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National Institute of Standards and Tectinology

Technology Administration, U.S. Department of Commerce

NIST Technical Note 1538

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NIST Technical Note 1538

Optimized Arbitrary Wireless Device Arrays

for Emergency Response Communications

William F. Young

Sandla National Laboratory

Networked Systems Survivability and Assurance Department

Albuquerque, MM 87185

Edward F. Kuester

Department of Electrical and Computing Engineering

University of Colorado

Boulder, CO 80309

Christopher L. Holloway

Electromagnetics Division

National Institute of Standards and Technology

Boulder, CO 80305

March 2005

^'^^reso*''

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

Technology Administration

Ptiillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Hratch G. Semerjian, Acting Director
Certain commercial entitles, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor Is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1538

Natl. Inst. Stand. Technol. Tech. Note 1538, 49 pages (March 2005)

CODEN: NTNOEF

U.S. Government Printing Office Washington: 2005

For sale by the Superintendent of Documents, U.S. Government Printing Office

Internet bookstore: gpo.gov Phone: 202-512-1800 Fax: 202-512-2250

Mail: Stop SSOP, Washington, DC 20402-0001
Contents

Executive Summary iv

1 Introduction 2

2 Basic Theory 3

3 Simulation Setup 9

4 Simulation Results 14

4.1 Average Gain with Elements in a Plane and Interior Observation Points 14

4.2 Effect of Current Magnitude on Optimized Gain 18

4.3 Effect of Current Phase on Optimized Gain 21

4.4 Average Gain with Elements in a Plane and Exterior Observation Points 23

4.5 Effect of Elements Located in a Volume versus a Plane 26

5 Conclusions and Future Work 37

6 References 38

Appendix A: Examination of Statistics Gathering Method 40

Executive Summary

Wireless communications are essential to emergency responders. Unfortunately, a

typical emergency response scenario involves communication into and within building

structures, which can severely interfere with or completely block the radio-frequency

channel. Several methods offer some potential relief to these problems, including the use

of multi-hop ad hoc networks for communication within the building, and the use of

smart antenna arrays at exterior locations to improve the communication between the

interior and exterior locations of the building. However, due to the enclosed nature of

buildings, and the loss in radio-frequency energy as a signal propagates through floors

and walls, no single solution completely solves the communication problem.

In this report, we consider another approach to improving the communication

capabilities of emergency responders. We investigate the possibility of creating antenna

arrays from wireless devices such as the emergency responders' radios, wireless sensor

networks, or arbitrarily placed wireless relay nodes. An antenna array allows the focusing

or directing of radiated electromagnetic energy in a desired direction. By intelligently

controlling the direction and energy level of the electromagnetic radiation, we can

increase the probability of establishing a communication link and improve the quality of

that link. The result improves the emergency responder's communication capabilities.

Our approach is based on antenna array theory. However, unlike the typical

assumption that the observation point or responder is located in the far field of the array, we assume the responder is located within or near the array volume. The array volume is

essentially the building. We also assume arbitrarily located wireless devices, which are in

effect the antenna array elements. This initial investigation shows gain is possible from

arbitrarily located wireless nodes, at frequencies between 1 00 MHz to 5 GHz. The results

also indicate gain is possible whether the emergency responder is moving through the

interior of the building, or moving around the perimeter of the building. This means the

array concept could provide benefit to emergency responders within the building, and to
emergency responders communicating with people on the exterior of the building. We

also point out our approach is not based on ad hoc networking or protocols. Although an

actual system will make use of such protocols, we are focusing on the electromagnetic

behavior of the system in aggregate.

The preliminary results discussed here are encouraging. We also continue to improve

our model of the real world environment by refining the approximation used for the

building materials, such as the floor of the building. In addition, in an actual

implementation, emergency responders would likely quickly place wireless devices as

they enter the building, or perhaps even robots could place the devices in the near future. We are investigating possible benefits from any inherent structure or pattern that arises

from the placement of the wireless devices, or from probable locations within the

building.

VI
Optimized Arbitrary Wireless Device Arrays

for Emergency Response Communications

William F. Young' Edward F. Kuester^ Christopher L. Holloway^

National Institute of Standards and Technology

325 Broadway, Boulder CO 80305

In typical emergency response situations wireless communications are essential to the

emergency responders. However, a typical scenario requires communicating into and

within building structures that may severely interfere with or completely block the radiofrequency channel. In this report, we are investigating the potential of utilizing arbitrarily

located wireless devices to increase the probability of maintaining a communication link

between two points and thus improve the emergency responder's communication

capabilities. The proposed approach employs methods based in antenna array and general

optimization theory. A combination of analytic and simulation results allows rapid and

efficient analysis of a variety of array or system configurations. In order to investigate

general trends, we approximate the floor of the building as a perfect electrical conductor

and the wireless devices as Hertzian dipole elements. Observation points are located

throughout the notional building volume, as well as on a perimeter zone. An important

aspect of our analytic solution considers the observation points as being in the far field of

the individual radiating elements, with the gain normalized by the system input power.

The initial analytic and simulation results lead to the following general conclusions. First, an appropriately controlled system of wireless devices can increase the communication

capability within a building by optimizing the directivity of electromagnetic radiation.

Second, the emergency responder can move throughout the building and surrounding

area, and receive similar benefit from the array of elements. Third, the optimized current phase information demonstrates a greater affect on the amount of achievable gain when

compared to the optimized current magnitude information.

Keywords: Emergency responder; random antenna array; optimization; wireless devices

' Sandia National Laboratories, PO Box 5800, MS 0785, Albuquerque, NM, 87185,

email: wfyoung@sandia.gov

^ University of Colorado, Boulder, CO

' Electromagnetics Division, NIST, MC 818.02, email: holloway@boulder.nist.gov
1. Introduction

Wireless communication represents a key supporting technology to the success of an emergency responder. Unfortunately, a typical emergency response scenario involves

communication into building structures, which can severely interfere with or completely

block the radio-frequency channel. One potential method of improving the radio

frequency channel within a building utilizes the intelligent confrol of the elecfromagnetic

radiation from wireless devices quickly placed in the building during entry by the

emergency responder. These devices would perform as antenna array elements to

improve communication capability for emergency responders both within the structure

and with external persons.

