MSOE MEMO To: [Dr. Tritt]

From: Simon, Nikolas A.

Date: February 16, 2016

Re: [Project (trilat2d.m,sort_pr.m,power_to_d_friis.m,distance_confirm.m,power_to_xy.m,circle.m,testscript.m) ]

Program Description:

This program serves at the main calculation tool for a more complex location tracking system. This

location tracking system seeks to determine the location of a node based on received power levels from

a fixed grid of beacons. The location tracking system is implemented such that the beacons are ceiling

mounted and the node can move in XY directions on the floor below. These beacons are strategically

placed in a 3 by 3 grid and are tagged such that it is clear from the program input which power level is

which. Each beacon broadcasts message payloads containing beacon number and a power level in units

of dBm. The node detects the associated RSSI (receives signal strength indicator) values of these

beacons which means each beacon in the grid has an identifying label and an associated received power

level from the perspective of the node.

The power values are stored in a list from 1 to 9 where they are then sorted into the three highest

power levels. Expected RSSI received input power levels are between 0 and -120 dBm. The known level

of transmit power Pt for all beacons is in the grid is approximately +10dbm. The known gains of the

transmit and receive antennas on the beacons and nodes are 9.6899 dB.

A method called trilateration is used to calculate the unknown node coordinate. Trilateration

determines an unknown location from multiple reference locations. The distances relative to these

reference locations are used to estimate the coordinate of the unknown location. At least three

reference locations must exist for trilateration to be implemented. If three reference locations are

considered, the origin of each reference location can be considered the center of a sphere where the

unknown location is the intersection of all three spheres.

For the trilateration calculation to work correctly, at least three beacon received power levels must be

known to determine the unknown location of the node in 2 dimensions. Because the node is

constrained to XY coordinates and can not move freely in the Z direction, a constant is introduced to

account for the difference in distance between the ceiling mounted beacons and the ground node in the

physical implementation of the program.

Equations of Interest:

Wavelength:

� = ��

Equation 1: Wavelength relates the speed of light to a desired frequency

�� = �� + � + �

Equation 2: 3d Pythagorean Theorem considers the Z direction

��� = � �� � �4����

Equation 3: Friis Equation relating received power, transmitted power, transmit/receive antenna gains,

wavelength, and range in an inverse square relationship of the three highest power levels

When solved for distance:

→ � = �4��

��� ���

�� = � ��

Equation 4: Linear path loss ratio which related transmitted power to received power. Path loss can be

determined by taking the ratio of transmitted power to received power. Depending on changes to the

orientation of the test environment, this value will change.

�� = � + 20 log � �4�� + 10� !" �1��

Equation 5: Equation for received power using the signal decay exponent which represents

environmental characterization

� = ��#$%& − ��#$&10 log(% � ��%�

Equation 6: Environmental calibration equation for the signal decay exponent where ��#$%& is a

reference received power level at 1 meter away from the source and ��#$& is the received power level at

a distance d away from the source

Solving for d:

→ 10�)log(%#�& − log(%#�%&* = ��#$%& − ��#$& → log(%#�& = ��#$%& − ��#$& + 10� !"(%#�%&10�

→ � = 10+,-#./&0,-#.&1(%23456/#$/&(%2 7

However exponential calculations like this can be computationally intensive and cause lag time in a

program implementation. To reduce computation time, the distance is calculated with a series

expansion to the needed accuracy.

� = �% 81 + 9(! + 9�! + 9;! + 9<! + 9=! + ⋯ ? , A = BC#(%&),-#./&0,-#.&*(%2

Received power can be determined with the environmental calibration equation which is dependent on

the signal decay exponent and with the Friis equation which is dependent on knowing the gains of the

transmit and receive antennas. Being able to calculate received power in two different ways allows for a

method of error checking because gain is not dependent on the signal decay exponent, it is a physical

characteristic of the antenna:

→ �� = 4��D��

Receiver gain equation where Ae represents effective aperture (area covered by the transmitter).

→ �D = � �$

The effective aperture is a function of power density Pd.

→ �$ + EF�7 = � � 4��

Power density is a function of transmit gain.

