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Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels
- 1. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels Ph.D. Defense Frank M. Schubert Navigation and Communications Section Department of Electronic Systems Aalborg University September 14, 2012 1 / 46
- 2. Ph.D. Studies Involved Institutions Aalborg University Navigation and Communications Section German Aerospace Center (DLR) Institute for Communications and Navigation in Weßling-Oberpfaffenhofen, Germany European Space Agency (ESA/ESTEC) Networking/Partnering Initiative (NPI): Establishing relations between ESA and universities/research institutes through joint supervision of Ph.D. students in Noordwijk, The Netherlands 2 / 46
- 3. Introduction and Problem Statement Multipath Propagation Deteriorates GNSS Positioning Performance Satellite 1 Satellite 2 Satellite 3 Receiver Wave propagation GPS/GNSS receivers track radio signals transmitted by satellites 3 / 46
- 4. Introduction and Problem Statement Multipath Propagation Deteriorates GNSS Positioning Performance Satellite 1 Receiver Wave propagation GPS/GNSS receivers track radio signals transmitted by satellites Best performance is achieved in clear sky conditions Objects like trees, forests scatter transmitted signals This multipath propagation impairs positioning performance Goals: Analyze wave scattering by trees Evaluate signal tracking in multipath-prone environments by simulation 3 / 46
- 5. Scattering by Vegetation – Previous Work Models Covering Vegetation Scattering in L-band (1-2 GHz) Measurements/simulations of single trees Caldeirinha (2001): outdoor and indoor measurements Fortuny & Sieber (1999): SAR imaging in anechoic chamber Models for terrestrial communications de Jong & Herben (2004) Models for land-mobile satellite communications: Goldhirsh & Vogel (1989) Cheffena & Pérez-Fontán (2010) Dedicated GNSS channel models Steingaß & Lehner (2005, 2007): Vehicles/pedestrians in urban/suburban environments Koh & Sarabandi (2002): Stationary receiver in forest ©Caldeirinha ©de Jong & Herben ©Steingaß & Lehner 4 / 46
- 6. Scattering by Vegetation – Previous Work and Motivation Summary Previous works cover Two-dimensional case (terrestrial models) Narrowband models, i.e. delay of individual components is not considered Urban/suburban environments for satellite navigation Stationary environments → We seek a non-stationary model of scattering by treetops Rural environments Moving receiver Suitable for satellite navigation applications Wideband model True propagation delays need to be reported ©Caldeirinha ©de Jong & Herben ©Steingaß & Lehner 4 / 46
- 7. Thesis Contents Overview GNSS signal generator GNSS receiver algorithm Channel Modeling Signal Processing Chain Ionospheric scintil- lations Narrowband Channel Model × Scattering volume Tree, forest Alley Wideband Channel Model Wideband Measure- ments Alley mea- surement Small forest mea- surement Single tree mea- surement GNSS satellite Atmospheric effects Multipath propagation GNSS receiver Electromagnetic wave propagation 5 / 46
- 8. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels Contents 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
- 9. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels Contents 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 7 / 46
