Question

Compute the energy and power of the given signals and state whether they are energy or power signals.

Answer 1

To determine if a signal is an energy or power signal, we can calculate its energy and power. The energy is calculated by integrating the squared absolute value of the signal over time, and the power is the energy divided by the time duration of the signal. If the power is zero, the signal is an energy signal.

Step-by-step explanation:

In order to determine if a signal is an energy or power signal, we need to calculate its energy and power.

The energy of a signal can be calculated by integrating the squared absolute value of the signal over time. This gives us the total energy contained in the signal.

The power of a signal can be calculated by dividing the energy by the total time duration of the signal. This gives us the average power of the signal. If the power of a signal is finite and non-zero, it is a power signal. If the power is zero, the signal is an energy signal.

For example, if we have a signal x(t) = A sin(ωt), where A is the amplitude and ω is the angular frequency, the energy can be calculated as:

E = ∫[A sin(ωt)]^2 dt = 1/2 A^2 T

where T is the duration of the signal. The power can then be calculated as:

P = E/T = 1/2 A^2.

Answer 2

To determine if a signal is an energy or power signal, one calculates the total energy or power over time. Energy signals tend to decay over time while power signals do not. The energy of electromagnetic waves is amplitude-dependent and for photons, it is frequency-dependent.

Step-by-step explanation:

To calculate the energy and power of signals and to determine whether they are energy or power signals, certain principles and formulas are used. The total energy (E) of a signal can be calculated by integrating the square of the signal over time (t). If the signal is not periodic, or if it decays to zero as time approaches infinity, it can be classified as an energy signal. However, periodic signals that do not decay over time are typically referred to as power signals, and their power (P) is defined as the average energy per time period.

To compute power, one can calculate changes in energy over time. Specifically, power is the rate at which work is done or energy is transferred. Calculating this involves examining power consumption and calculations of the cost of energy consumed. By analyzing the energy consumption over a set period, it is possible to estimate the operation cost for an electrical device.

For electromagnetic waves, the Poynting vector represents the power per unit area, and the energy intensity can be calculated by taking the average of this vector over a period. The energy of an electromagnetic wave is related to its amplitude; higher amplitude waves carry more energy. However, for photons, the individual packets of light, their energy is instead proportional to their frequency (E = hν, where h is Planck's constant and ν is the frequency), independent of amplitude.

The wave functions that correspond to these energies are known as stationary states or states of definite energy, and are a fundamental concept in quantum mechanics.