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time based ranging via uwb radios
- 1. Chapter-5: Time-Based RangingVia UWB Radios Prof. Jae-Young Pyun Dept. of Information and Communication Engineering Chosun University Submitted By: Sujan Shrestha (Student ID: 20157711)
- 2. Objective • Strategies toresolved Multipath Components (MPCs) in UWB • Due to the requirement of synchronization and complexity in AOA, TOA (or TDOA) is method of choice in UWB-Based Positioning Systems. In other side RSS has low ranging accuracy.
- 3. Outline 5.1 Time-Based Positioning 5.2Error Sources in Time-Based Ranging 5.3 Time-Based Ranging 5.4 Fundamental Limits for Time-Based Ranging 5.5 Maximum Likelihood (ML)-Based Ranging Techniques 5.6 Low-Complexity UWB Ranging Techniques Summary
- 4. 5.1 Time-Based Positioning NmReference Nodes (RNs) (xi, yi) Target Node (TN) (x, y) : Time of Flight estimate of the signal at the ith RN : speed of light : is the true distance between the TN and the ith RN : is the zero mean Gaussian Measurement Noise with variance, : is a non-negative distance bias introduced due to the obstructed line-of-sight (LOS)
- 5. To estimate positionof the TN, Using Non-linear Least Squares (NLS) technique
- 6. ,is Residual errorcorresponding to TN Location (x,y) ,characterizes the reliability of the measurement •Under NLOS propagation and a vast number of MPCs, computation may not be easy. •Further section shall focus on different error sources and formulation of Time- Based UWB Ranging problem
- 7. 5.2 Error Sourcesin Time-Based Ranging i. Multipath Propagation Figure: Illustration of TOA estimation problem in a multipath channel
- 8. •Effect due toNLOS signal Propagation or Antenna Effects. Figure: Different Scenarios for Channel Realization in LOS and NLOS Situations
- 9. Receiver Uses correlator (MatchedFilter) and perform spreading sequence of desired user Locks the correlation peak and identify the first MPC preceding the correlation Peak •Imperfect autocorrelation characteristics results correlation side-lobes between correlation peaks Figure: Illustration of Side-Lobe Interference (SLI)
- 10. •M-ary ternary orthogonalKeying (MTOK) sequence have optimal correlation characteristics when processed with a Bipolar Template (BPT) Figure : Periodic Code Correlations for MTOK-IR and TH-IR
- 11. ii. Multiple AccessInterference (MAI) •TOA ranging degrade in presence of MAI •Assigning orthogonal channels to different users either in Time, Frequency, Code or Space domains in a network can mitigate the problem. •Under Simultaneously Operating Network (SONs), we use non-linear filtering technique.
- 12. iii. Obstructed Lineof Sight propagation •NLOS is model as an exponentially, uniformly, or Gaussian distributed random variable •Standard Deviation , Hypothesis tests, Probability Density Functions (PDFs) of TOA measurements is performed. iv. Other error sources •Timing imperfections among reference devices •Clock drifting between Transmitter and receiver devices •Timing Jitter and Clock drifting effects •Sampling UWB signals at sub-Nyquist Rates
- 13. 5.3 Time-Based Ranging •Letthe received IR-UWB signal in multipath environment be represented as: , zero-mean additive white Gaussian noise (AWGN) with double sided power spectral density , a ranging signal , delay of the MPC , number of MPCs , channel coefficient
- 14. ,represents the energyof ranging symbol ,is the polarity code ,is time-hopping (TH) code ,denotes the received UWB pulse with unit energy ,is the frame duration , is number of chips per frame ,the chip duration ,is number of pulses (frames) per ranging symbol , represent width of received pulse , is assumption , represents duration of ranging symbol The Energy of UWB pulse is represented as:
- 15. Different ways toobtain the decision variables for TOA estimation is discussed further.i. Direct Sampling Receiver Sampled at or above the Nyquist rate for UWB system, increases cost and complexity of the Receiver.
- 16. ii. Matched Filter(MF) Receiver • If Received Pulse Shape is known at the receiver, a Matched Filter (MF) can be used for decision variables for TOA estimation. • Ranging accuracy is higher but receiving processes become complex. • It requires Nyquist Rate sampling, hence complex analog-to-digital converters (ADCs) is sampled at every , the MF outputs is obtained as: Where, (for is an integer multiple of )
- 17. iii. Energy Detection(ED) Receiver • Low complexity alternative is Energy Detection (ED) receiver, which does not assume the knowledge of received pulse shape. •ED is a non-coherent detection and simpler receiver structures • The integrator output samples for an ED receiver can be expressed as: Major Drawback is due to noise-squared and signal-cross-noise terms makes decision variable more noisy. ED ranging accuracy is low. Conversely, at Low sampling rate, ED receivers can have better energy capture compared to MF receiver.
