Luận án Location-aware multipath-based channel prediction for next generation wireless communication systems

5G and beyond-5G will be characterized by a wide variety of use cases, as well as
orders-of-magnitude increases in mobile data volume per area, number of connected
devices, and typical user data rate, all compared to current mobile communication
systems [1]. It is visioned that context information in general and location information
in particular can complement both traditional and disruptive technologies in addressing
several of the challenges in 5G and beyond-5G networks.
A majority of 5G devices will be able to rely on ubiquitous location awareness,
supported through several technological developments: a multitude of global navigation satellite systems (GNSS) are being rolled out, complementing the current global
positioning system (GPS). Combined with ground support systems and multiband operation, these systems aim to offer location accuracies around 1 m in open sky. In
scenarios where GNSS is weak or unavailable (in urban canyons or indoors), other
local radio-based technologies such as ultrawideband (UWB), Bluetooth, ZigBee, and
radio frequency identification (RFID), will complement current Wi-Fi-based positioning. Together, they will also result in submeter accuracy.
Accurate location information can be utilized across all layers of the communication
protocol stack [2] to improve the network performance. At higher layers, location
information is often employed directly in natural applications, such as location-aware
information delivery [3], or location-aware traffic-related services [4]. At lower layers,
i.e., PHY, MAC, network and transport layers, a channel quality metric (CQM) maps
between a channel performance indicator and the position is often used. Then at time
t, when the user position is predicted, from the CQM, channel qualities at new position
can be estimated before the user actually goes there. Certain system parameter such
as power, modulation type, etc can be adjusted to tailor to the channel condition at
the new position. 
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  1. [10] Chen X., Lu J., Fan P., and Letaief K.B. (2017). Massive mimo beamforming with transmit diversity for high mobility wireless communications. IEEE Access, 5:pp. 23032–23045. doi:10.1109/ACCESS.2017.2766157. [11] Kela P., Costa M., Turkka J., Koivisto M., Werner J., Hakkarainen A., Valkama M., Jantti R., and Leppanen K. (2016). Location based beamforming in 5g ultra- dense networks. In 2016 IEEE 84th Vehicular Technology Conference (VTC- Fall), pp. 1–7. doi:10.1109/VTCFall.2016.7881072. [12] Aviles J.C. and Kouki A. (2016). Position-aided mm-wave beam training un- der nlos conditions. IEEE Access, 4:pp. 8703–8714. doi:10.1109/ACCESS.2016. 2631222. [13] Wang Y., Klautau A., Ribero M., Soong A.C.K., and Heath R.W. (2019). Mmwave vehicular beam selection with situational awareness using machine learn- ing. IEEE Access, 7:pp. 87479–87493. doi:10.1109/ACCESS.2019.2922064. [14] Automotive sensors and electronics expo. [15] Wen N. and Berry R. (2006). Information propagation for location-based mac protocols in vehicular networks. In 2006 40th Annual Conference on Information Sciences and Systems, pp. 1242–1247. doi:10.1109/CISS.2006.286655. [16] Katragadda S., Ganesh Murthy C., Ranga Rao M., Mohan Kumar S., and Sachin R. (2003). A decentralized location-based channel access protocol for inter-vehicle communication. In The 57th IEEE Semiannual Vehicular Technology Conference, 2003. VTC 2003-Spring., volume 3, pp. 1831–1835 vol.3. doi:10.1109/VETECS. 