scattering model for vegetation canopies and simulation of satellite navigation channels

Scattering Model for Vegetation Canopies and Simulation of

Satellite Navigation Channels

Ph.D. Defense

Frank M. Schubert

Navigation and Communications SectionDepartment of Electronic Systems

Aalborg University

September 14, 2012

1 / 46

Ph.D. StudiesInvolved Institutions

Aalborg University Navigation and Communications

Section

German Aerospace Center (DLR) Institute for Communications and

Navigation in Weling-Oberpfaffenhofen,

Germany

European Space Agency(ESA/ESTEC) Networking/Partnering Initiative

(NPI): Establishing relationsbetween ESA anduniversities/research institutesthrough joint supervision of Ph.D.students

in Noordwijk, The Netherlands

2 / 46

Introduction and Problem StatementMultipath Propagation Deteriorates GNSS Positioning Performance

Satellite 1

Satellite 2

Satellite 3

ReceiverWave propagation

GPS/GNSS receivers track radiosignals transmitted by satellites

Best performance is achieved inclear sky conditions

Objects like trees, forests scattertransmitted signals

This multipath propagationimpairs positioning performance

Goals:

Analyze wave scattering bytrees

Evaluate signal tracking inmultipath-proneenvironments by simulation

3 / 46

Scattering by Vegetation Previous WorkModels Covering Vegetation Scattering in L-band (1-2 GHz)

Measurements/simulations of single trees

Caldeirinha (2001): outdoor andindoor measurements

Fortuny & Sieber (1999): SAR imagingin anechoic chamber

Models for terrestrial communications

de Jong & Herben (2004)

Models for land-mobile satellitecommunications:

Goldhirsh & Vogel (1989) Cheffena & Prez-Fontn (2010)

Dedicated GNSS channel models

Steinga & Lehner (2005, 2007):Vehicles/pedestrians inurban/suburban environments

Koh & Sarabandi (2002): Stationaryreceiver in forest

Caldeirinha

de Jong & Herben

Steinga & Lehner

4 / 46

Scattering by Vegetation Previous Work and MotivationSummary

Previous works cover

Two-dimensional case (terrestrialmodels)

Narrowband models, i.e. delay ofindividual components is notconsidered

Urban/suburban environments forsatellite navigation

Stationary environments

We seek a non-stationary model ofscattering by treetops

Rural environments Moving receiver Suitable for satellite navigation

applications Wideband model True propagation delays need to be

reported

Caldeirinha

de Jong & Herben

Steinga & Lehner

4 / 46

Thesis Contents Overview

GNSS signalgenerator

GNSSreceiver

algorithm

ChannelModeling

SignalProcessingChain

Ionosphericscintil-lations

NarrowbandChannelModel

Scatteringvolume

Tree, forest

Alley

WidebandChannelModel

?

WidebandMeasure-

ments

Alley mea-surement

Smallforest mea-surement

Singletree mea-surement

?

GNSSsatellite

Atmosphericeffects

Multipath propagation GNSSreceiver

Electromagnetic wave propagation

5 / 46


Satellite Navigation ChannelsContents

Simulation of Satellite Navigation Channels

Discriminator Function

Multipath Envelope

Satellite Navigation Channel Signal Simulator

SNACS Simulation Examples

Signal Model for Scattering Volumes

Single Scattering Volume

Channel System Functions

Time-Frequency Correlation Function

Several Scattering Volumes

6 / 46





Multipath Envelope








7 / 46

Simulation Using Discriminator FunctionGPS C/A Code Example

Satellites send spreading codes

Receiver correlates rx signal with locally generated code replica

Correlation function ss() =1Tc

Tc0 c(t)(t tsp/2 )dt

101

C/A code, Prompt

Cod

e

101

Early

Cod

e

0 2 4 6 8 10

101

Late

Cod

e

Time [s]2 1 0 1 2

2

1

0

1

2

3C/A code ACF, chip spacing 1

Time Delay [chips]