We here investigate the potential for creating directive arrays out of arbitrarily located

radio fransceivers. These fransceivers may take a variety of forms such as handheld

radios, wireless relay devices, or sensor nodes. The goal is to direct the electromagnetic

energy for particular radio-frequency channels within a large volume area, (e.g., a large

building), so as to improve both the coverage and/or data throughput of the wireless

communication channel between two points. This research is intended to directly improve

the design of wireless communication systems used in current and ftiture emergency

response scenarios.

Our approach applies optimization techniques and antenna array theory to arbifrarily

located radiating elements. Optimization techniques for arrays of discrete radiators have

a rich history, with References [1] through [3] detailing optimization techniques for

scalar radiation from discrete radiators and chapter 10 in Reference [4] providing a

discussion particularly relevant to this work. In addition, Reference [5] provides a highly

mathematical approach to the problem of optimizing electromagnetic radiation in a

particular direction. References [6] through [9] consider probabilistic approaches to

antenna array design, while References [10] and [11] investigate the effects due to random variation from initial optimized parameters such as element location and current
excitation. However, to the best of our knowledge, the appHcation of array theory

concepts to arbitrarily located wireless transceivers represents a new area of

investigation. In this preliminary report, we examine the feasibility of having the

observation point at a variety of locations within or near the array. The observation point

is analogous to an emergency responder trying to receive a transmission within a building

that also contains numerous wireless devices, which constitute the array elements.

Observation points on the perimeter of the facility also provide an important investigation

emphasis since communication will occur not only between emergency responders within

the building, but with persons outside of the building as well.

Our proposed topology of wireless devices resembles an ad hoc network, but our

focus is not on ad hoc networks or protocols. We are studying the possibility of

controlling the electromagnetic radiation of the wireless devices in aggregate so as to

form an antenna array. This differs from the typical power control of a single device,

which is used simply either to restrict or expand its radius of coverage, or to increase the

battery life by limiting usage. A complete system using our approach will require various

supporting protocols; however, our goal is to determine the necessary electromagnetic

behavior of the system.

The remainder of the report proceeds as follows. In Section 2, we outline the basic

underlying theory behind our approach; in Section 3, we describe the general simulation

setup and scenarios; in Section 4, we expand on further specifics for particular

simulations and draw important corresponding results; and finally, in Section 5, we

present some overall conclusions and ideas for future work.

2. Basic Theory

Basic concepts from antenna array theory serve as the starting point for the analytical

basis of optimizing the electromagnetic radiation from arbitrarily located transceivers or

elements. Optimizing the ratio of two characteristics associated with the elements is a
common approach used to determine the appropriate excitation and/or physical

orientation of the elements. This work seeks to maximize the electromagnetic energy at a

given observation point by determining the required current (magnitude and phase), on

each element. In this work, we assume the locations of the arbitrarily located elements are known. Figure 1 provides an example of the physical configuration considered in the

optimization process.

Figure 1 . Example configuration with four z-directed dipoles over a ground plane.

The example configuration in figure 1 consists of four z-directed dipole elements over

a perfect electric conductor (PEC) ground plane. In this analysis, the ground plane is

assumed infinite in extent in both the x and y directions. The observation point lies in the

vicinity of the elements ("vicinity" is defined as lying within the minimal volume

containing the elements, or in the perimeter region surrounding the array). Two

characteristics of the configuration are used in the optimization process: (1) the total power provided to the system, and (2) the z-directed radiation intensity at the observation

point over the ground plane. The optimization involves setting up the ratio of these two
quantities, and then finding the complex current excitations that produce the greatest

ratio.

Several typical measures of optimization apply to antenna arrays, (e.g., see Reference

[12]). However, we want to optimize the electromagnetic radiation intensity for a

particular direction, i.e., the z-direction, at a specific observation point within or near the

boundaries of the array volume. The specific ratio necessary to optimize takes the form

given below:

4^

4;r • (z - component of radiation intensity) 2 rj Psystem input power ~ -^-R^^E{r)-T{r)dv

where R^^ is the average distance between the elements and the observation point, E. is

the z-component of the electric field, //is the intrinsic impedance of free space, E is the

electric field vector, J is the current vector, r is the distance vector from the origin, and

dv is the volume differential. Distance i?^^ is used because the observation point is either

near or within the volume containing the array of elements, which suggests estimating the

distance to the observation point by means other than the distance from the origin or

center of the array to the observation point. Note that eq (1) is not exactly equivalent to

the classical definition of antenna directivity, as given in eqs (2-16) of Reference [13],

since that definition uses an R value corresponding to the far field of the array and

assumes all the array elements are located at the origin. In our case, the distance between

the observation points and the elements is not in the far field of the array, hence

using /?^^ provides a better mathematical representation of the overall behavior than

simply using some nominal R value, such as the distance to the farthest observation point.

Our goal is to maximize the value oip, which represents the gain in a specific direction.

The array of elements constitutes a discrete system, and both the numerator and

denominator take the form of matrices. In addition, the numerator matrix is Hermitian,
and the denominator is Hermitian positive definite. This allows the appHcation of classic

optimization theory to determine the maximum value ofp and the corresponding current

excitations on the elements. Derivation of the Green's function, G% for a z-directed

Hertzian dipole element over a PEC allows writing the volume integral in closed form. In

particular, the electric field is written as

where

G-'iF,FJ=

' + ; _ , (3)

'n, = ^lix-x„f+{y-y„f +{z-z„f , (m* dipole source location) (4)

m = ^|ix -x„y +{y- y^ f +{z + z„ f , (image of the m* dipole source) (5)

and

r = V777^, (6)

where F is the distance from the origin to the observation point. Other variables in eq (2)

are k, which is the wavenumber, j = V^ , and a, , which is the unit vector in the zdirection. The m' Hertzian dipole current source, J„, generating this electric field is

given as

Jm = -a, —: , (/)

where co is the radian frequency, //q it the permeability of free space, and d{) is the

impulse or delta function. By the sifting property of the delta fiinction in the current

source, and the dot product fi*om the vector multiplication, the volume integral in eq (1)

takes the form
-\Rc<^E-rdv^UJ-^f^f^G'ir^,FJ + ^^'iF^,r„)

COjU m=l n=\ k^ dz (8)

Equation (8) is a square jVxA'^ matrix, where A'^ is the number of source elements, i.e., four

source elements means A'^ = 4. Let B represent this denominator matrix. Now letting

R^ =|F^ -F„|and7?j =|^m ~'^n| ' ^^^ taking the real part, the terms in matrix eq (8) may be

written as

k sin(^/?J , sin(^i?,) , 1 3^

%7tco/iQ kR^ kR, k'dz„

sm{kRg) s\n{kR^)

kR„ kR,

(9)

Almost all the terms in eq (9) can be written in closed form, (except for the self terms

ofi?^ ); hence, the closed form expression for the matrix terms in B is as follows:

B^ =

v2/ 2-

^ (i+iz„-z„,rk') ^ 3{z„-z„,r

2d2

1 3(z„-zJ^

2d4 k'R

kR„ kRi

v2;,2-

^ (l + (z„+zJ-^0 ,3(z„+zJ2 d2

1 3(z„+zJkR, kRl

2 n4

yt'i?

