Signal Power and Energy (and Power Spectral Density)

The idea of "signal power" can be a bit abstract in the field of signal analysis. Why does it even make sense to coin the term "power" when signals can represent any physical measurement? If my signal represents temperature, what does it mean for this signal to have "power"? The answer to this question influences understanding a signal's power and its power spectral density (PSD). The below article is intended to walk through these questions and hopefully provide answers and insights.

Signal Power and Energy ≠ Physics Power and Energy

Before hopping into math, it's important to note that the definitions of signal power and energy are simpliy definitions and are not the same thing as the physics concepts of power and energy. Now they could be the same if the signals themselves represented the necessary physical quantities (e.g. current or voltage), but this is not necessarily always the case. Second, the definitions of signal power and energy are influenced by physical concepts so it will be easy to relate them to physical concepts. But remember - they are different things.

Signal Energy

The energy of a signal $x(t)$ is defined as the integral of the signal magnitude squared over time:

$$ E_s = \int_{-\infty}^{\infty} |x(t)|^2 \, dt $$

There are several observations we can make here:

The units of energy are {units of signal}². The fluidity in units should help distinguish itself from the definition of physical energy which has strict units of Joules
The time duration in which the signal is captured influences the amount of energy. If you were to capture the same signal for both a short duration and a long duration, the long duration capture would result in more energy (even though its the same signal!)
In addition to the time duration in which the signal was captured, the magnitude of the signal itself will influence the total energy.

Practically speaking, we can interpret energy as a useful metric to compare two similar signals to one another: "Does signal A have more energy than signal B?". The world "similar" suggests that we only want to compare signals that feature the same units. It would not make sense to ask "Does the pressure signal have more energy than the temperature signal?".

Another question that you might wonder is why do we square the magnitude of the signal? Why don't we just take the integrate the magnitude? Why is the square term necessary? Good question - we'll get to that eventually.

Signal Power

Signal power is defined as the rate of energy per unit time (note that this definition is exactly how you would think of power from a physics viewpoint). Per the previous definition of energy, power is calculated as below:

$$ P_s = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^{T} |x(t)|^2 \, dt $$

This definition of power should also be thought of "average power" since it consolidates all power information over all time into a single number. The result of this calculation would not allow you state that the signal posessed more power at one point in time versus another point. Further, this calculation integrates over all time would suggest that for any finite lengthed signal (e.g. pulse signal), the power will actually be zero.

Energy Signals and Power Signals

An energy signal is defined as any signal whose energy is finite over all time (i.e. $0 < E < \infty$). Intuitively this can be thought of as a transient or pulse-like signal that exists for a finite duration. The consequence of a signal with finite energy is that the average power is 0 since we're dividing a finite amount of energy by an infinite duration of time.

Alternatively, if a signal has finite power over all of time (i.e. $0 < P < \infty$) it is said to be a power signal. Note that finite power still results in infinite energy. Thus it is not possible for a signal to be both an energy signal and power signal. An example of a power signal would be a sinusoidal signal. This signal can easily be seen to have infinite energy (because it goes on forever) but a finite amount of energy per unit time (i.e. finite power).

Signals can also exist that are neither energy or power signals. Imagine a sinusoidal signal whose amplitude increases with time. The rate of energy per unit time of the signal continues to grow resulting in infinite power.

Why So Much Emphasis On Power?

Before continuing on, let's reflect on why there seems to be so much emphasis on signal power (versus signal energy).

First, the average signal power is a characterisic that is normalized against time. Unlike signal energy where the result is a dependent on the signal window size, the average signal power could be estimated by sampling the underly process for different time durations.

Second, the signal power allows us to characterize an infinite time signal into a finite quantity. This is incredibly useful since many signals are power signals and cannot be simply characterized by their energy (per the first point).

Parseval's Theorem

Parseval's Theorem states that the integral of signal sqaured and its transform sqaured are equal to one another. The Fourier transform of a signal satisfies this yielding:

$$ \int_{\infty}^{\infty} |X(f)|^2 \, df = \int_{\infty}^{\infty} |x(t)|^2 \, dt $$

This is actually very interesting since it shoud look familiar to the defintion of signal energy. One could even imagine that the signal energy defintition with influenced by Parseval's Theorem. Likewise, the power of a signal should satisfy Parseval's theorem.