Cor

rela

tion

earlylatemultipath contribution 1 ( = 0.4)multipath contribution 2 ( = 0.7)resulting discriminator function

8 / 46

Effects of Multipath Propagation on GNSS ReceiversTwo Components Model, cf. Hagerman (1973), Van Nee (1993), Braasch (1996)

Receiver reads line of sight signal (LOS) one additional multipath

component

GPS C/A error envelope top: component in-phase bottom: out-of-phase 1 chip early-late spacing 0.5 chip spacing

61 62 63 64 65Delay [ns]

0

0.5

1

Power

line of sightMP component

0 500 1000 1500Relative Delay [ns]

50

0

50

RangingError[m

]

Urban and rural areas: strong multipath propagation

Many echoes impinge within few nanoseconds after LOS: high error

Simulation needed for performance assessment9 / 46

GNSS Simulation Methods Correlation Domain

Channel assumed stationary duringintegration time (correlation)

e.g. Navsim, Furthner et al. (2000),Realization of an End-to-End SoftwareSimulator for Navigation Systems

Samples domain

Software defined receivers, e.g. Borreet al. (2007), A Software-Defined GPSand Galileo Receiver A Single-Frequency Approach

2 1 0 1 22

1

0

1

2

3C/A code ACF, chip spacing 1

Time Delay [chips]

Cor

rela

tion

earlylatemultipath contribution 1 ( = 0.4)multipath contribution 2 ( = 0.7)resulting discriminator function

101

C/A code, Prompt

Cod

e

101

Early

Cod

e

0 2 4 6 8 10

101

Late

Cod

e

Time [s]

New GNSS signals: longer integration times, channel non-stationary

Samples domain simulation needed10 / 46

The Satellite Navigation Channel Simulator (SNACS)Overview, Inputs, Outputs

Signalgeneration

Optional:AWGN,

up-conversion,

quantization,low-pass

filter

GNSS signalacquisition

and tracking

GNSS signalparameters

RangeestimationFIR filter

Interpolation

Channelmodel/ mea-surements

Parameters:scenery, tra-jectory, etc.

True range

11 / 46

Alley Drive Simulation Result

mean()= 28.4m

rms()= 32.2m

12 / 46

Simulation ExamplesDLR Urban Channel Model

Lehner & Steinga (2005), A

novel channel model for land

mobile satellite navigation

Scenery definition

Trajectory definition

0 20 40 60 80 100 12010

0

10

20

30

40

50

60

70

80

x [m]

y [m

]

Vehicle trajectory

start

vehicle position, one point per second

Channel response realization

13 / 46

Simulation ExamplesDLR Urban Channel Model, Noise Free Simulations

Trajectory

0 20 40 60 80 100 12010

0

10

20

30

40

50

60

70

80

x [m]

y [m

]

Vehicle trajectory

start

vehicle position, one point per second

Simulation parametersGPS C/A CBOC AltBOC

Sampling frequency 40MHz 60MHz 80MHz

Intermediate fre-

quency

10MHz 15MHz 20MHz

Precorrelation band-

width

8MHz 10MHz 10MHz

Correlation interval 1ms 4ms 1ms

Early-late spacing 1chips 0.4chips 0.5chips

C/A code CBOC(6,1,1/11) AltBOC(15,10)

14 / 46

ConclusionsPart I Simulation of Satellite Navigation Channels

GNSS signal

simulator

implementation

SNACS written in C++, multi-threading faster than Matlab-based implementations

Simulations of

scenarios

Measurements of drive through alley

DLR urban channel model, drive around

the block

Multipath propagation in rural

environments degrades positioning

performance

New GNSS signals Higher bandwidths: frequency-selective

channels

Longer integration times: non-stationary

channels

15 / 46





Multipath Envelope








16 / 46

Thesis Contents Overview

GNSS signalgenerator

GNSSreceiver

algorithm

ChannelModeling

SignalProcessingChain

Ionosphericscintil-lations

NarrowbandChannelModel

Scatteringvolume

Tree, forest

Alley

WidebandChannelModel

?

