---
title: "Z-transform of sin ωt"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:z-transform-sin-wt"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/z-transform-sin-wt
site: BCT Engineering
---

# Z-transform of sin ωt

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2077 Chaitra, 4 marks; 2075 Bhadra, 4 marks; 2075 Baisakh, 4 marks (asked 3 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-sin-wt
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Obtain the z-transform of x(t)={sin ωt, t≥ 0; 0, t<0}

## Short answer

The z-transform of sin(ωt) sampled with period T is derived using Euler's formula and geometric series. The result is X(z) = z sin(ωT) / (z² - 2z cos(ωT) + 1) for |z| > 1. Poles at e^(±jωT) on unit circle.

## Answer

Sampling with period $T$: $x(kT) = \sin\omega kT$ for $k \ge 0$. Use Euler's formula:

$$
\sin\omega kT = \frac{e^{j\omega kT} - e^{-j\omega kT}}{2j}
$$

#### Derivation

Since $\mathcal{Z}[a^k] = \sum_{k=0}^{\infty} a^k z^{-k} = \frac{1}{1-az^{-1}}$ for $|z| > |a|$:

$$
\begin{aligned}
X(z) &= \frac{1}{2j}\left[\frac{1}{1-e^{j\omega T}z^{-1}} - \frac{1}{1-e^{-j\omega T}z^{-1}}\right] \\
&= \frac{1}{2j}\cdot\frac{(e^{j\omega T} - e^{-j\omega T})z^{-1}}{1 - (e^{j\omega T}+e^{-j\omega T})z^{-1} + z^{-2}} \\
&= \frac{z^{-1}\sin\omega T}{1 - 2z^{-1}\cos\omega T + z^{-2}}
\end{aligned}
$$

using $e^{j\omega T} - e^{-j\omega T} = 2j\sin\omega T$ and $e^{j\omega T} + e^{-j\omega T} = 2\cos\omega T$.

Multiplying numerator and denominator by $z^2$:

$$
X(z) = \frac{z\sin\omega T}{z^2 - 2z\cos\omega T + 1}, \quad |z| > 1
$$

**Answer:** $\mathcal{Z}[\sin\omega t] = \dfrac{z\sin\omega T}{z^2 - 2z\cos\omega T + 1}$. The poles $z = e^{\pm j\omega T}$ lie on the unit circle, as expected for an undamped sinusoid.

## Check yourself

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

1. sin(wkT) is written as:
   A. (e^(jwkT) - e^(-jwkT))/(2j)
   B. (e^(jwkT) + e^(-jwkT))/2
   C. (e^(jwkT) - e^(-jwkT))/2
   D. e^(jwkT) x 2j
   Answer: A. Euler's formula. (Key point: Euler)

2. Z[a^k] = :
   A. 1/(1 + a z^-1)
   B. 1/(1 - a z^-1), for |z| > |a|
   C. z/(z+a)
   D. a/(z-1)
   Answer: B. Geometric series. (Key point: Basic transform)

3. Z[sin wt] =
   A. z cos(wT)/(z^2 - 2z cos(wT) + 1)
   B. sin(wT)/(z^2 + 1)
   C. z sin(wT)/(z^2 - 2z cos(wT) + 1)
   D. z sin(wT)/(z^2 + 2z cos(wT) + 1)
   Answer: C. Standard result. (Key point: Result)

4. Poles lie at:
   A. z = 0
   B. Inside the unit circle
   C. z = +-1
   D. z = e^(+-jwT) on the unit circle
   Answer: D. Undamped sinusoid. (Key point: Poles)

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