Final answer:
The signal x(t) = cos(3πt) is a power signal with finite average power and infinite total energy. Average power is the mean squared value over a period, while total energy for power signals is not a finite value because of their non-decaying, periodic nature.
Step-by-step explanation:
To determine the value of Average Power (P) and Total Energy (E) for the signal x(t) = cos(3πt), one needs to assess whether the signal is a power or energy signal. For a periodic signal like a cosine function, power can be calculated over one period of the signal. Since the cosine function is periodic, the average power can be represented as the mean squared value of the cosine function over one complete cycle. Thus, using Parseval's theorem for power signals which states that the average power of a periodic signal is the sum of the squared magnitudes of its Fourier coefficients divided by the signal's period, one would find that for x(t) = cos(3πt), only the fundamental frequency component contributes to the average power.
The signal x(t) = cos(3πt) is a power signal, as it is periodic and has a finite average power over one period. Therefore, the average power (P) of this signal would be half of the square of the amplitude (since cos²() averages to 1/2 over one period). As for total energy (E), because the signal is not time-limited and does not decay, its total energy extends to infinity; however, energy per unit time, or the power, is finite. In essence, the energy signal is characterized by finite energy over an infinite period, whereas the power signal has finite power over any interval, making x(t) = cos(3πt) a classic example of a power signal with infinite total energy.
The equation S(x, t) = cε0Ecos² (kx – wt) mentioned in the reference is related to the concept as it describes the intensity of electromagnetic waves, which is the average power per unit area. The time-averaged intensity is twice the average intensity, showing the relationship between energy and amplitude squared.