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Wireless Communications Systems M. LUISE 2

Basic Concepts (1/4)BasicBasic ConceptsConcepts (1/4)(1/4)

The position of a certain point in

space can be found from distances

(ranges) measured from this point to

some other known positions in space

2D user positioning problem

Transmitters O1 and O2 aresynchronized

The receiver issynchronized with both (withthe network)

The range dcan be derivedfrom a propagation-time

measurement , d=c(c=speed of light)

Both A and B are solutionsof the problem!

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AA thirdthird trasmittertrasmitter(O(O33)) wouldwould bebe requiredrequired toto solve thesolve the ambiguityambiguity::

Hypotheses:

Transmitters O1, O2 and O3are synchronized

The receiver issynchronized with the wholenetwork

B represents the trueposition of the receiver

PROBLEM: this approach cant be used as it is, because

the receiver is not synchronized with the network!

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TheThe receiverreceiver clockclock hashas anan offsetoffset tt

Hypotheses:

The receiver is synchronized

with the whole network

The receivermeasurements contain aconstant unknown time-shift

t = / c

The receiver position B can be obtained by

solving a nonlinear system of 3 equations

and 3 unknowns (xB, yB, ):

( ) ( )( ) ( )

( ) ( )

1 1

2 2

3 3

2 2

1

2 2

2

2 2

3

O B O B

O B O B

O B O B

d x x y y

d x x y y

d x x y y

+ = +

+ = + + = +

Transmitters O1, O2 and O3 aresynchronized

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Basic Concepts (4/4) (GPS in a nutshell)BasicBasic ConceptsConcepts (4/4) (GPS in a(4/4) (GPS in a nutshellnutshell))

In three dimensions: we need ranging from 4 reference points

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Wireless Communications Systems M. LUISE 6Receiver Coordinates and Time

((WeWellll seesee laterlater))

RecapRecapRecap

( ) ( ) ( )1

0

2N

k c

k

r t C c g t kT w t

=

= +

R c=

( ) ( ) ( )22 2

, , , , 1,2,3,4 x k y k z k k cx p y p z p R c k + + = =

ReceivedReceived signalsignal

DelayDelay EstimationEstimation

RangeRange EstimateEstimate

((PseudoPseudo--rangerange))

PositioningPositioning EquationsEquations (pkk-th satellite position, c RX clock bias)

c is the speed of ligth

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Systems of coordinatesSystemsSystems ofof coordinatescoordinates

Earth-centered

Spherical

Geocentric

Geodetic

Topocentric (local)

Cartesian

Fixed (ECEF)

Inertial

(ECI)

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The shape of the EarthTheThe shapeshape of theof the EarthEarth

GeodesyGeodesy isis the sciencethe science concernedconcernedwihwih thethestudystudy of theof theshapeshape andandsizesize of theof theEarthEarth in thein the

geometricgeometric sensesense asas wellwellasas withwith thetheformform of theof the equipotentialequipotentialsurfacessurfaces of theof thegravitygravity potentialpotential

F. R.F. R. HelmertHelmert (1880)(1880)The Earth as a Sphere

The founder of scientific geodesy was Eratosthenes (276-195 BC) of Alexandria who,assuming the Earth was spherical, deduced from measurements a radius for the Earth.

The Earth as an Ellipsoid

Towards the end of the 17th century, Newton demonstrated that the concept of a trulyspherical Earth was inadequate as an explanation of the equilibrium of the ocean surface,owing to the Earth rotation: he showed, by means of a simple theoretical model, that thehydrostatic equilibrium would be maintained if the equatorial axis were longer than thepolar axis. This is equivalent to the statement that the body is flattened towards the pole.

The Earth as a GeoidLaplace (1802), Gauss (1828), Bessels (1837) and others had already recognized that theassumption of an ellipsoidal model was not tenable when compared against high accuracyobservations. Listing (1873) had given the name geoidto the equipotential surface of theEarths gravity field which would coincide with the ocean surface, if the Earth were

undisturbed and without topography.

