Christian Gentner

German Aerospace Center (DLR), Köln, North Rhine-Westphalia, Germany

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Publications (18)4.37 Total impact

  • Christian Gentner, Thomas Jost
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    ABSTRACT: Positioning is next to communication the most important field of applications for wireless radio transmissions. This paper considers indoor positioning using wireless signals. Especially in indoor scenarios, multipath reception degrades the accuracy of the positioning device as long as the receiver is based on standard methods. Strategies to mitigate multipath effects on range estimates are in general based on the estimation of the channel impulse response (CIR). All these methods have in common that they determine the CIR in order to remove the influence on the estimate of the line-of-sight path delay. This paper focuses on multipath aided positioning by using the time difference of arrival between multipath components (TDoAbMC). Hence, the paper uses the multipath propagation of the wireless signal to allow positioning in cases of a insufficient number of transmitters or increase the accuracy otherwise. Measurements with a moving receive antenna showed, that multipath components are visible for several meters of receiver movement. To estimate and track the time-variant multipath components of the received signal, the paper uses a Kalman filter which utilizes maximum likelihood estimates as measurements. For positioning, the novel approach treats multipath components as signals from virtual transmitters which are time synchronized to the physical transmitter and fixed in their position. Additionally, using a time difference of arrival approach, the estimation of the user clock bias is not necessary. To use the information of the multipath components, the positioning algorithm has to estimate the user position and the position of the virtual transmitters simultaneously. Furthermore, the new approach does not rely on any prior information such as the room layout or a database for fingerprinting.
    2013 International Conference on Indoor Positioning and Indoor Navigation (IPIN); 10/2013
  • Ingmar Groh, Christian Gentner, Jesus Selva
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    ABSTRACT: In spread-spectrum navigation systems, the positioning error (or tracking accuracy) is determined from the jitter deviation of delay-locked loops (DLLs). However, jitter simulations are computationally complex, and conventional analytical methods provide only unsatisfactory and incomplete jitter results. Thus, our contribution presents a novel method for the analytical jitter computation by mapping the stochastic differential equation (SDE) of the DLL onto an Ornstein–Uhlenbeck (OU) SDE. Its solution is the well-known OU random process, which is a time-variant Gaussian distribution. The expectation value and the variance of the OU random process yield the jitter deviation. Thus, we derive the analytical time-variant jitter function with its transient response, which was to date unavailable. Contrary to previous jitter computations based on a loop transfer function, our method covers DLLs of any order. We obtain the loop parameters that minimise the jitter deviation analytically without computationally complex simulations. Above all, our method based on OU random processes enables for the first time an efficient joint analytical mean time to lose lock (MTLL) and jitter computation. Therefore, both computationally complex MTLL and jitter simulations become obsolete for many kinds of DLLs
    Transactions on Emerging Telecommunications Technologies. 12/2012; 23(2012-12-8):715-727.
  • Ingmar Groh, Christian Gentner, Jesus Selva
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    ABSTRACT: This paper presents a novel method for the analytical mean time to lose lock (MTLL) computation of coherent second-order Langevin delay-locked loops (DLLs). Analytical MTLL computation is a key task for DLLs, since the computational complexity of numerical MTLL simulations is far too high in many operating ranges of the second-order Langevin DLLs. To obtain the crucial MTLL values analytically without simulations, we rewrite the Langevin stochastic differential equation (SDE) as a vector-valued Ornstein-Uhlenbeck (OU) SDE. It includes a Gaussian noise term, which yields as a solution of the vector-valued OU SDE a time-variant Gaussian distribution. Thus, the complementary error function yields the loss of lock probability and thereby the MTLL. If we replace the complementary error functions by suitable exponential approximations, we obtain a simple MTLL expression with an exponential function as dominant term. The simple exponential MTLL expression yields the optimum loop parameters corresponding to the maximum MTLL. Simulation results confirm that the optimum loop parameters corresponding to our analytical MTLL computation method and to the simplified exponential approximation coincide. Besides the crucial analytical MTLL results, the OU random processes yield additionally the likewise crucial analytical jitter results.