- 18. iv. Delay-and-correlate (DaC)receiver •Does not require the knowledge of the received pulse shape to construct a local template •First arriving pulse is delayed and then used as a reference template to correlate later arriving pulse to obtain the decision variable, referred as Transmitted-Reference (TR) receiver. •Samples after correlating the received signal with delayed version of itself can be: •D, represents the delay between the pulse pairs. • in a Transmitted-Reference (TR) receiver becomes
- 19. Disadvantage: •Enhanced noise terms,noise-cross-noise terms and signal-cross-noise terms can make the decision variable noisy. Advantage: •DaC receiver can have better energy capture than the MF receiver at Low sampling rates Figure: Delay-and-correlate receiver
- 20. Comparative study ofThree Receivers •We consider a root-raised cosine (RRC) pulse with Tp = 1ns, of roll-off The RRC pulse is give by:
- 21. Figure: Received normalizedpulse shape and sampled outputs corresponding to MF, ED, and DaC receivers, 1ns pulse is sampled at 8 GHz and energy is collected within 1ns windows
- 22. S.N. MF RecieverED and TR (DaC) receiver 1 Uses RRC pulse as a template Collect energy within 1ns windows 2 Requires sampling rates on order of Nyquist rate to accurately capture the peak energy Can capture a sufficient amount of energy at lower sampling rates closer to the True TOA of the signal. 3 Can outperform ED and TR receiver below certain SNR values Enhanced noise terms at Low/Medium SNR regions become problematic •For TR receiver, it is assumed that half of the energy is spared for the reference pulse. •Performance of receiver depends on both the SNR and the sampling period.
- 23. 5.4 Fundamental Limitsfor Time-Based Ranging • Cramer-Rao Lower Bound (CRLB) are used for setting a lower bound on an estimator’s Mean Square Error (MSE) • Bounds other than CRLB have also been investigated as,
- 24. 5.4.1 Cramer-Rao LowerBounds for Single-Path Channels •From Chapter:2, CRLB for single-path AWGN Channels is given as: Where, is effective signal Bandwidth defined as, Where, is Fourier Transform of transmitted signal • CRLB for time-based ranging decreases with the square-root of the SNR and effective signal Bandwidth. • CRLB depends of Fourier Transform of the transmitted signal.
- 25. 5.4.2 Cramer-Rao LowerBounds for Multipath Channels • CRLB in multipath channels depends on the Pulse shape, Path gains, and SNR • For Ideal Auto correlation, CRLB for multipath channel converges to CRLB for single path channels. Disadvantage: • Sampling rates above the Nyquist rate are required in order to achieve the CRLB for UWB signals, which may not be possible practically. •CRLB is tight only at High SNR and is not accurate at Low and Moderate SNRs • Threshold effect of SNRs is not accounted by the CRLB
- 26. 5.4.3 Ziv-Zakai LowerBounds (ZZLB) for Single-path Channels •ZZLB is tight for a wide range of SNRs •ZZLB can be derived from following identity for the MSE of an estimator, , is identical to error probability of a binary hypothesis testing (BHT) with a sub-optimum decision rule given by,
- 27. Figure: ZZLBs andCRLBs in AWGN channels for different pulse widths.
- 28. In example weobserve that • ZZLBs and CRLBs overlaps in the high SNR region • At Lower SNR, ZZLB is much tighter than CRLB • The reason is at low SNRs, the received Signal is unreliable • Overall accuracy improves as shorter pulse duration are used
- 29. 5.4.4 Ziv-Zakai LowerBounds (ZZLB) for Multipath Channels •ZZLB on TOA estimation, the estimator has a-priori knowledge on multipath environment •Difficult for practical scenarios, so a Perfect Measurement Bound (PMB) is discussed and sets a lower-bound on any TOA estimator. •Error Probability for PMB is given as Where, the auto-correlation function for the multipath signal is given by, value can be plugged into ZZLB Lower Bound for single path channel so that average ZZLB for a particular environment can be obtained.
- 30. 5.5 Maximum Likelihood(ML)-Based Ranging Techniques • ML-based ranging techniques deals with varying a-priori information. 5.5.1 ML estimation with Full a-priori Information • TOA can be estimated by using MF that is perfectly matched to the received multipath signal. • The optimal template can be defined as: • Optimal receiver is not possible to implement in practice as due to unknown parameters to be estimated.