- 10. Simulation Using Discriminator Function GPS C/A Code Example Satellites send spreading codes Receiver correlates rx signal with locally generated code replica Correlation function φss(τ) = 1 Tc Tc 0 c(t)(t − tsp/2 − τ)dt −1 0 1 C/A code, Prompt Code −1 0 1 Early Code 0 2 4 6 8 10 −1 0 1 Late Code Time [µs] −2 −1 0 1 2 −2 −1 0 1 2 3 C/A code ACF, chip spacing 1 Time Delay [chips] Correlation ε early−late multipath contribution 1 (τ = 0.4) multipath contribution 2 (τ = 0.7) resulting discriminator function 8 / 46
- 11. Effects of Multipath Propagation on GNSS Receivers Two 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 Δτ 61 62 63 64 65 Delay τ [ns] 0 0.5 1 Power line of sight MP component 0 500 1000 1500 Relative Delay Δτ [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 assessment 9 / 46
- 12. GNSS Simulation Methods “Correlation Domain“ Channel assumed stationary during integration time (correlation) e.g. Navsim, Furthner et al. (2000), ”Realization of an End-to-End Software Simulator for Navigation Systems“ “Samples domain” Software defined receivers, e.g. Borre et al. (2007), “A Software-Defined GPS and Galileo Receiver – A Single- Frequency Approach” −2 −1 0 1 2 −2 −1 0 1 2 3 C/A code ACF, chip spacing 1 Time Delay [chips] Correlation ε early−late multipath contribution 1 (τ = 0.4) multipath contribution 2 (τ = 0.7) resulting discriminator function −1 0 1 C/A code, Prompt Code −1 0 1 Early Code 0 2 4 6 8 10 −1 0 1 Late Code Time [µs] New GNSS signals: longer integration times, channel non-stationary → Samples domain simulation needed 10 / 46
- 13. The Satellite Navigation Channel Simulator (SNACS) Overview, Inputs, Outputs Signal generation Optional: AWGN, up- conversion, quantization, low-pass filter GNSS signal acquisition and tracking GNSS signal parameters Range estimation FIR filter Interpolation Channel model/ mea- surements Parameters: scenery, tra- jectory, etc. True range Δ 11 / 46
- 14. Alley Drive Simulation Result mean(Δ)= 28.4m rms(Δ)= 32.2m 12 / 46
- 15. Alley Drive Simulation Result 12 / 46
- 16. Simulation Examples DLR Urban Channel Model Lehner & Steingaß (2005), “A novel channel model for land mobile satellite navigation“ Scenery definition Trajectory definition 0 20 40 60 80 100 120 −10 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
- 17. Simulation Examples DLR Urban Channel Model, Noise Free Simulations Trajectory 0 20 40 60 80 100 120 −10 0 10 20 30 40 50 60 70 80 x [m] y[m] Vehicle trajectory start vehicle position, one point per second Simulation parameters GPS C/A CBOC AltBOC Sampling frequency 40 MHz 60 MHz 80 MHz Intermediate fre- quency 10 MHz 15 MHz 20 MHz Precorrelation band- width 8 MHz 10 MHz 10 MHz Correlation interval 1 ms 4 ms 1 ms Early-late spacing 1 chips 0.4 chips 0.5 chips C/A code CBOC(6,1,1/11) AltBOC(15,10) 14 / 46
- 18. Conclusions Part 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
- 19. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels Contents 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 16 / 46
- 20. Thesis Contents Overview GNSS signal generator GNSS receiver algorithm Channel Modeling Signal Processing Chain Ionospheric scintil- lations Narrowband Channel Model × Scattering volume Tree, forest Alley Wideband Channel Model Wideband Measure- ments Alley mea- surement Small forest mea- surement Single tree mea- surement GNSS satellite Atmospheric effects Multipath propagation GNSS receiver Electromagnetic wave propagation 17 / 46