WidebandMeasure-

ments

Alley mea-surement

Smallforest mea-surement

Singletree mea-surement

?

GNSSsatellite

Atmosphericeffects

Multipath propagation GNSSreceiver

Electromagnetic wave propagation

17 / 46

Geometric Scenery Scenery in 3D: Top view:

0

(t) = 0 + t

V

T

= d1(r)T r

Scattering volume V

: filled with point-source scatterers r to model scatteringcenters

Fixed transmitter in T

Receiver moves on straight-line trajectory (t) = 0 + t

Distances

Transmitterscatterer: d1(r) Scattererreceiver: d2(t, r) Transmitterreceiver: dd(t)

18 / 46

Scattering Centers in TreetopsMade Visible by SAR Imaging

SAR imaging at 1-5.5 MHz of a fir tree in an anechoic chamber

Distinct scattering centers inside the treetop

Figures by Fortuny & Sieber (1999), Three-dimensional synthetic

aperture radar imaging of a fir tree: first results

19 / 46

Geometric Scenery Scenery in 3D: Top view:

0

(t) = 0 + t

Vr

T

dd(t)= T (t)

= d1(r)T r

(t) r = d2(t, r)

Scattering volume V: filled with point-source scatterers r to model scatteringcenters

Fixed transmitter in T

Receiver moves on straight-line trajectory (t) = 0 + t

Distances Transmitterscatterer: d1(r) Scattererreceiver: d2(t, r) Transmitterreceiver: dd(t)

20 / 46

Point-Source ScatterersModeled by Spatial, Marked Point Processes

0

(t) = 0 + t

Vr

T

dd(t)d1(r)

d2(t, r)

Effective scatterers not directly linked to tree constituents absorb system effects, e.g. antenna pattern

Scatterers are modeled by spatial point process {(r, r ) : r } R3 C: marked point process Points r, marks r Intensity function (r) with : V [0,) Conditional power Q(r) with Q : V [0,)

Marks have zero mean E{r} = 0

Marks are mutually uncorrelated E

r r

r, r

= Q(r)1

r = r

21 / 46

Transmitted and Received Signals

0

(t) = 0 + t

Vr

T

dd(t)d1(r)

d2(t, r) ds(t, r) = d1(r) + d2(t, r)

s(t, r) = ds(t, r)/c0

c0: speed of light

Transmitted signal in T can be written as st(t) = Re{st(t)exp(j2ct)} st(t): Baseband signal

Received signal in (t) is modeled as sum of delayed and attenuatedversions of st(t)

sr(t) =

rr

d2(t, r)

mplitde

st(t s(t, r))

Spherical wave propagation is assumed along r(t) path Waves amplitude dependent on distance to scatterer and its weight r

22 / 46

Channel System FunctionsTime-Variant Response

Integral form of the input-output relationship for an LTV channel

sr(t) =

st(t )h(t, )d Time-variant channel response h(t, ) consists of direct and scattered parts

h(t, ) = hd(t, ) + hs(t, ) hd(t, ) : first, no attenuation, magnitude normalized to 1

hs(t, ) =

r

sctterers

r

d2(t, r)

mpl.

exp

j2c

ds(t, r)

phse

( s(t, r))

dely

In the following: scattered part is considered

ds(t, r) = d1(r) + d2(t, r) Model: Measurement:

(t)

Vr

Tdd(t)

d1(r)

d2(t, r)

23 / 46

Time-Variant Response and Doppler-Delay Spread FunctionComparison of Measurements and Model, Single Tree

0 1 2 3 4Time t [s]

0

100

200

300

400

Delay[ns]

100 0 100Doppler Frequency [Hz]

