---
title: "Z-transform of t² e^(−at)"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:z-transform-t2-exp-at"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/z-transform-t2-exp-at
site: BCT Engineering
---

# Z-transform of t² e^(−at)

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2082 Kartik, 3 marks; 2081 Chaitra, 3 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/z-transform-t2-exp-at
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Find the Z-transform of x(t)=t²e⁻ᵃᵗ (for k≥ 0).

## Short answer

For x(t)=t²e^(-at), sampled as T²k²(e^(-aT))^k. Using Z[k²] = z(z+1)/(z-1)³ and scaling property, X(z) = T²e^(-aT)z(z+e^(-aT))/(z-e^(-aT))³.

## Answer

Sampled form: $x(kT) = (kT)^2 e^{-akT} = T^2 k^2 (e^{-aT})^k$. Let $r = e^{-aT}$.

#### Step 1: $\mathcal{Z}[k^2]$

From $\mathcal{Z}[1] = \frac{z}{z-1}$ and $\mathcal{Z}[k\,x(k)] = -z\frac{d}{dz}X(z)$:

$$
\mathcal{Z}[k] = \frac{z}{(z-1)^2}, \qquad \mathcal{Z}[k^2] = -z\frac{d}{dz}\frac{z}{(z-1)^2} = \frac{z(z+1)}{(z-1)^3}
$$

#### Step 2: multiply by $r^k$ (scaling: replace $z$ by $z/r$)

$$
\mathcal{Z}[k^2 r^k] = \frac{(z/r)(z/r+1)}{(z/r-1)^3} = \frac{rz(z+r)}{(z-r)^3}
$$

#### Result

$$
X(z) = \frac{T^2 e^{-aT} z\,(z + e^{-aT})}{(z - e^{-aT})^3}
$$

**Answer:** $\mathcal{Z}[t^2e^{-at}] = \dfrac{T^2e^{-aT}z(z+e^{-aT})}{(z-e^{-aT})^3}$ (equivalently $\dfrac{T^2e^{-aT}z^{-1}(1+e^{-aT}z^{-1})}{(1-e^{-aT}z^{-1})^3}$).

## Check yourself

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

1. Z[k x(k)] = :
   A. -z dX/dz
   B. z dX/dz
   C. -X(z)/z
   D. X(z/k)
   Answer: A. Used to get Z[k^2]. (Key point: Multiplication by k)

2. Z[k^2] =
   A. z/(z-1)^3
   B. z(z+1)/(z-1)^3
   C. z(z-1)/(z+1)^3
   D. z/(z-1)^2
   Answer: B. Derivative of z/(z-1)^2. (Key point: Z[k^2])

3. Z[t^2 e^-at] =
   A. T e^-aT z/(z - e^-aT)^2
   B. T^2 z (z+1)/(z-1)^3
   C. T^2 e^-aT z (z + e^-aT)/(z - e^-aT)^3
   D. e^-aT z/(z - e^-aT)
   Answer: C. Scaling z to z/r. (Key point: Result)