---
title: "z-transform of (1-e^{-at}); final value"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-1-minus-e-at-final"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-1-minus-e-at-final
site: BCT Engineering
---

# z-transform of (1-e^{-at}); final value

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2075 Baisakh, 2+3 marks; 2070 Bhadra, 5 marks (asked 2 times)
- Video: [VTU Engineering Maths 3 Z transform Final value theorem(statement & proof)](https://www.youtube.com/watch?v=TkEKsmAgQvg) — EASY MATHS EASY TRICKS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-1-minus-e-at-final
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Obtain the z-transform of x(t) = (1 - e^{-at}), a > 0 and hence evaluate x(∞) by using final value theorem.

## Short answer

X(z) = z(1 - e^{-aT}) / [(z-1)(z - e^{-aT})]. By the final value theorem, x(∞) = lim_{z→1} (z-1)X(z) = 1.

## Answer

#### z-transform
Take sampling period T. x(kT) = 1 − e^(−akT).

Z[1] = z/(z − 1)  (|z| > 1),  Z[e^(−at)] = z/(z − e^(−aT)).

X(z) = z/(z − 1) − z/(z − e^(−aT))
= z [ (z − e^(−aT)) − (z − 1) ] / [ (z − 1)(z − e^(−aT)) ]

**X(z) = z (1 − e^(−aT)) / [ (z − 1)(z − e^(−aT)) ]**

#### Final value x(∞)
Final value theorem: x(∞) = lim (z → 1) (z − 1) X(z). Since a > 0, e^(−aT) < 1, so the only pole outside the open unit disc is at z = 1, which is cancelled by (z − 1); the theorem applies.

x(∞) = lim (z → 1) z (1 − e^(−aT)) / (z − e^(−aT))
= (1 − e^(−aT)) / (1 − e^(−aT)) = 1

**Answer: x(∞) = 1**

This agrees with the direct limit: 1 − e^(−at) → 1 as t → ∞.

## Check yourself

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

1. What is Z[1 - e^{-at}]?
   A. z/(z-1) + z/(z-e^{-aT})
   B. 1/(z-1) - 1/(z-e^{-aT})
   C. z/(z-1)
   D. z/(z-1) - z/(z-e^{-aT})
   Answer: D. Linearity. (Key point: Transform)

2. Why can final value theorem be used?
   A. e^{-aT}<1, so only the pole at z=1 is on the boundary and is cancelled by (z-1)
   B. Because a<0
   C. Because there are no poles
   D. Because x(0)=0
   Answer: A. Other pole inside the unit disc. (Key point: Theorem applicability)

3. What is x(∞)?
   A. 0
   B. 1
   C. e^{-aT}
   D. ∞
   Answer: B. Agrees with the direct limit. (Key point: Result)

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