(10)

Self terms fori?„ =\r^ -^„| in eq (9), (when m = n), are dealt with as a limiting case since

direct calculation leads to indeterminate forms. Using a series expansion for the '—

terms, the self terms dependent on R^ become

8;r&jWo 6 J ^ttoj/Jq 6
To determine the closed form for the numerator of eq (1), we take the z-component of

the electric field derived in eq (2). Letting R^ =\^ -f^m\ ^"^ ^'m =\^ ~K\ results in the

following:

EAr).-

Ak

,-Mn.

kR

1--

kR„

{\ + {z-z„Ye ^ lAz-zJ" ^ \z-zj

2d2 k'R kRt k'Rl

-M'r, kRi

1-

kR'

{\ + {z + z„fe ^ 3jXz + z„y ^ 3iz + zj'

k'R'J kR'2

2 D'4 k'R:

(11)

The m subscript on the electric field refers to the element or current source responsible

for generating the field, as previously given in eq (7). Calling the numerator matrix A, we

can write A as the outer product of two vectors, i.e., A = ££* , where e is column vector

and f*is the complex conjugate row vector. Using eq (11), the elements of e are given

as

_ R

£ =

.* Rn £„(r)andr=-^£:(F) (12)

We want to maximize the ratio p = — , which equates to finding the largest

B

eigenvalue of

Aa = AjBa , (13)

where a represents a scaling of the current vector. Due to the structure of the matrices A

and B, only one nonzero eigenvalue exists for eq (13). We solve for the maximum of this

ratio using the following equation

max{p}=A^^=rB-'6. (14)

In addition, we can determine the magnitude and phase of the necessary current

scaling fi-om: a=B-'£ . (15)
Two important points to note are as follows. First, the scaled currents determine the

resulting gain at the observation point. Second, the existence of only a single nonzero

eigenvalue follows from the theory of eigenvalues and eigenvectors. The zero

eigenvalues correspond to eigenvectors orthogonal to the eigenvector associated with the

nonzero eigenvalue. Physically, the currents corresponding to the zero eigenvalues

produce no field at the observation point [14]. Only the currents associated with the

nonzero eigenvalue produce fields at the observation point. A succinct development of eq

(13) through eq (15) is found in Reference [15].

The sensitivity of the gain to the current magnitude and phase represents an important

part of this analysis. Using the vector |a| , whose elements are the magnitudes of the

optimized currents, the sensitivity of the gain to the optimized current magnitudes

follows from

«—;\ \a\ A\a\

\a\ B\a\

where D represents the gain at the particular observation point used in the optimization

process. In other words, we simply calculate the gain using only optimized current

magnitude information. Similarly, using Zo , the sensitivity of the gain to only the phases

of the optimized currents follows from

D(Ta)=^^ . (17)

Za BZa

With the basic theory in place, we now describe an appropriate simulation environment to

investigate some key attributes of the proposed approach.

3. Simulation Setup

Figure 2 below includes additional details on the basic simulation configuration

introduced in figure 1. Three important features in all of the simulation runs include:
1

.

ground plane modeled as a PEC, infinite in the x and y directions;

2. z-directed Hertzian dipoles, with current distribution defined in eq (7);

3. observation points chosen in the z = 1 .3 m plane.

In the simulation setup, the ground plane represents the floor of a room or building. The

elements equate to the wireless devices randomly placed throughout the room or building

as part of the communication system (hereafter called simply the system). A height of

1 .3 m for the observation points approximates the antenna height of the radio device on

the emergency response personnel, and the z-directed field corresponds to the vertically

polarized orientation of the antenna. Figure 2 provides an example of four elements at a

height of 1 .3 m over the ground plane.

Figure 2. Four z-directed elements above an infinite PEC ground plane. All dipole

elements located at a height of z = 1.3 m . The footprint of the volume containing the

elements is 25 m x 25 m

.

Each simulation run focuses on the effect of particular system parameters, which

include:

1

.

radio frequency range of 1 00 MHz-5.0 GHz;

2. x ,y , and z coordinates for observation points:

10
a. random x and y coordinates uniformly distributed over a [-8.0m, 8.0m]

square (see figure 3 below);

b. deterministic locations within the array footprint (see figure 4 below);

c. random x and y coordinates uniformly distributed in a 10 m perimeter

around the 25 m array footprint (see figure 5 below);

3. X ,y , and z coordinates for element locations:

a. random x and y coordinates uniformly distributed over [-12.5 m, 12.5 m]

square, height of z = 1 .3 m;

b. random x and y coordinates uniformly distributed over [-12. 5m, 12. 5m]

square, uniform random distribution for various ranges of z;

4. number of dipole elements ranging from 2 to 20 elements;

5. sensitivity to optimized current parameters:

a. magnitude sensitivity calculated from eq (16);

b. phase sensitivity calculated from eq (17).

11
a

a

"O" a

n

Xf" n

a

D

a

8.0m 4.5 m

< n n

a

a

a a ^

*^ Observation Locations

Figure 3. Example of random observation points within the array in the z = 1.3 m

plane. Array elements (not shown) are randomly located in either a plane or volume,

as indicated in number 3 above.

J3 a a Q jg

ta n a

-B s- -QE(

-eib s..

4.0m

-^ n

4.0m <

4.5 m

<m

Q Q £3 EJ m

*-* Observation Locations

Figure 4. Deterministic observation locations in the z = 1.3 m plane. Array elements

(not shown) are randomly located in either a plane or volume, as indicated in number 3.

12
n a a

a

n

u

n

u

a

n

u

n

n ° n

n

12.5m

n

lO.Om

n

a

n

n

a

»-' Observation Locations

Figure 5. Example of random observation points external to the array elements. Array

elements (not shown) are randomly located in either a plane or volume, as indicated in number 3 above.

