---
title: "z-transform of t²e^{-at}"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-t2-e-at"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-t2-e-at
site: BCT Engineering
---

# z-transform of t²e^{-at}

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 1.5 marks
- 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-t2-e-at
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the Z-transform of t² e^{-at}.

## Short answer

Set q=e^{-aT}. Applying the differentiation property twice yields Z[t^2 e^{-at}]=T^2 q z(z+q)/(z-q)^3. For T=1, Z=e^{-a} z(z+e^{-a})/(z-e^{-a})^3.

## Answer

Let the sampling period be T and q = e^(−aT).

**Step 1.** Z[e^(−at)] = z/(z − q).

**Step 2.** Use Z[t x(t)] = −T z d/dz X(z) twice.

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

Z[t² e^(−at)] = −T z d/dz [ T q z/(z − q)² ]
= −T² q z · [ (z − q)² − 2z(z − q) ]/(z − q)⁴
= −T² q z · [ (z − q) − 2z ]/(z − q)³
= T² q z (z + q)/(z − q)³

**Answer:**

Z[t² e^(−at)] = T² e^(−aT) z (z + e^(−aT)) / (z − e^(−aT))³,  |z| > e^(−aT)

For T = 1: e^(−a) z (z + e^(−a))/(z − e^(−a))³.

## Check yourself

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

1. With sampling period T, what is Z[e^(-at)]?
   A. z/(z - q) with q = e^(-aT)
   B. z/(z - e^(aT))
   C. z/(z + e^(-aT))
   D. T z/(z - e^(-aT))
   Answer: A. e^(-akT) = q^k, a geometric sequence with ratio q = e^(-aT). (Key point: Substitution)

2. Which property gives the factor t^2?
   A. Time shifting applied twice
   B. Z[t x(t)] = -T z dX/dz, applied twice
   C. Scaling X(z/a) applied twice
   D. Initial value theorem
   Answer: B. Multiplication by t corresponds to -Tz d/dz. (Key point: Rule used)

3. What is Z[t e^(-at)]?
   A. T z/(z - q)^2
   B. -T q z/(z - q)^2
   C. T q z/(z - q)^2
   D. T q z/(z - q)^3
   Answer: C. -Tz times d/dz[z/(z-q)] = -Tz * (-q)/(z-q)^2 = Tqz/(z-q)^2. (Key point: First step result)

4. Z[t^2 e^(-at)] equals
   A. T^2 q z (z - q)/(z + q)^3
   B. T^2 q z/(z - q)^3
   C. T q z (z + q)/(z - q)^2
   D. T^2 q z (z + q)/(z - q)^3 with q = e^(-aT)
   Answer: D. Second application gives T^2 q z (z+q)/(z-q)^3; ROC |z| > e^(-aT). (Key point: Final result)