When comparing the Friis distance values and the signal decay exponent distance values, they are no

more than 1 meter apart from each other for any given calculation. Any more than this would indicate a

calculation error.

Simple 2d Trilateration Equations:

�(� = #G( − G&� + #H( − H&�

��� = #G� − G&� + #H� − H&�

�;� = #G; − G&� + #H; − H&�

Equation Set 7:�(,�,; represents the distances between the reference beacons and the target unknown

node solved for with the Pythagorean Theorem.

G = �I;� + I(; + I�(2#G(H;� + G�H(; + G;H�(&

H = �J;� + J(; + J�(2#H(G;� + H�G(; + H;G�(&

Where:

� = G(� + H(� − �(�

= G�� + H�� − ���

= G;� + H;� − �;�

Equation Set 8: The system of Pythagorean equations is used to solve for the unknown x and y

coordinates which refer to K#G, H& in figure 1. The notation JLM and ILM means #GL − GM&and#HL − HM&.

Figure 1: Diagram of trilateration using three known reference points to determine an unknown

coordinate at the intersection of all three radii. R(,�,; represent the coordinates of the reference

locations.

When adjusting for the fact that the node and beacons are not in the same plane, a Z component must

be introduced into the trilateration equations so the 3d form of the Pythagorean Theorem must be used

for in the system of Pythagorean equations. The z value is always a constant in the implementation,

there is no z direction so it is assumes that the mobile node cannot move freely in the z direction:

�(� = #G( − G&� + #H( − H&� + #O&�

��� = #G� − G&� + #H� − H&� + #O&�

�;� = #G; − G&� + #H; − H&� + #O&�

There is no dependence in the z direction therefore the equation for Z reduces to Z=Z.

Program Inputs:

Received beacon power levels 1 to 9

Program Outputs:

X Y coordinates of unknown location, linear path loss, d error rate, power density, effective aperture

Intermediate Values:

PA: highest received power level

PB: second highest received power level

PC: third highest received power level

D1: transmit radius of beacon with the highest received power level

D2: transmit radius of beacon with the second highest received power level

D3: transmit radius of beacon with the third highest received power level

X1,Y1 : grid coordinates of the beacon with the highest received power level

X2,Y2 : grid coordinates of the beacon with the second highest received power level

X3,Y3 : grid coordinates of the beacon with the third highest received power level

Constants:

λ: 0.3279m wavelength equal to PQ where c is the speed of light and f is the desired center frequency of

915MHz

Gt: 9.6899dB, transmit antenna gain

Gr: 9.6899dB , receive antenna gain

Pt: ≈+10dbM , transmit power

Prd0: 1 dBm, reference received power level at 1m

H: vertical distance from node to ceiling mounted beacons

Discussion of Algorithm:

The user enters a list of 9 received power values in units of dBm. From 1 to 9 each number corresponds

with a fixed position in the 3x3 grid of beacons. The entered values are then sorted into the highest

three values, making sure to keep track of each power value’s specific location number. The three

highest power levels are then entered into the power distance relationship described by the Friis

equation. The three highest power levels must also determine the three individual XY coordinates

belonging to the three highest received power signals. These three results must then be weighted by

power strength to properly align the unknown node coordinate within the triangle of possible area

between the three highest received beacon power levels.

Figure 2: Illustration of possible position within three selected beacon locations. Based on highest,

middle, and lowest power level a weighting is applied which places the coordinate closer to the beacons

which output the highest power. This weighting affects the distance component of the calculations.

The three highest power levels must be processed and converted to three distinct XY coordinates

representing each beacon location. A point to point mapping of the beacon layout is predefined and

Cartesian coordinates are assigned to each beacon.

Figure 3: Mapping to Cartesian coordinates

In the original brainstorming of the algorithm development matrices were considered for data

processing and grid mapping. Storing values in matrices serves to further complicate the design

unnecessarily and the use of matrix calculations would increase calculation time which may affect

response time when the program is deployed.