0

100

200

300

400

Delay[ns]

t

0 1 2 3 4Time t [s]

0

100

200

300

400

Delay[ns]

100 0 100Doppler Frequency [Hz]

0

100

200

300

400

Delay[ns]

40 30 20 10 0Power [dB]

24 / 46

Time-Variant ResponseComparison of Measurements and Model, Group of Trees

Vehicles front camera

Measured channel response

Channel model visualization

Modeled channel response

25 / 46

Channel System FunctionsTime-Variant Transfer Function of the Scattered Part

|hs(t, )|2:

3 4 5 6 7Time t [s]

4300

4350

4400

4450

Delay[ns]

d(t)

30

20

10

0

Power[dB]

3 4 5 6 7Time t [s]

50

0

50

Frequency

[M

Hz]

30

20

10

0

Power[dB]

Time-frequency transfer function of the scattered part: Hs(t, ) = F {hs(t, )} =

rr

d2(t,r)exp

j2(c + ) ds(t,r)c0

26 / 46

Channel System FunctionsTime-Variant Response and Transfer Function of the Scattered Part

27 / 46

Lavf52.111.0

scene-httau-Htfs.mp4Media File (video/mp4)

Channel System FunctionsTime-Variant Transfer Function of the Scattered Part, Three Phases

|hs(t, )|2:

3 4 5 6 7Time t [s]

4300

4350

4400

4450

Delay[ns]

d(t)

30

20

10

0

Power[dB]

3 4 5 6 7Time t [s]

50

0

50

Frequency

[M

Hz]

30

20

10

0

Power[dB]

Time-frequency transfer function of the scattered part: Hs(t, ) = F {hs(t, )} =

rr

d2(t,r)exp

j2(c + ) ds(t,r)c0

28 / 46

First- and Second-Order Characterization of the Scattered PartMean and Time-Frequency Correlation Function

Hs(t, ) has zero mean:

E{Hs(t, )} = 0 Goal: time-frequency correlation

function R( , , t, t) =E

Hs(t, )Hs(t, )

Numerical estimation R( , , t, t) =

1K

K1k=0 H

s,k(t, )Hs,k(t

, )

Example of R( , , t, t) with K = 1000 t = 2.5s, = 0MHz

Long computation

Derive closed-form solution ofR( , , t, t)

|Hs(t, )|2:

3 4 5 6 7Time t [s]

50

0

50

Frequency

[M

Hz]

30

20

10

0

Power[dB]

R( , , t, t)

:

2 3Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.001

0.002

29 / 46

Time-Frequency Correlation FunctionClosed-Form Expression of R( , , t, t)

Hs(t, ) =

rr

d2(t,r)exp

j2(c + ) ds(t,r)c0

: spatial, marked point process (r): its intensity function

Goal: time-frequency correlation function R( , , t, t) = E

Hs(t, )Hs(t, )

R() = ErQ(r)g1(r, t, t, , , c, c0)

Campbells Theorem E

r (r)

=

R3 (r)(r)dr

Integral form R() = V Q(r)(r)g1(r, )dr

We define the probability density function (pdf) (r) 1Q(r)(r), =

Q(r)(r)dr

Time-Frequency Correlation FunctionClosed-Form Expression, Approximations

R() = EHs(t, )Hs(t, ) =E

g1(r, t, t, , , c, c0)

(r) 1Q(r)(r), =

Q(r)(r)dr

1. Decouple two factors in g1(r, ) Distance-dependent term is varying

slowly Phase term is varying rapidly

R() E {g2(r, )}E {g3(r, )}2. Assume plane wave propagation on

d1(r), d2(t, r)

R() g4(t, t, , , c, c0)

0

(t)

Vr

T

d1(r)

d2(t, r)

Approx. closed-form of R():

2 3Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.001

0.002

31 / 46

Time-Frequency Correlation FunctionComparison of Approximate Closed-From Expression and Monte Carlo Simulation

