indoor toa error measurement, modeling, and analysis

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Indoor TOA Error Measurement,Modeling, and Analysis

Ian Sharp and Kegen Yu, Senior Member, IEEE

Abstract— This paper presents a comprehensive investigationon the error characteristics of time-of-arrival (TOA) measure-ments obtained from positioning systems in indoor environmentswhere rich multipath and nonline-of-sight propagation exist.By careful analysis of the measured data from three different setsof measurements, a model for the range errors is developed. Themodel has two components, one associated with the TOA mea-surement in the receiver and another associated with the delayexcesses accumulated along the propagation path. By modelingmultipath signals mathematically, and application of the centrallimit theorem, it is shown that statistical shape of the leading edgeof a received pulse is essentially independent of the details of thescattering environment. Application of this statistical shape ofthe leading edge allows estimates of the statistical distributionof the TOA measurements to be calculated, either analyticallyor numerically from simulations. The second component of thedelay excess is a model based on the number of walls along thepath. Statistical performance based on this combined model isshown to be in good agreement with measured data collectedfrom three different systems that have different RF frequencies,signal bandwidths, and TOA detection algorithms. The insightsafforded by the theory assists in the design of more accuratepositioning systems.

Index Terms— Experimental verification, leading edgealgorithm, multipath and nonline-of-sight (NLOS) propagation,RF propagation through walls, TOA (time-of-arrival) errormodeling and analysis.

I. INTRODUCTION

THE development of mobile radio systems in recentyears has been dramatic, both for wide-area outdoor

use (data and position determination) as well as indoordata communications based on the IEEE 802.11 standards.While applications, such as tracking people and assets [1]–[5]have been suggested, indoor positioning is still mainly in itsinfancy, partly due to the difficult operating environment forradio transmissions. In such indoor environments the globalpositioning system does not work well since the satellitesignals are blocked by the buildings, and typically nonline-of-sight (NLOS) multipath conditions predominate. The design

Manuscript received September 10, 2013; revised January 23, 2014;accepted January 24, 2014. The work of K. Yu was supported in part bythe National Natural Science Foundation of China under Grant 61170252.The Associate Editor coordinating the review process was Dr. Dario Petri.

I. Sharp was with the Wireless and Networking Technologies Laboratory,Information and Communication Technologies Centre, Commonwealth Scien-tific and Industrial Research Organisation, Marsfield, NSW 2122, Australia.He is now with wireless systems (e-mail: [email protected]).

K. Yu was with the School of Civil and Environmental Engineering,University of New South Wales, Sydney, NSW 2052, Australia. He is withthe School of Geodesy and Geomatics, Wuhan University, Wuhan 430072,China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2014.2308995

and performance estimation of indoor positioning systems ischallenging as the rich multipath indoor radio propagationenvironment makes accurate range measurements difficult.In particular, the scattering of the radio signals results inranging measurement errors consisting of a biased componentas well as the random error typical of line-of-sight (LOS)outdoor positioning systems. The measurement and modelingof the ranging performance in both LOS and NLOS conditionshas been much reported in [6]–[18], but the results aredifficult to interpret in a generic sense due to the variations inmeasurement techniques, radio frequencies, signal bandwidths,range, and the characteristics of the building. Because of therandom nature of the received signal, the measurements aretypically modeled as range error statistical distributions, whichare the empirically fitted to a statistical model. However, suchdata are difficult to apply in other operating environments, asthe model parameters are only appropriate to the measurementtechnique and the particular indoor environment. In practicemeasured errors will have contributions associated with boththe scattering environment (a function of the building archi-tecture and the radio frequency) and the measurement methodfor determining the time-of-arrival (TOA) of the signal at thereceiver. However, by applying the statistical model proposedby this paper, it is shown that the measured statistical resultscan be explained without resorting to direct parameter fittingof the statistical models to the measured data. Note that themodel is intended to apply to indoor positioning systems withranges similar to that of NLOS Wi-Fi data transmissions usingIEEE 802.11 (say of the order of 50 m), and not short-range(say up to 10 m) typical of IEEE 802.15.4 (Zigbee).

The proposed method of estimating range errors consists ofsplitting the range errors into two components, namely thoseassociated with NLOS propagation from the transmitter to thereceiver, and second, the errors associated with the detectionof the signal within the receiver. While these two componentsare coupled in the measured data, this paper describes methodsof decoupling the relative contributions, which allows compar-isons of the model with the measured data. Furthermore, it willbe shown that in a NLOS environment the statistical character-istics of the leading edge of pulse detected by the receiver areessentially independent of the building characteristics, so thatparameters, such as the zero-range bias error and its associatedstatistical variation, can be predicted for any particular leadingedge TOA detection algorithm. Additionally, it is also shownrange errors associated with the propagation path can beexplained by propagation delay excesses associated with thenumber of walls along the path. As the number of walls alonga path is both a function of the architecture of the building and

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2 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

the range, the path component to the measured range errors canbe estimated (at least in theory) by use of a map of the buildingand the delay excess per wall. As a consequence, performanceof any particular positioning system, from relatively narrowbandwidth Wi-Fi based systems to ultrawide band (UWB), inany particular building can be estimated by application of thegeneral model proposed using the appropriate parameters.

Thus, in summary, the main contributions of this paper areas follows.

1) Presentation of measurements of range errors in typicalindoor operating conditions for two different actualpositioning systems, rather than using simulated data.

2) Analysis of the measured data to obtain insights into themechanisms causing range errors.

3) Description of a leading edge threshold TOA algorithmwith superior performance characteristics, and providinganalytical expressions to determine its probability den-sity function (PDF) of range errors.

4) Development of an analytical model for the shape of theleading edge in NLOS conditions.

5) Development of a statistical model of delays associatedwith propagation through walls.

6) Comparison of measured and model statistical data forthree different sets of measured data showing goodcorrelation.

The details of the TOA error analysis and statistical modelsare presented in the remainder of this paper. Section II brieflyreviews the theory and concepts of indoor multipath radiopropagation, and presents three sets of measured TOA data.Section III develops an analytical statistical model of theshape of the leading edge of the received multipath pulse;this shape model can be applied to estimate the performanceof any leading-edge TOA algorithm. Section IV applies thetheory of Section III to analyze the performance of theleading edge algorithms actually used for the measurementsdescribed in this paper. Section V develops a statistical modelof the ranging excess errors associated with internal walls ofbuildings. Section VI compares the measured TOA statisticalresults to the theory developed in the previous sections.Finally, Section VII provides a discussion and conclusion onthe consequences of the range error modeling and analysisdescribed in this paper.

II. INDOOR RADIO PROPAGATION MEASUREMENTS

The concept of this paper is to develop a generic statisticalmodel of NLOS TOA range errors that would be applicable toindoor mesh positioning systems with ranges similar to Wi-Fidata transmissions. To guide the development of the model,radio propagation and TOA measurements were performed ata number of RF frequencies and bandwidths from that typicalof Wi-Fi channels to 4 GHz UWB. By analyzing the measureddata documented in this section, the generic mathematicalmodel is developed in Sections III–V.

A. Indoor Propagation Characteristics

Indoor radio propagation is complex, with multiple scatter-ing, diffraction, and signal attenuation associated with walls

Fig. 1. Example of the impulse response of a 2–6 GHz UWB signal in abuilding (with time relative to the straight-line delay). The NLOS path lengthis 21 m. The nominal resolution of the measurements of individual signals isabout 1 ns. Notice the curved leading edge.

and other obstacles along the path from the transmitter tothe receiver. Because of this complexity, analysis requiresconsiderable simplifications. One possible approach is to adoptthe uniform diffraction theory [19], so that the radio signalscan be represented as rays, incorporating both reflected anddiffracted paths. With this interpretation of the scattered signal,the received signal can be expected to be composed of multipleindividual components with varying amplitudes, phases, anddelays depending on the path from the transmitter to thereceiver.