Program Functions:

1) Sort Received Powers(sort_pr.m)

Inputs: 9 Pr values in units of dBm

Saves inputs to a structure and sorts out the three highest values and their associated beacon number

Outputs: 3 greatest Pr values in the possible range of between 0 and -120 dBm, and their associated

beacon number

H

L M

2) Coordinate Mapping(power_to_xy.m)

Inputs: beacon number associated with 3 highest Pr values

Maps beacon number to Cartesian in the pattern described in Figure 3.

Outputs: x y coordinates for each of the three beacons

3) Power to distance conversion(power_to_d_friis.m)

Inputs: three highest power levels

Implementation of Equation 3

Outputs: three distances representing the radius of each reference point for trilateration

4) Error Checking(distance_confirm.m)

Inputs: 3 highest received powers, reference received power

The second method for distance calculation to compare with the results from the first distance

calculation (implementation of the series form of Equation 5).

Outputs: 3 distances representing the radius of each reference point for trilateration

5) Trilateration(trilat2d.m)

Inputs: x y coordinates of the three reference beacons, distances of each reference point

Implementation of trilateration equations

Outputs: unknown x y coordinate

6) Calculation Plotting(circle.m)

Inputs: x y coordinates and d for each reference location, the unknown coordinate, plot formatting

Plots reference points and the circle representing their individual distances represented in Figure 1 as

well as the unknown coordinate with the plot text function

Outputs: figure displaying unknown coordinate graphically

7) Test Fixture(testscript.m)

Inputs: User entered received power values

Calls all of the previously described functions and prompts the user for input data in the command

window.

Outputs: Figure displaying calculation results graphically

Figure 4: Concept flow diagram of program operation

Error Detection:

The radii represented by the three distances must also intersect for all three beacon locations. This can

be tested geometrically by comparing distances based on the origin of each beacon where overlap

indicates a successful basis for trilateration.

Figure 5: Illustration of all three radii not intersecting indicating that a valid trilateration calculation

cannot be performed

Due to the fact that received power can be calculated in two different ways (by knowing the gain, and by

knowing the signal decay exponent), program inputs can be checked for validity. The system equipment

uses 915MHz patch antennas with a gain of 9.6899 dB and the expected path loss exponent for line of

sight is approximately 1.7 .

RSSI is influenced by a few different factors. Fading is how the signal weakens with the distance of the

received power by the inverse square law. Shadowing is where there is no direct line of sight path

between the receiver and the transmitter. Diffraction occurs off of sharp edges in a room. Scattering

off of objects in a room.

Figure 6: Example desired input where the filled circles represent the beacons with the three highest

power levels and the unknown location is assumed to be somewhere between the three.

Figure 7: Example undesired input where the three highest power levels are scattered to the outskirts of

the grid which represents a much larger area where the unknown location could be. Furthermore this

input data does not make sense physically because the node will always be closer to three adjacent

beacons surrounding it than a beacon located on the other side of the grid.

Because for any given three highest power values the location of the unknown coordinate must be

somewhere in the area created by the implied triangle, a larger area indicates a larger possible error. If

beacons on both sides of the grid exhibit highest power levels, this indicates that there is an

inconsistency with the input RSSI values to the program because the node is always physically closest to

the nearest three beacons that are around it as illustrated in figure 5 (where the node is assumed to be

somewhere in the area between beacons 1, 2, and 5).

Program Example Calculations:

Figure 8: Sample figure generated by the program where the intersection of the three circles represents

the approximate area of the unknown point and the plotted asterisk marks the exact calculated

unknown coordinate. The plotted unknown coordinate is somewhat offset from the intersection of the

three signals. The plot text function is used to mark the unknown coordinate with an asterisk. This

slight shifting could be caused by the circle plotting or the text plot function so that the intersection is

not perfectly aligned with the point.

Figure 9: Received power inputs used to generate figure 8

Figure 10: Example of an invalid input data set where the trilateration calculation cannot be performed

and no unknown coordinate is plotted.

Figure 11: Input power values for figure 10 where NaN is printed to the command window for the XY

trilateration calculation results.