2003.1207140. [17] Va V., Choi J., Shimizu T., Bansal G., and Heath R.W. (2018). Inverse multipath fingerprinting for millimeter wave v2i beam alignment. IEEE Transactions on Vehicular Technology, 67(5):pp. 4042–4058. doi:10.1109/TVT.2017.2787627. [18] Va V., Shimizu T., Bansal G., and Heath R.W. (2017). Position-aided millime- ter wave v2i beam alignment: A learning-to-rank approach. In 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Com- munications (PIMRC), pp. 1–5. doi:10.1109/PIMRC.2017.8292679. [19] Klautau A., Batista P., González-Prelcic N., Wang Y., and Heath R.W. (2018). 5g mimo data for machine learning: Application to beam-selection using deep learning. In 2018 Information Theory and Applications Workshop (ITA), pp. 1–9. doi:10.1109/ITA.2018.8503086. [20] Wang Y., Narasimha M., and Heath R.W. (2018). Mmwave beam prediction with situational awareness: A machine learning approach. In 2018 IEEE 19th Inter- 69
  2. for low-rate wireless personal area networks (lr-wpans): Amendment to add alter- nate phy (amendment of ieee std 802.15.4). IEEE Approved Std P802.15.4a/D7, Jan 2007 . [31] Cassioli D., Win M.Z., and Molisch A.F. (Aug 2002). The ultra-wide bandwidth indoor channel: from statistical model to simulations. IEEE Journal on Selected Areas in Communications, 20(6):pp. 1247–1257. ISSN 1558-0008. doi:10.1109/ JSAC.2002.801228. [32] Fuschini F., Vitucci E.M., Barbiroli M., Falciasecca G., and Degli-Esposti V. (2015). Ray tracing propagation modeling for future small-cell and indoor ap- plications: A review of current techniques. Radio Science, 50(6):pp. 469–485. doi:10.1002/2015RS005659. [33] Liang G. and Bertoni H. (1998). A new approach to 3-d ray tracing for propagation prediction in cities. IEEE Transactions on Antennas and Propagation, 46(6):pp. 853–863. doi:10.1109/8.686774. [34] Athanasiadou G.E., Nix A.R., and McGeehan J.P. (March 2000). A microcellular ray-tracing propagation model and evaluation of its narrow-band and wide-band predictions. IEEE Journal on Selected Areas in Communications, 18(3):pp. 322– 335. doi:10.1109/49.840192. [35] Kloch C., Liang G., Andersen J., Pedersen G., and Bertoni H. (2001). Comparison of measured and predicted time dispersion and direction of arrival for multipath in a small cell environment. IEEE Transactions on Antennas and Propagation, 49(9):pp. 1254–1263. doi:10.1109/8.947016. [36] Degli-Eposti V., Lombardi G., Passerini C., and Riva G. (2001). Wide-band mea- surement and ray-tracing simulation of the 1900-mhz indoor propagation channel: comparison criteria and results. IEEE Transactions on Antennas and Propaga- tion, 49(7):pp. 1101–1110. doi:10.1109/8.933490. [37] Degli-Esposti V., Guiducci D., de’Marsi A., Azzi P., and Fuschini F. (2004). An advanced field prediction model including diffuse scattering. IEEE Transac- tions on Antennas and Propagation, 52(7):pp. 1717–1728. doi:10.1109/TAP.2004. 831299. [38] Rossi J.P. and Gabillet Y. (2002). A mixed ray launching/tracing method for full 3-d uhf propagation modeling and comparison with wide-band measurements. IEEE Transactions on Antennas and Propagation, 50(4):pp. 517–523. doi:10. 1109/TAP.2002.1003388. [39] Didascalou D., Dottling M., Geng N., and Wiesbeck W. (2003). An ap- proach to include stochastic rough surface scattering into deterministic ray-optical 71
  3. [50] Leitinger E., Meissner P., Rüdisser C., Dumphart G., and Witrisal K. (2015). Evaluation of position-related information in multipath components for indoor positioning. IEEE Journal on Selected Areas in Communications, 33(11):pp. 2313–2328. doi:10.1109/JSAC.2015.2430520. [51] Leitinger E. (2016). Cognitive indoor positioning and tracking using multipath channel Information. Ph.D. thesis, Graz University of Technology. [52] Leitinger E., Meyer F., Hlawatsch F., Witrisal K., Tufvesson F., and Win M.Z. (2019). A belief propagation algorithm for multipath-based slam. IEEE