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The World Geodetic System 1984 (WGS-84)The WorldThe World GeodeticGeodetic SystemSystem 1984 (WGS1984 (WGS--84)84)

0.0818191908426Eccentricity

Flattening

6378137 maSemi-major axis

ValueSymbolParameter

=p a -be a

WGS-84 Reference Ellipsoid

=2 2

e 2

a -be

a

The most used and probably the most accurate reference frame is the

World-Geodetic System1984 (WGS-84) Reference Ellipsoid

1298.277223563

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Earth-Centered Inertial (ECI) coordinatesEarthEarth--CenteredCentered InertialInertial (ECI)(ECI) coordinatescoordinates

Origin:

Z-Axis:

X-Axis:

Y-Axis:

Earths center of mass

direction of mean rotational axis of Earth

direction of the vernal equinox (i.e. the

intersection bewteen the ecliptical plane

and the plane of the equator)

direction orthogonal to Z-Axis and X-

Axis

cos sin 0

sin cos 0

0 0 1

ECI ECEF

ECI ECEF

ECI ECEF

x x

y y

z z

=

( )0 0E t t = +

angle between the vernal equinox and the

Greenwich meridian at reference time

E=

0 =

0t

7.292115 10-5

rad/s (WGS-84)

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Geocentric coordinatesGeocentricGeocentric coordinatescoordinates

Assuming the Earth as a sphere:

2 2 2

ECEF ECEF ECEF r x y z = + +

distance:

latitude:

2 2arctan ECEFgeoc

ECEF ECEF

zx y

= +

longitude:

arctan ECEFgeoc

ECEF

y

x

=

altitude:

geoc E h r r=

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Geodetic coordinatesGeodeticGeodetic coordinatescoordinates

Assuming the Earth as an ellipsoid:

( )

2

3

2 3

2sin

1arctan

2 cos

p p

ECEF

p

geod

p p

e ez a

e

p e e a

+

=

latitude:

longitude:

arctan ECEF geod geoc

ECEF

y

x

= =

altitude:

cosgeod

geod

ph =

0.0818191908426Eccentricity

1/298.277223563Flattening

6378137 maSemi-major axis

ValueSymbolParameter

=pa -b

ea

WGS-84 Reference Ellipsoid

=2 2

e 2

a -be

awhere:

( )2 2

2 2, arctan ,

1 1 sin

ECEF ECEF ECEF

pe geod

z a p x y

p e e

= + = =

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Topocentric reference system (1/2)TopocentricTopocentric referencereference system (1/2)system (1/2)

Such a spherical reference system can be useful when locating a celestial body with respect to the

observer position (e.g.: estimating the elevation of a SV, computing carrier-to-noise ratio):

Origin:

u-Axis:

n-Axis:

e-Axis:

observers position

direction of local vertical

direction of the North pole

direction orthogonal to u-Axis and n-

Axis

ENU (East

North

Up) coordinates

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Topocentric reference system (2/2)TopocentricTopocentric referencereference system (2/2)system (2/2)

ENU (EastNorthUp) coordinates

sin cos 0

sin cos sin sin cos

cos cos cos sin sin

geod geod

geod geod geod geod geod

geod geod geod geod geod

E x

N y

U z

=

elevation:

azimuth:

range:

( )2 2arctan U E N = +

( )arctan E N =

2 2 2 E N U = + +

{, , ,C O ECEF ECEF ECEF x y z = =

where:

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Range and Pseudorange (2/2)RangeRange and Pseudorange (2/2)and Pseudorange (2/2)

( ) ( ) ( )at n nm atmu sus r rsuc T T t r tt tct t ct tt t = + + + + + = + + + +

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The impact of measurement noise (1/2)The impact ofThe impact of measurementmeasurement noisenoise (1/2)(1/2)

The measurement of geometric range is affected by several noise sources:

Ephemeris data:errors in the transmitted location of the satellite

Satellite clock:errors in the transmitted clock

Ionosphere: errors in the corrections of pseudorange caused by ionosphericeffects (after applying ionospheric model)

Troposphere:errors in the corrections of pseudorange caused by troposphericeffects (after applying tropospheric model)

Multipath:errors caused by reflected signals entering the receiver antenna

Receiver: errors in the receivers measurement of range caused by thermalnoise, software accuracy, interchannel biases

Selective Availability (SA): intentional degradation introduced by the system(removed since May 2000)

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The impact of measurement noise (2/2)The impact ofThe impact of measurementmeasurement noisenoise (2/2)(2/2)

18.00.018.0Selective Availability

9.09.00.5Receiver

1.41.01.0Multipath

0.70.50.5Troposphere

4.00.54.0Ionosphere

2.10.72.0Satellite clock

2.10.02.1Ephemeris data

TotalRandomBiasError source

One-sigma error (m)

This kind of errors cannot be corrected by any Positioning algoritm: in order to mitigate tn, it can be

modeled as a Gaussian R.V. and reduced by using a statistical approach (e.g. the Extended Kalman Filter):

( ) ( ) N = n2 2 2 2 2 2 2 2 2 2

t n2.1 + 2.1 + 4.0 + 0.7 + 1.4 + 9.0 m 110 m t 0, 110 m

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The Sagnac effect (1/2)TheThe SagnacSagnac effecteffect (1/2)(1/2)

During the propagation time ofthe signal transmission, a clock

on the surface of the Earth will

experience a finite time-shift with

respect to the resting referenceframe at the geocenter.