- 21. Geometric Scenery Scenery in 3D: Top view: 0 (t) = 0 + t V T = d1(r) T − r Scattering volume V Fixed transmitter in T Receiver moves on straight-line trajectory (t) = 0 + t 18 / 46
- 22. Scattering Centers in Treetops Made 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
- 23. 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 scattering centers Fixed transmitter in T Receiver moves on straight-line trajectory (t) = 0 + t Distances Transmitter–scatterer: d1(r) Scatterer–receiver: d2(t, r) Transmitter–receiver: dd(t) 20 / 46
- 24. Point-Source Scatterers Modeled 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, βr ) : r ∈ } ⊂ R3 × C: marked point process Points r, marks βr Intensity function ϱ(r) with ϱ : V → [0, ∞) Conditional power Q(r) with Q : V → [0, ∞) Marks have zero mean E {βr } = 0 Marks are mutually uncorrelated E β∗ r βr r, r = Q(r)1 r = r 21 / 46
- 25. 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) τs(t, r) = ds(t, r)/c0 c0: speed of light Transmitted signal in T can be written as ˜st(t) = Re {st(t) exp(j2πƒct)} st(t): Baseband signal Received signal in (t) is modeled as sum of delayed and attenuated versions of st(t) ˜sr(t) = r∈ βr d2(t, r) mplitde ˜st(t − τs(t, r)) Spherical wave propagation is assumed along r–(t) path Wave’s amplitude dependent on distance to scatterer and its weight βr 22 / 46
- 26. Channel System Functions Time-Variant Response Integral form of the input-output relationship for an LTV channel sr(t) = st(t − τ)h(t, τ) dτ Time-variant channel response h(t, τ) consists of direct and scattered parts h(t, τ) = hd(t, τ) + hs(t, τ) hd(t, τ) : first, no attenuation, magnitude normalized to 1 hs(t, τ) = r∈ sctterers βr d2(t, r) mpl. exp −j 2π λc ds(t, r) phse δ(τ − τs(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
- 27. Time-Variant Response and Doppler-Delay Spread Function Comparison of Measurements and Model, Single Tree 0 1 2 3 4 Time t [s] 0 100 200 300 400 Delayτ[ns] −100 0 100 Doppler Frequency ν [Hz] 0 100 200 300 400 Delay[ns] t→ν 0 1 2 3 4 Time t [s] 0 100 200 300 400 Delayτ[ns] −100 0 100 Doppler Frequency ν [Hz] 0 100 200 300 400 Delay[ns] −40 −30 −20 −10 0 Power [dB] 24 / 46
- 28. Time-Variant Response Comparison of Measurements and Model, Group of Trees Vehicle’s front camera Measured channel response Channel model visualization Modeled channel response 25 / 46
- 29. Channel System Functions Time-Variant Transfer Function of the Scattered Part |hs(t, τ)|2 : 3 4 5 6 7 Time t [s] 4300 4350 4400 4450 Delayτ[ns] τd(t) −30 −20 −10 0 Power[dB] ¢ τ → ƒ 3 4 5 6 7 Time t [s] −50 0 50 Frequencyƒ[MHz] −30 −20 −10 0 Power[dB] Time-frequency transfer function of the scattered part: Hs(t, ƒ) = Fτ {hs(t, τ)} = r∈ βr d2(t,r) exp −j2π(ƒc + ƒ) ds(t,r) c0 26 / 46
- 30. Channel System Functions Time-Variant Response and Transfer Function of the Scattered Part 27 / 46
- 31. Channel System Functions Time-Variant Transfer Function of the Scattered Part, Three Phases |hs(t, τ)|2 : 3 4 5 6 7 Time t [s] 4300 4350 4400 4450 Delayτ[ns] τd(t) −30 −20 −10 0 Power[dB] ¢ τ → ƒ 3 4 5 6 7 Time t [s] −50 0 50 Frequencyƒ[MHz] −30 −20 −10 0 Power[dB] Time-frequency transfer function of the scattered part: Hs(t, ƒ) = Fτ {hs(t, τ)} = r∈ βr d2(t,r) exp −j2π(ƒc + ƒ) ds(t,r) c0 28 / 46
