Final answer:
The signal f(t) = e⁻ᵗ cos(t) is an energy signal because it has finite energy, indicated by the exponentially decaying term, and its average power tends to zero as time approaches infinity.
Step-by-step explanation:
Energy and Power Signals
To determine whether the signal f(t) = e-t cos(t), for t ≥ 0, is an energy signal or a power signal, we use the definitions of energy and power signals. An energy signal has finite energy, and its power tends to zero as time approaches infinity. Conversely, a power signal has finite power over an infinite interval.
Energy of the Signal
The energy E of the signal is given by the integral over all time of the absolute square of the signal:
E = ∫ |f(t)|2 dt
For the given signal, f(t) = e-t cos(t), this becomes:
E = ∫ (e-t cos(t))2 dt
Given the exponential term in the signal, which decays to zero as t approaches infinity, it suggests that the integral will converge, indicating finite energy.
Power of the Signal
The average power P over a period T is given by:
P = ⅔ /T ∫0T |f(t)|2 dt
Since the energy is finite and the power tends to zero over an infinite interval, the signal is an energy signal.