Step 1: Understand ROC for two-sided signals

   Two-sided signals have ROC between poles, which may or may not include the unit circle.

2. Step 2: Analyze the statement's generality

   The statement is incorrect because ROC depends on pole locations; it is not guaranteed to include |z|=1.

3. Final Answer:

   ROC may or may not include the unit circle depending on poles -> Option C

4. Quick Check:

   ROC depends on poles, not fixed unit circle [OK]

Hint: ROC inclusion of unit circle depends on poles [OK]

Common Mistakes:

• Assuming ROC always includes unit circle
• Ignoring pole positions
• Confusing two-sided with causal signals

5.

You have a signal with Z-transform X(z) = \frac{z}{z - 2}. Which ROC ensures the signal is stable?

hard

A. |z| < 2

B. |z| = 2

C. |z| > 2

D. |z| >= 0

Solution

1. Step 1: Identify the pole location

   The pole is at z = 2.

2. Step 2: Recall stability condition

   For stability, ROC must include the unit circle |z|=1, so ROC must be inside the pole: |z| < 2.

3. Step 3: Choose ROC that includes unit circle

   Only ROC |z| < 2 includes |z|=1, ensuring stability.

4. Final Answer:

   |z| < 2 -> Option A

5. Quick Check:

   Stable ROC includes unit circle inside poles [OK]

Hint: Stable ROC must include unit circle inside poles [OK]

Common Mistakes:

• Choosing ROC outside pole for stability
• Ignoring unit circle inclusion
• Confusing stability with causality