    IEEE Transactions on Communications 11/2012; · 1.75 Impact Factor
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    ABSTRACT: Global navigation satellite systems (GNSSs) can deliver very good position estimates under optimum conditions. However, especially in urban canyons and indoor scenarios with severe multipath propagation and blocking of satellites by buildings the accuracy loss can be very large. Often, positioning with GNSSs is impossible in these scenarios. On the other hand, cellular wireless communication systems such as the third generation partnership project (3GPP) long-term evolution (LTE) provide excellent coverage in urban and most indoor environments. Thus, this paper researches timing based positioning algorithms, in this case time difference of arrival (TDoA), using 3GPP-LTE and GPS measurements. This paper considers a particle filter for 3GPP-LTE TDoA positioning and the fusion of 3GPP-LTE signals with GPS measurements. To obtain better positioning results, a 3GPP-LTE TDoA error model is derived, which splits the TDoA errors in slow varying and fast varying errors. The slow varying error model is included in the prediction model and the fast varying error model in the likelihood function of the particle filter. The last part of this paper, evaluates the positioning performances of the developed particle filter in an indoor scenario. These evaluations show clearly the possibility of using 3GPP-LTE measurements for indoor positioning. Additionally, it shows the advantage of fusing 3GPP-LTE with GPS measurements.
    Proceedings of the 25th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2012); 09/2012
  • Conference Paper: LTE Positioning Methods
    Armin Dammann, Christian Gentner, Emanuel Staudinger
    North American LTE Forum 2012; 04/2012
  • Ingmar Groh, Christian Gentner, Stephan Sand
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    ABSTRACT: Orthogonal frequency division multiplexing (OFDM) became a popular transmission approach since it suppresses intersymbol interference (ISI) due to large channel delays. However, intercarrier interference (ICI) for high mobility receivers yields corrupted channel estimates for pilot-aided OFDM channel estimation. Thus, this paper presents a novel time-variant channel estimation approach to mitigate this system impairment. We linearize the time-variant channel and determine the expansion point by channel estimates corresponding to the current OFDM symbol, and we get the unknown channel slopes by an iterative data and channel estimation. Our algorithm combines a least squares or a minimum norm channel slope estimation and the detection of the channel paths. It sets any channel path power equal zero once the corresponding power estimate is smaller than a given threshold. Our algorithm exploits the large channel correlations between the cyclic prefix and the successive OFDM symbol optimally. These maximum correlations reason the superiority of ICI mitigation with cyclic prefixes compared to channel slope estimation with adjacent OFDM symbols, which needs at least twice the number of pilot symbols. Additionally, our iterative ICI mitigation reduces noise impairments.
    76th IEEE Vehicular Technology Conference (VTC-Fall); 01/2012
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    ABSTRACT: Services and applications based on accurate knowledge of the mobile terminal (MT) location play a fundamental role in current and future wireless communications systems. In addition, the United States Federal Communications Commission (FCC) has stated accuracy requirements on the location determination process of enhanced 911 (E-911) emergency callers. Global navigation satellites systems (GNSSs) based positioning provides a sufficient accuracy in rural and suburban environments, where a sufficient number of satellites are visible in line-of-sight conditions. However, in GNSS critical environments, such as dense urban, urban canyon or even indoors, view to sky is limited. In these environments, low signal power, bad satellite constellations, severe multipath and non-line-of-sight propagation causes erroneous and biased position estimates. Especially in these environments, cellular wireless communication systems provide good coverage and can be used for position determination of the MT. Mobile radio communications systems like GSM, UMTS or the currently deployed 3GPP-LTE primarily target on optimizing communication performance figures such as bandwidth efficiency or data throughput. Availability, signal strength or even signal bandwidths, however, make them interesting for positioning. However, the correlation properties of the synchronization signals of the 3GPP LTE system limit the positioning performances. Moreover, while non-line-of-sight propagation improves the communication performance, it degrades the navigation performance due to the additional distance the signal might travel. Hence, this paper shows an indoor positioning approach with the 3GPP-LTE mobile communication standard, which is currently deployed in many countries. Moreover, it shows the benefit of using the 3GPP-LTE mobile communication system for indoor positioning. Therefore, this paper describes a novel real-time mobile radio based positioning system using time-difference-of-arrival (TDOA) measurements. The paper considers an indoor scenario, where the transmitters are located outdoors and the MT is moving in an office building. The position estimation is done by a