---
title: "z-transform of t·e^{-at}"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-te-at"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-te-at
site: BCT Engineering
---

# z-transform of t·e^{-at}

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2074 Bhadra, 3 marks; 2072 Asoj; 2070 Bhadra (asked 3 times)
- 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-te-at
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of t e^{-at}.

## Short answer

Using the multiplication-by-t property, Z[t e^{-at}] = T e^{-aT} z / (z - e^{-aT})^2. For T=1 it simplifies to e^{-a} z / (z - e^{-a})^2, with region of convergence |z| > e^{-aT}.

## Answer

Let the sampling period be T, so x(kT) = kT e^(−akT) and Z[x(t)] = Σ x(kT) z⁻ᵏ.

Start with the standard result (geometric series, |z| > e^(−aT)):

Z[e^(−at)] = Σ (e^(−aT)/z)ᵏ = z/(z − e^(−aT))

Use the multiplication-by-t property, Z[t x(t)] = −T z d/dz X(z):

Z[t e^(−at)] = −T z d/dz [ z/(z − e^(−aT)) ]

d/dz [ z/(z − q) ] = [ (z − q) − z ]/(z − q)² = −q/(z − q)², where q = e^(−aT).

So Z[t e^(−at)] = −T z · (−q)/(z − q)² = T q z/(z − q)².

**Answer:**

Z[t e^(−at)] = T e^(−aT) z / (z − e^(−aT))²

For T = 1 this is e^(−a) z/(z − e^(−a))². Region of convergence |z| > e^(−aT).

## Check yourself

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

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

2. Which property gives Z[t e^{-at}]?
   A. Z[t x(t)] = T z X(z)
   B. Z[t x(t)] = X(z)/z
   C. Z[t x(t)] = d/dz X(z)
   D. Z[t x(t)] = -T z d/dz X(z)
   Answer: D. Multiplication by t. (Key point: Property)

3. What is Z[t e^{-at}]?
   A. T e^{-aT} z/(z - e^{-aT})²
   B. T z/(z - e^{-aT})
   C. e^{-aT} z/(z - e^{-aT})
   D. T e^{-aT}/(z - e^{-aT})²
   Answer: A. -Tz × (-q)/(z-q)². (Key point: Result)