- 31. 5.5.2 ML estimationwith No prior Information •In presence of Gaussian Noise, ML solution is equivalent to a minimum mean square error (MMSE) solution given as, Where, are the samples of reconstructed received signal, given by, • ML estimator achieves the CRLB asymptotically 5.5.3 Ranging with Generalized Maximum Likelihood(GML) ratio test • Searches only the paths prior to the strongest MPC • Received signal can be re-written as sum of first path, remaining paths and noise as
- 32. Disadvantages: • High computationalcomplexity since a search of unknown parameter set is required. • Requires very High sampling rates at or above the Nyquist rate 5.5.4 Sub-Nyquist sampling ML estimation with different levels of a-priori information • ML estimators that can operate at Low Sampling Rates with different levels of a-priori information are described. • To obtain the decision variables, an Energy Detection (ED) receiver is considered. i. Multiple Hypothesis Testing System Model • Different Hypotheses can be written as follows:
- 33. , is desiredsignal , is the nth element of z , is the noise after BPF , is the true hypothesis ii. Maximum Energy Selection (MES) •To determine TOA estimation from these samples, we use MES from the sample vector z, by neglecting the information in the neighboring samples, which give, •Disadvantages: MES is susceptible to noise, MES may not provide high time resolution because of large delay between the first path and the strongest path.
- 34. iii. Maximum EnergySum Selection (MESS) • It exploits the energy in the neighboring MPCs. • There exists an optimum window length that depends on the channel realization and SNR • Window Shift that captures Largest energy determine the TOA of received signal • Optimum sliding window size increases as the SNR increases Figure: Simulated MAEs corresponding to different lengths of sliding windows at different SNRs
- 35. iv. Weighted MaximumEnergy Sum Selection (W-MESS) •If knowledge of channel energies is available, the TOA estimate can be obtained as, •But it may be impractical to obtain the perfect knowledge of channel vector v. Double-Weighted Maximum Energy Sum Selection (DW-MESS) • For correct , the mean and variance of are minimized. • It yields the following TOA estimate,
- 36. v. Bayesian Estimation •If distribution of is known a-priori for each energy block m, the noise variance is known accurately, the TOA estimate can be obtained using a Bayesian approach. The leading energy block can be estimated as, •Where the Probability Distribution Function expressed as, • It serves as a benchmark for other sub-optimal estimators.
- 37. 5.6 Low-Complexity UWBRanging Techniques • Due to a-priori knowledge requirement and implementation complexities, ML techniques discussed in earlier section are not very practical. 5.6.1. Ranging with largest- peak-detection techniques • To improve the performance of the peak detector is to consider the largest correlation peaks. • Algorithms involve the detection of the N largest positive and negative values of MF output, where N is number of paths considered in the search • Three algorithms are proposed as a. Single Search b. Search and Subtract c. Search, Subtract and Readjust
- 38. a. Single Search •It calculates Absolute values of Match Filter (MF) output. • If time indices of strongest MPCs are represented by , the TOA of received signal is estimated as, • Delay and amplitude vectors are estimated with a single look • Where, denotes the sampling period of the receiver • Efficient for resolvable channels (multipath are separable) Figure: Single search TOA estimator
- 39. b. Search andSubtract • In order to improve TOA estimation performance in non-resolvable channels (non separable channel), we have to modify single search algorithm. •After estimating TOA corresponding to the strongest MPC ( ) , this MPC is regenerated using the received pulse shape and subtracted from the received signal. • The TOA of second strongest MPC ( ) is estimated using the updated received signal. Again this MPC is reconstructed and subtracted from the signal. •The same procedure iterates times, TOA of the received signal is given by the minimum of the TOA values
- 40. c. Search, Subtractand readjust • Improve the performance of the search and subtract algorithm by joint estimation of the channel coefficients at each iteration of the algorithm. • The channel coefficient for the second strongest MPC is calculated as, •According to trade-off between accuracy and complexity, value should be optimized Figure: Search, Subtract and Readjust TOA estimator
- 41. Comparison of ThreeAlgorithms • Single search algorithm has lowest complexity but yields worst accuracy as compared to two techniques. It gives better result in “direct LOS” and “high SNR” cases •Later Two algorithms, can perform better in non-resolvable channels and require matrix inversion operations, their implementation may be computationally intensive at large values of . They are superior in “extreme-low SNR” and “low SNR” cases due to the Larger presence of overlapped paths.
- 42. 5.6.2. Ranging withTwo-Step TOA estimators • Two-step TOA estimators can be used to relax the sampling rate requirement . • At First Step, a rough timing estimate is obtained using Low Sampling Rates • Second Step refines the TOA estimate using higher sampling rates Figure: Block Diagram for Two step TOA estimator
- 43. a. First Step •A low-complexity receiver with a low sampling rate is employed so as to obtain a rough estimate of the TOA. • Energy Detection (ED) receiver can be used to provide a rough TOA estimate and to reduce uncertainty region for the TOA. • Critical parameter is the selection of the sampling interval Tsmp for ED receiver. •If Tsmp is selected very large, ED can accurately lock desired signal but ambiguity region remains very large • If Tsmp is selected very small, ambiguity region is narrowed but first MPC may be missed.