The simulation environment basically consists of using mathematical software to

randomly locate radiating elements and observation points, and then compute the closed

form expressions previously discussed in Section 2. Each particular simulation

configuration undergoes multiple trials, e.g., 1000 trials with frequency = 100 MHz,

observation point x and y coordinates chosen at random, and four dipole elements.

Computation of the A and B matrices follows from eq (10) and eq (12), while

optimization of the resulting gain is calculated from eq (14). Note in all the simulation

runs, the radiating or array elements are located within the notional building volume,

whereas the observation points may be either interior or exterior to the building volume,

as depicted in the three previous figures. Further details on each set of simulations

precede the figures illustrating the corresponding results in Section 4, along with a

discussion on each set of figures. Section 5 contains the overall conclusions.

13
4. Simulation Results

4.1 Average Gain with Elements in a Plane and Interior Observation Points

Each simulation run typically includes 1000 trials, (the only exception is described in Appendix A). A trial consists of first placing the radiating elements in the notional

building space, then locating the observation points in accordance with one of the layouts

discussed in Section 3, and finally calculating the gain for each observation point by

optimizing the current magnitude and phase on each Hertzian dipole element. Thus, each

trial generates a number of optimized gains equal to the number of observation points,

(usually 24 in these simulations, with the exception described in Appendix A). The

average gain is computed by taking the average of the average gain per trial determined

at the observation points. A mathematical representation takes the form:

o...=((A))4i:[|t«.l. ('8)

where L is the number of trials, / is the number of observation points per trial, D, is the

gain at one observation point, and Dave is the average gain. The (•) notation represents the

calculation of the average.

Figures 6 to 9 depict simulated gain versus number of elements for five different

frequencies, with the number of elements ranging from 2 to 20. Figure 6 shows the

average gain calculations based on the x and y coordinates of the observation points

uniformly distributed over the interval [-8.0m, 8.0m], (figure 3 depicts an example

distribution). Figure 7 shows similar simulation results, but with the deterministic

observation points chosen to follow the layout presented previously in figure 4. Figures 8

and 9 show the standard deviation corresponding to the average gain of figures 6 and 7,

respectively. This represents the standard deviation in the average gain per trial as given

by the equation

14
a = ^{{D„,Y)-{{D^))\ (19)

where o is the standard deviation. Based on this series of plots, the method of selecting x

and y observation coordinates, such as random versus periodic, in a specific plane of

observation does not appear to significantly influence the statistical behavior of the

optimization process.

Figures 6 through 9 provide three general insights. First, the average gain

increases as the number of elements increases. The gain increases approximately 3 dB

when increasing fi-om 4 to 8 elements or from 8 to 16 elements; or in general, a doubling

of the elements causes a 3 dB increase in gain. This is not an obvious conclusion since the

ratio under consideration as given by eq (1), normalizes the gain by the total system input

power. (Note that gain does not simply result from an increasing number of elements;

compare to the results in figures 12, 14, and 19.) Second, the standard deviation appears

to settle below 0.75 dB in almost all cases. The anomaly for the 100 MHz behavior in

figures 8 and 9 is still under investigation, but we note that the same behavior occurs for

both the random and deterministic observation locations.

The third insight arises when we examine the behavior due to the electrical height

of the elements above the ground plane. Nearly idenfical plots of the average gain occur

for the frequencies of 1.0 GHz, 2.5 GHz, and 5.0 GHz in all four of the figures. However,

the 100 MHz plot indicates approximately an improvement of 1 dB over these three

frequencies and a 2 dB improvement over the 500 MHz case, for all numbers of

elements. This result is due to the height over the PEC ground plane relative to a

wavelength at a specific frequency resulting in constructive or destructive interference.

The use of the PEC in the simulations represents a crude approximation to the real world

surface, such as a concrete floor, so these differences due to height may be less

pronounced in the implementation of a system. However, since the actual implementation

can be expected to have devices located less than 2 m above the ground, the choice of

frequencies could significantly impact the actual gain achieved.

15
22

20|-

isl16

CD 14 C

ro 12 O

(DO) 10

03

u.

o 100 MHz

X 500 MHz

+ 1.0 GHz

a 2.0 GHz

V 5.0 GHz

10 12 14 16 18 20 22 Number of Elements

Figure 6. Average gain using 1000 trials and 24 random observation positions over 16m X 16m square area.

m 14 o

c

ro 12 CD

a>O) 10

ffl (U> R <

6

4

2

o o

o ®

O ffl

o °

ffl o

o 100 MHz

X 500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

ffl

-

8 10 12 14 Number of Elements16 18 20 22

Figure 7. Average gain using 1000 trials and 24 deterministic observation positions of

figure 4.

16
1.8

1 1 1 1 1

-

1.6 -e- 100 MHz

--- 500MHz

m I'* -+- 1.0 GHz - 2, ><. -B 2.0 GHz

1 ^-2

\ -V- 5.0 GHz -

CD s > K

Q) 1

v , D s o

^ 1— X

ro 0.8 [S . TJ /-vV ^ c ^A

ro ^ V ^ "^ ,

^ \ •- W 0.6 "^ 0-. - y^^y__^"---^----o-^- ^;;-0"--0---0-"^

0.4

~ ^?'"'----s-------_^---_.

'~ ~*'"~ - ~ -,

0.2

1 1 1 . 1 1

1 1 1 1 1 8 10 12 14 Number of Elements

16 18 20 22

Figure 8. Standard deviation of average gain plotted in figure 6.

1.8 -

1.6 - -

-e- 100 MHz

--- 500MHz

8 1.2 -+- 1.0 GHz

- -B- 2.0 GHz .

ro

+

-V- 5.0 GHz

-

rd

Dev

Standa

o

o

b>

bo

" ^ '^ ^ ^

0.4

0.2

n

G -

•

--'•:^----.--_ --e-

-V--©--

------^_-

,,-G----°----o-

~ ~ ~ ~ ~'^~ - - - —,_ ___-*<

"'-'-=p----------^---------S--------«---------H

-

8 10 12 14 Number of Elements

16 18 20 22

Figure 9. Standard deviation of average gain plotted in figure 7.

17
4.2 Effect of Current Magnitude on Optimized Gain

Figures 10 through 12 illustrate the effects of using only the optimized current

magnitudes in computing the gain results. Figures 10 and 1 1 result from taking only the

magnitude of the optimized current vector for the same configurations used to generate

figures 6 and 7. After determining the optimized current scaling from eq (15), application

of eq (16) provides the gain using only the optimized current magnitudes. All the

elements are excited in phase, i.e., with a zero phase difference between elements.