Trans- actions on Wireless Communications, 18(12):pp. 5613–5629. doi:10.1109/TWC. 2019.2937781. [53] Molisch A.F. (2009). Ultra-wide-band propagation channels. Proceedings of the IEEE, 97(2):pp. 353–371. doi:10.1109/JPROC.2008.2008836. [54] Meissner P. and Witrisal K. (March 2012). Analysis of position-related informa- tion in measured uwb indoor channels. In Antennas and Propagation (EUCAP), 2012 6th European Conference on, pp. 6–10. [55] Greenstein L., Michelson D., and Erceg V. (1999). Moment-method estimation of the ricean k-factor. IEEE Communications Letters, 3(6):pp. 175–176. doi: 10.1109/4234.769521. [56] Greenstein L.J., Ghassemzadeh S.S., Erceg V., and Michelson D.G. (2009). Ricean k-factors in narrow-band fixed wireless channels: Theory, experiments, and statistical models. IEEE Transactions on Vehicular Technology, 58(8):pp. 4000–4012. doi:10.1109/TVT.2009.2018549. [57] Tepedelenlioglu C., Abdi A., Giannakis G., and Kaveh M. (2001). Performance analysis of moment-based estimators for the k parameter of the rice fading distri- bution. In 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), volume 4, pp. 2521–2524 vol.4. doi:10.1109/ICASSP.2001.940514. [58] Tepedelenlioglu C., Abdi A., and Giannakis G. (2003). The ricean k factor: estimation and performance analysis. IEEE Transactions on Wireless Commu- nications, 2(4):pp. 799–810. doi:10.1109/TWC.2003.814338. [59] Marzetta T. (1995). Em algorithm for estimating the parameters of a multivariate complex rician density for polarimetric sar. In 1995 International Conference on Acoustics, Speech, and Signal Processing, volume 5, pp. 3651–3654 vol.5. doi: 10.1109/ICASSP.1995.479778. 73
  4. [71] Gustafsson F. and Gunnarsson F. (July 2005). Mobile positioning using wireless networks: possibilities and fundamental limitations based on available wireless network measurements. IEEE Signal Processing Magazine, 22(4):pp. 41–53. [72] Kumar S., Gil S., Katabi D., and Rus D. (2014). Accurate indoor localization with zero start-up cost. In MobiCom. [73] Poutanen J., Salmi J., Haneda K., Kolmonen V., and Vainikainen P. (Jan 2011). Angular and shadowing characteristics of dense multipath components in indoor radio channels. IEEE Transactions on Antennas and Propagation, 59(1):pp. 245–253. doi:10.1109/TAP.2010.2090474. [74] Karedal J., Tufvesson F., Czink N., Paier A., Dumard C., Zemen T., Mecklen- brauker C.F., and Molisch A.F. (July 2009). A geometry-based stochastic mimo model for vehicle-to-vehicle communications. IEEE Transactions on Wireless Communications, 8(7):pp. 3646–3657. doi:10.1109/TWC.2009.080753. [75] Virk U.T., Haneda K., and Wagen J. (April 2015). Dense multipath components add-on for cost 2100 channel model. In 2015 9th European Conference on An- tennas and Propagation (EuCAP), pp. 1–5. [76] Oestges C., Clerckx B., Raynaud L., and Vanhoenacker-Janvier D. (Nov 2002). Deterministic channel modeling and performance simulation of microcellular wide-band communication systems. IEEE Transactions on Vehicular Technol- ogy, 51(6):pp. 1422–1430. doi:10.1109/TVT.2002.804846. [77] Kanatas A.G., Kountouris I.D., Kostaras G.B., and Constantinou P. (Feb 1997). A utd propagation model in urban microcellular environments. IEEE Transactions on Vehicular Technology, 46(1):pp. 185–193. doi:10.1109/25.554751. [78] Bello P. (1963). Characterization of randomly time-variant linear channels. IEEE Transactions on Communications Systems, 11(4):pp. 360–393. [79] Durgin G. (2002). Space-time Wireless Channels. Prentice Hall Press, Upper Saddle River, NJ, USA, first edition. ISBN 0-13-065647-X. [80] Rasmussen C. and Williams C. (January 2006). Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA, USA. [81] Krall C. (2008). Signal Processing for Ultra Wideband Transceivers. Ph.D. thesis, Graz University of Technology. [82] Eyceoz T., Duel-Hallen A., and Hallen H. (1998). Deterministic channel modeling and long range prediction of fast fading mobile radio channels. IEEE Communi- cations Letters, 2(9):pp. 254–256. doi:10.1109/4234.718494. 75