- 32. First- and Second-Order Characterization of the Scattered Part Mean and Time-Frequency Correlation Function Hs(t, ƒ) has zero mean: E {Hs(t, ƒ)} = 0 Goal: time-frequency correlation function R(ƒ, ƒ , t, t ) = E Hs ∗ (t, ƒ)Hs(t , ƒ ) Numerical estimation ˆR(ƒ, ƒ , t, t ) = 1 K K−1 k=0 H∗ s,k (t, ƒ)Hs,k(t , ƒ ) Example of ˆR(ƒ, ƒ , t, t ) with K = 1000 t = 2.5 s, ƒ = 0 MHz Long computation → Derive closed-form solution of R(ƒ, ƒ , t, t ) |Hs(t, ƒ)|2 : 3 4 5 6 7 Time t [s] −50 0 50 Frequencyƒ[MHz] −30 −20 −10 0 Power[dB] ˆR(ƒ, ƒ , t, t ) : 2 3 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.001 0.002 29 / 46
- 33. Time-Frequency Correlation Function Closed-Form Expression of R(ƒ, ƒ , t, t ) Hs(t, ƒ) = r∈ βr d2(t,r) exp −j2π(ƒc + ƒ) ds(t,r) c0 : spatial, marked point process ϱ(r): its intensity function Goal: time-frequency correlation function R(ƒ, ƒ , t, t ) = E Hs ∗ (t, ƒ)Hs(t , ƒ ) R(·) = E r∈ Q(r)g1(r, t, t , ƒ, ƒ , ƒc, c0) Campbell’s Theorem E r∈ ƒ(r) = R3 ƒ(r)ϱ(r) dr Integral form R(·) = V Q(r)ϱ(r)g1(r, ·) dr We define the probability density function (pdf) γ(r) −1Q(r)ϱ(r), = Q(r)ϱ(r) dr < ∞ R(·) = Eγ {g1(r, ·)} → Introduce approximations to be able to proceed Hs(t, ƒ) ↓ E {βr } = 0, E β∗ r βr r, r = Q(r)1 r = r ↓ Campbell’s Theorem ↓ R(·) 30 / 46
- 34. Time-Frequency Correlation Function Closed-Form Expression, Approximations R(·) = E Hs(t, ƒ)Hs ∗ (t , ƒ ) = Eγ g1(r, t, t , ƒ, ƒ , ƒc, c0) γ(r) −1Q(r)ϱ(r), = Q(r)ϱ(r) dr 1. Decouple two factors in g1(r, ·) Distance-dependent term is varying slowly Phase term is varying rapidly → R(·) ≈ Eγ {g2(r, ·)} Eγ {g3(r, ·)} 2. Assume plane wave propagation on d1(r), d2(t, r) → R(·) ≈ g4(t, t , ƒ, ƒ , ƒc, c0) 0 (t) Vr T d1(r) d2(t, r) Approx. closed-form of R(·): 2 3 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.001 0.002 31 / 46
- 35. Time-Frequency Correlation Function Comparison of Approximate Closed-From Expression and Monte Carlo Simulation Approximate closed-form expression 2 3 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.001 0.002 4.9 5 5.1 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.03 0.06 0.09 0.12 Monte Carlo Simulation (K = 1000) 2 3 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.001 0.002 Rx far away: t = 2.5 s, ƒ = 0 MHz 4.9 5 5.1 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.03 0.06 0.09 0.12 Rx close: t = 5 s, ƒ = 0 MHz 32 / 46
- 36. Time-Frequency Correlation Function vs. Transfer Function Correlation Function R Reveals Characteristics of Hs(t, ƒ) Hs(t, ƒ): 1 2 3 4 5 6 7 8 Time t [s] −50 0 50 Frequencyƒ[MHz] −30 −20 −10 0 Power[dB] ➀ ➁ ➂ R(ƒ, ƒ = 0, t, t = const): Œ t = 2.5 s 2 2.5 3 Time t [s] 0 0.001 0.002 −50 0 50 Frequencyƒ[MHz] t = 5 s 4.9 5 5.1 Time t [s] 0 0.03 0.06 0.09 0.12 −50 0 50 Ž t = 7.5 s 7 7.5 8 Time t [s] 0 0.001 0.002 −50 0 50 33 / 46
- 37. Time-Frequency Correlation Function vs. Transfer Function Correlation Function R Reveals Characteristics of Hs(t, ƒ) Hs(t, ƒ): 4.9 5 5.1 Time t [s] −50 0 50 Frequencyƒ[MHz] −30 −20 −10 0 Power[dB] ➁ R(ƒ, ƒ = 0, t, t = const): Œ t = 2.5 s 2 2.5 3 Time t [s] 0 0.001 0.002 −50 0 50 Frequencyƒ[MHz] t = 5 s 4.9 5 5.1 Time t [s] 0 0.03 0.06 0.09 0.12 −50 0 50 Ž t = 7.5 s 7 7.5 8 Time t [s] 0 0.001 0.002 −50 0 50 33 / 46