As anticipated from the previous results, figures 10 and 11 exhibit no significant

differences. However, the overall gain is substantially decreased from the gain depicted

in figures 6 and 7. For instance, for 10 elements in figures 6 and 7, all frequencies exhibit

a gain in excess of 12 dB, while for 10 elements all frequencies are below 8dB in both

figures 1 and 1 1 . Also, the rate of gain increase, approximately 1 dB per doubling of

elements in figures 10 and 11, represents a substantial decrease from the 3dB per

doubling of elements depicted in figures 6 and 7.

Figure 12 illustrates an important result because it provides a baseline with which to compare current optimization techniques. For comparison purposes, a random uniform

distribution for the current magnitudes with identical phases is used in place of the

optimized current scaling vector in eq (16). The gain is computed from

^(^)~. (20) a Ba

where a is a vector with a uniform random distribution for current magnitudes and zero

phase. The results are shown in figure 12, where the gain remains fairly constant as the

number of elements increases. The gain in figures 10 and 1 1 exceeds the gain in figure 12

for all five frequencies and number of elements. This indicates the optimized current

magnitude information alone provides gain, even though the amount of gain may be

substantially reduced.

18
CD 14 C

ro 12 CD

0)O) 10

S5

T

1 T

"

o 100 MHz

500 MHz

+D

1.0 GHz

2.0 GHz

-

V 5.0 GHz

S I? o J o

o 9

o o 9

o

. o o

ffl a

"

X

1 8 10 12 14 Number of Elements

16 18 20 22

Figure 10. Average gain using only the magnitude of the optimized currents for 1000 trials and 24 random positions over 16 m x 16 m square area in the z = 1 .3 m plane.

19
22

20

18 -

16 - m 14 c

ro 12 o

(DO) 10

(1)

6 -

-

o 100 MHz

« 500 MHz

+ 1.0 GHz

2.0 GHz

-

V 5.0 GHz

-

o ^

H

g o o

S3O

9

o

9

o o

. X . ^

8 10 12 14 Number of Elements

16 18 20 22

Figure 11. Average gain using only the magnitude of the optimized currents for 1000 trials and 24 deterministic positions of figure 4.

22

20

18

16 m 14 o

c

ro 12 O

a>o> 10 m

a)> R <

6

4

2

-

o 100 MHz

+ 500 MHz

1.0 GHz

. D 2.0 GHz .

• V 5.0 GHz-

•

O

-

o o o O (D o o o o

g s

»

s ^ f§ Eb ^ ^ ?

8 10 12 14 Number of Elements

16 18 20 22

Figure 12. Average gain using uniformly distributed random current magnitudes and

identical phase for 1000 trials and 24 random positions over 16 m x 16 m square area in the z= 1.3 m plane.

20
4.3 Effect of Current Phase on Optimized Gain

The next two figures illustrate the effect of optimized phase on the gain. Figure 13

illustrates the gain obtained from using only the optimized phase information from

eq (15) with random observation points. This figure uses the same points as the average

gain plots in figures 6 and 10. These results demonstrate the significant effect of phase in

achieving the maximum possible gain for a given configuration. In comparing the gain

results achieved using optimized current phase information only versus optimized current

magnitude information only, figure 13 versus figure 10 indicates a higher average rate of

increase in gain. For example, at 100 MHz, the rate of increase in gain is approximately

2 dB per doubling of the elements when using the optimized current phase information

(see figure 13), as compared to approximately 1 dB per doubling of the elements when

using the optimized magnitude information (see figure 1 0).

Finally, figure 14 shows the results of using a uniform random distribution for the

phase with unity magnitude for the current vector, a , in eq (20). The use of unity

magnitude currents with uniformly distributed random phases over [0, 27i] depicted in

figure 14 is nearly identical to the results depicted in figure 12. Figures 12 and 14 show

that the optimization and/or control of the excitation currents (either current phase or

magnitude), is required to achieve gain at a given observation point.

21
22

20 -

18 -

16

00 14- C

TO 12|-

O

TO 10 - E

(U

6 -

4

2

o 100 MHz

X 500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

6 8 10 12 14 Number of Elements16 18 20 22

Figure 13. Average gain using only phase of optimized currents for 1000 trials and 24

random positions over 16 m x 16 m square in the z = 1.3 m plane.

m 14 cS 12 o

<1> O) 10

0)

- —1

-

:

•

o 100 MHz

500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

-

-

o O o o o o o o o o

-

- 6

ft 6 6 ffl s a ffl $

^ -

8 10 12 14 Number of Elements

16 18 20 22

Figure 14. Average gain using uniformly distributed random phase with unity magnitude

currents for 1 000 trials and 24 random positions over 1 6 m x 1 6 m square area in the z = 1.3 m plane.

22
4.4. Average Gain with Elements in a Plane and Exterior Observation Points

Communication into a building can also be difficult, so the next results focus on the

possible improvement in gain using this multiple element system and optimization

process when the observation point lies within a 10 m perimeter of the building. Figure 5

illustrates an example distribution of the observation points. Figures 1 5 and 1 6 depict the

average gain and corresponding standard deviation for the case where the observation

point lies in a 10 m perimeter of the array boundary. The rate of gain increase in Figure

15 indicates a gain of approximately 3 dB per doubling of the number of elements, which

matches the gain per element increase exhibited in figures 6 and 7. The general trend in

the standard deviations plotted in Figure 1 6 also mimics the trend shown in figures 8 and

9 (although o is higher for some specific cases).

Figures 1 7 and 1 8 show gain results based on only the optimized current magnitudes

and phases, respectively. Figure 19 depicts the use of unity magnitude current with

uniformly distributed random phases over [0, 2n] in calculating the gain. (The case of

uniform random current magnitudes with equal phases is effectively the same as the

results in figure 19, and thus omitted.) While using either the optimized current

magnitudes or phases provides a gain increase over the random current case, the

optimized phases enable a larger increase. The rate of increase using the optimized

phases tracks the rate of approximately 2 dB per doubling of the elements when using the

interior observation points. However, the optimized magnitudes begin to flatten out for

the 14-element configuration.

23
m 14 c

0)cn 10

(0

k_

1 1 1

1

,

9

9 9

9

9

9

#

+

+

-

e

+

+ "

V +

100 MHz

500 MHz

- + 1.0 GHz - D + n 2.0 GHz

V V 5.0 GHz

.

