Common Z-transform pairs in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Z-transform of the unit impulse sequence \( \delta[n] \)?

The Z-transform of \( \delta[n] \) is 1. This is because \( \delta[n] \) is 1 at \( n=0 \) and 0 elsewhere, so \( X(z) = \sum_{n=-\infty}^{\infty} \delta[n] z^{-n} = 1 \).

beginner

What is the Z-transform of the unit step sequence \( u[n] \)?

The Z-transform of \( u[n] \) is \( \frac{1}{1 - z^{-1}} \) for \( |z| > 1 \). This comes from summing the geometric series \( \sum_{n=0}^\infty z^{-n} \).

intermediate

What is the Z-transform of the sequence \( a^n u[n] \) where \( |a| < |z| \)?

The Z-transform is \( \frac{1}{1 - a z^{-1}} \) for \( |z| > |a| \). This is a geometric series with ratio \( a z^{-1} \).

intermediate

What is the Z-transform of the sequence \( n u[n] \)?

The Z-transform of \( n u[n] \) is \( \frac{z^{-1}}{(1 - z^{-1})^2} \) for \( |z| > 1 \). This is derived by differentiating the Z-transform of \( u[n] \) with respect to \( z^{-1} \).

beginner

What is the Z-transform of the delayed sequence \( x[n-k] \) in terms of \( X(z) \)?

The Z-transform of \( x[n-k] \) is \( z^{-k} X(z) \). This means delaying the sequence by \( k \) shifts the Z-transform by multiplying by \( z^{-k} \).

What is the region of convergence (ROC) for the Z-transform of \( a^n u[n] \)?

A |z| = |a|

B |z| < |a|

C All z

D |z| > |a|

The ROC for \( a^n u[n] \) is outside the circle of radius \( |a| \), so \( |z| > |a| \).

What is the Z-transform of the sequence \( \delta[n-3] \)?

A 1

B z^{-3}

C z^{3}

D 0

Delaying the impulse by 3 multiplies the Z-transform by \( z^{-3} \).

Which sequence has the Z-transform \( \frac{1}{1 - z^{-1}} \)?

A Unit step \( u[n] \)

B Unit impulse \( \delta[n] \)

C Sequence \( n u[n] \)

D Sequence \( a^n u[n] \)

The unit step \( u[n] \) has Z-transform \( \frac{1}{1 - z^{-1}} \).

What operation in time domain corresponds to multiplying by \( z^{-k} \) in Z-domain?

A Delaying the sequence by k

B Differentiating the sequence

C Scaling the sequence

D Reversing the sequence

Multiplying by \( z^{-k} \) corresponds to delaying the sequence by \( k \) steps.

What is the Z-transform of the sequence \( n u[n] \)?

A \( \frac{1}{1 - z^{-1}} \)

B \( z^{-1} \)

C \( \frac{z^{-1}}{(1 - z^{-1})^2} \)

D 1

The Z-transform of \( n u[n] \) is \( \frac{z^{-1}}{(1 - z^{-1})^2} \).

Explain the Z-transform of the unit step sequence and its region of convergence.

Think about summing a geometric series starting at n=0.

Describe how delaying a sequence by k affects its Z-transform.

Consider the shift property of the Z-transform.

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