  5. Communications Surveys Tutorials, 20(2):pp. 1124–1148. doi:10.1109/COMST. 2017.2785181. [94] Perera C., Zaslavsky A., Christen P., and Georgakopoulos D. (2014). Context aware computing for the internet of things: A survey. IEEE Communications Surveys Tutorials, 16(1):pp. 414–454. doi:10.1109/SURV.2013.042313.00197. [95] Wymeersch H., Seco-Granados G., Destino G., Dardari D., and Tufvesson F. (2017). 5g mmwave positioning for vehicular networks. IEEE Wireless Commu- nications, 24(6):pp. 80–86. doi:10.1109/MWC.2017.1600374. [96] Hult R., Campos G.R., Steinmetz E., Hammarstrand L., Falcone P., and Wymeer- sch H. (2016). Coordination of cooperative autonomous vehicles: Toward safer and more efficient road transportation. IEEE Signal Processing Magazine, 33(6):pp. 74–84. doi:10.1109/MSP.2016.2602005. [97] Karlsson R. and Gustafsson F. (2017). The future of automotive localization algorithms: Available, reliable, and scalable localization: Anywhere and any- time. IEEE Signal Processing Magazine, 34(2):pp. 60–69. doi:10.1109/MSP. 2016.2637418. [98] Koivisto M., Hakkarainen A., Costa M., Kela P., Leppanen K., and Valkama M. (2017). High-efficiency device positioning and location-aware communications in dense 5g networks. IEEE Communications Magazine, 55(8):pp. 188–195. doi: 10.1109/MCOM.2017.1600655. [99] Chen L., Thombre S., Järvinen K., Lohan E.S., Alén-Savikko A., Leppäkoski H., Bhuiyan M.Z.H., Bu-Pasha S., Ferrara G.N., Honkala S., Lindqvist J., Ruot- salainen L., Korpisaari P., and Kuusniemi H. (2017). Robustness, security and privacy in location-based services for future iot: A survey. IEEE Access, 5:pp. 8956–8977. doi:10.1109/ACCESS.2017.2695525. [100] Mazuelas S., Bahillo A., Lorenzo R.M., Fernandez P., Lago F.A., Garcia E., Blas J., and Abril E.J. (2009). Robust indoor positioning provided by real-time rssi values in unmodified wlan networks. IEEE Journal of Selected Topics in Signal Processing, 3(5):pp. 821–831. doi:10.1109/JSTSP.2009.2029191. [101] Fadzilla M.A., Harun A., and Shahriman A.B. (2018). Localization assessment for asset tracking deployment by comparing an indoor localization system with a possible outdoor localization system. In 2018 International Conference on Com- putational Approach in Smart Systems Design and Applications (ICASSDA), pp. 1–6. doi:10.1109/ICASSDA.2018.8477602. 77
  6. [112] Cadger F., Curran K., Santos J., and Moffett S. (2013). A survey of geographical routing in wireless ad-hoc networks. IEEE Communications Surveys Tutorials, 15(2):pp. 621–653. doi:10.1109/SURV.2012.062612.00109. [113] Muppirisetty L., Svensson T., and Wymeersch H. (Feb. 2016). Spatial wireless channel prediction under location uncertainty. 15(2):pp. 1031–1044. ISSN 1536- 1276. doi:10.1109/TWC.2015.2481879. [114] Meissner P., Leitinger E., Lafer M., and Witrisal K. (June 2014). Real-time demonstration of multipath-assisted indoor navigation and tracking (mint). In 2014 IEEE International Conference on Communications Workshops (ICC), pp. 144–149. ISSN 2164-7038. doi:10.1109/ICCW.2014.6881187. [115] Ulmschneider M. (2021). Cooperative Multipath Assisted Positioning. Ph.D. thesis, Hamburg University of Technology. [116] Rath M., Kulmer J., Leitinger E., and Witrisal K. (2020). Single-anchor posi- tioning: Multipath processing with non-coherent directional measurements. IEEE Access, 8:pp. 88115–88132. doi:10.1109/ACCESS.2020.2993197. [117] Großwindhager B., Rath M., Kulmer J., Bakr M.S., Boano C.A., Witrisal K., and Römer K. (2018). Salma: Uwb-based single-anchor localization system using multipath assistance. Proceedings of the 16th ACM Conference on Embedded Networked Sensor Systems. [118] P. Stoica A.N. (1995). On the concentrated stochastic likelihood function in array signal processing. Circuits, Systems and Signal Processing. [119] Venus A., Leitinger E., Tertinek S., and Witrisal K. (May 2021). A message pass- ing based adaptive pda algorithm for robust radio-based localization and tracking. In IEEE Radar Conference. Atlanta, GA, USA. doi:10.1109/RadarConf2147009. 