particle filter. Furthermore, to improve the positioning accuracy, this paper derives a time-variant error model for indoor positioning. Using this time-variant error model, the positioning error of the MT can be decreased significantly. The downlink of 3GPP-LTE is based on Orthogonal Frequency Division Multiplexing (OFDM), which allows a spectral efficient and flexible usage of the available frequency spectrum. The 3GPP-LTE standard specifies signal parts, dedicated to time and frequency synchronization. For our investigations we use these synchronization signals for TDOA based position estimation. The 3GPP-LTE signal structure, in particular the properties of the synchronization properties, will be discussed in detail in the final paper. For the 3GPP-LTE positioning we apply a particle filter in order to process TDOA measurements of the 3GPP-LTE base stations. The TDOA measurements are taken from DLR´s 3GPP-LTE positioning test-bed. The test-bed consists of up to four synchronous transmitters. From each transmitter site a predefined 3GPP-LTE OFDM frame can be transmitted periodically. The test-bed operates at 2.4 - 2.5 GHz, providing signal bandwidths of up to 20 MHz. At the receiver we convert the received signal down to base band and sample both the inphase and quadrature component. The sampled signal is stored on a hard disk, which allows both offline and real-time processing. These sampled data are processed by the TDOA estimation algorithm and consists of two steps. The first step estimates the TDOA roughly by correlating the narrow band synchronization signals with the received sequence and the second step performs subsample based estimation with the wideband pilot symbols. These algorithms use a first peak and maximum peak detection algorithms, for detecting the first and maximum arriving signal. However, the 20 MHz bandwidth limits the rough synchronization and allows only a sample based estimation within 15 meters, which hinders accurate positioning in indoor environments. Thus, an oversampling approach is used, to tackle this issue which results in a significant error reduction. However, due to hardware imperfections of the test-bed and channel errors, such as non-line-of-sight and multipath propagation, the TDOA measurements are noisy and biased. Thus, to obtain better positioning results, this bias has to be predicted and mitigated. Several approaches exist to model time of arrival based ranging in indoor environments. All of these statistical models depend on bandwidth, carrier frequency and are time-invariant. However, for navigation applications an evaluation of the multipath and non-line-of-sight error for a moving receiver is essential. Hence, to improve the positioning accuracy, we derive in this paper a time-variant TDOA error model based on a measurement campaign of an outdoor-to-indoor channel. The evaluation of this measurement campaign yields an autoregressive error model which allows us to predict the TDOA error for a moving MT. By using this model within the particle filter yields promising results in terms of error mitigation. Each particle itself models the multipath and non-line-of-sight error according to the obtained time-variant error model. Additionally, we compare these results to the more general approach by assuming an uncorrelated error. In the final paper, we will provide a detailed description of the 3GPP-LTE downlink signal structure, which we use for TDOA based positioning. We will discuss and describe the applied algorithms for timing (pseudo-range) estimation. Furthermore, we will describe the 3GPP-LTE test-bed and the scenario in detail. Additionally, the particle filter used for the positioning estimation will be described in detail. Especially, we will analyze the measurement results and the derived time-variant model to predict and mitigate the multipath and non-line-of-sight error in the particle filter.
    IEEE/ION PLANS 2012; 01/2012
  • Christian Gentner, Stephan Sand, Armin Dammann
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    ABSTRACT: Global navigation satellite systems (GNSSs) can deliver very good position estimates under optimum conditions. However, in urban and indoor scenarios positioning with GNSS is often impossible. On the other hand, cellular wireless communication systems, e.g., the new, orthogonal frequency division multiplexing (OFDM) based third generation partnership project’s long-term evolution (3GPP-LTE), provide excellent coverage in urban and most indoor environments. Thus, this paper researches timing based positioning algorithms for OFDM using time difference of arrival (TDOA) measurements and the 3GPP-LTE signals. Therefore, it introduces synchronization, TDOA estimation, and signal–to–noise ratio (SNR) estimation algorithms. To solve the navigation equation for TDOA, this paper considers the static Gauss-Newton algorithm, positioning Kalman filter and particle filter. Further, this paper derives new Cram´er-Rao lower bounds (CRLBs) to analyze the obtained algorithms. First, new CRLBs are derived for TDOA estimation and pairwise synchronized or fully synchronized transmitters. Afterwards, static and novel dynamic recursive Bayesian CRLBs are derived for position estimation. The CRLBs are compared to real estimated TDOAs and positions. Improvements of the positioning algorithms are still possible compared to the CRLBs. Nevertheless, this paper demonstrates that indoor positioning with TDOAs from OFDM based on 3GPP-LTE is possible.
