2070 Magh · 3+1 marks

Energy and power of (3+4j)e^(2t)u(−t)

Short answer

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

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.

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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