2021.9455311. [120] Wilding T., Leitinger E., and Witrisal K. (2021). Multipath-based localization and tracking considering off-body channel effects. ArXiv, abs/2110.09932. [121] Meissner P., Leitinger E., Fröhle M., and Witrisal K. (2013). Accurate and robust indoor localization systems using ultra-wideband signals. In European Navigation Conference. . European Navigation Conference, ENC ; Conference date: 23-04- 2013 Through 25-04-2013. [122] Santos T., Karedal J., Almers P., Tufvesson F., and Molisch A.F. (2010). Model- ing the ultra-wideband outdoor channel: Measurements and parameter extraction method. IEEE Transactions on Wireless Communications, 9(1):pp. 282–290. doi: 10.1109/TWC.2010.01.090391. 79
  7. APPENDICES A. Description of channel measurement campaigns A.1. Measurement campaign 1 Frequency Domain Measurements - Vector Network Analyzer Frequency-domain measurements have been obtained with a Rhode & Schwarz ZVA- 24 VNA. The frequency range has been chosen as the full FCC bandwidth from 3.1 to 10.6 GHz (corresponding to a wavelength range of 9.67 cm to 2.83 cm), resulting in a delay resolution of 0.1333 ns and a spatial resolution of 4 cm. At the l-th trajectory position, a sampled version Hl[k] of the CTF Hl(f) with a frequency spacing of ∆f is measured. The VNA has been calibrated up to (but not including) the antennas with a through-open-short-match (TOSM) calibration. The FCC bandwidth has been measured for different discrete frequencies with a frequency resolution of 1.5 MHz.The transmit power has been set to 15 dBm. Measurement Post Processing For the VNA measurements, the major system influences on the measured CTF H(f) have already been removed by the previously mentioned TOSM calibration. This includes cables and connectors, but not the antennas, which are considered as part of the transmission channel. The necessary post-processing tasks reduce to a filtering of the signal to select a desired frequency band out of the FCC range and to downconvert the signal transformed to time domain to obtain a baseband signal. The filtering is done with a baseband pulse s(t) that covers the desired bandwidth. The CTF is measured at Nf discrete frequencies fk = k∆f +fmin, k = 0, , Nf 1, − where fmin is the lowest measured frequency. This sampled CTF H[k] corresponds to a Fourier series representation of the time-domain CIR h(τ) [121], which is periodic with a period of τmax. With f0 and fc denoting the lower band edge and the center frequency −1 of the extracted band, respectively, and using an IFFT with size NFFT = (∆f∆τ) , ⌈ ⌉ where ∆τ is the desired delay resolution, the time domain equivalent baseband signal is obtained as −j2π(fc−f0)t r(t) = IFFTN H[k]S[k] e . (A.1) FFT { } Here, S[k] is the discrete frequency domain representation of the pulse s(t) in the desired frequency range. This procedure is similar to [122]. 81
  8. 6 uwin 4 A2 2 p rwin 0 ym −2 lwall lwin −4 −6 −8 A278 −5 0 5 10 15 20 xm (a) Overview of floorplan 5 4 A2 cpill 3 p 2 ym 1 pl 0 −1 −2 2 4 6 8 10 12 xm (b) Close-up of floorplan Figure A.1: Scenario floor-plan: a physical anchor is located at position a1 and an examplary VA is at position a . The gray grid with positions p indicates the measurement grid with 5 5 cm spacing; 2 ℓ × the red dot indicates its center position p, the actual mobile agent position used in the illustration. Blue lines depict specular reflections at wall segments. 83