The measured range is rather

than .

r

r

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The ideal approachThe idealThe ideal approachapproach

Pseudorange: ( )a stmu nrr c c tt t t t = + + + +

rt = st = atm u nt tr c ct = + +

Each block requires at least a

coarse user position estimation

An (iterative) estimation algorithm is needed!

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The Position Estimation iterative algorithm (1/4)The PositionThe Position EstimationEstimation iterativeiterative algorithmalgorithm (1/4)(1/4)

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The computation of user position is based upon a non-linear

system, as shown in the following

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The tropopheric model is based upon statistical estimation of temperature,

pressure and humidity (e.g.: models proposed by Niell (1996) and Collins &

Langley (1997) for WAAS)

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The Position Estimation iterative algorithm (summary)The PositionThe Position EstimationEstimation iterativeiterative algorithmalgorithm ((summarysummary))

The tropopheric model is based uponstatistical estimation of temperature,

pressure and humidity (e.g.: models

proposed by Niell (1996) and Collins

& Langley (1997) for WAAS)

The computation of user position isbased upon a non-linear system, as

shown in the following

GPS Message

An accurate description of the algorithms for satellite position computation, clock offset and relativistic effects

correction and ionospheric delay estimation can be found in NAVSTAR GPS Interface Control Document (IDC)

ICD-GPS-200, Rev. C-PR (http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm )

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Computation of user position (1/3)ComputationComputation ofof useruser position (1/3)position (1/3)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1 1

2 2 2

3 3 3

4 4 4

2 2 2

1

2 2 22

2 2 2

3

2 2 2

4

s u s u s u u

s u s u s u u

s u s u s u u

s u s u s u u

x x y y z z c t

x x y y z z c t

x x y y z z c t

x x y y z z c t

= + + +

= + + + = + + +

= + + +

NonNon--linearlinear system of 4system of 4 equationsequations in 4in 4 unknownsunknowns ((xxuu,, yyuu,, zzuu, t, tuu))

( )= f

( ) ( )1 2 3 4, , , , , , ,T T

u u u ux y z c t

( )1 2 3 4( ) ( ), ( ), ( ), ( )T

f f f f f

( ) ( ) ( ) ( )2 2 2

i i ii s u s u s u uf x x y y z z c t + + +

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Computation of user position (3/3)ComputationComputation ofof useruser position (3/3)position (3/3)

2nd step: iteration of the solution

0

=

1, ,4:i =

( ) ( ) ( )2 2 2

i i ii s u s u s u

r x x y y z z = + +

i i u i

r c t = +

,1 ,2 ,3 ,4

, , , 1

i i i s u s u s u

i i i i

i i i

x x y y z z a a a a

r r r

= = = =

1 = A

?

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GPS constellationGPSGPS constellationconstellation

24+424+4 satellitessatellites aboutabout 11

tonton eacheach

Average height 20,192Average height 20,192

km (periodkm (period aboutabout 12 h)12 h)

Average speed 3874Average speed 3874

m/s (14,000 km/h)m/s (14,000 km/h)66 orbitsorbits

55 deg55 deg wrtwrt thethe EquatorEquator

AtAt leastleast 66 satellitessatellites arearevisiblevisible at theat the samesame timetime

in openin open fieldfield onon anyany

pointpoint ofof thethe EarthEarth

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Satellite selectionSatelliteSatellite selectionselection

NN> 4> 4 satellitessatellites areare visiblevisible andand trackedtracked

Dilution of Precision (DOP):amidst all the available satellites, the four ones

which guarantee the best solution accuracy

(according to a statistical analysis) are chosen.

