---
title: "Energy and power of (3+4j)e^(2t)u(−t)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-complex-exponential"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-complex-exponential
site: BCT Engineering
---

# Energy and power of (3+4j)e^(2t)u(−t)

Signals and Systems · Chapter 1: Signals

- Asked in: 2070 Magh, 3+1 marks
- Video: [Energy and Power Signals | Solved Problems / Examples](https://www.youtube.com/watch?v=bZgQD_7d5VM) — Shrenik Jain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-complex-exponential
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Calculate the total energy and total average power of the signal given below: x(t) = (3+4j) e^(2t) u(−t). Also state whether the signal is energy signal, power signal or neither.

## Short answer

Total energy is 6.25 J, and average power is 0 W. Since energy is finite, the signal is an energy signal.

## Answer

Given $x(t) = (3+4j)e^{2t}u(-t)$, which is non-zero only for $t \le 0$ and decays to 0 as $t \to -\infty$.

**Magnitude squared:**

$$
|x(t)|^2 = |3+4j|^2 \, e^{4t}\, u(-t) = 25\, e^{4t}\, u(-t) \qquad (|3+4j| = \sqrt{9+16} = 5)
$$

**Total energy:**

$$
\begin{aligned}
E &= \int_{-\infty}^{\infty}|x(t)|^2\,dt = \int_{-\infty}^{0} 25\,e^{4t}\,dt \\
&= 25\left[\frac{e^{4t}}{4}\right]_{-\infty}^{0} = 25\left(\frac{1}{4} - 0\right) = 6.25\ \text{J}
\end{aligned}
$$

**Total average power:**

$$
P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2\,dt = \lim_{T\to\infty}\frac{1}{2T}\cdot\frac{25}{4}\left(1 - e^{-4T}\right) = 0
$$

**Answer:** $E = 6.25$ J and $P = 0$ W. Since the energy is finite and non-zero and the power is zero, $x(t)$ is an **energy signal**.

## Check yourself

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

1. What is |x(t)|^2 for x(t) = (3+4j)e^(2t)u(-t)?
   A. 7 e^(4t) u(-t)
   B. 5 e^(2t) u(-t)
   C. 25 e^(4t) u(-t)
   D. 25 e^(2t) u(-t)
   Answer: C. |3+4j| = 5, so squared it is 25, and |e^(2t)|^2 = e^(4t). (Key point: Magnitude squared)

2. What is the total energy?
   A. 25 J
   B. 1.25 J
   C. Infinite
   D. 6.25 J
   Answer: D. E = 25 ∫ from -∞ to 0 of e^(4t) dt = 25/4 = 6.25. (Key point: Energy)

3. What is the total average power?
   A. 0
   B. 6.25 W
   C. 25 W
   D. Infinite
   Answer: A. Finite E divided by 2T as T → ∞ gives 0. (Key point: Power)

4. How is the signal classified?
   A. Power signal
   B. Energy signal
   C. Neither
   D. Periodic signal
   Answer: B. E is finite and non-zero while P = 0. (Key point: Classification)

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