Source Code:

%Nikolas Simon

%trilat2d.m

function[ x, y, z ] = trilat2d(x1, x2, x3, y1, y2, y3, d1, d2,

d3, h)

%2-D trilateration function

% a constant of z is included to account for beacons and node not

being

% in the same 2d plane %system of three pythagorean equations solved for x and y

A=x1^2+y1^2-d1^2;

B=x2^2+y2^2-d2^2;

C=x3^2+y3^2-d3^2; xnum=A*(y3-y2)+B*(y1-y3)+C*(y2-y1);

xdem=2*(x1*(y3-y2)+x2*(y1-y3)+x3*(y2-y1)); ynum=A*(x3-x2)+B*(x1-x3)+C*(x2-x1);

ydem=2*(y1*(x3-x2)+y2*(x1-x3)+y3*(x2-x1));

x=xnum/xdem;

y=ynum/ydem; z=h;

%Nikolas Simon

%sort_pr.m

%sorting function where the nineinput power levels are accepted

as inputs

%and Pa Pb Pc represent the three highest power levels. Ba Bb Bc

represent

%the three locations in the structure associated with the three

highest

%power levels function [Pa Ba Pb Bb Pc

Bc]=sort_pr(Pr1,Pr2,Pr3,Pr4,Pr5,Pr6,Pr7,Pr8,Pr9)

%structure containing recieved power input values

beacon.p1=Pr1;

beacon.p2=Pr2;

beacon.p3=Pr3;

beacon.p4=Pr4;

beacon.p5=Pr5;

beacon.p6=Pr6;

beacon.p7=Pr7;

beacon.p8=Pr8;

beacon.p9=Pr9; M=[beacon.p1 beacon.p2 beacon.p3 beacon.p4 beacon.p5 beacon.p6

beacon.p7...

beacon.p8 beacon.p9]; sortedValues = sort(M(:)); %sort values

maxValues = sortedValues(end-2:end); %find the 3 largest values Pa=maxValues(3);

Pb=maxValues(2);

Pc=maxValues(1);

%series of switch case statements to determine the power level's

original

%location in the grid

switch Pa

case beacon.p1

Ba=1;

case beacon.p2

Ba=2;

case beacon.p3

Ba=3;

case beacon.p4

Ba=4;

case beacon.p5

Ba=5;

case beacon.p6

Ba=6;

case beacon.p7

Ba=7;

case beacon.p8

Ba=8;

case beacon.p9

Ba=9;

end switch Pb

case beacon.p1

Bb=1;

case beacon.p2

Bb=2;

case beacon.p3

Bb=3;

case beacon.p4

Bb=4;

case beacon.p5

Bb=5;

case beacon.p6

Bb=6;

case beacon.p7

Bb=7;

case beacon.p8

Bb=8;

case beacon.p9

Bb=9;

end switch Pc

case beacon.p1

Bc=1;

case beacon.p2

Bc=2;

case beacon.p3

Bc=3;

case beacon.p4

Bc=4;

case beacon.p5

Bc=5;

case beacon.p6

Bc=6;

case beacon.p7

Bc=7;

case beacon.p8

Bc=8;

case beacon.p9

Bc=9;

end end

%Nikolas Simon

%power_to_d_friis.m

%friis distance equation implemented for each reference of the

three

%reference beacons

%function to calculate the radii to the unknown coordinate from

the

function[d1 d2 d3] =

power_to_d_friis(Pa,Pb,Pc,wavelength,gain,transmit_p) %convert to miliwatts

pr1=(10.^(Pa/10));

pr2=(10.^(Pb/10));

pr3=(10.^(Pc/10)); pt=(10.^(transmit_p/10)); d1=wavelength/(4*pi*sqrt((pr1/pt)/(gain*gain))); d2=wavelength/(4*pi*sqrt((pr2/pt)/(gain*gain))); d3=wavelength/(4*pi*sqrt((pr3/pt)/(gain*gain)));

end

%Nikolas Simon

%distance_confirm.m

%function to calculate distance based on signal decay exponent

dependence

function[Ed1 Ed2 Ed3] =

distance_confirm(Pa,Pb,Pc,ref_recieved_pwr) %1m refrence distance

d0=1; %expected signal decay exponent for line of sight

n=1.7; %taylor series calculation for each reference distance

k1=(log(10)*(ref_recieved_pwr-Pa))/(10*n);

k2=(log(10)*(ref_recieved_pwr-Pb))/(10*n);

k3=(log(10)*(ref_recieved_pwr-Pc))/(10*n); Ed1=d0*(1+((k1)/(1))+((k1^2)/(factorial(2)))+((k1^3)/(factorial(3

)))+...

