Antenna arrays are preferable in certain applications because of their beam steering ability, but analyzing a design for tens or hundreds of beam patterns is unwieldy. One way to condense this data into a single metric is to measure the cumulative distribution function (CDF) of effective isotropic radiated power (EIRP) for numerous beam patterns.

## Definitions

**EIRP** is a function of direction and is defined as the gain of a transmitting antenna in a direction multiplied by the power delivered to the antenna from the transmitter [1].

EIRP is equivalent to the power that must be delivered to an isotropic antenna in order to produce the same signal level in the given direction. For example, if an antenna accepts 2 mW (≈3 dBm) from the transmitter and the antenna's gain is 5 dB in a given direction, then the EIRP in that direction is 8 dBm. This means that the signal in that direction is equal to that of an isotropic antenna fed with a power of 8 dBm.

Typically, antenna gain ($G$) and EIRP ($E$) are expressed as a function of direction $G\left(\theta, \phi\right)$, $E\left(\theta, \phi\right)$ and for practical antennas this is a continuous function with minimum and maximum values

\begin{eqnarray} 0 \lt G_{min} &\lt& G_{max} \lt \infty \\ 0 \lt E_{min} &\lt& E_{max} \lt \infty \end{eqnarray}or in dB

\begin{eqnarray} -\infty \lt G_{min} &\lt& G_{max} \lt \infty \\ -\infty \lt E_{min} &\lt& E_{max} \lt \infty \end{eqnarray}Equality, where

\begin{eqnarray} G_{min} &=& G_{max} \\ E_{min} &=& E_{max} \end{eqnarray}is for an ideal isotropic antenna only.

A probability density function $f\left(E(\theta,\phi)\right)$ , over all directions $(0\leq\theta\leq\pi, 0\leq\phi\lt2\pi)$ is defined as

\begin{equation} \int^\infty_{-\infty} f(E)dE = \int_{E_{min}}^{E_{max}}f(E)dE = 1 \end{equation}and the probability over all directions of the EIRP, being between any two values $E_1$ and $E_2$ is

\begin{equation} P=\int^{E_2}_{E_1}f(E)dE \end{equation}In a typical polar 3-D gain or EIRP plot, the magnitude at each $(\theta,\phi)$ is plotted as a radius from the origin. For this plot type, the EIRP of an antenna system is contained in the closed region between a sphere centered on the origin of radius $E_{min}$ and a second equal or larger sphere centered on the origin of radius $E_{max}$.

The **CDF**, or distribution function, of a probability density function $f(x)$ is [2]

and gives the probability that $x\leq x_1$.

For the EIRP probability density function, $f(E)$, the corresponding CDF, $F_E(x)$, is the probability that EIRP is $\leq x$. When $F_E(x\lt E_{min})$ the probability is $0$, and when $F_E(x\geq E_{max})$ the probability is $1$. For $E_{min}\leq x\leq E_{max}$ $F_E(x)$ gives the fraction of all possible directions, or fraction of 4$\pi$ steradians, for which $E\leq x$. Additionally, $(1-F_E)$ gives the fraction for which $E \gt x$. In other words, the fraction of the polar plot of $E$ that is extending beyond the sphere of radius $x$.

When EIRP over the sphere is sampled at a finite number of directions, such as in a measurement or simulation, the CDF is approximated by

\begin{equation} F_E(x)\cong\frac{\text{# directions such that }E\leq x}{\text{total # directions}} \end{equation}The sampling rate over the sphere is discussed below.

#### References

- IEEE standard for definitions of terms for antennas. "IEEE Std. 145-2013", 2013.
- William Feller. "An Introduction to Probability Theory and Its Applications", vol. I. John Wiley & Sons, Inc., 1959.