  9. 11 segment 1 10 segment 2 segment 3 segment 4 9 (1) segment 5 a5 segment 6 F E 8 D segment 7 C B A phys. anchor 7 VA 6 (1) a(1) D† a(1) a 5 4 p 1 2 4 C† -direction in meter y 3 B† 2 φk 1 (2) (2) (2) A† a a a4 1 2 C∗ 0 B∗ A∗ 1 − (2) a3 2 − 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 − − − − − − − − − − − − − x-direction in meter Figure A.3: Floor plan of the evaluation scenario. Bold black lines denote walls, thick gray lines represent glass windows, other lines illustrate other materials. Two blue crosses represent the physical anchors; orange circles denote virtual anchors (VAs) which were considered in the experimental vali- dation. An agent moves along a trajectory segmented into seven parts indicated with distinct colors. Capital letters (with or without mark or ) refer to sub-segments of different materials along each ∗ † wall. 85
  10. Figure A.5: Calibration setup for time domain measurements up to (but not including) the antennas. Hence, the influence of the device internal transfer functions and the measurement cables and connectors, combined in the transfer function Hsys,i(f) for the i-th RX channel, as well as the crosstalk between TX channel and i-th RX channel, Hcross,i(f), have to be compensated. For the further description, we will drop the channel index. To achieve this, two types of measurements are necessary. First, to determine the crosstalk, the TX antenna is unmounted and the TX port is terminated with a 50Ω match and the crosstalk signals are measured. Second, also the RX antennas are unmounted and TX and RX cables are connected. In this way, Hmeas(f) = Hsys(f) + Hcross(f) are measured. Using the measurement configuration with all the antennas as depicted in Figure A.5 yields Hmeas(f) = H(f)Hsys(f)+Hcross(f). Hence, a calibrated version of the radio channel transfer function is obtained as Hmeas(f) Hcross(f) H(f) = − . (A.2) Hsys(f) Hcross(f) − To avoid excessive noise gain, we use a thresholding on the time-domain representation of the denominator in (A.2) and set samples below the threshold to zero. The time domain signal is obtained by an inverse Fourier transformation. Finally, the time- domain signal within the desired frequency range around the center frequency fc can be computed using a suitable baseband pulse shape s(t) as  j2πfct −j2πfct r(t) = h(t) s(t)e e δ(t τshift) (A.3) ∗ ∗ − Here, τshift is a time shift that accounts for the delays of connectors in the calibration 87
  11. C. Predicted Variance ∗ ∗ The predicted variance V[y(ϕk(p )) k, θk] at position p approximates the variance |D of the SMC amplitude values, which relates to the DMC term in (2.7) for position p∗. ∗ Assuming the Gaussian approximation to hold for αˆk(p ) , we have | | ∗ 1 ∗  ∗ 2 V[ψ(ϕk(p )) k, θk] Sν(τk; p )Tp + N0 dk(p ) . (C.8) |D ≈ 2 ∗ 1 While the PDP Sν(τk; p ) decreases with delay τk = c dk(p), the squared distance ∗ 2 dk(p ) will be increasing, implying a counteracting effect. The left-hand side is independent of the distance. We can thus re-write the equation to illustrate the shape ∗ of the PDP as a function of τk at some angle ϕk, 2 S (τ ; ϕ∗)T [ψ(ϕ∗) , ] N . ν k k p 2 2 V k k θk 0 (C.9) ≈ τk c |D − This result indicates a reciprocal squared decay in contrast to the usually assumed exponential decay. On the other hand, from Appendix B, by neglecting measurement noise, we have:  2  2 E νk(p) + wk(p) E νk(p) = TpSν(τk; p) (C.10) | | ≈ | | So, the predicted variance obtained from GPR can be validated by comparing with the DMC power, c.f. equations (C.8-C.9). 89