Chapter-5: Time-Based Ranging Via UWB Radios

Prof. Jae-Young PyunDept. of Information and Communication Engineering

Chosun University

Submitted By: Sujan Shrestha (Student ID: 20157711)

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.

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

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)

To estimate position of the TN, Using Non-linear Least Squares (NLS) technique

,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

5.2 Error Sources in Time-Based Ranging

i. Multipath Propagation

Figure: Illustration of TOA estimation problem in a multipath channel

•Effect due to NLOS signal Propagation or Antenna Effects.

Figure: Different Scenarios for Channel Realization in LOS and NLOS Situations

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)

•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

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.

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

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

,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:

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.

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 )

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.

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

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

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:

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

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.

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,

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.

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

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,

Figure: ZZLBs and CRLBs in AWGN channels for different pulse widths.

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

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.

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.

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

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:

, is desired signal

, is the nth element of z

, is the noise after BPF

, is the true hypothesisii. 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.

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

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,

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.

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

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

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

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

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.

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

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.

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.

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.

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

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

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

, 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

•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

Figure: Illustration of search back scheme: (a) single cluster (b) multiple clusters

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.

Figure: MAE performances of different algorithms for the optimal thresholds that

minimize the MAE

• 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

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.

감사합니다 Thank You