---
title: "z-transform of e^{-at} sin wt"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-e-at-sin-wt"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-e-at-sin-wt
site: BCT Engineering
---

# z-transform of e^{-at} sin wt

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2070 Magh
- 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-e-at-sin-wt
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of e^{-at} sin wt.

## Short answer

By the damping rule, Z[e^{-at} sin wt] = z e^{-aT} sin wT / (z^2 - 2z e^{-aT} cos wT + e^{-2aT}), with region of convergence |z| > e^{-aT}.

## Answer

Take sampling period T, so x(kT) = e^(−akT) sin(wkT).

**Step 1.** Z[sin wt] = z sin wT/(z² − 2z cos wT + 1).

**Step 2.** Damping rule: Z[e^(−at) x(t)] = X(z e^(aT)). Replace z by z e^(aT):

Z[e^(−at) sin wt] = (z e^(aT)) sin wT / ( z² e^(2aT) − 2 z e^(aT) cos wT + 1 )

Multiply numerator and denominator by e^(−2aT):

**Answer:**

Z[e^(−at) sin wt] = z e^(−aT) sin wT / ( z² − 2z e^(−aT) cos wT + e^(−2aT) ),  |z| > e^(−aT)

(Direct check: write sin wt = (e^(iwt) − e^(−iwt))/(2i) and sum the two geometric series with ratio e^(−aT ± iwT)/z.)

## Check yourself

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

1. Which standard result is the starting point?
   A. Z[sin wt] = z sin wT/(z^2 - 2z cos wT + 1)
   B. Z[sin wt] = z cos wT/(z^2 - 2z sin wT + 1)
   C. Z[sin wt] = z sin wT/(z^2 + 2z cos wT + 1)
   D. Z[sin wt] = sin wT/(z - 1)
   Answer: A. This is the transform of the sampled sine, with denominator z^2 - 2z cos wT + 1. (Key point: Base transform)

2. How does multiplying by e^(-at) change the transform?
   A. Replace z by z e^(-aT)
   B. Replace z by z e^(aT)
   C. Multiply X(z) by e^(-aT)
   D. Replace z by z - a
   Answer: B. Z[e^(-at) x(t)] = X(z e^(aT)). (Key point: Damping rule)

3. After simplifying, Z[e^(-at) sin wt] is
   A. z e^(aT) sin wT/(z^2 - 2z e^(aT) cos wT + e^(2aT))
   B. z sin wT/(z^2 - 2z e^(-aT) cos wT + 1)
   C. z e^(-aT) sin wT/(z^2 - 2z e^(-aT) cos wT + e^(-2aT))
   D. z e^(-aT) cos wT/(z^2 - 2z e^(-aT) sin wT + e^(-2aT))
   Answer: C. Multiplying top and bottom by e^(-2aT) gives this form. (Key point: Final result)

4. What is the ROC?
   A. |z| > 1
   B. |z| < e^(-aT)
   C. |z| > e^(aT)
   D. |z| > e^(-aT)
   Answer: D. The poles are at e^(-aT) e^(+-jwT), magnitude e^(-aT); the causal ROC lies outside. (Key point: ROC)

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