Now consider a typical example of a UWB measurementin an indoor environment, as shown in Fig. 1. The data inthis figure was measured using a Network Analyzer method[27, Appendix B], whereby the windowed broadband spectrumis converted to the impulse response using an inverse Fouriertransform. The impulse response shows that signal scatteringresults in a large number of received signals, with a generaltrend of decreasing amplitude as a function of the delay excess.With these high-bandwidth measurements, individual scatteredsignals down to the measurement resolution (in this case 1 ns)can be observed. Clearly to accurately measure the TOA, thesystem should use the first detectable signal, namely using theleading edge of the impulse. Similar measurements reported in[11], [13], and [14] used the first detectable peak instead of theleading edge, but the resulting standard deviations (STDs) inthe range-errors for the same bandwidth are 2–4 times worsecompared with the examples in Section II-B. Indeed, usingmeasured data in a NLOS environment it is shown in [20]that leading edge algorithms significantly outperform othercommon methods using peak data, including super-resolutionmethods, such as MUSIC.

B. Measured Range Error Data

To provide a variety of measured TOA data for lateranalysis, three separate data sets will be used, covering awide variety of RF frequencies, signal bandwidth, and TOAdetection algorithms. Because of the extensive nature of thedata sets, only a limited sample is included in the followingsubsections. Unlike most reported TOA measurements, twoof the data sets relate to actual positioning systems, while


SHARP AND YU: INDOOR TOA ERROR MEASUREMENT, MODELING, AND ANALYSIS 3

TABLE I

SUMMARY OF SYSTEMS AND PARAMETERS

the third UWB case is extracted from extensive measurementsdescribed in [21]. An overall summary of the systems andtheir parameters is presented in Table I. For more details onthe UWB, wireless ad hoc system for positioning (WASP),and the precision location system (PLS), refer to Section VI.

The results shown in Fig. 2(a)–(c) are representative ofthe larger set of measurements, all of which have similarcharacteristics. The data in the figures show the scatter ofrange errors plotted as a function of range. Also shown isthe linear trendline (solid line) and the two lines ± one pulserise-time relative to the trendline (dashed lines). The data areanalyzed to determine the zero-range intercept bias error (δr0),the range error STD relative to the trendline (σr ), and theslope (λ) of the trendline. Note also that all the measurementshave NLOS conditions, i.e. there is at least one wall betweenthe transmitter and the receiver. Observe that when the dataare presented in this manner, despite the different parametersdescribed above, the generic characteristics are quite similar,suggesting common underlying principles.

1) UWB Measurements: The UWB measurement data inFig. 2(a) are extracted from an extensive series of measure-ments reported in [21]. This reference contains data, whichincludes the range errors with an associated range, and for theNIST building a map of the measurement locations, allow-ing later comparisons with theory. The measurement method[27, Appendix B] is based on Gaussian windowing the spec-trum from a Network Analyzer, with the pulse determined byan inverse Fourier transform. The TOA is determined using athreshold set at a multiple (usually 12) of the noise root-mean-squared (RMS) level. Other similar but less comprehensiveUWB data are given in [22].

The zero-range intercept bias for the NIST building is0.41 ± 0.21 m. This value seems unusually large for UWB,and probably implies some opaque walls rather than the cinderblock construction stated. Data for two other buildings given in[21] have intercept biases of 0.21 ± 0.17 and 0.08 ± 0.17 m.Note that the large statistical variation in these estimates showsthat the zero-range intercept parameter is difficult to determineaccurately from measured data. Similar bandwidth data in[22] also reports the NLOS bias increases linearly range, witha zero-range bias of 0.019 m and a slope of 0.022.

2) WASP Measurements: The measurements in Fig. 2(b)are based on the WASP [23]–[25]. This experimental systemhas been designed to achieve an indoor positioning accuracy

of 1 m (or better) based on a mesh network of nodes.The concept is based on using Wi-Fi analog chip radios, butgenerating an effective wider 125 MHz channel by quicklyscanning over time eight 20 MHz Wi-Fi channels [25], andthen generating a pulse with a rise-time of 13.5 ns byappropriate signal processing of the spread-spectrum signal.The measured range data are from internodal round-trip-timemeasurements [25], [26], [28] using TOA detection based ona threshold set at a level of 18 dB below the local peak nearthe leading edge, as described in [24] and Section IV-A.

3) PLS Measurements: The measurements in Fig. 2(c) arebased on the CSIRO developed PLS, which operates in the2.4-GHz ISM band. This system was designed for outdoorsporting applications, and in particular horse racing. Thedesigned positional accuracy outdoors is 0.5 m. The system isnot intended for operation indoors, but testing was performedto assess its performance indoors.1 The pulse with a rise-time of 25 ns is generated by despreading a 40 Mchips persecond direct sequence spread-spectrum signal. In the testing,range was determined using a round-trip-time technique [26]between two base stations, similar to that used by the WASP.The TOA measurements are based on the Ratio Algorithmdescribed in Section IV-C.

C. Interpretation of Measured Data

While the measurement systems described in Section II-Bare quite different, it is evident that the overall characteristicsof the range error data are quite similar. In particular, note thefollowing characteristics.

1) Nearly all the data lie within the two boundary linesdefined relative to the least-squares (LSs) fitted lineartrendline and the pulse rise-time. This characteristic is tobe expected from a good leading-edge algorithm whichis not affected by signal delays greater than the rise-timeof the pulse.

2) The trendline delay excesses are a linear function ofrange. This characteristic is interpreted as the direct pathsignal being below the detection threshold of the leadingedge TOA algorithm due to signal attenuation (suchas through walls) and scattering along the straight-lineNLOS path from the transmitter to the receiver.

1The unsatisfactory PLS performance indoors prompted the development ofthe WASP for indoor applications.



Fig. 2. (a) UWB range errors (NIST building). Frequency 5 GHz, Gaussianwindow channel-width 500 MHz (250 MHz 3-dB bandwidth), and pulse rise-time 3 ns (0.9 m). Range error STD (detrended) 0.42 m, slope 0.0355, andzero-range bias 0.41 m. Number of samples is 38 (2 bad). Cinder block walls.(b) WASP range errors (Marsfield Campus buildings, CSIRO). Frequency5.8 GHz, channel (reconstructed) bandwidth 125 MHz, and pulse rise-time13.5 ns (4.0 m). Range error STD (detrended) 1.14 m, slope 0.0209, andzero-range bias 0.87 m. Number of samples 191 (1 bad). (c) PLS range errors(Pawsey building, CSIRO). Frequency 2.4 GHz, channel bandwidth 60 MHz,and pulse rise-time 25 ns (7.5 m). Range error STD (detrended) 3.0 m, slope0.156, and zero-range bias 0.84 m. Number of samples 128 (3 bad).

3) The scatter about the linear trendline is broadly sim-ilar, and largely independent of range, although someextreme range errors (termed bad data in Fig. 2) tendto be at longer ranges. This characteristic is interpreted

as a range-independent variation associated with themeasurement of TOA in the receiver, and the pathscattering of errors being independent of range.

4) The zero-range intercept is nonzero. This characteristicis interpreted as being associated with the TOA detectionalgorithm as it is independent of the delay excess, whichincreases linearly along the path.