- 44. b. Second Step •Uses Higher sampling rates and more accurate techniques in order to precisely determine the TOA • For this it uses search back algorithms, correlation-based techniques, method-of- moments estimator. •Advantage: Narrows down the TOA search space in its low-complexity first step and smaller time interval in second step.
- 45. 5.6.3. Ranging withDirty Templates • Dirty-template receiver operates on symbol-rate samples. • Received Signal can be used as a correlator template, which is noisy (“dirty”) • TOA is estimated by cross-correlations of the symbol-length portions of received signal • For dirty-template scheme, both non-data aided (blind) and data-aided approaches can be considered. • In non-data aided case, symbols are equiprobable where as for a data-aided case, special training sequences is considered Advantage: • It has unique multipath energy collection capability • No multipath parameter estimation is required Disadvantage: • Performance degradation since signal itself is noisy, • TOA estimation will have an ambiguity.
- 46. 5.6.4. Threshold-based Ranging •Compare individual signal samples with certain threshold in order to identify the first arriving MPC • Advantage: Ranging can be implemented in the analog domain • For illustration, we consider the figure as, Figure: Illustration of Threshold-based first path detection where denotes a threshold and denotes the length of a search-back window
- 47. a. Max • Basedon the selection of strongest sample. • Multiplication of it’s time index with sampling time will give TOA of received signal • But it suffers from performance degradation under NLOS propagation where strongest path is not necessarily the first path b. Peak-Max • Based on the selection of earliest sample among the strongest. • TOA estimation has to be optimized according to channel characteristics. c. Simple Thresholding (ST) • Takes an estimate of First arriving path • Threshold-to-noise Ratio (TNR) is defined and TOA is estimated as the first threshold crossing event
- 48. d. Threshold-based rangingwith Jump Back and Search Forward (JBSF) algorithm • It considers an ED receiver • Assumption is that the receiver is synchronized to the strongest path. • First, algorithm jumps to a sample prior to the strongest path and searches for the leading edge in the forward direction by comparing the samples against a threshold. •Search proceeds until the sample-under-test is above a threshold Figure: Illustration of JBSF algorithm and SBS algorithm using ED receiver
- 49. , denotes searchback window length in samples , is the index of strongest sample , is the index of first arriving path’s sample , is the index of first sample within the search back window , is the delay between first arriving path’s sample and the strongest sample , is the delay between the index of the first sample within search window and first arriving path’s sample • Threshold is set base upon the standard deviation of the noise
- 50. •Setting a thresholdto a very small value, yield early false alerts •Using a larger threshold, Mean Absolute Error (MAE) may be minimized by the detection of a stronger sample later than the first sample. e. Threshold-based ranging with Serial Backward Search (SBS) algorithm • The paths/samples can be searched one-by-one in backward direction • SBS handle the existence of noise that is cause due to time delay between two clusters, gaps between the MPCs of same cluster for accurate leading edge detection. •Two different cases are considered for SBS algorithm as, e.i. Case 1 A single cluster channel is considered, where there is no noise-only region between the strongest sample and the leading edge sample e.ii. Case 2 A multiple-cluster channel structure is considered where there may be noise-only regions between the strongest path and first path
- 51. Figure: Illustration ofsearch back scheme: (a) single cluster (b) multiple clusters
- 52. e.i. Case 1:dense single cluster (SC) analysis • The leading block estimate for SBS-SC is give by, e.ii . Case 2: multiple clusters (MCs) with noise-only region analysis • Typical UWB channels arrive at the receiver in Multiple Clusters (MCs) i.e. groups of MPCs that are separated by noise-only samples.
- 53. Figure: MAE performancesof different algorithms for the optimal thresholds that minimize the MAE
- 54. • The accuracyof the SBS-MC algorithm is observed to be inferior to that of the JBSF algorithm •The MAEs for JBSF are plotted for different threshold settings as, •If the threshold is set low, Probability of False Alarm in the noise-only region of the signal may be larger
- 55. Summary Treatment of timebased ranging via UWB radios includes, Potential error sources Quantification of fundamental performance limits via Cramer-Rao and Ziv-Zakai lower bounds Emphasis on importance of accurate ranging for precise positioning and different error sources in time-based ranging are discussed. Time-based ranging are formulated through various transceiver types We investigated accuracy and Maximum Likelihood based techniques Finally, an alternative Low-complexity ranging algorithms for UWB systems are discussed.
- 56.