---
title: "Inverse Z-transform of function"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-z-transform-of-function"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-z-transform-of-function
site: BCT Engineering
---

# Inverse Z-transform of function

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Chaitra (new course), 2 marks
- Video: [Inverse Z-Transform | Inverse Z-Transform Using Partial Fraction | Signals and System | Mathspedia |](https://www.youtube.com/watch?v=ZnB7lGg9YF0) — Mathspedia
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-z-transform-of-function
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse z-transform of z/((z − 1)(z − 2)).

## Short answer

Use partial fractions on X(z)/z to get X(z)=z/(z-2)-z/(z-1). Apply inverse Z-transform of z/(z-a) being a^k to obtain x(k)=2^k - 1 for k=0,1,2,…

## Answer

Use partial fractions on X(z)/z.

```text
X(z)/z = 1/((z − 1)(z − 2)) = 1/(z − 2) − 1/(z − 1)
X(z)   = z/(z − 2) − z/(z − 1)
```

Using Z⁻¹[z/(z − a)] = aᵏ:

**Answer: x(k) = 2ᵏ − 1, k = 0, 1, 2, ...**

**Check by long division:** X(z) = z/(z² − 3z + 2) = z⁻¹ + 3z⁻² + 7z⁻³ + ..., so x(0) = 0, x(1) = 1, x(2) = 3, x(3) = 7. These equal 2ᵏ − 1 ✓.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. To invert z/((z-1)(z-2)) by partial fractions, one expands
   A. X(z)/z
   B. X(z) directly
   C. X(z) z
   D. 1/X(z)
   Answer: A. Dividing by z puts the answer in terms of z/(z-a). (Key point: Method)

2. X(z)/z = 1/((z-1)(z-2)) equals
   A. 1/(z-1) - 1/(z-2)
   B. 1/(z-2) - 1/(z-1)
   C. 1/(z-2) + 1/(z-1)
   D. 1/2(z-2) - 1/(z-1)
   Answer: B. A at z = 2 is 1; B at z = 1 is -1. (Key point: Decomposition)

3. Z^-1[z/(z-a)] =
   A. k a
   B. a^(-k)
   C. a^k
   D. k^a
   Answer: C. Standard geometric sequence. (Key point: Pair used)

4. x(k) =
   A. 2^k + 1
   B. 1 - 2^k
   C. 2^(k-1)
   D. 2^k - 1
   Answer: D. Check x(1) = 1, x(2) = 3. (Key point: Answer)

## Related in Mathematics III

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