Based on the above discussion and Fig. 2(a)–(c), indoorranging errors have two main components, namely the errorsthat accumulate along the path associated with scatteringand passing through walls, and errors associated with thedetermination of the TOA in the receiver. Thus, NLOS rangingerrors can be expressed as the sum of these two types of errors,namely

εNLOS = εTOA(τpulse) + εpath(R, Nwalls; fc, BW). (1)

The errors from TOA determination will be a function of theTOA detection algorithm and the rise-time of the leading edgeof the receiver baseband pulse (τpulse). The path errors due toNLOS propagation will be a function of the path length (R)and it is proposed by the number of walls through which thepath passes (Nwalls), while the RF signal carrier frequency ( fc)and the signal bandwidth (BW) will be constant parameters forany given positioning system. Note that often in the literature(such as in [11], [12], [14], and [19]), the bias errors areaveraged over all data ranges, so this range-related effectcannot be inferred from the published results.

To assist in the understanding of the detailed developmentof the range error model (1) in the following sections, Fig. 3summarizes the flow of the mathematical development of themodel in terms of the TOA detection algorithm, and the pathdelay excesses through walls in Sections III–V. In general, themathematical development determines the mean (bias) rangingerrors and the random variations in the delays expressed interms of the PDFs, and hence the associated means and STDs.The complete mathematical model is then compared with themeasured performance derived from the three sets of rawrange error data summarized in Section II-B. Also note thatthe ∼ symbol signifies a random variable or a functionassociated with a random process throughout the followingtext and equations.

III. NLOS LEADING EDGE ANALYSIS

This section provides a theoretical analysis of the shape ofthe leading edge of the received pulse in a NLOS environment.The aim of the analysis is to obtain a more general insightinto the characteristics of the leading edge of the NLOSmultipath pulse shape. In the following analysis, it is assumedthat the nominal (no multipath) received pulse is triangularin shape due to bandlimiting. For direct-sequence spread-spectrum systems (such as the WASP and the PLS) thisis a good approximation to the despread signal, while forthe inverse fast Fourier transform (FFT) method of detection(UWB case in Section II) the shape depends on the weightingfunction, which is Gaussian in this case. The triangular shapeassumption greatly simplifies the mathematics without greatlyaffecting the results due to the consequences of the central



Fig. 3. Flow diagram of the development of the range error model summarized by (1).

limit theorem (CLT) associated with the summing of the manymultipath signals.

A. Equi-Amplitude With Random Phase Case

The initial (over simplified) case considered is where allthe multipath signals are of equal amplitude but with randomphase; this simplifying assumption is used in the followinganalysis, but variation in the amplitude is considered later inSection III-B.

In a severe multipath environment there will be a largenumber of interference signals, the amplitude of which tendsto decrease with increasing delay excess (Fig. 1). For theo-retical analysis the indoor NLOS multipath signal amplitudedistribution can be approximated as an exponential functionof delay excess, with the exponential parameter defining therate of decay. If the pulse rise-time is much less than thisdecaying period, then to a first-order approximation they canbe considered of equal amplitude. For example, the WASPsystem described in Section II-B2 has a pulse rise-time of13.5 ns, which is small compared with the delay spread shownin Fig. 1.

Now consider the signal phase characteristics. As the pathlengths are much greater than the wavelength, even smallchanges in the path length will result in a rapid changein phase. Thus, it is reasonable to assume that the interferencesignals will have random phase, typically assumed to havea uniform distribution over [0, 2π].

As only interference signals with delays up to aboutthe pulse rise-time (τpulse) can affect the performance ofa leading edge algorithm, without loss of generality onlyinterference signals with delays up to τpulse are considered.

Furthermore, these signals are assumed to be uniformly dis-tributed in time throughout this period with a separation ofδ × τpulse. Thus in the following analysis, it is convenientto normalize the time by τpulse, so the time separation ofindividual signal components is δ in normalized time. Inaddition, it is convenient to normalize the pulse amplitude tounity. While the delay excesses are assumed to be equallyspaced, simulations showed that other random distributions(such as statistically uniform distribution of delays) have littleeffect on the characteristics of the leading edge of the pulse.

First consider that all the UWB delta function componentsof the signal (similar to that shown in Fig. 1) are in phaseand of unit amplitude. The actual transmitted signal will be ofnarrower bandwidth, so the received signal will be the UWBsignal (delta functions) convolved by the impulse responseof the bandlimited baseband signal, which as stated in theintroduction to this section is assumed to be triangular inshape. For this special case the convolution is easy to calculate,namely the summation of the delayed unit-amplitude signals,so that starting at the leading edge the cumulative pulseamplitude at the normalized times 0, δ, 2δ, 3δ, . . . will be

0, δ, 2δ + δ, 3δ + 2δ + δ, . . . δ

N∑

n=1

n (2a)

where there are N delta functions in the UWB signal withinthe bandlimited pulse rise-time. Note that the amplitude offirst (associated index n = 1) convolved delta functioncomponent has an amplitude of N at time index position N ,and conversely the delta function component at n = N hasunit amplitude. This is a consequence of the convolutionprocess, which has a index N−n associated with the triangular



Fig. 4. Example (thick line) of a simulated leading edge pulse, based on the parameters of the PLS, and normalized to unit amplitude. Also shown is thenominal triangular pulse (with the peak at normalized time 2). There are 20 multipath signals in the simulated pulse, three of which are also shown. In thisexample the SNR is 30 dB. A typical threshold value for TOA detection is also shown.

convolution process. These concepts are also shown in Fig. 4,showing the components of a (simulated) NLOS pulse. If thenth UWB signal has a random phase φn , resolving the signalinto real and quadrature components the magnitude of themultipath pulse at pulse time τ = Nδ is given by

MN = δ

√√√√√[

N∑

n=1

n cos φN−n

]2

+[

N∑

n=1

N sin φN−n

]2

. (2b)

The calculation of the statistics of the magnitude of pulse atindex position N using (2b) is difficult due to the nonlinearityof the square-root function, so an alternative approximatemethod is used. Because the components in (2b) are thesummation of a number of random variables, each componentwill have an approximate Gaussian distribution due to the CLT.Furthermore, the expected value of the summation will bezero, as the expected values of the cosines and sines arezero. Under such circumstances the magnitude of the multipathpulse at the N th index position will exhibit approximatelya Rayleigh statistical distribution, provided N is sufficientlylarge; this simplification is later confirmed through simula-tions. At this stage, it is also worth noting that because ofthe consequences of the CLT this result also applies to anystatistical (or nonstatistical) amplitude distribution, due to therandomizing effect of the random phase. Using this Rayleighdistribution approximation the mean (μ) and the STD (σ ) ofthe amplitude are sought. These are given by definition

μ = E[MN ]σ 2 = E[M2

N ] − E[MN ]2. (3)

By expanding the summations in (2), and noting that theexpectation of the cross-product terms will be zero due to their

random phases, the expected value of the magnitude-squaredis given by

E[M2

N

] = δ2N∑

n−1

n2 = δ2S(N) ≈ δ2 N3

3(N � 1) (4)

where S(N) = N(N + 1)(2N + 1)/6.While the expected value of the magnitude cannot be easily

calculated, by assuming a Rayleigh statistical distribution thevariance to the mean-squared ratio has a known value of4/π − 1. Thus

σ 2

μ2 = 4

π− 1 = E

[M2

N

]

E[MN ]2− 1. (5)

Combining (3) and (5) gives the mean of the magnitude as

μ(N)≈√

π

4E[MN ]=δ

√π

2

√S(N)≈ δ

2

√π

3

√N3 (N � 1).