+ -

8 10 12 14 Number of Elements16 18 20 22

Figure 15. Average gain for 1000 trials of 24 random observation points in the 10m

perimeter of the array.

2.5

2 -

CD2,c

.0 1.5 >Q

oc

(0 w

0.5

1 1 1 1 1 —

1

1 r

-0- 100 MHz

+

X 500 MHz

\ -+- 1.0 GHz

-B 2.0 GHz

-V- 5.0 GHz

^-^

-

V

N

-+„ a

V H- ^ \ " -t^ ^- ^\ ~^~+---+,

, ~\^^ ^+----+--_

-^.v.^^e--, ~^ - ----~--^---:S:.-.^

--. - --*----9----=.-§-v.. .4,,.,^,_,^

Q -© 0-- 0- -0-

8 10 12 14 Number of Elements16 18 20 22

Figure 16. Standard deviation of average gain for 1000 trials of 24 random observation

points in the 10 m perimeter of the array.

24
CQ 14 2,cS 12 CD

(UO) 10

—1

- o 100 MHz

+ 500 MHz

1.0 GHz

• D 2.0 GHz - V 5.0 GHz

.

o o o o o o o

-f- 9 9

o

ra + B

^ i g a

9

+ "

« «

,

X *

1

, ,

, _i 1

, , 8 10 12 14 Number of Elements

16 18 20 22

Figure 17. Average gain using only the magnitudes of the optimized currents for 1000

trials and 24 random observation points in the 10 m perimeter of the array.

££

20 - . O 100 MHz

X 500 MHz

18 +D

1.0 GHz

2.0 GHz

16 V 5.0 GHz

^^

CD 14 . o

o , 2, o o C

TO 12 - o o . O c) D V

S) 10 2

o

D

DV

DV

V

- O D V

* > 8 c] V X

" + < O DV

7 X + +

+

6 DV

i +

+

4 D

+

2

n

+

8 10 12 14 Number of Elements16 18 20 22

Figure 18. Average gain using only the phase of the optimized currents for 1000 trials and 24 random observation points in the 10 m perimeter of the array.

25
22

20h

18

16 m 14 C

ro 12 O

O) 10 2

(U

o 100 MHz

^ 500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

9 9 9 e

8 10 12 14 Number of Elements16 18 20 22

Figure 19. Average gain using uniformly distributed random phase with unity magnitude

currents for 1000 trials and 24 random observation points in the 10 m perimeter of the

array.

4.5 Effect of Elements Located in a Volume versus a Plane

This next series of plots (figures 20 through 31), depicts the gain behavior when the

radiating elements are not constrained to the same plane as the observation point.

Specifically, coordinates of the radiating elements are uniformly distributed along the

three coordinate axes, x, y, and z, while the observation points all lie in the z = 1.3 m

plane. The first six plots illustrate the gain and standard deviation for three different

volumes with observation points within the building boundaries, while the next six plots

result from observation points in the 10 m perimeter. In all six cases (three each for the

interior and exterior observation points), the x and y coordinates of the elements are

random uniformly distributed in the interval [-12.5 m, 12.5 m]. The z coordinates are

random uniformly distributed as well, over one of the three possible intervals.

26
[0.1m, 0.5m], [0.1m, 1.0m], or [0.1m, 1.5m]. Note, these volume intervals start at

z = 0. 1 m in anticipation of the elements being some finite height above the ground in a

practical system.

In the first three plots (figures 20, 21, and 22), the average gain does not change

significantly as the volume of possible element locations is increased. Also, the gain

exhibits the same rate of increase as when the elements all lie in the z = 1.3 m plane,

giving approximately 3 dB per doubling of the elements (compare to figures 6 and 7).

22

201-

18

16 m 14 C

(5 12 CD

(DO) 10

ro

0)

o 100 MHz

X 500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

8 10 12 14 Number of Elements16 18 20 22

Figure 20. Average gain using 1 000 trials and 24 random observation positions over

16 m X 16 m square area; random radiating elements in a volume

25 m X 25 m x 0.4 m; z positions range over the interval [0. 1 m, 0.5 m].

27
22

201-

18

16

CD 14 C

S 12 CD

O) 10 o 100 MHz

» 500 MHz

+ 1.0 GHz

D 2.0 GHz

V 5.0 GHz

8 10 12 14 16 Number of Elements

18 20 22

Figure 21. Average gain using 1000 trials and 24 random observation positions over 16m X 1 6 m square area; random radiating elements in a volume

25 m X 25 m x 0.9 m; z positions range over the interval [0. 1 m, 1 .0 m].

28
m 14 cS 12 CD

Oiat 10 2

.

1 -1 1 1

.

•

o

o

o 9

o5

o

1

-

- o

o1

o1

S

1

-

• o

1

W

- o 100 MHz

500 MHz

+ 1.0 GHz

o D 2.0 GHz - V 5.0 GHz

i

.

-

8 10 12 14 Number of Elements16 18 20 22

Figure 22. Average gain using 1000 trials and 24 random observation positions over

16 m X 16 m square area; random radiation elements in a volume

25 m X 25 m x 1 .4 m; z positions range over the interval [0. 1 m, 1.5 m].

These next three plots (figures 23, 24, and 25), illustrate the standard deviation

associated with the three previous gain plots. In all three case, the standard deviation falls below 1 dB for all frequencies after only six elements. Placing the elements in a volume,

as opposed to a single plane appears to smooth the standard deviation. The combination

of the gain and standard deviation indicate an overall smoothing effect from allowing the

elements to reside at a variety of heights. An actual system will likely utilize elements at

a variety of heights, so this smoothing behavior without sacrificing gain represents an

added benefit.

29
1.8

1

+

- -©- 100 MHz

1.6 --- 500MHz -

' -+- 1.0 GHz

m 1-4

\ ^ - B- 2.0 GHz . a > ^ -V- 5.0 GHz

c . _ v\ 1 ^ O 1.2 \ ^ -

*4_r ni ^ ^' ^v >

"* ^ S ^

^ \><^ ^^ -

-g \ °. -..

CD 0.8 .

\ N + ::

.

. -o ^ -^ - V c 1 ". ^ = ,^

W 0.6

0.4

0.2

n

o

1 1 1 1 1 1 1 1 1

-

8 10 12 14 Number of Elements16 18 20 22

Figure 23. Standard deviation of average gain results shown in figure 20.

