---
title: "z-transform of t²aᵗ"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-t2-a-t"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-t2-a-t
site: BCT Engineering
---

# z-transform of t²aᵗ

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Bhadra, 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-a-t
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Obtain the z-transform of t² aᵗ.

## Short answer

Let b=a^T. Using Z[t x(t)]=-Tz dX/dz twice gives Z[t^2 a^t]=T^2 b z(z+b)/(z-b)^3. With T=1, it reduces to a z(z+a)/(z-a)^3.

## Answer

Let the sampling period be T and put b = aᵀ, so that aᵗ at t = kT equals bᵏ.

**Step 1.** Z[aᵗ] = Σ (b/z)ᵏ = z/(z − b).

**Step 2.** Multiplication by t: Z[t x(t)] = −T z d/dz X(z). Apply it twice.

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

Z[t² aᵗ] = −T z d/dz [ T b z/(z − b)² ] = −T² b z · [ (z − b)² − 2z(z − b) ]/(z − b)⁴

= −T² b z · [ (z − b) − 2z ]/(z − b)³ = −T² b z (−z − b)/(z − b)³

**Answer:**

Z[t² aᵗ] = T² b z (z + b)/(z − b)³, with b = aᵀ

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

For T = 1: a z (z + a)/(z − a)³.

## Check yourself

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

1. How is a^t handled at t = kT?
   A. Put b = a^T so a^t = b^k and Z[a^t] = z/(z - b)
   B. Put b = a/T so Z[a^t] = z/(z - b)
   C. Z[a^t] = z/(z - a) regardless of T
   D. Z[a^t] = T z/(z - a)
   Answer: A. At t = kT, a^(kT) = (a^T)^k = b^k, a geometric sequence with ratio b. (Key point: Sampling substitution)

2. Which rule produces t x(t) from X(z)?
   A. Z[t x(t)] = T z dX/dz
   B. Z[t x(t)] = -T z dX/dz
   C. Z[t x(t)] = -T dX/dz
   D. Z[t x(t)] = X(z/T)
   Answer: B. Differentiating X(z) = sum x(kT) z^-k brings down -kT z^-k-1, giving -Tz X'(z). (Key point: Multiplication by t rule)

3. How is Z[t^2 a^t] obtained from Z[a^t]?
   A. Apply -T z d/dz once and square the result
   B. Differentiate once and multiply by t^2
   C. Apply the operator -T z d/dz twice
   D. Integrate twice with respect to z
   Answer: C. Each factor of t needs one application of -Tz d/dz. (Key point: Repeated application)

4. What is Z[t^2 a^t] with b = a^T?
   A. T^2 b z (z - b)/(z + b)^3
   B. T b z/(z - b)^2
   C. T^2 z (z + b)/(z - b)^2
   D. T^2 b z (z + b)/(z - b)^3
   Answer: D. Differentiating T b z/(z-b)^2 and multiplying by -Tz gives T^2 b z (z+b)/(z-b)^3. (Key point: Final result)

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