Using CDF to Assess 5G Antenna
Directionality
Scott Langdon

Remcom Inc., State College, Pa.

Three of the goals for 5G mobilecommunications networks are toincrease data capacity, decrease latency and connect many more devices. mmWave frequencies with large channel bandwidths are being used to help meetthese needs. Indeed, release 15 of the 3GPP
5G specification includes frequencies in the28 and 38 GHz bands. Drawbacks of thesehigher frequencies are increased path loss inclear air, in air with precipitation and in environments with higher reflectivity—especiallywith larger cells outside of dense urban environments.1
Fortunately, the shorter mmWave wavelengths enable more directional antennas inboth the base station and user equipment(UE) than is practical at lower frequencies.
Antenna arrays for which the radiation pattern can be steered are particularly interesting because higher gain in the desired
direction makes up for some of the added
path loss, and the narrower beamwidth canreduce same-cell interference. At mmWave,
arrays become more practical, even for amobile phone.2
A significant metric for the performanceof a mobile phone antenna is the gain in
the direction of the base station. Since theorientation of a phone and direction toward
the tower can vary greatly, the phone should
to be able to point its maximum gain in any
direction. Hence, characterizing the ability ofan antenna system to accomplish this is an
important metric; one way to do this is predicting or measuring the effective or equivalent isotropic radiated power (EIRP) over all
possible directions.
EIRP AND CDF
EIRP, which is a function of direction, isthe gain of a transmitting antenna in that
direction multiplied by the power delivered
to the antenna from the transmitter.3 EIRP
can be thought of as the equivalent power
required to be delivered to an isotropic antenna to produce the same signal level. For
example, if an antenna is driven by 2 mW (3dBm) from the transmitter and the antenna
has 5 dB gain in a given direction, the EIRPin that direction is 8 dBm; the signal in that
direction would be the same as if the antenna were isotropic and driven with an 8 dBm
signal.
Antenna gain, G, and EIRP, E, are usually expressed as a function of direction, i.e.,G(θ,ϕ) and E(θ,ϕ). For practical antennas,gain and EIRP are typically continuous func-

Reprinted with permission of MICROWAVE JOURNAL® from the August 2019 issue.
©2019 Horizon House Publications, Inc.

TechnicalFeaturetions with minimum and maximum
values:0 < Gmin < Gmax < ∞ and0 < Emin < Emax < ∞(1)

In dB,−∞ < Gmin < Gmax < ∞ and−∞ < Emin < Emax < ∞(2)For an ideal isotropic antenna,Gmin = Gmax and Emin = Emax.
We can define a probability density function, f (E(θ,ϕ)), over all directions (0 ≤ θ ≤ π, 0 ≤ ϕ < 2π) asEmax∞∫ f (E)dE = ∫ f (E)dE = 1−∞(3) Fig. 1 Realized gain CDF at the 3

λ/2 resonance of a 76 mm printed circuitdipole.

 Fig. 3 Realized gain of the 8 × 8patch, fed for maximum gain normal tothe plane of the patches.

Emin

The probability over all directionsof the EIRP being between any twovalues E1 and E2 (inclusive) isP =∫E2

E1f (E)dE(4)In a typical polar 3D gain or EIRPplot, the magnitude at each θ and ϕis plotted as a radius from the origin.
For this type of plot, the EIRP of anantenna system will be contained in
the closed region between a sphereof radius Emin centered on the origin
and an equal or larger sphere centered on the origin of radius Emax.
A cumulative distribution function (CDF) or, simply, the distribution function of the probability density function f (x) isx1

F ( x1 ) = ∫ f ( x ) dx

(5)

−∞

and gives the probability that x ≤x1.4
For the EIRP probability density
function f (E), the corresponding
CDF, FE(x), gives the probability thatthe EIRP will be ≤ x:
FE ( x < Emin ) = 0 andFE ( x ≥ Emax ) = 1(6)For Emin ≤ x ≤ Emax, FE(x) gives
the fraction of all possible directions (i.e., fraction of 4π steradians)
for which E ≤ x and (1 − FE) gives
the fraction for which E > x, i.e., thefraction of a polar plot of E which is“poking out” of a sphere of radius
x. For example, Figure 1 shows asphere representing the realized Fig. 2 8 × 8 patch antenna array on a52.5 mm x 52.5 mm substrate.

gain of the 3λ/2 resonance of a 76
mm printed circuit dipole, showing
a magnitude of 2 dBi. Approximately 10 percent of the directions havegain more than 2 dBi; in this case,F(2 dBi) ≅ 0.9, so about 90 percentof the gain pattern is contained
within the sphere.
When the EIRP over the sphere issampled at a finite number of directions, such as in a measurement or
simulation, the CDF can be approximated by
FE ( x ) ≅#Directions with E ≤ x
Total # of Directions(7)

PATCH ARRAY
To illustrate, the CDF of the EIRP
will be calculated for a 64 element
patch array antenna at 28 GHz.5
All simulations and processing are
performed using XFdtd®.6 The geometry is an 8×8 element patch
array on a 52.5 mm × 52.5 mm ×0.254 mm substrate with a groundplane (see Figure 2). The electricalparameters of the substrate are εr =

 Fig. 4 Realized gain of the 8 × 8patch, fed for maximum gain at θ = 37
degrees and φ = 90 degrees.

