Region of convergence in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute the Z-transform of a sequence.

Signal Processing

import numpy as np

x = [1, 2, 3]
z = np.array([1, -1j, 1j])
X = sum(x[n] * z[1]n for n in range(len(x)))
print(X)

Drag options to blanks, or click blank then click option'

A**

B*

C+

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using '*' instead of '**' causes a multiplication error.

Using '+' or '-' changes the meaning and causes wrong results.

The Z-transform uses powers of z, so we raise z to the power n using '**'.

2fill in blank

medium

Complete the code to check if the magnitude of z is inside the region of convergence (ROC).

Signal Processing

z = 0.5 + 0.5j
roc_radius = 1.0
if abs(z) [1] roc_radius:
    print('Inside ROC')
else:
    print('Outside ROC')

Drag options to blanks, or click blank then click option'

A>

B==

C<

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '>' reverses the condition and gives wrong results.

Using '==' is too strict and rarely true.

The ROC includes points where the magnitude of z is less than the radius.

3fill in blank

hard

Fix the error in the code to compute the ROC for a causal sequence with pole at p.

Signal Processing

p = 0.8
roc = {z for z in np.linspace(0, 2, 100) if abs(z) [1] abs(p)}
print(len(roc))

Drag options to blanks, or click blank then click option'

A>

B<

C<=

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '<' selects inside the pole, which is incorrect for causal sequences.

Using '<=' or '>=' changes the boundary inclusion incorrectly.

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

4fill in blank

hard

Fill both blanks to create a dictionary of z values and their ROC status for a pole at p.

Signal Processing

p = 0.6
z_values = np.linspace(0, 1.5, 10)
roc_dict = {z: 'inside' if abs(z) [1] abs(p) else 'outside' for z in z_values if z [2] 0}
print(roc_dict)

Drag options to blanks, or click blank then click option'

A<

B>

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '>' in the first blank reverses the ROC condition.

Using '<=' or '>=' changes the strictness of the condition incorrectly.

We check if |z| < |p| for inside ROC and filter z > 0 to avoid zero or negative values.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps z to magnitude and ROC status for poles p1 and p2.

Signal Processing

p1, p2 = 0.5, 1.2
z_vals = np.linspace(0, 2, 15)
roc_info = {z: (abs(z), 'inside' if abs(z) [1] abs(p1) and abs(z) [2] abs(p2) else 'outside') for z in z_vals if z [3] 0}
print(roc_info)

Drag options to blanks, or click blank then click option'

A>

B<

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Reversing inequalities causes wrong ROC classification.

Not filtering positive z values includes zero or negatives.

The ROC is between poles, so |z| < |p1| and |z| <= |p2|, filtering z > 0.