---
title: "Z-transform of k a^(k−1)"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:z-transform-k-a-k-1"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/z-transform-k-a-k-1
site: BCT Engineering
---

# Z-transform of k a^(k−1)

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2082 Kartik, 3 marks
- 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-k-a-k-1
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Find the Z-transform of x(k)=k·aᵏ⁻¹.

## Short answer

Z-transform of k·a^(k-1) using Z[a^k]=z/(z-a) and multiplication by k property. Result: Z[k a^(k-1)] = z/(z-a)² for |z|>|a|.

## Answer

Use $\mathcal{Z}[a^k] = \frac{z}{z-a}$ and the property $\mathcal{Z}[k\,x(k)] = -z\frac{d}{dz}X(z)$.

$$
\begin{aligned}
\mathcal{Z}[k a^k] &= -z\frac{d}{dz}\left(\frac{z}{z-a}\right) = -z\cdot\frac{(z-a) - z}{(z-a)^2} = \frac{az}{(z-a)^2}
\end{aligned}
$$

Dividing by $a$:

$$
\mathcal{Z}[k a^{k-1}] = \frac{z}{(z-a)^2}, \quad |z| > |a|
$$

**Alternative (differentiate with respect to $a$):**
$\sum_{k=0}^{\infty} a^k z^{-k} = \frac{z}{z-a}$; differentiating both sides with respect to $a$ gives $\sum k a^{k-1}z^{-k} = \frac{z}{(z-a)^2}$.

**Check:** $\frac{z}{(z-a)^2} = z^{-1} + 2az^{-2} + 3a^2z^{-3} + \cdots$, i.e. coefficients $k a^{k-1}$ ✔.

**Answer:** $\mathcal{Z}[k a^{k-1}] = \dfrac{z}{(z-a)^2} = \dfrac{z^{-1}}{(1-az^{-1})^2}$.

## Check yourself

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

1. Which property is used to derive Z[k a^(k-1)] from Z[a^k]?
   A. Real translation
   B. Initial value
   C. Multiplication by k: -z d/dz X(z)
   D. Convolution
   Answer: C. Multiplying by k gives -z dX/dz. (Key point: Property used)

2. What is Z[k a^(k-1)]?
   A. z(z+a)/(z-a)^3
   B. z/(z-a)^2
   C. az/(z-a)^2
   D. z/(z-a)
   Answer: B. Z[k a^k] = az/(z-a)^2, dividing by a gives z/(z-a)^2. (Key point: Result)

3. Differentiating sum a^k z^-k = z/(z-a) with respect to a gives?
   A. Z[a^(k+1)]
   B. Z[k^2 a^k]
   C. Z[k a^k]
   D. Z[k a^(k-1)] = z/(z-a)^2
   Answer: D. d/da of a^k is k a^(k-1), and d/da of z/(z-a) is z/(z-a)^2. (Key point: Alternative method)