2.2 and loss tangent = 0.0009. This
example is restricted to the antenna
array structure to illustrate the method. In practice, the antenna would
be evaluated by itself in the initialstages of the design, then simulated
in a more complex environment, including the complete mobile phonegeometry and, possibly, other arrays added for better diversity ofcoverage and polarization.
The realized gain patterns for
maximum signal in two directions
are shown in Figures 3 and 4. Figure
3 shows the radiation pattern of thearray with all elements fed in phase,so the principal lobe is perpendicular to the plane of the antenna, i.e.,θ = 90 degrees. The maximum gainis 24.2 dBi. Figure 4 shows the radiation pattern of the array with allelements fed with a phase taper
across the 2D array, so the principle
lobe is along the direction θ = 37
degrees and θ = 90 degrees. Themaximum realized gain in this caseis 23 dBi. A single element of thisTechnicalFeature
can be used to estimate the ability of multiple arrays or subarrays
to work in combination to provide
higher EIRP in all directions and reduce blind spots. This measure canalso be evaluated using multiple antennas for diversity and coverage inall polarizations.

1.00.9Fractional Area0.82×21×8
4×4
8×8

0.7
0.6
0.5
0.4
0.3
0.2
0.1
005

1015

202530EIRP (dBmW)

354045

 Fig. 5 EIRP CDFs for the full 8 × 8 element patch array and several subarrays.
array will have reduced gain toward
the back (i.e., the ground planeside), so the array will not have gain
as high in that hemisphere. Also, integrating the array into a phone will
yield additional effects on the radiation pattern.
Determining how well the array
performs in every direction is useful.
What is the distribution of EIRP overthe sphere? Since this is a simulation, it is straightforward to characterize the maximum gain in many
directions. However, to capture themany variations we expect on thegain pattern due to the geometry,the number of sample directions
needs to be large. The CDF of EIRP
provides a useful, one-dimensional
function to characterize the array’s
performance over all possible directions.
The far zone radiation due toeach element in the array is computed for the full geometry. These
patterns can be combined in postprocessing to compute the gainand EIRP for any combination ofelements, enabling the power andphase delivered to each element tobe set independently. To computethe CDF, the phase of each elementin the array is adjusted to provide
maximum EIRP in each of a suitably
large number of sample directions
representing the sphere of all possible directions. The CDF of EIRP
function can then be computedfrom Equation 7.

A CDF of the EIRP for the full 8×8
element array and several subarraycombinations is shown in Figure 5.
For all cases, the transmitter is assumed to have a power of 23 dBm,
which is typical for a mobile phone.
For the 8×8 element case, the EIRP
is about 37 dBm at a fractional areaof 0.5, which means that half of thedirections have an EIRP larger than
37 dBm. For the 4×4 subarray, elements from one quarter of the arrayare used to determine the CDF ofEIRP. This CDF curve has a similar
shape but shifted down in EIRP, asexpected for an array with one quarter as many elements. Elements fromone corner of the array are used in
the CDF for the 2×2 subarray, andthe CDF has a similar drop in EIRP
compared to the 4×4 case. Finally, a
1×8 arrangement of elements fromone side of the array shows the EIRP
lies between the 2×2 and 4×4 element subarrays, again as expected.
This type of array might be used ina mobile phone, with arrays on twosides to provide more complete
coverage.
As seen from these curves, about
half of the possible directions havesignificantly lower EIRP due to theground plane, which emulates theplacement of the array in a phone.
One way to provide broader coverage is to place multiple antenna arrays in the device, such as on eitheredge of the mobile phone, as mentioned. By properly combining thedistributions, the CDF of the EIRPCONCLUSION
Steerable array antennas are ofsignificant interest to help meet thegoals of 5G mobile communications. At mmWave frequencies, suchas 28 and 38 GHz, fairly large anddirectional arrays become practical
for relatively small devices such as
mobile phones; however, these frequencies have higher path loss thanat microwave frequencies whichhave been used in previous generations. For a given power level, theability of antenna arrays to control
the direction of maximum radiation
will allow for much better EIRP in
the direction of communications.
The CDF of EIRP, computed froma suitably large number of sample
directions, may be used to assessthe directionality and effective coverage of an antenna array. The article used a simple example of an
8×8 patch antenna array to demonstrate the usefulness of the CDFof EIRP to characterize the ability ofan array to provide good EIRP in all
directions.■
References

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Zhao, Y. Azar, K. Wang, G. N. Wong, J.
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