Common Z-transform pairs in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

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

Signal Processing

import sympy as sp
z = sp.symbols('z')
n = sp.symbols('n', integer=True, nonnegative=True)
x = sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result is [1]

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

A1 / (1 - z**-1)

B1 / (z - 1)

Cz / (z - 1)

Dz / (1 - z)

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing z and z^-1 in the formula.

Using z/(z-1) instead of 1/(1-z^-1).

The Z-transform of the unit step sequence u[n] is 1 / (1 - z^-1) for |z| > 1.

2fill in blank

medium

Complete the code to find the Z-transform of the sequence x[n] = a^n u[n].

Signal Processing

a = sp.symbols('a')
x = a**n * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result is [1]

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

A1 / (1 - a * z**-1)

B1 / (1 - z**-1 / a)

Cz / (z - a)

Dz / (1 - a)

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping a and z in the denominator.

Forgetting the unit step function u[n].

The Z-transform of x[n] = aⁿ u[n] is 1 / (1 - a z^-1) for |z| > |a|.

3fill in blank

hard

Fix the error in the code to compute the Z-transform of the sequence x[n] = delta[n - k].

Signal Processing

k = sp.symbols('k', integer=True, nonnegative=True)
x = sp.KroneckerDelta(n - k)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result should be [1]

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

Az**(k-1)

Bz**k

C1 / z**k

Dz**-k

Attempts:

3 left

💡 Hint

Common Mistakes

Using z^k instead of z^-k.

Confusing the sign in the exponent.

The Z-transform of delta[n - k] is z^-k.

4fill in blank

hard

Fill both blanks to compute the Z-transform of x[n] = n * a^n * u[n].

Signal Processing

x = n * a**n * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = [1] / ([2])**2

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

Aa * z**-1

B1 - a * z**-1

Cz - a

D1 - z * a

Attempts:

3 left

💡 Hint

Common Mistakes

Forgetting to square the denominator.

Mixing up numerator and denominator terms.

The Z-transform of n aⁿ u[n] is (a z^-1) / (1 - a z^-1)².

5fill in blank

hard

Fill all three blanks to compute the Z-transform of x[n] = cos(ω₀ n) u[n].

Signal Processing

omega0 = sp.symbols('omega0', real=True)
x = sp.cos(omega0 * n) * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = ([1] - [2] * z**-1) / (1 - 2 * [3] * z**-1 + z**-2)

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

Az

Bcos(omega0)

Dsin(omega0)

Attempts:

3 left

💡 Hint

Common Mistakes

Using sin(ω₀) instead of cos(ω₀).

Mixing up powers of z in numerator and denominator.

The Z-transform of cos(ω₀ n) u[n] is (z - cos(ω₀) z^-1) / (1 - 2 cos(ω₀) z^-1 + z^-2).

Practice

1. Which of the following is the Z-transform of the sequence x[n] = a^n u[n], where u[n] is the unit step?

easy

A. X(z) = z / (z - a), |z| < |a|

B. X(z) = 1 / (1 - a z^{-1}), |z| > |a|

C. X(z) = (1 - a z^{-1}) / z, |z| > |a|

D. X(z) = a / (1 - z^{-1}), |z| > 1

Solution

1. Step 1: Recall the definition of the Z-transform for geometric sequences

   The Z-transform of a^n u[n] is a standard pair given by the sum of a geometric series: \( \sum_{n=0}^\infty a^n z^{-n} = \frac{1}{1 - a z^{-1}} \) for \( |z| > |a| \).

2. Step 2: Match the formula with the options

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| matches the formula exactly with the correct region of convergence.

3. Final Answer:

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| -> Option B

4. Quick Check:

   Geometric sequence Z-transform = 1/(1 - a z^{-1}) [OK]

Hint: Geometric sequences transform to 1/(1 - a z^{-1}) [OK]

Common Mistakes:

• Confusing region of convergence
• Mixing numerator and denominator terms
• Using wrong inequality for |z|

2. Which of the following is the correct Z-transform expression for the unit impulse sequence δ[n]?

easy

A. X(z) = z / (z - 1)

B. X(z) = z^{-1}

C. X(z) = 1 / (1 - z^{-1})

D. X(z) = 1

Solution

1. Step 1: Understand the unit impulse sequence

   The unit impulse δ[n] is 1 at n=0 and 0 elsewhere.