    ICL GNSS 2012; 01/2012
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    ABSTRACT: Indoor positioning is an extremely challenging task for GNSS positioning. Thus, we propose to use terrestrial communications systems such as 3GPP-LTE as a complementary positioning system. We estimate the expected positioning performance of 3GPP-LTE indoor positioning by assessing link budgets and calculating the Cram´er-Rao lower bounds for pseudo-range and 2D position estimation. An experiment shows that the combination of mobile radio based positioning can achieve a positioning accuracy in the range of a few meters, indicating that such technologies are suitable for complementing GNSS indoors.
    ION GNSS 2011; 09/2011
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    ABSTRACT: Time Based (TB) localization in terrestrial mobile radio as an augmentation for global navigation satellite systems has recently gained plenty of interests. As an essential tool to develop suitable algorithms for positioning applications in mobile radios, channel models for wireless transmissions have growing significance. Currently there is a lack of investigations on comparing the propagation characteristics at 2.45 GHz and 5.2 GHz for positioning applications. Therefore, we present a statistic evaluation based on a channel measurement campaign. Several propagation characteristics are researched like the received power and the delay spread. While some measures like the received power is carrier frequency dependent, most of the measures like the non line-of-sight bias or the delay spread are independent of this measurement parameter. Instead they are more influenced by the location and the environment.
    The 5th European Conference on Antennas and Propagation; 04/2011
  • Emanuel Staudinger, Christian Gentner
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    ABSTRACT: This paper presents a novel maximum likelihood (ML) time difference of arrival (TDoA) estimation algorithm for subsample delays. We introduce a new initial acquisition and cell search algorithm with implicit multiple access interference (MAI) cancelation. A joint carrier frequency offset (CFO) estimation and subsample delay estimation is used to compensate Doppler spreads and oscillator drifts. Analytical derivations show how the CFO influences the TDoA subsample delay estimation and vice versa. Furthermore, we show through numerical simulations that the geographic base station mapping and the used synchronization codes influence the estimation accuracy. Additionally, we introduce a new successive interference cancelation (SIC) to improve the overall accuracy.
    8th Workshop on Positioning, Navigation and Communication 2011 (WPNC'11); 04/2011
  • Christian Gentner, Stephan Sand
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    ABSTRACT: In global navigation satellite systems, the mobile location is calculated by measuring the line-of-sight (LOS) signal propagation time between the mobile terminal and at least 4 satellites. However, in an urban area, the direct path may be blocked. Thus, the satellite signals propagate an additional distance due to reflection and diffraction. This effect, where no direct path is available, is called non-line-of-sight (NLOS) propagation and adds a positive bias to the geometric LOS (GLOS) propagation time. This can cause an estimation error in the order of hundreds of meters [1]. If enough measurements to identify the position are available, it is possible to discard the NLOS signals or weight the signal with a lower priority. However it poses a great challenge to mitigate the NLOS impact through processing at the receiver. To handle the NLOS propagation various NLOS mitigation algorithms and methods have been proposed, cf. [1], [2]. The authors of [3] summarize for example the NLOS detection algorithms for ultra wide–band (UWB) systems and compare them to each other. The main contribution of this paper is the detection and mitigation of NLOS scenarios. Most mitigation algorithms try to detect and mitigate the NLOS effects at the same time. Therefore they try to estimate the NLOS error. This paper separates the NLOS detection and mitigation into two parts. The first part describes an algorithm to detect NLOS signals. A lot of approaches have been published in the recent years to detect NLOS paths for example by the received signal strengths (RSS), cf. [1], [2]. These approaches are usable, but the accuracy depends directly on the noise and the fading effects. Therefore we analyze a more robust algorithm to detect such signals. The authors of [3] proposed an algorithm for UWB called confidence metric which considers the multipath propagation effects. Therefore it detects NLOS signals by comparing the power of the first incoming path to the multipath with the highest power under consideration of the each propagation time. This approach is useful for small distance propagation. In case of GNSS this method is not applicable because of the long propagation time and thus almost equivalent measurements. Thus we adapt this algorithm to the case of GNSS. Instead of considering the whole propagation time, we consider only the additionally propagation