(6)

Note that (6) also applies at any position on the leading edge(namely at the nth position) as well as the N th position.In the context of Fig. 1 where the high-resolution impulsehas N distinguishable impulses (delta functions) in the time-normalized leading edge, then δ = 1/N and τn = nδ = n/N .As the pulse amplitude does not affect the TOA measurement,the N th position can be used to normalize the amplitude ofthe nth position of the multipath pulse, so the expected valueof the magnitude of the normalized pulse is given by

μ(τ ) = μ(n)

μ(N)=

[ n

N

]3/2 ≡ τ 3/2 (7)

where the last expression in (7) is the equivalent continuousanalytical function, so that it can be used in theoretical calcula-tions later. Finally, assuming a Rayleigh statistical distribution,



the ratio of the STD to the mean can be used to estimatethe STD of the leading edge normalized shape, namely from(5) and (7)

σ (τ ) ≈√

4

π− 1τ 3/2. (8)

An important consequence of the above analysis on the designof a leading edge TOA algorithm is the rapid increase in theSTD of the multipath pulse noise as a function of delay fromthe epoch of the pulse. Clearly, the best quality measurementsare near the epoch of the pulse, and the worst data are nearthe peak. This observation explains the large TOA variationfor algorithms based on the peak of the first detectable signal.

B. Random Amplitude and Phase Case

While the equi-amplitude analysis provides useful estimatesof the shape of the leading edge, actual propagation conditionswill have varying amplitudes (Fig. 1), so that the statisticalcharacteristics with varying signal amplitudes and phases areanalyzed in this section. From the analysis in Section III-A,the expected value of the magnitude-squared is

E[M2

N

] = δ2 E

[N∑

n=1

n2a2N−n

]= δ2

N∑

n=1

n2 E[a2]. (9)

The expected value of the signal amplitude-squared in (9) canbe evaluated if the statistics of the signal amplitude are known.For example, with a Rayleigh distribution of the amplitude thecorresponding magnitude-squared is

E[M2

N

] = 1

3N(2N + 1)(N + 1)(δσ )2 ≈

(2

3(δσ )2

)N3. (10)

If (10) is compared with (4) it can be observed that the onlydifference is the constant term, which is also true for anyother amplitude statistical distributions, albeit with a differentconstant. Thus, if the multipath pulse is normalized in the samemanner as described in Section III-A, the resulting mean shapewill be the same, namely as defined by (7). Thus providedthere is a sufficiently large number of scattering sources suchthat statistics of large numbers is reasonably valid (about fivesources, see [27 Sec. 2.1.4]), the mean shape of the leadingedge is independent of the signal amplitude statistics. The STDin the shape of the leading edge can also be calculated in amanner similar to that described in Section III-A.

C. Simulation of Multipath Pulses

To show the above concepts, Fig. 4 shows the results ofa simulation using the concepts described in Section III-B.The figure shows the nominal triangular pulse shape, and themultipath pulse corrupted by noise. The multipath signals havea Rayleigh amplitude distribution, while the phases have auniform distribution over [0, 2π]. The time axis is normalizedby the nominal pulse rise-time of the triangular pulse, and themultipath pulse amplitude is normalized to unity. Observe theconcave curvature of the multipath pulse, which is also delayedrelative to the nominal pulse. Each random pulse will have adifferent shape, but the above analysis shows the on averagethe shape is defined by (7). The TOA detection algorithm task

is to estimate the arrival time with minimal errors comparedwith the idea pulse shape, despite the multipath pulse distor-tion and the effects of receiver noise.

IV. PERFORMANCE OF TOA ALGORITHMS

Section III established the shape of the leading edge pulse,plus its statistical variation; this model is now applied to twoTOA algorithms as used in the actual measurement systemsdescribed in Section II. Note, however, that the merits ofthese algorithms are not being assessed, but rather the analysisallows comparisons with measured data.

Before considering particular TOA algorithms, it is usefulto define desirable characteristics of leading edge algorithms.Ideally, the measurement of TOA for a positioning systemshould:

1) scan from the noise region towards the signalregion to locate the first detectable signal for furtherprocessing;

2) be independent of the pulse amplitude;3) be independent of the noise level before and in the signal

pulse;4) should try to minimize both the effects of noise and mul-

tipath interference; this typically requires a compromisein performance between these two effects;

5) at high signal-to-noise ratio (SNR) and low multipath,the TOA should approach a constant value, called theepoch of the pulse;

6) all nodes in a network should have similarcharacteristics.

There are many possible TOA detection algorithms, but thetwo described in the following subsections are based on thethree particular measurement examples used in this paper.The merits of each algorithm are not particularly relevantfor this paper. Nevertheless, all the algorithms are based ona small section of the impulse response near the leadingedge. In all three sets of data described in Section II, theTOA pulse is derived using an inverse Fourier transformfrom various wideband/spread-spectrum signals. As an FFTis very computationally efficient, the pulse generation is alsovery efficient. In addition, interpolation of the bandlimitedsignal can be efficiently performed using the FFT method.The actual determination of the TOA of the pulse requiresa trivial additional amount of time compared with the basicpulse generation.

A. Threshold Algorithm

The concept of a TOA Threshold Algorithm [9], [12], [20]is simple—the epoch of the pulse is defined by the positionwhere the leading edge of the (normalized) pulse crosses(exceeds) a specified threshold level. In the nominal pulsewithout multipath or noise corruption, the threshold amplitude(ath) can be defined by the relationship to the peak (A) of thepulse, namely

ath = αA (11)

where α is a fixed parameter of the algorithm so that thethreshold time position (pulse epoch) is independent of the



pulse amplitude. The corresponding reference epoch is ατpulsefrom which ranging errors can be determined for a multipathpulse. Hence, in the ideal case the errors will be zero. However,in the presence of receiver noise and multipath signals definingthe threshold level is more difficult. In particular, in a multipathenvironment, the peak signal can be remote from the leadingedge (Fig. 1), so that applying (11) will result in a thresholddifferent from that in the error-free case, resulting in rangemeasurement errors. Second, receiver noise before the leadingedge can be falsely interpreted as the leading edge of the pulse.Clearly, from the above analysis of the variation in the shapeof the leading edge the threshold should be as low as possible,but conversely should be well above the noise level. Further-more, the multipath noise will result in noise-related errors indetermining the peak amplitude (A), which in turn will resultin errors in defining the threshold level and consequentially thepulse epoch. Based on this discussion and Fig. 4, the proposedThreshold Algorithm is summarized as follows.

1) Using the position of the pulse peak amplitude as atime reference guide, the noise section of the pulse(well before the peak) is located, and the noise (σn)determined. An initial noise threshold is defined as kσn ,where k is typically 3 or 4. The value of k is used bythe system designer to select between better multipathperformance (smaller k), or better rejection (larger k) offalse detection because of noise before the signal pulse.See also Fig. 4.

2) The time position where the pulse leading edge exceedsthe noise threshold is determined. To minimize falsedetection, a total of three adjacent samples is checked,and the search continues if all three do not exceed thethreshold.

3) Due to the curved nature of the multipath leading edge[(7) and Fig. 4], this position will be somewhat greaterin time than the true start of the leading edge, and willalso be a function of the pulse SNR.

4) Using this noise threshold time, define a time positionadvanced by the known rise-time of the nominal pulse.The peak of the pulse within this time interval is located,and defined as the nominal peak A for use in (11). Notethat ath > kσn is assumed to hold; otherwise define thethreshold amplitude as ath = kσn .

5) The scan as in step (2) above is repeated with theupdated threshold level to locate the pulse epoch.

This somewhat complex Threshold Algorithm significantlyoutperforms an algorithm with a fixed threshold in a multipathenvironment [20].

B. Random Variation in the TOA Performance

While the TOA performance analysis can be based on themean pulse shape described in Section III, this approach failsto account for the statistical variations in the leading edge.Thus, a more rigorous theoretical approach must apply thevariability in the leading edge to the TOA algorithm, whichresults in the statistical variation in the epoch of the pulse;from this statistical variation (as defined by its PDF) the truemean performance can be calculated.