2. Step 2: Calculate the Z-transform

   By definition, \( X(z) = \sum_{n=-\infty}^\infty δ[n] z^{-n} = z^{0} = 1 \).

3. Final Answer:

   X(z) = 1 -> Option D

4. Quick Check:

   Impulse Z-transform = 1 [OK]

Hint: Impulse transforms to 1 always [OK]

Common Mistakes:

• Confusing impulse with step function
• Adding z terms unnecessarily
• Forgetting impulse is zero except at n=0

3. What is the Z-transform of the sequence x[n] = n u[n]?

medium

A. X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1

B. X(z) = 1 / (1 - z^{-1})^2, |z| > 1

C. X(z) = z / (z - 1), |z| > 1

D. X(z) = 1 / (1 - z)^2, |z| > 1

Solution

1. Step 1: Recall the Z-transform of the sequence n u[n]

   The sequence n u[n] has Z-transform \( X(z) = \frac{z^{-1}}{(1 - z^{-1})^2} \) with region of convergence \( |z| > 1 \).

2. Step 2: Match the formula with the options

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 matches the formula exactly including the region of convergence.

3. Final Answer:

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 -> Option A

4. Quick Check:

   Linear sequence Z-transform = z^{-1}/(1 - z^{-1})^2 [OK]

Hint: Linear sequences transform to z^{-1}/(1 - z^{-1})^2 [OK]

Common Mistakes:

• Confusing powers of z
• Using (z - 1) instead of (1 - z^{-1})
• Ignoring region of convergence

4. Identify the error in the following Z-transform expression for x[n] = a^n u[n]: X(z) = 1 / (1 - z a^{-1})

medium

A. The variable z and a are swapped in the denominator

B. The region of convergence is missing

C. The exponent on a should be negative

D. The formula is correct as is

Solution

1. Step 1: Recall the correct Z-transform formula for a^n u[n]

   The correct formula is \( X(z) = \frac{1}{1 - a z^{-1}} \), not \( 1/(1 - z a^{-1}) \).

2. Step 2: Analyze the given expression

   The given expression swaps z and a in the denominator, which changes the meaning and is incorrect.

3. Final Answer:

   The variable z and a are swapped in the denominator -> Option A

4. Quick Check:

   Correct formula uses a z^{-1}, not z a^{-1} [OK]

Hint: Check order of variables in denominator carefully [OK]

Common Mistakes:

• Swapping z and a in denominator
• Ignoring negative powers of z
• Missing region of convergence

5. Given the sequence x[n] = (1/2)^n u[n] + 3 δ[n], what is the combined Z-transform X(z)?

hard

A. X(z) = 3 * z / (z - 1/2), |z| > 1/2

B. X(z) = 3 z^{-1} + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

C. X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

D. X(z) = 3 + 1 / (1 - 2 z^{-1}), |z| > 2

Solution

1. Step 1: Find Z-transform of each sequence part

   The Z-transform of (1/2)^n u[n] is \( \frac{1}{1 - (1/2) z^{-1}} \) with \( |z| > 1/2 \). The Z-transform of 3 δ[n] is 3.

2. Step 2: Add the transforms for combined sequence

   Since Z-transform is linear, add both: \( X(z) = 3 + \frac{1}{1 - (1/2) z^{-1}} \).

3. Final Answer:

   X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2 -> Option C

4. Quick Check:

   Sum of transforms = 3 + 1/(1 - 0.5 z^{-1}) [OK]

Hint: Sum transforms for combined sequences linearly [OK]

Common Mistakes:

• Multiplying instead of adding transforms
• Incorrect region of convergence
• Wrong coefficient placement