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

# Z-transform property proof

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Chaitra (new course), 2 marks
- Video: [Properties of Z-Transform (Final Value Theorem)](https://www.youtube.com/watch?v=HINKNjZlb7U) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/z-transform-property-proof
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find x(0) and x(∞) of the Z-transform x(t) = 1 − e^(−t).

## Short answer

Using initial and final value theorems on X(z)=z/(z−1) − z/(z−e^(−T)), x(0)=lim_{z→∞} X(z)=0, x(∞)=lim_{z→1} (z−1)X(z)=1. Directly, 1−e^0=0 and 1−e^(−∞)=1.

## Answer

Take x(t) sampled at t = kT, so x(k) = 1 − e^(−kT). (Here "x(0)" and "x(∞)" are found by the initial and final value theorems.)

**Z-transform:**

```text
X(z) = z/(z − 1) − z/(z − e^(−T))
```

**Initial value:** x(0) = lim_{z→∞} X(z) = 1 − 1 = 0.

**Final value:** x(∞) = lim_{z→1} (z − 1) X(z).

```text
(z − 1) X(z) = z − z(z − 1)/(z − e^(−T))  →  1 − 0 = 1   as z → 1
```

**Answer: x(0) = 0 and x(∞) = 1.** (Directly: 1 − e^0 = 0 and 1 − e^(−∞) = 1.)

## Check yourself

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

1. x(0) is obtained from X(z) by
   A. lim as z to infinity of X(z)
   B. lim as z to 1 of X(z)
   C. lim as z to 0 of X(z)
   D. lim as z to 1 of (z-1)X(z)
   Answer: A. Negative powers of z vanish as z to infinity. (Key point: Initial value theorem)

2. x(infinity) is obtained as
   A. lim as z to infinity of X(z)
   B. lim as z to 1 of (z-1)X(z)
   C. lim as z to 1 of X(z)
   D. lim as z to 0 of z X(z)
   Answer: B. Valid when poles of (z-1)X lie inside the unit circle. (Key point: Final value theorem)

3. The Z-transform of 1 - e^(-kT) is
   A. z/(z-1) + z/(z - e^(-T))
   B. 1/(z-1) - 1/(z - e^(-T))
   C. z/(z-1) - z/(z - e^(-T))
   D. z/(z-1) - z/(z + e^(-T))
   Answer: C. Z[a^k] = z/(z-a) with a = 1 and e^(-T). (Key point: X(z))

4. x(0) and x(infinity) are
   A. 1 and 0
   B. 0 and 0
   C. 1 and 1
   D. 0 and 1
   Answer: D. 1 - e^0 = 0 and 1 - e^(-infinity) = 1. (Key point: Result)

## Related in Mathematics III

- [Z-transform property proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Az-transform-property-proof-two) — Mathematics III, chapter 3
- [Z-transform property proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Az-transform-property-proof-three) — Mathematics III, chapter 3
- [Laplace transform of derivative/shift property](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transform-of-derivative-shift-property) — Mathematics III, chapter 3
- [Laplace transform of derivative/shift property](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transform-of-derivative-shift-property-two) — Mathematics III, chapter 3

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- [Derive time shifting property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-time-shifting-derive) — Digital Signal Analysis and Processing, chapter 2
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