Question

The signal X (t) = e^-4t u (t) is _______________

(a) Power signal with P∞ = \(\frac{1}{4}\)

(b) Power signal with P∞ = 0

(c) Energy signal with E∞ = \(\frac{1}{4}\)

(d) Energy signal with E∞ = 0

This question was addressed to me during an interview.

The question is from Periodic and Non-Periodic Signals in section Signals and Systems Basics of Signals and Systems

Answer 1

Right choice is (c) Energy signal with E∞ = \(\frac{1}{4}\)

Easy explanation: If a signal has E∞ as ∞ and P∞ as a finite value, then the signal is a power signal. If a signal has E∞ as a finite value and P∞ as ∞, then the signal is an energy signal.

|x (t)| < ∞, E∞ = \(\int_{-∞}^∞ |x(t)|^2 \,dt\)

= \(\int_∞^∞ e^{-4t} u(t) \,dt \)

= \(\in_∞^∞ e^{-4t} \,dt = \frac{1}{4}\)

So, this is not a power signal but an energy signal.

\(P_∞ = lim_{T→∞} \frac{1}{2T} \int_{-T}^T |x(t)|^2 \,dt = ∞.\)