Approximate closed-form expression

2 3Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.001

0.002

4.9 5 5.1Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.03

0.06

0.09

0.12

Monte Carlo Simulation (K = 1000)

2 3Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.001

0.002

Rx far away: t = 2.5s, = 0MHz

4.9 5 5.1Time t [s]

50

0

50

Frequency

[M

Hz]

0

0.03

0.06

0.09

0.12

Rx close: t = 5s, = 0MHz32 / 46

Time-Frequency Correlation Function vs. Transfer FunctionCorrelation Function R Reveals Characteristics of Hs(t, )

Hs(t, ):

1 2 3 4 5 6 7 8Time t [s]

50

0

50

Frequency

[M

Hz]

30

20

10

0

Power[dB]

R( , = 0, t, t = const): t = 2.5s

2 2.5 3Time t [s]

0

0.001

0.002

50

0

50

Frequency

[M

Hz]

t = 5s

4.9 5 5.1Time t [s]

0

0.03

0.06

0.09

0.12

50

0

50

t = 7.5s

7 7.5 8Time t [s]

0

0.001

0.002

50

0

50

33 / 46

Time-Frequency Correlation Function vs. Transfer FunctionCorrelation Function R Reveals Characteristics of Hs(t, )

Hs(t, ):

4.9 5 5.1Time t [s]

50

0

50

Frequency

[M

Hz]

30

20

10

0

Power[dB]

R( , = 0, t, t = const): t = 2.5s

2 2.5 3Time t [s]

0

0.001

0.002

50

0

50

Frequency

[M

Hz]

t = 5s

4.9 5 5.1Time t [s]

0

0.03

0.06

0.09

0.12

50

0

50

t = 7.5s

7 7.5 8Time t [s]

0

0.001

0.002

50

0

50

33 / 46

Time-Frequency Correlation FunctionApproximate Closed-From Expr. R( , , t, t) Is Stationary With Respect to for t = t

R( , , t, t) shows: the process Hs(t, ) is non-stationary For t = t: Hs(t, ) becomes stationary R( , , t, t) = R( , t), =

R( , , t, t)

, t = 2.5s, = 0Hz:

2 3Time t [s]

10050

0

50

100

Frequency

[M

Hz]

0

0.001

0.002

R( , , t, t)

t=t=2.5s R( , t = 2.5s)

How well do the approximationswork?

R( , t)

:

4 5 6Time t = t [s]

10050

0

50

100

Frequency

[M

Hz]

0

0.03

0.06

0.09

0.12

Stationarity with respect to :

symmetry along = 0MHz

34 / 46

Time-Frequency Correlation FunctionApproximate Closed-From Expr. R( , , t, t) Is Stationary With Respect to for t = t

far: t = 2.5s, = 0MHz

2 3Time t [s]

10050

0

50

100

Frequency

[M

Hz]

0

0.001

0.002

close: t = 5s, = 0MHz

4.9 5 5.1Time t [s]

10050

0

50

100

Frequency

[M

Hz]

0

0.03

0.06

0.09

0.12

Stationary for t = t = 2.5s, = 0MHz:

0

0.001

0.002

100 0 100Frequency [MHz]

R()MC

t = t = 5s, = 0MHz:

0

0.03

0.06

0.09

0.12

100 0 100Frequency [MHz]

R()MC

Comparison with Monte Carlo Sim. (K = 100000)

35 / 46

Future Application of Time-Frequency Correlation Function IPower Delay Profiles, S(, t) = F {R( , t)}

100 80 60 40 20 0 20 40 60 80 10070

60

50

40

30

20

10PDP of scattered part, RC_LOW_C, tree01

Delay [ns]

Pow

er [d

B]

100 80 60 40 20 0 20 40 60 80 10040

35

30

25

20

15

10

5

0

5PDP of scattered part, scaled to distance, RC_LOW_C, tree01

Delay [ns]

Pow

er [d

B]