time of the first incoming signal power and the signal with the highest power. This adapted algorithm is more stable against varying signal to noise ratio and outputs the probability whether the received signal is LOS or NLOS. If we detect a NLOS signal and more than 3 satellites are LOS, we can easily extract the NLOS signals. But for example in urban scenarios, not always 4 LOS GNSS satellite signals are available. Thus we have to use the NLOS signals, mitigate this error or use other sensors. Therefore the second part of this paper adapts the extended Kalman filter by using the NLOS probability information. The EKF is a well-known algorithm for positioning and tracking. This adapted EKF (AEKF) estimates the position by using the probability information of the previous algorithm. For low LOS probabilities, the AEKF either extracts the signals or estimates the position with the highest probability. Additionally we consider in the further simulation the TDOA measurements of the LTE (long time evolution) network. If more than 3 LOS satellite signals are available, the LTE measurements generally cannot improve our position estimate. But if less than 4 LOS satellites are available, the AEKF uses this information to mitigate the NLOS effect. To verify the derived NLOS detection algorithm and the AEKF, we consider a typical urban scenario with high buildings and street canyons. We analyzed this scenario with a raytracing tool for a fixed satellite constellation and fixed LTE base stations. The simulation results compare the position estimation of the AEKF to the EKF and the static method with Gauss-Newton. Additionally we consider different scenarios with GPS only, GPS + Galileo and GPS + Galileo + LTE. Especially the simulations show that if more than 3 LOS GNSS satellite signals are available, we cannot improve the positioning error with additionally using the LTE signals. But in scenarios where less than 4 satellites are LOS we can improve the position estimation. Additionally the simulations show the improvement of the described NLOS detection algorithm compared to general detection algorithms. With a varying signal to noise ratio, we are still able to detect more than 90% of the LOS signals correct. To conclude, this paper shows an algorithm for NLOS detection and mitigation. The NLOS detection algorithm is based on the idea of [3] where the multipath power is compared to the propagation time. Especially this algorithm provides probability information if a received signal is LOS or NLOS. The proximate adapted EKF uses this information, mitigates the NLOS effect or uses the additional LTE network. With these algorithms we are able to obtain a precise position fix even if less than 4 LOS satellites signals are available. In the final paper, we will present the derivation of the NLOS detection algorithm and the AEKF. Additionally we provide information about the false alarm probability, detection probability of NLOS signals versus signal to noise ratio. Furthermore we provide additional the simulation results which show the advantage of using these algorithms compared to EKF and the static method with Gauss-Newton. [1] J. a. Figueiras and S. Frattasi, Mobile positioning and tracking: from conventional to cooperative techniques, ser. New ecologies for the twenty-first century. Wiley, 2010. [2] K. Yu, I. Sharp, and Guo. (2009) Groundbased wireless positioning [3] J. Schroeder, S. Galler, K. Kyamakya, and K. Jobmann, "NLOS detection algorithms for ultra-wideband localization," in Workshop on Positioning, Navigation and Commun. (WPNC), Mar. 2007
    The ION 2011 International Technical Meeting; 01/2011
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    Alexander Zeh, Christian Gentner, Daniel Augot
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    ABSTRACT: The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an (n,k) Reed-Solomon code, a multiplicity s and a list size l , our algorithm has time complexity O(ls4n2).
    IEEE Transactions on Information Theory 01/2011; 57(2011-09-9):5946-5959. · 2.62 Impact Factor
  • Christian Gentner, Ingmar Groh, Stephan Sand, Armin Dammann
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    ABSTRACT: This paper presents a novel analytical derivation of the false alarm probability (FAP) and detection probability (DP) for non-line-of-sight (NLOS) detection based on LTE signals. Position estimation in indoor or urban area with cellular wireless communication systems is getting more and more important. However, especially in these environments NLOS propagation causes positioning errors in the order of hundreds of meters. Thus, detection and mitigation of NLOS signals is a critical task. In this paper, we derive the FAP and DP on power-scaled detectors based on LTE multipath signals. The derivation considers frequency-selective fading channels taking sub-sample multipath interference, channel dynamics and initial frequency offsets into account. The results show a simple way to calculate the DP versus FAP. Simulation results verify the analytical probabilities for the WINNER II C2 channel model. Additionally, the simulations show the improved FAP of an antenna array at the receiver.