- 38. Time-Frequency Correlation Function Approximate Closed-From Expr. R(ƒ, ƒ , t, t ) Is Stationary With Respect to ƒ for t = t R(ƒ, ƒ , t, t ) shows: the process Hs(t, ƒ) is non-stationary For t = t : Hs(t, ƒ) becomes stationary → R(ƒ, ƒ , t, t) = R(Δƒ, t), Δƒ = ƒ − ƒ R(ƒ, ƒ , t, t ) , t = 2.5 s, ƒ = 0 Hz: 2 3 Time t [s] −100 −50 0 50 100 Frequencyƒ[MHz] 0 0.001 0.002 R(ƒ, ƒ , t, t ) t=t =2.5 s → R(Δƒ, t = 2.5 s) → How well do the approximations work? R(Δƒ, t) : 4 5 6 Time t = t′ [s] −100 −50 0 50 100 FrequencyΔƒ[MHz] 0 0.03 0.06 0.09 0.12 Stationarity with respect to ƒ: symmetry along ƒ = 0 MHz 34 / 46
- 39. Time-Frequency Correlation Function Approximate Closed-From Expr. R(ƒ, ƒ , t, t ) Is Stationary With Respect to ƒ for t = t far: t = 2.5 s, ƒ = 0 MHz 2 3 Time t [s] −100 −50 0 50 100 Frequencyƒ[MHz] 0 0.001 0.002 close: t = 5 s, ƒ = 0 MHz 4.9 5 5.1 Time t [s] −100 −50 0 50 100 Frequencyƒ[MHz] 0 0.03 0.06 0.09 0.12 Stationary for t = t = 2.5 s, ƒ = 0 MHz: 0 0.001 0.002 −100 0 100 Frequency ƒ [MHz] ≈ R(·) MC t = t = 5 s, ƒ = 0 MHz: 0 0.03 0.06 0.09 0.12 −100 0 100 Frequency ƒ [MHz] ≈ R(·) MC Comparison with Monte Carlo Sim. (K = 100000) 35 / 46
- 40. Future Application of Time-Frequency Correlation Function I Power Delay Profiles, S(τ, t) = FΔƒ→τ {R(Δƒ, t)} −100 −80 −60 −40 −20 0 20 40 60 80 100 −70 −60 −50 −40 −30 −20 −10 PDP of scattered part, RC_LOW_C, tree01 Delay [ns] Power[dB] −100 −80 −60 −40 −20 0 20 40 60 80 100 −40 −35 −30 −25 −20 −15 −10 −5 0 5 PDP of scattered part, scaled to distance, RC_LOW_C, tree01 Delay [ns] Power[dB] 36 / 46
- 41. Future Application of Time-Frequency Correlation Function II Bayesian 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-Fold Marginalized Bayesian Filter for Multipath Estimation in Satellite Navigation Receivers” 37 / 46
- 42. Several Scattering Volumes So far: only single scattering volume considered Now: several scattering volumes → Extend model to cover attenuation of direct component hd(t, τ) = 10−d,dB(t)/10 attenuation exp −j 2π λ dd(t) phase δ(τ − τd(t)) delay τd(t) = dd(t)/c0 d,dB(t) = ηdp(T, V, t) η is specific attenuation in dB/m Goal: geometric-stochastic channel model Definition of scenery needed deterministic stochastic 0 (t) V r T d1(r) d2(t, r) dd(t) dp(T,V, t) 38 / 46
- 43. Several Scattering Volumes Deterministic Definition of Scenery ©Google Define locations of trees with a GIS tool (e.g. Google Earth) Convert long., lat., alt. coordinates to Cartesian Transform coordinates: vehicle starts in origin and moves in z = 0 plane Define trajectory by points and interpolate it with cubic splines 39 / 46
- 44. Several Scattering Volumes Deterministic Definition of Scenery, Comparison of Measurement and Model ©Google Measurement Model 40 / 46
- 45. Several Scattering Volumes Stochastic Generation of Scenery 1. Define trajectory 2. Draw forward df and sideward ds displacements for each street side constant uniform, exponential, or Gaussian distributions 3. Draw tree shape and dimensions (t) O y dfr,0 dsr,0 bt,0 dfr,1 dsr,1 bt,1 dfr,2 dsr,2 bt,2 dfl,0 dsl,0 bt,3 dfl,1 dsl,1 bt,4 dfl,2 dsl,2 bt,5 Example Winding trajectory Four different segments 41 / 46
- 46. Several Scattering Volumes Stochastic Generation of Scenery 1. Define trajectory 2. Draw forward df and sideward ds displacements for each street side constant uniform, exponential, or Gaussian distributions 3. Draw tree shape and dimensions Example Winding trajectory Four different segments 41 / 46