36 / 46

Future Application of Time-Frequency Correlation Function IIBayesian Receiver Algorithms

Receivers are unlikely to

generate virtual scenarios

Correlation function of

scattered part provides

average channel

characteristics

Krach et al. (2010), An Efficient Two-FoldMarginalized Bayesian Filter for Multipath

Estimation in Satellite NavigationReceivers

37 / 46


So far: only single scattering volume considered

Now: several scattering volumes

Extend model to cover attenuation of direct component

hd(t, ) = 10d,dB(t)/10

attenuation

exp

j2dd(t)

phase

( d(t))

delay

d(t) = dd(t)/c0 d,dB(t) = dp(T,V , t) is specific attenuation in dB/m

Goal: geometric-stochastic

channel model Definition of scenery needed

deterministic stochastic

0

(t)

Vr

T

d1(r)

d2(t, r)

dd(t)dp(T ,V , t)

38 / 46

Several Scattering VolumesDeterministic Definition of Scenery

Google

Define locations of trees with a GIStool (e.g. Google Earth)

Convert long., lat., alt. coordinates toCartesian

Transform coordinates: vehicle startsin origin and moves in z = 0 plane

Define trajectory by points andinterpolate it with cubic splines

39 / 46

Several Scattering VolumesDeterministic Definition of Scenery, Comparison of Measurement and Model

Google

Measurement

Model

40 / 46

Several Scattering VolumesStochastic Generation of Scenery

1. Define trajectory

2. Draw forward df and sideward dsdisplacements for each street side constant uniform, exponential, or Gaussian

distributions

3. Draw tree shape and dimensions

(t)

Oy dfr,0

dsr,0bt,0

dfr,1

dsr,1bt,1

dfr,2

dsr,2bt,2

dfl,0

dsl,0bt,3

dfl,1

dsl,1bt,4

dfl,2

dsl,2bt,5

Example Winding trajectory Four different segments

41 / 46

Several Scattering VolumesStochastic Generation of Scenery

1. Define trajectory

2. Draw forward df and sideward dsdisplacements for each street side constant uniform, exponential, or Gaussian

distributions

3. Draw tree shape and dimensions

Example Winding trajectory Four different segments

41 / 46

Channel Model C++ ImplementationInput Files, Output Files, Processing, External Tools

Receiver parameters Vehicle speed Trajectory Antenna pattern

Scenery definitionsStochasticScenery Generator

Scenery parameters Trees, forests Geometry Avg. # of scat.

(t)

Oy dfr,0

dsr,0bt,0

dfr,1

dsr,1bt,1

dfr,2

dsr,2bt,2

dfl,0

dsl,0bt,3

dfl,1

dsl,1bt,4

dfl,2

dsl,2bt,5

Channel Model Engine

Transmitter parame-ters Position Frequency

calculates time-variant responseh(t, ) = hd(t, ) + hs(t, )

for all simulation times

Channel Response

Process with SNACS,MATLAB, Python

Scene description files

POV-Ray: ImagesFFmpeg: Videos

42 / 46

SNACS Simulation of Channel Model ResultCombination of Part I & II

C/A code, 1 chip

spacing, 45 dBHz

Scenery stochastically

generated Comparison with an

actual scenariorequires modelcalibration Scattered energy Treetops specific

attenuations

43 / 46

ConclusionsPart II Model of Scattering Volumes

Observations Conclusions

Comparison of derivedchannel system functionsand measurements: goodfit

Model based on point-source scatterers is realistic

Derivation oftime-frequencycorrelation function of thescattered part

Derivation of closed-form expression is possible

Tools of the theory of point processes permitrigorous derivations

Identification of stationarity regions

Comparisons with MonteCarlo simulations: good fit

Indication: assumptions can be justified

Geometric channelmodels require scenerydefinition

Stochastic generation of scenery: convenientgeneration of many trees

44 / 46

Outlook Improve model

downsides Multiple scattering, non-isotropic scattering

Scatterers are static

Diffraction effects, buildingtree interactions

Model calibration Determine scattering coefficients frommeasurements

Directional dependencies

Make use ofR( , , t, t)