    Multi-Carrier Systems Solutions (MC-SS) 2011; 01/2011
  • Christian Gentner, Ingmar Groh
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    ABSTRACT: This paper presents a novel analytical derivation of the false alarm probability (FAP) and detection probability (DP) for non-line-of-sight (NLOS) detection of GNSS signals. The NLOS propagation in for example urban environments causes positioning errors in the order of hundreds of meters in navigation systems. Thus, detection and mitigation of NLOS signals is a critical task for high accuracy navigation receivers. In this paper, we derive the FAP and DP on power–scaled detectors based on the multipath signals. The derivation considers frequency–selective fading channels taking sub–chip multipath interference, channel dynamics and initial frequency offsets into account. The results show a simple way to calculate the DP versus FAP. Simulation results verify the analytical probabilities and show the trade–off between temporal non–coherent versus coherent averaging. Clearly, the optimum noise averaging depends on the length of the symbol sequence and channel dynamics. Additionally, the simulations show the improved DP of an antenna array at the receiver. Furthermore we see the influence of sub–chip interference in the simulation results.
    VTC Spring 2011; 01/2011
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    Ingmar Groh, Armin Dammann, Christian Gentner
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    ABSTRACT: Motivated by the possibility of decreasing the intersymbol interference (ISI) which is due to large delays of a multipath mobile radio channel, orthogonal frequency division multiplexing (OFDM) became very popular. However, the timevariance of the mobile radio channel induces intercarrier interference (ICI) yielding substantial channel estimation errors and thereby tremendous transmission impairments. Contrary to previous algorithms which resort to a linearization of the time-variant channel, we combat the ICI using eigenspaces of time-domain covariance matrices defined by the autocorrelation function of the Doppler spread. We perform a basis expansion using Slepian sequences and determine the basis coefficients of the time-variant channel by channel estimates from previous OFDM symbols. Once we know these basis coefficients, we obtain the necessary time-variant channel estimation by the Slepian sequences. These time-variant channel estimates allow a symbol detection in frequency domain which eliminates the ICI almost completely. Simulation results investigate both the signal to interference ratio (SIR) and the bit error ratio (BER) of our new ICI mitigation methods and reveal the superiority compared to previous algorithms for ICI reduction.
    VTC Spring 2011; 01/2011
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    A. Zeh, C. Gentner, M. Bossert
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    ABSTRACT: In this paper we propose a new algorithm that solves the Guruswami-Sudan interpolation step for Reed-Solomon codes efficiently. It is a generalization of the Feng-Tzeng approach, the so-called fundamental iterative algorithm. From the interpolation constraints of the Guruswami-Sudan principle it is well known that an improvement of the decoding radius can only be achieved, if the multiplicity parameter s is smaller than the list size l. The code length is n and our proposed algorithm has a complexity (without asymptotic assumptions) of O(ls<sup>4</sup> n<sup>2</sup>).
    Information Theory Workshop, 2009. ITW 2009. IEEE; 11/2009
  • Christian Gentner, Ingmar Groh, Stephan Sand
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    ABSTRACT: These days, the global positing system (GPS) is the most common position system in the world. To determine a position, at least 4 satellites are necessary. Therefore it measures the signal propagation time between the mobile terminal and the satellites. However, in indoor or urban area, the direct path may be blocked. Thus, the satellite signals propagate an additional distance due to reflection and diffraction. This effect, where no direct path is available, is called non-line-of-sight (NLOS) propagation and adds a positive bias to the geometric LOS (GLOS) propagation time. This can cause an estimation error in the order of hundreds of meters, cf. [1], [2]. Therefore to obtain better positioning results it is necessary to identify NLOS signals and mitigate this additional bias by reconstructing the LOS path. If enough measurements to identify the position are available, it is possible to discard the NLOS signals or weight the signal with a lower priority. However it poses a great challenge to mitigate the NLOS impact through processing at the receiver. This paper discusses a novel analytical derivation of the false alarm probability (FAP) and detection probability (DP) for GNSS NLOS detection using a dual frequency receiver. Generally, dual frequency receivers are used to obtain better positioning results