The Threshold Algorithm as defined above is complicated,so a rigorous analytical mathematical analysis is difficult; thussome simplifications are necessary. The determination of theepoch of the signal using the Threshold Algorithm is based onscanning the leading edge of the pulse from the noise sectionuntil the amplitude exceeds a threshold. As the leading edge ina multipath environment is random, the statistical performanceshould be based on the analysis of many such random pulses.However, an alternative equivalent approach that simplifies themathematics is to reverse the order of the analysis, so thatthe statistical variation at each point on the leading edge isanalyzed first, and then the probability that the statistical pulseexceeds the threshold is calculated.

The analysis in Section III showed that the pulse amplitudeat each point on the leading edge has approximately Rayleighstatistics, so using (8) the complementary cumulative distri-bution function of the probability that a threshold amplitudea will be exceeded at normalized time τ is given by

Ce(τ ) = exp

[− πa2

2 (4 − π) τ 3

](τ ≥ 0) (12a)

and differentiating (12a) the corresponding PDF is

pe(τ, a) = 3πa2

2 (4 − π) τ 4 exp

[− πa2

2(4 − π)τ 3

]. (12b)

The threshold value a, as given by (11), depends on thepeak amplitude (A), which in this case is a statistical randomvariable at τ = 1. As before, A (and hence a) will haveapproximately a Rayleigh statistical distribution, so the PDFof the threshold amplitude is

pa(a, α) = πa

(4 − π) αexp

[− πa2

2 (4 − π) α2

]. (13)

Using this definition of the threshold amplitude, the estimateof the PDF of the pulse epoch is given by

Pe(τ, α) =∞∫

0

pa(a, α) pe(τ, a)da = 3α2τ 2

(α2 + τ 3)2 . (14)

The corresponding mean and STD of the TOA using (14) aregiven by

με (α) = τpulse

(2π

3√

3α2/3 − α

)(15a)

σε (α) = τpulse

(2

3

√π

(√3 − π/3

)α2/3

). (15b)

These analytical results are compared with measurements inSection VI.

C. Performance of Ratio Algorithm

This section considers the performance of another typeof leading edge TOA algorithm (used by the PLS inSection II-B3), which is based on the concept that the shape ofleading edge is largely independent of the pulse amplitude andthe receiver noise. In particular, the Ratio Algorithm describedin detail in [27, Ch. 4] is based on determining the ratio (ρ)of the pulse amplitude at two points on the leading edge,



and compares it to a value ρ0, which represents a zero-errorset-point. The set-point value (typically about 0.4) is chosento optimize performance (a compromise between noise andmultipath tracking errors). A feedback loop adjusts the phaseof a local clock to try to maintain the measured ratio at theset-point, namely ρ = ρ0. The local clock then gives the TOA.The statistical performance analysis of the Ratio Algorithm isgiven in Appendix A.

To show an alternative to the analytical methods usedfor the Threshold Algorithm, the performance of the RatioAlgorithm will be evaluated by a Monte-Carlo simulationusing the statistical characteristics of the leading edge estab-lished in Section III; simulation methods are appropriate whenanalytical calculations are difficult. The resulting cumulativedistribution function (CDF) using parameters appropriate forthe PLS is shown in Fig. 9.

V. PATH PROPAGATION

The analysis in Sections III and IV shows that in a heavilymultipath NLOS environment the shape of the leading edge isdistorted, resulting in TOA measurement errors. These effectsare essentially independent of the length of the propagationpath, but additional errors occur due to the characteristics ofthe multipath scattering along the path, resulting in a delayexcess above the straight line path delay. This section analysesthis delay excess based on the number of walls along the path,with each wall contributing an increment in the delay excess.

The path propagation error can be considered as having amean (bias) error and a random component. The mean biascomponent will be a function of the range and the mean num-ber of walls (Nwalls) along the path of range R. The randomcomponent for a given range would be associated with thestatistical variation in the number of walls for that particularrange, and the variation in the excess delay associated with thesignal scattering at each wall. The path range errors associatedwith walls can thus be expressed in the form

εpath = (γ0 + �γ )(N0(R) + �Nwalls) (16)

where γ0 is the mean delay excess per wall along a pathin the particular indoor environment, N0(R) is the expectednumber of walls along a path of length R, �γ is a ran-dom variable (with zero mean) which represents the varia-tion in mean delay excess per wall along various paths oflength R, and �Nwalls is the random variation in the numberof walls along various paths of length R. These statisticscan be analyzed both theoretically and from measured data.Also, the statistical variation in the number of walls alonga path in a particular environment can be determined fromarchitectural maps of the building.

The overall statistics for the range error along indoor pathscan determined by expanding (16), and applying the statisticaldistribution of the two random components, namely �γ and�Nwall. However, the analysis is considerable simplified byobserving that in both cases the random component is muchsmaller than the mean of the component, so that the productof the random components can be ignored to a good order of

accuracy, namely

εpath ≈ γ0 N0(R) + ⌊N0(R)�γ + γ0�Nwalls

⌋

= N0(R)�γ + γ0 N (R). (17)

Thus, the range error associated with the path can be expressedas weighted sum of two random variables, one associated withthe variation in the number of walls along the path and anotherassociated with the variation in delay excess per wall. ThePDF of the sum of these random components will be theconvolution of their individual PDFs. The statistical analysisof each component is given in the following subsections.

A. Range Error Bias Models

The bias error model considered below (see measuredresults in Section II and elsewhere [29]–[34]) is that the meanbias errors can be modeled as a linear function of range.This model assumes that the scattering effects increase thebias linearly with range, but does not explicitly consider anyspecific mechanisms, such as the effects of walls and otherlarge objects. This model can be expressed as

εbias = δr0 + λR (18)

where δr0 is bias at zero range and λ is the model parameterdetermined by a LS fit to the measured data, and will varyfrom building to building due to variations in the architecture.An alternative building architecture-based model uses thenumber of walls along a path and the apparent delay excessper wall as a proxy for the more complex scattering along thepropagation path and through walls. The data in Section V-Bsuggests a mean bias model as

εbias = ε0 + γ Nwalls(R) (19)

where the mean number of walls are calculated in range binsas explained in Section V-B. It will be shown in the nexttwo sections that models (18) and (19) are in fact differentmanifestations of the same underlying statistical processes;model (19) is preferable theoretically as the parameters ofthe model are directly related (at least in principle) to thebuilding architecture, while model (18) is practically easy toapply using measured data, as shown in [33].

B. Walls-Based Statistics Along a Path

Consider determining the parameters in (19) for a particularoperating environment and positioning system. The mean biaserrors can be determined from measurements (for example, asshown in Section II), and a map of the building can be usedto determine the number of walls along each measurementpath, so the γ parameter in (19) can be estimated by a LSfit to the data. To reduce the statistical variation, it is usefulto group the data into range bins, and analyze the mean biaserror in a bin versus the mean number of walls associatedwith a range bin, as shown in the example in Fig. 5. The totalmeasurement noise is reduced considerably by averaging thedata in each range bin (about 30 samples per bin in this case),so the variation is reduced by about a factor of 1/

√30. As can



Fig. 5. WASP data for the mean bias error versus the mean number of walls in a range bin are shown. The linear model (19) parameters are intercept bias0.59 m, slope 0.160 m per wall, and with an STD of 0.17 m.

be observed in Fig. 5, there is apparently a linear variation asdefined by (19).

