Signal power and energy calculation

The most common signal characteristics are energy and power. In signal theory, these terms require additional comments because they are a bit different from these what we are using in AC or DC systems.

What are power and energy? When we connect R resistor to voltage U, then the resistor will dissipate some power which is equal P=U2/R. During time T the energy loss on this resistor will be E= TU2/R.

Now let us say that we add some signal s() instead of DC voltage. In this case, the power will depend on time as the signal is time dependent. The term is called instantaneous power: p(s)=s(t)2/R

to calculate energy loss during time T we need to integrate:

E = \int_{0}^{T}p(t)dt = \frac{1}{R}\int_{0}^{T}s^{2}(t)dt

Sometimes it is more convenient to evaluate average power during some time T:

P_{AV} = \frac{E}{T} = \frac{1}{RT}\int_{0}^{T}s^{2}(t)dt

When we talk about signal power, we don’t care about R load. Therm Signal power usually is used for comparing different signals. For this it is agreed to use R=1, then we exclude resistance from the formulation, and then we can talk about signal power and energy in signal theory:

E = \int_{0}^{T}s^{2}(t)dt

p(t) = s^{2}(t)

P_{AV} = \frac{1}{T}\int_{0}^{T}s^{2}(t)dt

Signal energy may be finite and infinite. For instance, a finite signal will have limited length energy while its level won’t go to infinite. Any periodical signal has endless energy. If signal energy is unlimited, then we can only talk about average power over all time axis:

P_{AV} = \lim_{T\rightarrow \infty} \frac{1}{T}\int_{-T/2}^{T/2}s^{2}(t)dt

The square root of average power gives the root mean square (RMS) of signal:

\sigma_{s} =\sqrt{P_{AV}}  = \sqrt{\lim_{T\rightarrow \infty} \frac{1}{T}\int_{-T/2}^{T/2}s^{2}(t)dt}

For signals with period T, you don’t have to calculate average power or RMS using a LIM function. This is enough to calculate average values during one period. Read more on [Electrical signal power and energy calculations by example].

3 Comments:

  1. I have a doubt.if the energy of the signal is infinite it is said yo be a power signal.not an energy signal.also if the energy is finite it is not a power signal.how can it be possible for a power signal not to an energy signal(since pwr=energy/time)

  2. Power of a signal is average power. So for a periodic input we can calculate the average power only within a period, not considering total time duration. So the average power is a finite one, like sine wave. But the energy of the signal is not finite because the signal is not approaching towards zero as time approaches towards infinity.

  3. I want an equation to calculate power based mainly on distance
    also can include the SF(Spreading factor) and frequency that the signal sent on?
    thanks in advance

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