((k1^4)/(factorial(4)))+((k1^5)/(factorial(5)))); Ed2=d0*(1+((k2)/(1))+((k2^2)/(factorial(2)))+((k2^3)/(factorial(3

)))+...

((k2^4)/(factorial(4)))+((k2^5)/(factorial(5)))); Ed3=d0*(1+((k3)/(1))+((k3^2)/(factorial(2)))+((k3^3)/(factorial(3

)))+...

((k3^4)/(factorial(4)))+((k3^5)/(factorial(5)))); end

%Nikolas Simon

%power_to_xy.m

%point to point mapping functionto establish the reference grid

function [x1 y1 x2 y2 x3 y3]=power_to_xy(Ba,Bb,Bc) %assumed layout mapping to cartesian with 7 as (1,1) in a 3x3

grid

%1_2_3

%4_5_6

%7_8_9 %scaling set to 1

scalePa=1;

scalePb=1;

scalePc=1;

%depending on the location inputs, the switch statements will

determine the

%cartiesian coordinates

switch Ba

case 1

x1=1*scalePa;

y1=3*scalePa;

case 2

x1=2*scalePa;

y1=3*scalePa;

case 3

x1=3*scalePa;

y1=3*scalePa;

case 4

x1=1*scalePa;

y1=2*scalePa;

case 5

x1=2*scalePa;

y1=2*scalePa;

case 6

x1=3*scalePa;

y1=2*scalePa;

case 7

x1=1*scalePa;

y1=1*scalePa;

case 8

x1=2*scalePa;

y1=1*scalePa;

case 9

x1=3*scalePa;

y1=1*scalePa;

end switch Bb

case 1

x2=1*scalePb;

y2=3*scalePb;

case 2

x2=2*scalePb;

y2=3*scalePb;

case 3

x2=3*scalePb;

y2=3*scalePb;

case 4

x2=1*scalePb;

y2=2*scalePb;

case 5

x2=2*scalePb;

y2=2*scalePb;

case 6

x2=3*scalePb;

y2=2*scalePb;

case 7

x2=1*scalePb;

y2=1*scalePb;

case 8

x2=2*scalePb;

y2=1*scalePb;

case 9

x2=3*scalePb;

y2=1*scalePb;

end switch Bc

case 1

x3=1*scalePc;

y3=3*scalePc;

case 2

x3=2*scalePc;

y3=3*scalePc;

case 3

x3=3*scalePc;

y3=3*scalePc;

case 4

x3=1*scalePc;

y3=2*scalePc;

case 5

x3=2*scalePc;

y3=2*scalePc;

case 6

x3=3*scalePc;

y3=2*scalePc;

case 7

x3=1*scalePc;

y3=1*scalePc;

case 8

x3=2*scalePc;

y3=1*scalePc;

case 9

x3=3*scalePc;

y3=1*scalePc;

end

end

%Nikolas Simon

%circle.m

%plotting function to show the trilateration calculation

graphically

%accepts the three reference xy values and thier respective

distances, the

%input variables a,b,c represent which subplot is being plotted,

theLabel

%is an input string for each subplot title, ux and uy are the

calculated

%coordinates of the unknown node indicated graphically by an

asterisk function circle(x1,y1,r1,x2,y2,r2,x3,y3,r3,a,b,c,theLabel,ux,uy) hold on subplot(a,b,c); th = 0:pi/50:2*pi; %circle 1 xunit1 = r1 * cos(th) + x1; yunit1 = r1 * sin(th) + y1; %circle 2 xunit2 = r2 * cos(th) + x2; yunit2 = r2 * sin(th) + y2; %circle 3 xunit3 = r3 * cos(th) + x3; yunit3 = r3 * sin(th) + y3; plot(xunit1, yunit1,xunit2, yunit2,xunit3, yunit3);