The data, such as shown in Fig. 5, can provide importantinformation about the nature of the bias errors. Observe thatthe delay excess for the first wall (about 0.6 m) is muchmore than the delays for subsequent walls (about 0.15 m).The interpretation is that the intercept bias ε0 at walls = 0 isnot related to walls, but is due to some other process, namelythe characteristics of the leading edge algorithm previouslyanalyzed in Sections III and IV. Furthermore, it is concludedthat the noise in the raw range error data is associated withboth the TOA measurement variation and the variation in thedelay excess associated with walls.

The data in Fig. 5 can also give further insight into thenature of the causes of delay excesses as a function ofpropagation along the path from the transmitter to the receiver.From Fig. 5, the estimated delay excess per wall is about0.16 m, which is large compared with the thickness of thewalls (about 5 cm), but small compared with the sizes of roomsand the distance separation between walls. These observationsinferred from multiple measurements over a wide area areconfirmed by specific individual measurements through oneor two walls described in [28]. For example, the median delayexcess (measured using the WASP hardware) through a glasswall was 0.22 m, and through a double-brick wall 0.15 m.For propagation through two internal walls the measured delayexcess was 0.4 m, or about twice that of a single wall, andin agreement with the data in Fig. 5. Clearly, these measureddelay excesses show that the main component of the excessis not associated with the delay expected from a single rayand the electrical properties of the material of the walls,but is a more complex relationship associated with multiplepaths. The exact mechanisms associated with walls are beyondthe scope of this paper, except the observation that the path

delays are closely related to the number of walls along thepath.

The number of walls for a path between two points can bedetermined from the bit map (black/white) generated from thearchitectural drawings of the building. A wall is a black bit (orbits) along the path, so that the number of walls can be easilycounted. By averaging over a large number of random pathswithin the building (typical of paths from a mobile deviceto a base station), the building-wide average walls per unitlength can be computed. Using this average parameter, thestatistical variation in the number of walls for a given pathlength can be estimated. This is the classic description ofa Poisson statistical problem, namely the actual number ofwalls along a path will be a random variable, which exhibitsa Poisson statistical distribution.

An example of the statistical distribution of the number ofwalls in a range bin is shown in Fig. 6, together with thePoisson distribution with the same mean number of walls asin the range bin data set. The similarity of the two distributionssupports the assumption of Poisson statistics.

C. Statistics of Path Range Errors

This section provides a statistical analysis of the path rangeerrors based on the model (19). This model has two randomvariables, namely the number of walls along a path of length Rand the delay excess per wall. The number of walls perunit length (η) is considered to be a constant throughout thecoverage area of the system, and can be determined from amap of the building. As a consequence the expected numberof walls along the path of length R is N0 = ηR, but the actualnumber of walls N(R) is expected to have a Poisson statisticaldistribution, namely

P(n, N0) = e−N0 Nn0 /n! (n = 0, 1, 2, . . .). (20)



Fig. 6. Figure compares the distribution on the number of walls for therange bin 25–30 m (with the average number of 7.1 walls) with the Poissondistribution using this mean value.

The delay excess at each wall will be random variable ofunknown statistical variation, but as the delay accumulatesat each wall along the path the overall average delay perwall for the path is expected to be approximately a Gaussianrandom variable due to the consequences of the CLT. Thus,the �γ random variable in (17) is assumed to have the normaldistribution N(0, σγ ), where the STD σγ is determined fromthe measured data. Note, however, that as the delay excess isalways positive the STD in the normal distribution should belimited to about σγ ≤ γ0/3, where γ0 is the mean delay perwall. Now consider the overall statistics of the range errorsassociated with a path. From (20) the expected (mean) erroris

E[εpath] = γ0 N0(R) ≈ ηγ0 R = λR. (21)

As from (1) there is also a contribution to the range errorsfrom the TOA algorithm, the expected value of the rangeerrors cannot be modeled by (21) alone. However, as the TOAalgorithm statistics are independent of the path range, with anoffset bias of ε0 and a zero-mean random component, the totalrange error can be modeled as

εNLOS(R) = ε0 + ηγ0 R + Noise(R) (22)

where the Noise(R) has zero expectation at all ranges. Observethat (22) is the same as model (18), which shows that therange-based model (18) can be predicted from the wall-basedmodel. Thus, plotting the measured range error against rangeis expected to show a noisy linear trendline, which indeedis what is seen in the measured data in Section II and in[31], [33], and [34]. Further, performing a LS fit to the datausing the linear model (22) allows estimates of parametersε0 and λ = ηγ0. Additionally, as the η parameter can beestimated from the map of the building, an estimate of themean delay excess per wall (γ0) can also be determined fromthe measured range error data.

Now consider determining the PDF of the random compo-nent of the path errors εpath based on (17). From the abovediscussion, and using ε as a dummy variable in the following

Fig. 7. Examples of the PDFs of NLOS range error along a path, withγ0 = 0.15 m, σγ = 0.03 m, and the expected number of walls along the path(N0) as indicated in the legend.

analysis, the statistical distribution of the first component in(17) is a Normal (Gaussian) distribution

g(ε; σγ ) = N(0, N0(R)σγ ). (23)

The PDF of the second component is a Poisson distribution,so the mean delay excess through n walls is εwalls(n) = γ0nwith a probability of P(n, N0). Finally, as from (17) therandom component of total delay excess error is the sum ofthe Gaussian and Poisson distributed components, the PDF ofthe total path error is the convolution of the two PDFs givenby (20) and (23), namely

C (ε) =∞∫

−∞P (x, N0)g (ε − x; σλ) dx . (24)

To evaluate this integral, note that the P(x) is zero unlessx = nγ0, so the integral in (24) becomes for the nth case

C(ε, n) = P(n, N0)g(ε − nγ0; σγ ). (25)

The total PDF is obtained by summing (25) over all n, namely

ppath(ε) =∞∑

n=0

P(n, N0)g(ε − nγ0; σγ ) (26)

which is effectively the weighted sum of shifted Gaussiandistributions. Note that this distribution includes the LOS case(n = 0); if only NLOS cases are included, then the summationin (26) will commence at n = 1. The expectation (mean) ofthe ranging error is

E[ε]

=∞∫

0

ε ppath(ε)dε=∞∑

n=0

P(n, N0)

∞∫

−nγ0

(x +nγ0)g(x; σγ )dx .

(27a)

Now as the usual case is that nγ0 � N0σγ (see abovecomment on limiting σγ ) the lower limit can effectively be



extended to −∞ with little error, so the last integral in(27a) can be evaluated to be nγ0, and the expectation thenreduces to

E[ε] ≈ γ0

∞∑

n=0

n P(n, N0) = γ0 N0 (27b)

as the summation in (27b) can be recognized as the mean ofthe Poisson distribution. The result in (27b) is in agreementwith (21). A further consequence is that the combined distrib-ution will have a peak near N0γ0 = ηγ0 R, as shown in Fig. 7.A similar analysis shows that the variance of the range errorsis approximately

var[ε] ≈ N0γ20 + N2

0 σ 2γ . (28)

Examples of the distribution with typical parameters is shownin Fig. 7. As for a Poisson distribution the mean number ofwalls is linearly related to the range, the mean and STD aregiven by

με = ηγ0 R

σε = ηR

√γ 2

0

ηR+ σ 2

γ ≈ γ0√

ηR (29)

where the last approximation applies as (typically) γ0 �σγ

√N0. In this case the STD of the range error along the

path only increases slowly with range, but this variation canbe masked by a larger contribution to the STD from the TOAalgorithm, which is independent of range.

Fig. 8 shows the STD typically for the three measurementdata sets (UWB, WASP, and PLS) described in Section II. Theresults depend on the relative effects of the TOA algorithm andthe path statistics. Observe in particular that the WASP has aSTD which is practically independent of range. This effectis evident in Fig. 2(b), and also has been noted previously[33], [34] for measured data from the WASP system, contraryto a common assumption that the STD is proportional torange. This agreement between theory and measured data lendssupport to the analysis described above.

