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

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

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2069 Poush
- Video: [Z-Transform Of Cosinusoidal Signal | Z-Transform | Signals and Systems](https://www.youtube.com/watch?v=3SrkDIZxGs4) — Ekeeda
- 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-cos-wt
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of e^{at} cos wt for t ≥ 0.

## Short answer

Sample at t=kT: x(k)=e^{akT} cos(wkT). Apply Euler's formula and Z[b^k]=z/(z-b) to get X(z)=z(z - e^{aT} cos wT)/(z^2 - 2z e^{aT} cos wT + e^{2aT}), |z| > e^{aT}.

## Answer

**Setup.** Sample at t = kT. Then x(k) = e^{akT} cos(wkT), k ≥ 0.

**Method.** Write cos(wkT) = (e^{jwkT} + e^{−jwkT})/2 and use Z[bᵏ] = z/(z − b):

```text
Z[e^{akT} cos wkT]
 = ½ [ z/(z − e^{(a+jw)T}) + z/(z − e^{(a−jw)T}) ]
 = ½ · z [ 2z − e^{aT}(e^{jwT} + e^{−jwT}) ]
       / [ z² − z e^{aT}(e^{jwT} + e^{−jwT}) + e^{2aT} ]
```

Since e^{jwT} + e^{−jwT} = 2 cos wT:

**Answer:** X(z) = z (z − e^{aT} cos wT) / (z² − 2z e^{aT} cos wT + e^{2aT}), for |z| > e^{aT}.

With T = 1 this reads z(z − eᵃ cos w)/(z² − 2z eᵃ cos w + e²ᵃ). (Equivalent to the shifting rule: replace z by z e^{−aT} in Z[cos wkT].)

## Check yourself

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

1. With t = kT, e^(at) cos wt becomes
   A. e^(akT) cos(wkT)
   B. e^(ak) cos(wk)
   C. e^(aT) cos(wkT)
   D. e^(-akT) cos(wkT)
   Answer: A. Substitute t = kT. (Key point: Sampling)

2. The poles used are
   A. e^(-(a+jw)T) and e^(-(a-jw)T)
   B. e^((a+jw)T) and e^((a-jw)T)
   C. e^(aT) only
   D. e^(jwT) and e^(-jwT) only
   Answer: B. Writing cos as a sum of exponentials multiplies each by e^(akT). (Key point: Exponential form)

3. X(z) =
   A. z(z - cos wT)/(z^2 - 2z cos wT + 1)
   B. z e^(aT) sin wT/(z^2 - 2z e^(aT) cos wT + e^(2aT))
   C. z(z - e^(aT) cos wT)/(z^2 - 2z e^(aT) cos wT + e^(2aT))
   D. z(z + e^(aT) cos wT)/(z^2 + 2z e^(aT) cos wT + e^(2aT))
   Answer: C. Uses e^(jwT)+e^(-jwT) = 2 cos wT. (Key point: Result)

4. The ROC is
   A. |z| > e^(-aT)
   B. |z| < e^(aT)
   C. |z| > 1
   D. |z| > e^(aT)
   Answer: D. Poles have magnitude e^(aT). (Key point: ROC)

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## Related in other subjects

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