- 47. Channel Model C++ Implementation Input Files, Output Files, Processing, External Tools Receiver parameters Vehicle speed Trajectory Antenna pattern Scenery definitions Stochastic Scenery Generator Scenery parameters Trees, forests Geometry Avg. # of scat. (t) O y dfr,0 dsr,0 bt,0 dfr,1 dsr,1 bt,1 dfr,2 dsr,2 bt,2 dfl,0 dsl,0 bt,3 dfl,1 dsl,1 bt,4 dfl,2 dsl,2 bt,5 Channel Model Engine Transmitter parame- ters Position Frequency calculates time-variant response h(t, τ) = hd(t, τ) + hs(t, τ) for all simulation times Channel Response Process with SNACS, MATLAB, Python Scene description files POV-Ray: Images FFmpeg: Videos 42 / 46
- 48. SNACS Simulation of Channel Model Result Combination of Part I & II C/A code, 1 chip spacing, 45 dBHz Scenery stochastically generated Comparison with an actual scenario requires model calibration Scattered energy Treetops’ specific attenuations 43 / 46
- 49. Conclusions Part II – Model of Scattering Volumes Observations Conclusions Comparison of derived channel system functions and measurements: good fit Model based on point-source scatterers is realistic Derivation of time-frequency correlation function of the scattered part Derivation of closed-form expression is possible Tools of the theory of point processes permit rigorous derivations Identification of stationarity regions Comparisons with Monte Carlo simulations: good fit Indication: assumptions can be justified Geometric channel models require scenery definition Stochastic generation of scenery: convenient generation of many trees 44 / 46
- 50. Outlook Improve model downsides Multiple scattering, non-isotropic scattering Scatterers are static Diffraction effects, building–tree interactions Model calibration Determine scattering coefficients from measurements Directional dependencies Make use of R(ƒ, ƒ , t, t ) Measurement processing, power delay profiles Bayesian receiver algorithms Cheffena & Ekman (2008), “Modeling the Dynamic Effects of Vegetation on Radiowave Propagation” 45 / 46
- 51. Outlook Improve model downsides Multiple scattering, non-isotropic scattering Scatterers are static Diffraction effects, building–tree interactions Model calibration Determine scattering coefficients from measurements Directional dependencies Make use of R(ƒ, ƒ , t, t ) Measurement processing, power delay profiles Bayesian receiver algorithms Enhance GNSS simulation 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
- 52. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels Thank you very much for your attention! 3 4 5 6 7 Time t [s] 4300 4350 4400 4450 Delayτ[ns] τd(t) −30 −20 −10 0 Power[dB] 46 / 46
- 53. Scattering Model for Vegetation Canopies and Simulation of Satellite Navigation Channels 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
- 54. Wave Propagation 0 (t) = 0 + t Vr T dd(t) d1(r) d2(t, r) Wave propagation is described by Maxwell’s equations, possible solutions are Spherical wave, assumed along d2(t, r): ˜sr(, t) = Re β r − exp −j 2π λc r − sr(t) exp(j2πƒct) Plane wave, assumed along d1(r) and dd(t): ˜sr(, t) = Re{β exp(−jk) sr(t) exp(j2πƒct)} ƒc: carrier frequency, λc: wave length, wave vector: k = 2π λc ek received signal: ˜sr(, t) is bandpass version of low-pass sr(t) 48 / 46