1.8

1.6 m 1.4 o

co 1.2

'^^ ro>

(U 1 Q

T7

ro 08 o

c

ro CO 0.6

0.4

0.2 -

\

\

\ -o- 100 MHz

-«- 500MHz

o \> -+- 1.0 GHz

B 2.0 GHz

• \ -V- 5.0 GHz .

>K

*

^ . t3 .

^ ~+ , - . _

- o.

o o o e e %

8 10 12 14 Number of Elements

16 18 20 22

Figure 24. Standard deviation of average gain results shown in figure 21.

30
1.8

1.6 m 1.4

C

g 1.2

«3 0.8 T3C

TO W 0.6

0.4

0.2

^

-

^ \ -e- 100 MHz

--- 500MHz

-+- 1.0 GHz

Q \\ -B- 2.0 GHz

\

* ^ -V- 5.0 GHz

^

^ \

' \\

""s \v;\

^ \ V \ ^ ^ '^^ ^o

-

^^«- -^®. ,;;|:;;^:V:^-^.. ^mm -

8 10 12 14 Number of Elements16 18 20 22

Figure 25. Standard deviation of average gain results shown in figure 22.

The next series of plots (figures 26, 27, and 28), shows the average gain and the

corresponding standard deviation when the observation point resides in the 10 m

perimeter of the element volume. All the observation points lie in the z = 1.3 m plane and

are randomly distributed. The behavior is very similar to the case with interior

observation points, namely a smoothing effect on the standard deviation while still

maintaining the same rate of gain increase as when the elements all reside in the z = 1 .3 m

plane. The spread of the average gain between frequencies decreases from approximately

6dB in the planar case to less than 4dB in the volume case. As mentioned for the

previous six plots, this indicates a potential added benefit for an actual system.

31
22

20 . _

18

16

+ + O

+ O

+O

9

+O

+O

m 14 C

O

- +O

+O

9

+ o

o 9

9

9

0)en 10 o 100 MHz

S

X D

X 500 MHz

3? o V +D

1.0 GHz

2.0 GHz

-

6 V 5.0 GHz

D

4 V

-

2

, , , , ,

[) 2 4 6 8 10 12 14 16 18 20 2 Number of Elements

Figure 26. Average gain using 1000 trials and 24 random observation positions over 10 m perimeter; random radiating elements in a volume

25 m X 25 m x 0.4 m; z positions range over the interval [0.1 m, 0.5 m].

32
DQ 14 cO

0)a> 10

ro

6

a

a o 100 MHz

+V

X 500 MHz

D +DV

1.0 GHz

2.0 GHz

5.0 GHz

8 10 12 14 Number of Elements

16 18 20 22

Figure 27. Average gain using 1000 trials and 24 random observation positions over 1 m perimeter; random radiating elements in a volume

25 m X 25 m x 0.9 m; z positions range over the interval [0.1 m, 1.0 m].

33
m 14

;dc

CD

O) 10 2

6|-

4

2

r-

• o

O

O o

¥

o

-

- o ^ ^

- o X ¥

¥

o " ¥

o f

f

- o 100 MHz

X 500 MHz - o f + 1.0 GHz

. D 2.0 GHz . V 5.0 GHz

f

I '

-

8 10 12 14 Number of Elements

16 18 20 22

Figure 28. Average gain using 1000 trials and 24 random observation positions over 1 m perimeter random radiation elements in a volume

25 m X 25 m x 1.4 m; z positions range over the interval [0.1 m, 1.5 m].

These next three plots (figures 29, 30, and 31), illustrate the standard deviation

associated with the three previous gain plots. In all three plots, the standard deviation

drops below 1 dB for greater than four elements. The spread between frequencies is

slightly greater than when the observation points reside on the interior of the building, but

the difference is not large.

34
2.5

CO2,

C

g 1.5

ro >

0) Q

V.

TO 1 D

C

ro

0.5

?

^

^

o 100 MHz

« 500 MHz

-+ 1.0 GHz

-B- 2.0 GHz

-V- 5.0 GHz

1 1 1 1 1

•+ - - "+ - - . .+ -©--— -~-®- ——---^ .@ g)

1 1 1 1 1 8 10 12 14 Number of Elements

16 18 20 22

Figure 29. Standard deviation of average gain results shown in figure 26.

35
2.5

GO^

C

g

ro >

0) Q

"S

CD oc

ro W

2 •

1.5 -

0.5 -

+ ^

-e 100 MHz

< 500 MHz

-+ 1.0 GHz

-a- 2.0GHz

V 5.0 GHz

~~°—

-0- —e-- - -e — p,

1 1 1 1 1

1 1 1. 1. 1 8 10 12 14 Number of Elements

16 18 20 22

Figure 30. Standard deviation of average gain results shown in figure 27.

2.5

m

2,c

q 1.5

'>

a> Q

g

ra 1 T3cB

w

0.5

Q

1 II 1 1

\

\

o 100 MHz

--- 500 MHz

+ - 1.0 GHz

-B- 2.0 GHz

-V 5.0 GHz

\

^ D- ~

"" ~x- --

-

1 1 1 1

~ ~ +

V- - - -V~~---^SS - -g- - - - -D

8 10 12 14 Number of Elements

16 18 20 22

Figure 31. Standard deviation of average gain results shown in figure 28.

36
5. Conclusions and Future Work

Several preliminary conclusions related to communication systems for emergency

responders may be drawn from these initial analytic and simulation results. > An appropriately controlled system of wireless devices can theoretically

increase the communication capability within a building by effectively

functioning as an antenna array to support optimized directivity of the

electromagnetic radiation. This can improve the probability of receiving a

radio-frequency signal, as well as the quality of the signal in terms of signalto-noise ratio. > The emergency responder can move throughout the building and surrounding

area, and receive a similar benefit from the array. > The average gain increases by approximately 3 dB per doubling of the

elements when using optimized current magnitudes and phases. In addition,

the overall average gain is greater than 14 dB, with a standard deviation of less

than 0.75 dB for all cases involving 20 elements.

> The use of the optimized current phase information only provides

approximately 2 dB of gain increase per doubling of the elements. On the

other hand, using only the optimized magnitudes provides 1 dB or less of gain

per doubling of the number of radiating elements. This implies the achievable

gain is affected more by the accuracy of the optimized phases than the

optimized magnitudes.