Least-Squares solution:

all the N pseudorange measurements are used

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Dilution of Precision (1/3)Dilution of Precision (1/3)Dilution of Precision (1/3)

Dilution of Precision (DOP)=Propagation to the final coordinates ofmeasurement errors on the range

Assume that the range measurement (so called pseudorange) is affected by an

error c= + = +

( ) ( )21 2 3 4, , , , 0,i N

( ) ( ){ } 2( )TE c =C I

Receiver noise

AtmosphereSatellite positionMultipath

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Dilution of Precision (2/3)DilutionDilution of Precision (2/3)of Precision (2/3)

We can compute the variance (accuracy) of the user position/time:

2 1 2( ) ( )Tc

=C A A V

11 12 13 14

21 22 23 242

31 32 33 34

4

2

2

2

21 42 4 4

23 4

C C C

C C C

C C C

C C C

u

u

u

u u u u u u

u u u u u u

u u u u u u

u u u u u u u

x y x z x t

x y y z y t

x z y z z t

x t y t

y

t

x

z

z t

c v v v v

c v v v v

v v v vc

v v v vc c c c

= =

C

1 = A

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Dilution of Precision (3/3)DilutionDilution of Precision (3/3)of Precision (3/3)

The user position standard deviation is minimized (accuracy maximized) bychoosing the 4-satellite set that minimizes one of the following quantities (tipically,GDOP):

Vertical Dilution of Precision(VDOP) 33

uz

n

VDOP v

=

Horizontal Dilution of Precision(HDOP)

2 2

11 22

u ux y

n

DOP v v

+= +

Position Dilution of Precision(PDOP)

2 2 2

11 22 33

u u u x y z

n

PDOP v v v

+ += + +

Time Dilution of Precision

(TDOP)

44ut

n

cTDOP v

=

Geometrical Dilution ofPrecision (GDOP)

2 2 2 2 2

11 22 33 44

u u u u x y z t

n

cGDOP

v v v v

+ + + =

= + + +

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Least-Squares solutionLeastLeast--SquaresSquares solutionsolution

This approach utilizes all theNpseudorange measurementsthat are available; the algorithm is very similar to the iterative

solution with 4 satellites just seen.

Using the usual definitions for and A, and considering N>4satellites, we find

( )1

T T

= A A A

where now A is an Nx4 matrix, is the (4x4)

pseudo-inverse of A, and where we use this into the sameiterative procedure as before

( )1T T

A A A

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General Architecture of a Positioning RXGeneral Architecture of a Positioning RXGeneral Architecture of a Positioning RX

LO

RF Filter IF Filter AGC

AD

AD

RC

RC

RF

Front-End

Carrier

NCO

A-to-DConverter

Real-to-ComplexConverter

Code

NCO

Discriminators

Code Tracking(DLL)

Phase & FreqTracking

(PLL / FLL)

Nav Data BitsRe-Generation

Loop

AGC Control

Loop

Pseudo-rangeGeneration

Nav DataDecoding

NavigationNavigationProcessingProcessing

Receiver

Monitoring

Data I/OCommunication

DSP Application

RF Filter

Antenna

LNA

Mixer

RF Cable

Local Oscillator

CarrierCounter-rotation

CodeDespreading

Digital-domain processing

Digital Front-End

USB

PC

GPS Antenna

The SW Radio !http://www.gnu.org/software/gnuradio/

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Experimental Results (1/3)ExperimentalExperimental ResultsResults (1/3)(1/3)

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Altitude(m)

90858075706560555045403530252015105

Time epoch n

43.78150

43.78140

43.78130

43.78120

43.78110

43.78100

43.78090

43.78080

43.78070

43.78060

43.78050

Latitude(

)

10.3987510.3985010.3982510.39800

Longitude ()

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150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Altitude(m)

8075706560555045403530252015105

Time epoch n

43.78150

43.78140

43.78130

43.78120

43.78110

43.78100

43.78090

43.78080

43.78070

43.78060

43.78050

Latitude()

10.3987510.3985010.3982510.39800

Longitude ()

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43.78150

43.78140

43.78130

43.78120

43.78110

43.78100

43.78090

43.78080

43.78070

43.78060

43.78050

Latitude()

10.3987510.3985010.3982510.39800

Longitude ()

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Altitude(m)

858075706560555045403530252015105

Time epoch n

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PulchraPulchra suntsunt quaequae videmusvidemus

QuaeQuae scimusscimus pulchriorapulchriora

Longe pulcherrima quaeLonge pulcherrima quae ignoramusignoramus......

Niels Stensen (Niccol Stenone, 1638-86)

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