- 55. First- and Second-Order Characterization of the Channel Mean and Time-Frequency Correlation Function Hs(t, ƒ) = r∈ βr d2(t,r) exp −j2π(ƒc + ƒ) ds(t,r) c0 Hs(t, ƒ) has zero mean E {Hs(t, ƒ)} = 0 Time-frequency correlation function, autocorrelation function (acf) R(ƒ, ƒ , t, t ) = E Hs(t, ƒ)Hs ∗ (t , ƒ ) R(·) = E r∈ Q(r) d2(t,r)d2(t ,r) exp j2π c0 (ƒc + ƒ)ds(t, r) − (ƒc + ƒ )ds(t , r) Campbell’s Theorem E r∈ ƒ(r) = R3 ƒ(r)ϱ(r) dr R(·) = V Q(r)ϱ(r) d2(t,r)d2(t ,r) exp j2π c0 (ƒc + ƒ)ds(t, r) − (ƒc + ƒ )ds(t , r) dr We define the pdf γ(r) −1Q(r)ϱ(r), = Q(r)ϱ(r) dr < ∞ R(·) = Eγ 1 d2(t,r)d2(t ,r) exp j2π c0 d1(r), d2(t, r), d2(t , r) · ¯ƒ ¯ƒ ƒ − ƒ , ƒ + ƒc, −(ƒ + ƒc) T 49 / 46
- 56. Time-Frequency Correlation Function Approximation, Closed-Form Solution R(·) = E Hs(t, ƒ)Hs ∗(t , ƒ ) = Eγ 1 d2(t,r)d2(t ,r) exp j2π c0 d1(r), d2(t, r), d2(t , r) · ¯ƒ γ(r) −1Q(r)ϱ(r), = Q(r)ϱ(r) dr, ¯ƒ ƒ − ƒ , ƒ + ƒc, −(ƒ + ƒc) T R(·) ≈ Eγ 1 d2(t, r)d2(t , r) E1 Eγ exp j2π c0 d1(r), d2(t, r), d2(t , r) · ¯ƒ E2 R(·) ≈ 1 d,μ(t)d,μ(t ) 1 + e((t), μγ)Tγe((t ), μγ) d,μ(t)d,μ(t ) ≈E1 exp j2π c0 dT,μ, d,μ(t), d,μ(t ) · ¯ƒ () ≈E2 Center of gravity: μγ = Eγ {r} = R3 rγ(r) dr Covariance matrix: γ = Eγ ˜r ˜rT (t, t , ƒ, ƒ ) = c−1 0 [e(T, μγ)(ƒ − ƒ ) + e((t), μγ)(ƒ + ƒc) − e((t ), μγ)(ƒ + ƒc)] () Eγ {exp(j2π ˜r · )} = R3 exp(j2π ˜r · )γ(˜r) d˜r 0 (t) Vr T d1(r) d2(t, r) μγ d,μ(t) ˜r dT,μ e(T, μγ) e((t), μγ) 50 / 46
- 57. Time-Frequency Correlation Function Closed-Form Expression, Approximation R(·) = E Hs(t, ƒ)Hs ∗ (t , ƒ ) = Eγ g1(t, t , ƒ, ƒ , r, ƒc, c0) γ(r) −1Q(r)ϱ(r), = Q(r)ϱ(r) dr R(·) ≈ g2(t, t , ƒ, ƒ , ƒc, c0) Center of gravity: μγ = Eγ {r} = R3 rγ(r) dr Plane wave approximations d1(r) ≈ dT,μ + e(T, μγ) · ˜r d2(t, r) ≈ d,μ(t) + e((t), μγ) · ˜r R(·) ≈ g3(t, t , ƒ, ƒ , ƒc, c0) 0 (t) Vr T d1(r) d2(t, r) μγ d,μ(t) ˜r dT,μ e(T, μγ) e((t), μγ) Approx. closed-form expr. of R(·): 2 3 Time t [s] −50 0 50 Frequencyƒ[MHz] 0 0.001 0.002 51 / 46
- 58. SNACS Implementation Software 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
- 59. SNACS Implementation GNSS Signal Generation 53 / 46
- 60. A New C/N0 Estimation Method Comparison of Standard Method and New Approach Standard method by van Dierendonck (?) Proposed method SNRW,k = M =1 (2 + Q2 ) k SNRN,k = M =1 2 k + M =1 Q 2 k SNRW,k = M =1 | + jQ| − π 2 2 k SNRN,k = M =1 | + jQ| − π 2 2 k Common calculation of C/N0: M = 10, K = 50 μP = 1 K K k=1 SNRW,k SNRN,k C/N0 = 10 log10 1 Tc μP−1 M−μP 54 / 46
- 61. A New C/N0 Estimation Method Simulation Results Channel response C/N0 simulation result 10 15 20 25 30 35 15 20 25 30 35 40 C/N0 Estimation Results C/N0[dB-Hz] standard method new method 10 15 20 25 30 35 -10 -5 0 5 10 Reference Trajectory, Speed Time [s] ReferenceSpeed[m/s] GPS C/A code 0.1 chip spacing AWGN: 35 dbHz New C/N0 estimation method is less susceptible to Doppler 55 / 46
- 62. DLR Land Mobile Satellite Channel Model Measurement 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 − 1560 MHz (L-band) bandwidth: 100 Mhz power: 10 W (EIRP) 56 / 46
- 63. Time-Variant Channel Impulse Responses (CIR) Using channel model data: CIR → FIR coefficients interpolation −1 0 1 2 3 4 5 6 x 10 −8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 delay τ [s] magnitude CIR impulses sinc for CIR impulse 1 sinc for CIR impulse 2 sum of sinc functions FIR coefficients Time-continuous CIR impulses must be interpolated to time-discrete FIR coefficients Low-pass interpolation: FR(t) = m k=0 αk · sin[ωmax(t − τk )] ωmax(t − τk ) ωmax = 2π ƒsmpl 2 Example: ƒsmpl = 100 MHz 57 / 46
- 64. Acronyms acf autocorrelation function GSCM geometric-stochastic channel model SNACS Satellite Navigation Channel Signal Simulator pdf probability density function 58 / 46