VI. STATISTICAL COMPARISON BETWEEN

MEASUREMENTS AND THEORY

This section describes the comparison between theoryand measurements of ranging errors statistical distribu-tions, and are based on the raw range error measurementsdescribed in Section II and the statistical models described inSections III–V. As the model has two components, one asso-ciated with the TOA algorithm and one associated with thepropagation path, the combined model (as shown in Fig. 3)will include a particular TOA PDF [such as (14) for theThreshold Algorithm] and the Path model (24) to obtain thecombined statistical model (PDF) of the ranges errors minusthe bias errors. The two PDFs are combined by a convolutionof the PDFs of these two statistically independent processes.

A. Leading Edge TOA Measurements Comparison

The measured raw range errors presented in Section II havecontributions from both the leading edge detection method and

Fig. 8. Variation of the STD of the range error along a path for threedifferent systems. The variation with range depends on the characteristics ofeach system, but generally the STD increases slowly with range.

Fig. 9. CDF of the simulation of TOA errors for the Ratio Algorithmcompared with measured PLS range error data (50 samples) for ranges upto 10 m. The Monte-Carlo simulation results are based on 10 000 randompulses. Negative ranging errors are possible due to the statistical variations inthe shape of the leading edge.

errors associated with path propagation. These two randomeffects cannot be decoupled, but the random errors will bemainly associated with leading edge detection provided thepath length is not too long (say ranges up to 10 m).

A summary of the short-range leading edge data is given inTable II, comparing the measured and theoretical values basedon the model in Section III. The mean and STD of the rangeerrors contributable to the TOA are given (and the associated90% confidence interval), as well as the model predictions.

For the UWB case the match is somewhat poor as patherrors dominate, and thus in this case a comparison withthe leading edge theoretical results is not possible. However,the WASP theoretical and measured mean and STD data inTable II agree well, as the path errors are comparativelysmall in this case. For the PLS case Fig. 9 shows theCDF of the measured range errors and the correspondingMonte-Carlo simulation results for the Ratio Algorithm basedon the leading edge theory in Section III. The results are quite



TABLE II

SUMMARY OF THE UWB (NIST), CSIRO WASP AND PLS PERFORMANCE DATA, AND PARAMETERS. THE 90% CONFIDENCE INTERVALS

(INDICATED BY ±∗) ARE GIVEN FOR THE MODEL PARAMETERS ESTIMATED FROM THE MEASURED DATA

similar, showing that the Ratio Algorithm simulation based onthe theory in Section III adequately describes the statistical dis-tribution of measured data. The measured and theoretical meanand STD for the PLS given in Table II are also quite similar.

A Chi-squared goodness-of-fit analysis was also performedto check if the measured range errors are from the TOAmodel distribution. The Chi-squared test [34] is based onk range-bins counts in the histogram bins of the measuredrange errors, and the associated counts using the modelPDF distribution. Then, it can be shown [34] that thesummation V = ∑k

i=1 (Measuredi − Modeli )2/Modeli hasapproximately a Chi-squared statistical distribution χ2

k−1.If V is not greater than the Chi-squared distribution at the1−α level of significance, then the measured data are deemedto be consistent with the model statistical distribution. For thetwo distributions in Fig. 9, at the α = 0.1 level of significancethe values are [V , χ2

k−1,1−α] = [8.00, 10.65], so the Chi-squared test confirms that the measured range error data areconsistent with the model statistical distribution.

B. Comparison of Statistical Variations in Range Errors

The theory in Sections III–V suggests that range errors havecontributions from two sources, namely the leading edge TOA

detection process, and secondly from delay excesses associatedwith NLOS path propagation through walls. The statisticalmodel for the first process has no parameters, which need tobe determined from measured data, while the second processhas two parameters (γ0, σγ ), which require matching to themeasured data; a third parameter (η) for the path propagationmodel can be determined from an architectural map of thebuilding.

The following statistical results are obtained after theremoval of linear trendline bias effects. In this case the residualerrors are limited to those defined by the random componentof the leading edge theory of Section III for a particular TOAalgorithm in Section IV, and the path statistical distribution,namely as given by (24). It is shown in [33] that for mesh posi-tioning systems the bias errors can be estimated and removed,so that the residual statistical ranging errors will effectivelydefine the positional accuracy. The parameters (γ0, σγ ) aredetermined from a LS fit to the measured data, but there islittle variation in the fit due to σγ (changing this parametermainly affects the tails of the distribution). In the model themean number of walls (N0) is set at the overall mean for thebuilding.

As with the comparisons of the statistical distributionsin Fig. 9, the measured and model distributions shown



Fig. 10. (a) CDF of the bias-corrected range errors for the NIST building.The Path model parameters are γ0 = 0.092 m, σγ = 0.03 m, and N0 =4.5 walls. The RMS error in the CDF between the measured and model is0.041, and [V, χ2

k−1,1−α ] = [2.80, 6.25]. (b) CDF of the bias-corrected rangeerrors for the WASP system. The Path model parameters are γ0 = 0.160 m,σγ = 0.05 m, and N0 = 4.3 walls. The RMS error in the CDF between themeasured and model is 0.020, and [V, χ2

k−1,1−α ] = [15.1, 16.0]. (c) CDFof the bias-corrected range errors for the PLS. The Path model parametersare γ0 = 0.74 m, σγ = 0.12 m, and N0 = 3.9 walls. The RMS error inthe CDF between the measured and model is 0.024, and [V, χ2

k−1,1−α ] =[9.52, 10.65].

in Fig. 10(a)–(c) are tested using the Chi-squared analysismethod. However, in this case as there are two parametersderived from the measured data, so the degrees of freedom inthe Chi-squared distribution must be reduced by two.

1) UWB Statistical Results: The measured and the theoreti-cal CDFs of the combined statistical model for the randomcomponent of the errors are compared in Fig. 10(a). Thelimited number of samples (38) means that close matchingcannot be expected, but the two distributions are quite similar,with the model correctly predicting the overall shape and scopeof the range errors. Furthermore, the Chi-squared test passes,so the measured range errors data are consistent with the modelstatistical distribution.

2) WASP Statistical Results: The measured and statisticalmodel CDFs for the random components of the WASP errorsare shown in Fig. 10(b). The large number of samples (190)means that a close match should be possible, and indeed themodel and measured distributions match very well, and theChi-squared test passes; these good results provide furtherconfidence in the validity of the model.

3) PLS Statistical Results: The theoretical and measuredresults for the random component of the PLS errors areshown in Fig. 10(c). The relatively large number of samples(129) means that good statistical estimates should be possiblefrom the measured data. Overall, the model predicts thescope and general statistical characteristics quite well, and theChi-squared test passes.

4) Summary of Statistical Results: The residual range errorstatistical distributions for the three different systems generallyshow a good match between the measurements and the theory.The residual errors have contributions from both the TOAalgorithm and the path propagation, but the relative contri-butions depend on the specific parameters of the particularsystem. Nevertheless, the residual statistics are quite similardespite the bandwidths varying from 40 to 250 MHz. Onenoticeable characteristic common to all is that the statisticaldistributions are not Gaussian, with a noticeable upper tailto the distribution. However, the leading edge algorithmsconstrain the tail to about the pulse rise-time in each case,although the relative tail size somewhat decreases with thesignal bandwidth.

