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.