Ultra Wide Band Radios Ranging

1. Chapter-5: Time-Based Ranging Via UWB Radios
Prof. Jae-Young Pyun
Dept. of Information and Communication Engineering
Chosun University
Submitted By: Sujan Shrestha
(Student ID: 20157711)

2. Objective
• Strategies to resolved 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
5.2 Error 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
Nm Reference 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 position of the TN, Using Non-linear Least Squares (NLS) technique

6. ,is Residual error corresponding 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 Sources in Time-Based Ranging
i. Multipath Propagation
Figure: Illustration of TOA estimation problem in a multipath channel

8. •Effect due to NLOS signal Propagation or Antenna Effects.
Figure: Different Scenarios for Channel Realization in LOS and NLOS Situations

9. Receiver Uses correlator
(Matched Filter) 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 orthogonal Keying (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 Access Interference (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 Line of 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
•Let the 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 energy of 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 to obtain 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 of Three Receivers
•We consider a root-raised cosine (RRC) pulse with Tp = 1ns, of roll-off The
RRC pulse is give by:

21. Figure: Received normalized pulse 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 Reciever ED 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 Limits for 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 Lower Bounds 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 Lower Bounds 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 Lower Bounds (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 and CRLBs in AWGN channels for different pulse widths.

28. In example we observe 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 Lower Bounds (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 estimation with 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 computational complexity 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 desired signal
, 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 Energy Sum 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 Maximum Energy 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 UWB Ranging 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 and Subtract
• 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, Subtract and 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 Three Algorithms
• 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 with Two-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 with Dirty 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
• Based on 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 ranging with 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 search back 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 threshold to 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 of search 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 performances of different algorithms for the optimal thresholds that
minimize the MAE

54. • The accuracy of 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 time based 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. 감사합니다
Thank You