grid on

title(num2str(theLabel)); text(ux,uy,'*'); hold off

%Nikolas Simon

%testscript.m

%test fixture for the calculation functions

%display two different figures using the same circle plotting

function, the

%trilateration results using the friis equation for distance and

the signal

%path exponent equation for distance %define constant values

h=5;%set room height to 5m

wavelength=0.3279;%meters

transmit_p=10;%dBm

gain=9.6899;%antenna gain dB

ref_recieved_pwr=1;%dBm %user entered recieved power levels

% sample values used in documentation -11,-14,-13,-20,-4,-1,-9,-

3,-9

p1 = input('Enter Power Value 1: ');

p2 = input('Enter Power Value 2: ');

p3 = input('Enter Power Value 3: ');

p4 = input('Enter Power Value 4: ');

p5 = input('Enter Power Value 5: ');

p6 = input('Enter Power Value 6: ');

p7 = input('Enter Power Value 7: ');

p8 = input('Enter Power Value 8: ');

p9 = input('Enter Power Value 9: '); %call to the sorting function

[Pa Ba Pb Bb Pc Bc]=sort_pr(p1,p2,p3,p4,p5,p6,p7,p8,p9); %call to the mapping function

[x1 y1 x2 y2 x3 y3]=power_to_xy(Ba,Bb,Bc); %friis distance calculation

[d1 d2 d3] =

power_to_d_friis(Pa,Pb,Pc,wavelength,gain,transmit_p);

%signal decay exponent distance calculation

[Ed1 Ed2 Ed3] = distance_confirm(Pa,Pb,Pc,ref_recieved_pwr); %calls to the trilateration function using

[ x_1, y_1, z_1 ] = trilat2d(x1, x2, x3, y1, y2, y3, d1, d2, d3,

h);

[ x_2, y_2, z_2 ] = trilat2d(x1, x2, x3, y1, y2, y3, Ed1, Ed2,

Ed3, h); %plot the xy point on the graph circle(x1,y1,d1,x2,y2,d2,x3,y3,d3,1,2,1,'Trilateration Using

Friis Distance',x_1,y_1); circle(x1,y1,Ed1,x2,y2,Ed2,x3,y3,Ed3,1,2,2,'Trilateration Using

Signal Path Exponent Distance',x_2,y_2); %display relevant distance and xy values to the command window disp('x y z coordinates Friis')

disp(x_1);

disp(y_1);

disp(z_1); disp('x y z coordinates signal decay exponent')

disp(x_2);

disp(y_2);

disp(z_2); disp('Friis d values')

disp(d1);

disp(d2);

disp(d3); disp('path exponent d values')

disp(Ed1);

disp(Ed2);

disp(Ed3);

Works Referenced:

http://cdn.intechweb.org/pdfs/13525.pdf

http://www.es.ewi.tudelft.nl/msc-theses/2013-Kocsi.pdf

https://books.google.com/books?id=mdQlBAAAQBAJ&pg=PA404&lpg=PA404&dq=signal+decay+expone

nt+rssi&source=bl&ots=7aKCjETv38&sig=ex_vF6Ye1jf31wJ6waNMQRO7ZKs&hl=en&sa=X&ved=0ahUKE

wiO5sWgs-HKAhXFdT4KHesYBSkQ6AEILzAC#v=onepage&q&f=false

http://www.ijser.org/researchpaper%5CWi-Fi-Indoor-Positioning-System-Based-on-RSSI-

Measurements.pdf

http://www.antenna-theory.com/basics/aperture.php

http://www.tscm.com/pwr-dens.pdf

http://cial.csie.ncku.edu.tw/publication/pdf/conference_papers/A%20RSSI-

based%20Algorithm%20for%20Indoor%20Localization%20Using%20ZigBee%20in%20Wireless%20Senso

r%20Network.pdf