Measurement processing, power delay profiles

Bayesian receiver algorithms

Cheffena & Ekman (2008), Modeling theDynamic Effects of Vegetation on Radiowave

Propagation

45 / 46

Outlook Improve model

downsides Multiple scattering, non-isotropic scattering

Scatterers are static

Diffraction effects, buildingtree interactions

Model calibration Determine scattering coefficients frommeasurements

Directional dependencies

Make use ofR( , , t, t)

Measurement processing, power delay profiles

Bayesian receiver algorithms

Enhance GNSSsimulation

SNACS is open-source software

Research, Academics

Compare SNACS simulations of

Channel measurements Developed Model Requires model calibration

Ranging to multiple satellites, position domain

45 / 46



Thank you very much for your attention!

3 4 5 6 7Time t [s]

4300

4350

4400

4450

Delay[ns]

d(t)

30

20

10

0

Power[dB]

46 / 46



Additional Slides

Wave Equations

Derivation of Second Moment

SNACS Implementation

SNACS Signal Generation

C/N0 Estimation Method

DLR Measurement Campaign

SINC Interpolation

Acronyms

47 / 46

Wave Propagation

0

(t) = 0 + t

Vr

T

dd(t)d1(r)

d2(t, r)

Wave propagation is described by Maxwells equations, possiblesolutions are Spherical wave, assumed along d2(t, r):

sr(, t) = Re

r exp

j2cr

sr(t)

exp(j2ct)

Plane wave, assumed along d1(r) and dd(t):

sr(, t) = Re{exp(jk)

sr(t)

exp(j2ct)}

c: carrier frequency, c: wave length, wave vector: k =2cek

received signal: sr(, t) is bandpass version of low-pass sr(t)

48 / 46

First- and Second-Order Characterization of the ChannelMean and Time-Frequency Correlation Function

Hs(t, ) =

rr

d2(t,r)exp

j2(c + ) ds(t,r)c0

Hs(t, ) has zero mean E{Hs(t, )} = 0

Time-frequency correlation function, autocorrelation function (acf) R( , , t, t) = E

Hs(t, )Hs(t, )

R() = En

rQ(r)

d2(t,r)d2(t,r) exp

j2c0

(c + )ds(t, r) (c + )ds(t, r)o

Campbells Theorem E

r (r)

=

R3 (r)(r)dr

R() = VQ(r)(r)

d2(t,r)d2(t,r) exp

j2c0

(c + )ds(t, r) (c + )ds(t, r)

dr

We define the pdf (r) 1Q(r)(r), =

Q(r)(r)dr

Time-Frequency Correlation FunctionApproximation, Closed-Form Solution

R() = EHs(t, )Hs(t , ) = E

1d2 (t,r)d2 (t

,r) exp

j2c0

d1(r), d2(t, r), d2(t , r)

(r) 1Q(r)(r), =

Q(r)(r)dr,

, + c ,( + c)T

R() E

1

d2(t, r)d2(t , r)

E1

E

exp

j2

c0

d1(r), d2(t, r), d2(t , r)

E2

R() 1d,(t)d,(t)

1+e((t),)Te((t),)

d,(t)d,(t)

!