by correcting for example the ionosphere errors. We derive in this paper the FAP and DP for power–scaled detectors based on the multipath signals for detecting NLOS signals. The concept of this paper bases on the idea of [3]. The authors in [3] present a derivation of the FAP and DP of detecting pilot bursts of W-CDMA systems. Instead of considering an antenna array where the long-term channel statistics are identical, we consider receiver for simultaneous reception of two GNSS frequency bands. The derivation considers frequency–selective fading channels taking sub–chip multipath interference, channel dynamics and initial frequency offsets into account. Each antenna element consists of a correlator bank with Ns correlators which partitions the received symbol sequence in M non–overlapping subsequences of length Nc. The last part of the receiver normalizes the power and searches for the maximum of all correlator outputs. If the output is above a threshold the signal is considered as LOS or NLOS if blow. Thus the FAP defines the probability that the detection is beyond the threshold whereas no signal is on the GLOS path, which is caused by noise or by strong multipaths. The DP states that the detection is beyond the threshold and the received signal is LOS. To determine DP, we compute either the threshold corresponding to a defined FAP or vice versa and calculate afterwards the corresponding DP. To obtain these probabilities and since the received signal is zero-mean complex Gaussian distributed, the correlator outputs are also zero-mean Gaussian distributed. Therefore the test characteristics can be characterized by its covariance matrix for each frequency. We calculate the cumulative density function (CDF) of the FAP and DP by the eigenvalues of the covariance matrix for each delay and each partitioned sequence. Contrary to [3], the eigenvalues for the FAP are not equivalent. Hence we have to use the lemma of Gil–Pelaez (cf. [4]) to calculate the CDF. We can evaluate this CDF numerically for the FAP to obtain either the FAP for a defined threshold or the threshold for a defined FAP. Afterwards, we can calculate with these values the DP in the same way, also with the lemma of Gil–Pelaez. Especially in this paper we have to calculate both probabilities with the lemma of Gil–Pelaez, because of different eigenvalues. If all eigenvalues of the covariance matrix are identical, the lemma of Gil–Pelaez reduces to the chi–square distribution [3], where the probability computation simplifies. To verify the derived results, we consider the L1 and L5 frequencies of GPS with a chip rate of 1024 kHz which is scrambled by a Gold–code. We consider rectangle pulse–shaped symbols and a frequency selective Rayleigh fading channel. The simulations consider a receiver velocity of 100 km/h and are done for each single frequency and both frequencies together which use the coherent channel knowledge. The simulation results show clearly the advantages of dual frequency receivers. Additionally we see the influence of sub–chip interference to the FAP and DP. Especially it shows also the trade–off between temporal non–coherent versus coherent averaging. Due to the high receiver mobility of 100 km/h, the symbol partitioning into more blocks performs best, since the number of chips of coherent correlation is smallest. To conclude, the results of this paper show a simple way to calculate the DP versus FAP. Simulation results verify the analytical probabilities and show the trade–off between temporal non–coherent versus coherent averaging. Clearly, the optimum noise averaging depends on the length of the symbol sequence and channel dynamics. Additionally, the simulations show the improved DP of dual frequency receiver. Furthermore we see the influence of sub–chip interference in the simulation results. Thus the final paper will present the derivation of the FAP and the DP in more detail. Especially we state the covariance matrix and the derivation of the eigenvalues where the influence of the sub–chip interference and the dual frequencies are obvious. Furthermore we will provide additional simulation results which show the FAP/DP versus SIR and like in [3] the optimal partitioning of the symbol sequence depending on the velocity. [1] J. a. Figueiras and S. Frattasi, Mobile positioning and tracking: from conventional to cooperative techniques, ser. New ecologies for the twenty-first century. Wiley, 2010. [2] K. Yu, I. Sharp, and Guo. (2009) Groundbasedwireless positioning. [3] L. Schmitt, V. Simon, T. Grundler, C. Schreyoegg, and H. Meyr, “Initial Synchronization of W-CDMA Systems Using a Power-Scaled Detector with Antenna Diversity in Frequency-Selective Rayleigh Fading Channels,” in Proceedings of the IEEE Global Communications Conference (GLOBECOM 2003), San Francisco, USA, December 2003. [4] J. P. Imhof, “Computing the distribution of quadratic forms in normal variables,” Biometrika, vol. 48, no. 3/4, pp. 419–426, 1961.
    The ION 2011 International Technical Meeting; 01/2001