---
title: "Inverse transform of −3.894z/(z²+0.6065)"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:inverse-minus-3894"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/inverse-minus-3894
site: BCT Engineering
---

# Inverse transform of −3.894z/(z²+0.6065)

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2073 Bhadra, 4 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/inverse-minus-3894
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Obtain the inverse transform of the function X(z)=-3.894z/(z²+0.6065).

## Short answer

Match to damped sine pair with r=0.7788, Ω=π/2. Scaling X(z) yields x(k) = −5 (0.7788)^k sin(kπ/2) for k ≥ 0.

## Answer

The denominator has complex poles on the imaginary axis. So match $X(z)$ to the damped-sine pair:

$$
\mathcal{Z}\left[r^k\sin k\Omega\right] = \frac{rz\sin\Omega}{z^2 - 2rz\cos\Omega + r^2}
$$

#### Step 1: Find the poles

$$
z^2 + 0.6065 = 0 \;\Rightarrow\; z = \pm j\sqrt{0.6065} = \pm j\,0.7788
$$

So $r = 0.7788$ and $\Omega = \dfrac{\pi}{2}$. Note that $0.7788 = e^{-0.25}$.

#### Step 2: Match coefficients

With $\Omega = \pi/2$: $\cos\Omega = 0$ and $\sin\Omega = 1$. The pair becomes

$$
\mathcal{Z}\left[r^k\sin\frac{k\pi}{2}\right] = \frac{rz}{z^2 + r^2} = \frac{0.7788z}{z^2 + 0.6065}
$$

Rewrite $X(z)$:

$$
X(z) = -\frac{3.894}{0.7788}\cdot\frac{0.7788z}{z^2 + 0.6065} = -5.0\cdot\frac{0.7788z}{z^2+0.6065}
$$

since $3.894/0.7788 = 5.000$.

#### Step 3: Inverse transform

$$
x(k) = -5\,(0.7788)^k\sin\frac{k\pi}{2} = -5\,e^{-0.25k}\sin\frac{k\pi}{2}, \quad k \ge 0
$$

#### Step 4: Check by long division

$X(z) = \dfrac{-3.894z^{-1}}{1 + 0.6065z^{-2}} = -3.894z^{-1} + 2.3617z^{-3} - 1.4324z^{-5} + 0.8687z^{-7} - \dots$

| $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| $x(k)$ | 0 | −3.894 | 0 | 2.362 | 0 | −1.432 | 0 | 0.869 |

The formula gives the same values; for example, $k = 3$: $-5(0.4724)(-1) = 2.362$.

The poles have magnitude $0.7788 < 1$, so the oscillation decays.

**Answer:** $x(k) = -5(0.7788)^k\sin\dfrac{k\pi}{2}$, i.e. $0, -3.894, 0, 2.362, 0, -1.432, \dots$

## Check yourself

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

1. Where are the poles of -3.894z/(z^2+0.6065)?
   A. z = ±j0.7788
   B. z = ±0.7788
   C. z = ±j0.6065
   D. z = ±0.6065
   Answer: A. z^2 = -0.6065 gives z = ±j sqrt(0.6065) = ±j0.7788. (Key point: Pole locations)

2. Which pair is used, and what are r and Omega?
   A. r^k sin(k Omega) with r = 0.7788, Omega = pi/2
   B. r^k cos(k Omega) with r = 0.7788, Omega = pi
   C. r^k sin(k Omega) with r = 0.6065, Omega = pi/4
   D. a^k with a = 0.7788
   Answer: A. With cos Omega = 0 the denominator is z^2 + r^2, so Omega = pi/2. (Key point: Matching to standard pair)

3. What is x(k)?
   A. -5(0.7788)^k sin(k pi/2)
   B. 5(0.7788)^k sin(k pi/2)
   C. -3.894(0.7788)^k cos(k pi/2)
   D. -5(0.6065)^k sin(k pi/2)
   Answer: A. 3.894/0.7788 = 5, so x = -5 r^k sin(k pi/2); x(1) = -3.894. (Key point: Final sequence)

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