E1

exp

j2

c0

dT, , d,(t), d,(t)

()

E2

Center of gravity: = E {r} =

R3 r(r)dr

Covariance matrix: = E

r rT

(t, t , , ) = c10 [e(T,)( ) +e((t),)( + c) e((t),)( + c)]

() E {exp(j2r )} =

R3 exp(j2r )(r)dr

0

(t)

Vr

T

d1(r)

d2(t, r)

d,(t)

r

dT ,

e(T ,)

e((t),)

50 / 46

Time-Frequency Correlation FunctionClosed-Form Expression, Approximation

R() = EHs(t, )Hs(t, ) =E

g1(t, t, , , r, c, c0)

(r) 1Q(r)(r), =

Q(r)(r)dr

R() g2(t, t, , , c, c0)

Center of gravity:

= E {r} =

R3r(r)dr

Plane wave approximations

d1(r) dT, + e(T,) rd2(t, r) d,(t) + e((t),) r

R() g3(t, t, , , c, c0)

0

(t)

Vr

T

d1(r)

d2(t, r)

d,(t)

r

dT ,

e(T ,)

e((t),)

Approx. closed-form expr. of R():

2 3Time t [s]

50

0

50Frequency

[M

Hz]

0

0.001

0.002

51 / 46

SNACS ImplementationSoftware Structure

Modular object-oriented approach, written in C++ Parallel processing, pipeline approach

Every processing module runs as its own thread Convolution and correlation expand to multiple threads

Modules are connected with circular buffers for asynchronous access

52 / 46

SNACS ImplementationGNSS Signal Generation

53 / 46

A New C/N0 Estimation MethodComparison of Standard Method and New Approach

Standard method by van Dierendonck(?)

Proposed method

SNRW,k =

M=1(

2 +Q

2 )

k

SNRN,k =

M=1

2

k+

M=1Q

2

k

SNRW,k =

M=1

| + jQ |

2

2

k

SNRN,k =h

M=1

| + jQ |

2

i2

k

Common calculation of C/N0:

M = 10, K = 50

P =1K

Kk=1

SNRW,kSNRN,k

C/N0 = 10 log10

1Tc

P1MP

54 / 46

A New C/N0 Estimation MethodSimulation Results

Channel response C/N0 simulation result

10 15 20 25 30 3515

20

25

30

35

40

C/N0 Estimation Results

C/N

0[dB-H

z]

standard methodnew method

10 15 20 25 30 35-10

-5

0

5

10

Reference Trajectory, Speed

Time [s]Reference

Speed[m

/s]

GPS C/A code

0.1chip spacing

AWGN: 35dbHz

New C/N0 estimation method is

less susceptible to Doppler

55 / 46

DLR Land Mobile Satellite Channel ModelMeasurement campaign

Multipath reception cause errors

in GNSS receivers

Perform channel sounding

measurements DLR conducted measurements in

2002 for urban, sub-urban, rural,and pedestrian scenarios frequency: 1460 1560MHz

(L-band) bandwidth: 100Mhz power: 10W (EIRP)

56 / 46

Time-Variant Channel Impulse Responses (CIR)Using channel model data: CIR FIR coefficients interpolation

1 0 1 2 3 4 5 6

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

delay [s]

mag

nitu

de

CIR impulsessinc for CIR impulse 1sinc for CIR impulse 2sum of sinc functionsFIR coefficients

Time-continuous CIR impulses

must be interpolated to

time-discrete FIR coefficients

Low-pass interpolation:

FR(t) =m

k=0k

sin[max(t k )]max(t k )

max = 2smpl2

Example: smpl = 100MHz

57 / 46

Acronyms

acf autocorrelation function

GSCM geometric-stochastic channel model

SNACS Satellite Navigation Channel Signal Simulator

pdf probability density function

58 / 46

Simulation of Satellite Navigation ChannelsDiscriminator FunctionMultipath EnvelopeSatellite Navigation Channel Signal SimulatorSNACS Simulation Examples

Signal Model for Scattering VolumesSingle Scattering VolumeChannel System FunctionsTime-Frequency Correlation FunctionSeveral Scattering Volumes

AppendixAdditional SlidesWave EquationsDerivation of Second MomentSNACS ImplementationSNACS Signal GenerationC/N0 Estimation MethodDLR Measurement CampaignSINC InterpolationAcronyms

scattering model for vegetation canopies and simulation of satellite navigation channels

Science