VII. CONCLUSION

This paper investigated the effects of NLOS multipathscattering in an indoor environment on range measurements.Based on observations from measured data, a generic math-ematical statistical model was developed, and was tested atthree different RF frequencies, signal bandwidths, and TOAdetection algorithms; good matches between the model and themeasured data were observed. Based on the observed charac-teristics of the scattered signal with NLOS propagation it wasshown that the leading edge of the pulse can be predicted(both the mean shape and its statistical variation) withoutany need for information about the propagation environment.Furthermore, a statistical model of scattering and propagationthrough walls correctly predicts a mean linear increase in thedelay excess along the path, and also predicts the observedstatistical variation in the measured data.

The consequences of the analysis in this paper have impactson the performance of indoor positioning systems. First,although range bias errors have long been observed indoors,it was not appreciated that the expected zero-range bias is



similar for all paths. The consequence of this observation isthat pseudorange [range plus an (unknown) constant rangeoffset] positioning will be largely unaffected, as this bias canbe incorporated into the pseudorange constant. Second, it isshown that the linear bias with range previously observed in[31], [33], and [34] is associated with internal walls and assuch is expected to be common to most indoor situations. It isshown in [33] that if this bias effect is correctly compensatedfor, the positional errors associated with range-based positionfixing are approximately halved. Finally, these results meanthat designers of positioning systems can make a prioriperformance predictions in particular operating environmentswithout having to resort to measurements or complicatednumerical simulations. However, the current model has oneparameter (mean delay per wall) which must be matched to themeasured data, although the data in this paper could be usedto estimate this parameter. In theory, this wall delay parametercould be estimated by theoretical radio-wave analysis, and issuggested as a topic for further investigation.

APPENDIX A

STATISTICAL PERFORMANCE OF THE RATIO ALGORITHM

This Appendix considers the performance of the RatioAlgorithm introduced in Section IV-C, and described in detailin [27, Ch. 4]. This TOA algorithm is based on determiningthe ratio of two points on the leading edge, with the signalepoch defined when this ratio is equal to the required set-point value ρ0. For a bandlimited signal used in real systemsthe leading edge needs only to be sampled at a period of abouthalf the pulse rise-time, so that nominally only two points arelocated above the noise on the leading edge, although the pulsecan be interpolated without error due to the Sampling theorem.Thus, the Ratio Algorithm can efficiently use all the availableinformation about the leading edge.

The analysis of the performance of the Ratio Algorithmis based on the statistics of the amplitude of samples (suffi-ciently separated in time) as established in Section III, namelyRayleigh statistics. Thus, the statistics of the ratio is deter-mined by deriving its PDF from two statistically independentRayleigh samples

ρ = A1

A2⇒ R(a, σ1)

R(a, σ2). (A-1)

The determination of the required PDF is based on themethod described in [35, Ch. 9]. Consider two transformationsY1 = A1/A2 = ρ and Y2 = A2 such that random variablesare a1 = y1 y2 and a2 = y2. Then, the joint probability of theproduct is given by

fY1Y2 (y1, y2) = |J| fa1a2 (y1y2, y2) (A-2)

where |J| = |y2| is the Jacobian of transform equations.As the two variables have Rayleigh statistics their jointprobability is

fA1 A2 (a1, a2) = a1a2

σ 21 σ 2

2

exp

[−

(a2

1

2σ 21

)−

(a2

2

2σ 22

)](A-3a)

and thus from (A-2)

fY1Y2 (y1, y2)=|y2| y1y22

σ 21 σ 2

2

exp

[−1

2

{(y1y2)

2

σ 21

+ y22

σ 22

}].

(A-3b)

Finally, the required PDF can be determined by integratingout the dummy y2 variable in (A-3b), namely

fY1 (y1) = y1

σ 21 σ 2

2

∞∫

−∞|y2| y2

2 exp

[−1

2

{(y1 y2)

2

σ 21

+ y22

σ 22

}]dy2

or

f (ρ) = y1

σ 21 σ 2

2

∞∫

0

x3e−αx2dx = 2ω2ρ

(ρ2 + ω2)2 (ρ ≥ 0) (A-4)

where α = 1/2[(ρ/σ1)2 + (1/σ2)

2], ω = σ1/σ2, andρ = y1 from the original transformation. The correspondingCDF obtained by integrating (A-4) is

CDFρ (ρ) = ρ2

ρ2 + ω2 (ρ ≥ 0). (A-5)

The PDF defined by (A-4) is zero at ratios of both zeroand infinity, but has a large slowly converging upper tail,characteristic of measured range error data. Using (A-4) itcan be shown that the mean ratio is μρ = (π/2) ω, but theSTD does not exist (infinite) due to the slow convergence inthe upper tail. The 90% CDF occurs when ρ = 3ω and the99% when ρ = 10ω.

While the above analysis is based on the NLOS perfor-mance, the design of the epoch tracking function must bebased on the ideal LOS case where by definition the trackingerror is zero. Thus, the ratio set-point ρ0 = (τ1/τ2)0 =((τ2 − �) /τ2)0 defines the two set-point locations (τ1 and τ2)on the leading edge, where � is the separation between the twopoints in time. The set-point should be chosen such that thelocation of the first point is not too close to the noise threshold,and the second point is not too close to the peak where theeffects of multipath interference are greatest. Typical valuesof ρ0 [27 Sec. 4.6] are in the range 0.3–0.5 with a separationof � = 0.4.

For the NLOS it would be advantageous if the meanratio ρ was the set-point ratio ρ0, which would result inan unbiased estimate of ρ. Applying (7) to obtain the val-ues of the Rayleigh parameters, the requirement for thiscondition is

ρ0 = ρ = π

2ω = π

2

σ1

σ2= π

2

(τ1

τ2

)1.5

= π

2ρ1.5

0 (A-6)

which has the solution ρ0 = 4/π2 = 0.405, and is in thedesirable range described above.

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Ian Sharp is a Senior Consultant on wirelesspositioning systems. He has more than 30 yearsof engineering experience in radio systems. Hisinitial involvement in positioning technology was inaviation, and with the Interscan microwave landingsystem in 1980. From 1980 to 1990, he was theResearch and Development Manager for the Quik-trak covert vehicle tracking system. This systemis now commercially operating worldwide. From1990 to 2007, he was with Commonwealth Scien-tific and Industrial Research Organisation (CSIRO),

Melbourne, VIC, Australia, mainly on developing experimental radio systems.He was the Inventor and Architect Designer with CSIROs precision locationsystem (PLS) for sports applications. The PLS has been successfully trailed inAustralia and the USA. He holds a number of patents relating to positioningtechnology. He is a co-author of the book Ground-Based Wireless Positioning(Wiley and IEEE Press, 2009).

Kegen Yu (SM’12) received the Ph.D. degree inelectrical engineering from the University of Sydney,Sydney, NSW, Australia, in 2003.

He is currently a Professor with the School ofGeodesy and Geomatics, Wuhan University, Wuhan,China. He has worked for Jiangxi Geologicaland Mineral Bureau, Nanchang, China, Departmentof Electrical Engineering at Nanchang University,CWC at the University of Oulu, CSIRO ICT Centre,Department of Electronic Engineering at MacquarieUniversity, and School of Surveying and Geospatial

Engineering (now integrated within the School of Civil and EnvironmentalEngineering) and the Australian Centre for Space Engineering Research withinthe University of New South Wales. He has been an Adjunct Professor withMacquarie University since 2011.

Dr. Yu is currently on the editorial boards of the EURASIP Journal onAdvances in Signal Processing, the IEEE TRANSACTIONS ON AEROSPACE

AND ELECTRONIC SYSTEMS, and the IEEE TRANSACTIONS ON VEHIC-ULAR TECHNOLOGY. He is the Lead Guest Editor for a Special Issue ofPhysical Communication on Navigation and Tracking and for a Special Issueof the EURASIP Journal on Advances in Signal Processing on GNSS RemoteSensing.

indoor toa error measurement, modeling, and analysis

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