> Placing the elements in a volume versus a single plane may actually benefit a

system utilizing several frequencies by smoothing out the gain statistics. One of the challenges of implementing a system based on this optimization process is

the control of the relative phase between elements. An important ongoing investigation

examines the effect on gain as the phases deviate from the optimal case. As noted in the

fourth bullet above, gain is still achieved when only the optimized current magnitudes are

37
used, so even if the phase cannot be tightly controlled, gain should be achievable since

the current magnitude is fairly straightforward to control.

These preliminary results are encouraging, but further refinement of the simulation

environment is necessary. One improvement currently under investigation is the

modeling of the ground plane as other than a PEC. Another point of investigation is the

effect of mutual coupling between radiating elements. Location sensitivity analysis will

help determine how often the optimization process would require updating in reaction to

changes in the topology of the system of wireless devices. Finally, the impact of losing some of the array elements must be studied, as the actual practical implementation will

operate in a harsh environment.

6. References

[1] E. K. Krupitskii, On the Maximum Directivity of Antennas Consisting of Discrete

Radiators. Technical Physics, Vol. 7, No. 3, pp 257-259, September 1962.

[2] J. L. Synge, Directivity for Scalar Radiation. Quarterly of Applied Mathematics, Vol.

24, No. 1, pp. 90-92, 1966.

[3] L. Susman and W. K. Kahn, Measures of Beam Concentration for Scalar Radiation.

Radio Science, Vol, 5, No. 7, pp. 1091-1097, July 1970.

[4] R. F. Harrington, Field Computation by Moment Methods, The Macmillan Company,

New York, pp. 189-212, 1968.

[5] T.S. Angell and A. Kirsch, Optimization Methods in Electromagnetic Radiation.

Springer-Verlag New York, Inc., New York, 2004.

[6] Y. T. Lo, A Probabilistic Approach to Large Aperiodic Antenna Arrays. Proceedings

of the Application Forum on Antenna Research, pp. 580-610, January 1964.

[7] Y. S. Shifrin, Statistical Antenna Theory. The Golem Press, Boulder, CO, 1971.

[8] Y.T. Lo and S.W. Lee, Editors, Antenna Handbook: Theory, Applications, and

Design. Van Nostrand Reinhold Co. Inc., New York, NY, Chapter 14, 1988.

38
[9] W. J. L. Queiroz and M.S. Alencar, Project of Antenna Arrays with Random

Parameters. Proceedings Brazilian Microwave and Optoelectronics Society/Microwave

Theory and Techniques Society, International Microwave and Optoelectronics

Conference 2003, pp. 33-38.

[10] F. Tseng and D. K. Cheng, Gain Optimization for Arbitrary Antenna Arrays Subject

to Random Fluctuations. IEEE Transactions of Antennas and Propagation, Vol. AP-15,

No. 3, pp. 356-366, May 1967.

[11] E. N. Gilbert and S. P. Morgan, Optimum Design of Directive Antenna Arrays

Subject to Random Variations. The Bell System Technical Journal, Vol. 34, pp. 637-663,

May 1955.

[12] T.S. Angell and A. Kirsch, Optimization Methods in Electromagnetic Radiation, pp. 106-107,2004.

[13] C. A. Balanis, Antenna Theory: Analysis and Design. John Wiley & Sons, Inc., p. 39, 1997.

[14] R. F. Harrington, Field Computation by Moment Methods. Macmillan, New York,

pp. 197-198, 1968.

[15] D.K. Cheng and F.I. Tseng, Gain Optimization for Arbitrary Antenna Arrays. IEEE

Transactions on Antennas and Propagation, Communications, pp 973-974, 1965.

39
Appendix A: Examination of Statistics Gathering Metliod

This series of plots demonstrates the effect of choosing 1 000 observation points and

24 trials versus 24 observation points and 1000 trials. Figures A-1 and A-2 depict the

average optimized gain (i.e., using optimized magnitude and phase for element currents

to achieve gain at a specific observation point, as performed in Section 4.1), when the

observation points reside in the interior and the exterior of the building, respectively. A

uniform random distribution on both the x- and y-axis, and the z = 1.3 m plane provide

the observation point coordinates. Compare these first two plots to figures 7 and 15,

respectively. The results are nearly identical between figure A-1 and figure 7, and figure

1 5 and figure A-2.

Figures A-3 and A-4 illustrate the corresponding standard deviation of the average

gain in figures A-1 and A-2. These two plots compare to figures 9 and 16. All the trends

are similar, including the fi^equency, which shows the greatest standard deviation, namely

500 MHz for figures 34 and 9, and 1.0 GHz for figures A-3 and 16. Since computing the

standard deviation of the average per trial generates these plots, some differences are

expected. However, statistically, the results are almost identical, which implies either

method of collecting first-order statistics will provide similar results.

40
22

20 -

18 -

16 - m 14 cO

0)D) 10 2

6

4

2

o ^ ?

^

o 100 MHz

>< 500 MHz

+ 1.0 GHz

2.0 GHz

V 5.0 GHz

8 10 12 14 Number of Elements16 18 20 22

Figure 32. Average gain using 24 trials and 1000 random observation positions over 16 m X 16 m square area in the z = 1.3 m plane.

m 14 2,c

ro 12 CD

0)O) 10 2

1 —I —1

1 1

o

°

o

o

o

o

O

9

o

DV

9

9 ^

9

+

+ D X o V +

• nV

+

o

D X o 100 MHz

V

+

X 500 MHz

jj + 1.0 GHz

D 2.0 GHz

" DV

V 5.0 GHz

.

+

1 8 10 12 14 18 20 22 Number of Elements

Figure 33. Average gain for 24 trials of 1000 random observation points in the

10 m perimeter of the array.

41
1.8

1.6 m 1.4

cg 1.2

ro 0.8 D

cW 0.6

0.2

1 1 T 1

-

-o- 100 MHz

--- 500MHz

-+ 1.0 GHz

-B- 2.0 GHz

V- 5.0 GHz

-

\

0. X'-" ^

1 1 1 1 1 1 J 1 i 1 8 10 12 14 Number of Elements

16 18 20 22

Figure 34. Standard deviation for average gain results in figure 32.

1.8

1.6 -

m 1.4

cg 1.2

m 0.8 D

cmW 0.6 -

0.4

0.2

-

—

1

1

1

1 1 1

-

+ -0- 100 MHz

' --- 500MHz

1 -+- 1.0 GHz

• -B- 2.0 GHz -

\ -V- 5.0 GHz

-

\

- -

- V. \-

8 10 12 14 Number of Elements16 18 20 22

Figure 35. Standard deviation of average gain results shown in figure 33.

42
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