Large Error Performance of UWB Ranging in Multipath and Multiuser ...

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006

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Large Error Performance of UWB Ranging in Multipath and Multiuser Environments Joon-Yong Lee, Member, IEEE, and Sungyul Yoo

Abstract—Accurate detection of direct path signal is essential for ranging to the full capabilities of ultra-wideband (UWB) signals. However, the possibility of a large error can be a challenging problem for accurate ranging and positioning. Three different kinds of potential errors encountered in correlation-based time-of-arrival (TOA) estimation are introduced, and an approximate analysis of them is posed. Upper and lower bounding technique is applied to the variance of one of them, namely an early false alarm error. Probability of occurrence of the other two kinds of error is found using the channel statistics, and the mean square of them is evaluated by simulation method. Discussion of the effect of multiple-access schemes on the accuracy of UWB TOA estimation is initiated. Index Terms—Delay estimation, multipath channel, multiuser channel, ranging, ultra-wideband (UWB).

I. INTRODUCTION PERSISTENT problem in the design of ranging and positioning systems for the application in multipath and multiuser environments is the accurate estimation of the time of arrival (TOA) of the direct path signal. Ultra-wideband (UWB) radio is regarded as one of the most viable solutions for this application because of its extremely fine time resolution. In the presence of a clear line-of-sight (LOS) path, accurate ranging using UWB signals is highly feasible [1], [2]. However, in multipath environments, time-delay estimation of the direct path signal is not a simple task because of the presence of noise, interference, and stronger multipath components than direct path [3]. Algorithms to detect the direct path signal have been proposed by several researchers [3]–[6]. It is well known that, in additive white Gaussian noise (AWGN) channels, a correlation detector is a maximum-likelihood time-delay estimator and is optimum in the sense that the minimum error variance is achieved. The Cramer–Rao lower bound (CRLB) of the variance of estimation errors is known to be [7]

A

(1)

Manuscript received July 31, 2005; revised January 9, 2006. This work was supported by the Korea Research Foundation under Grant R05-2004-000-12640-0. This work was presented in part at the IEEE International Conference on Ultra-Wideband, Zurich, Switzerland, September 2005. J.-Y. Lee is with the School of Computer Science and Electrical Engineering, Handong University, Pohang 791-708, Korea (e-mail: [email protected]). S. Yoo was with the Ultra-Wideband Communication Laboratory, Handong University, Pohang 791-708, Korea. He is now with the Department of Electronics Engineering, Kookmin University, Seoul 136-702, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.871997

where denotes the standard deviation of the estimation errors, is the root mean square (rms or Gabor) bandwidth of the signal, and is signal-to-noise ratio (SNR). However, the CRLB is achieved only in the presence of small errors [8]. In low-SNR scenarios, the variance of estimation errors deviates from the values given by the small error analysis [9] and is difficult to examine thoroughly. Ianniello [9] presented an analysis for the probability of an anomalous estimate in correlation-based timedelay estimation. An attempt was made by Zhang et al. [10] to evaluate the CRLB for the time-delay estimation of time-hopped UWB signals. In this paper, we shall attempt to analyze the large error performance for UWB time-delay estimation in multipath and multiuser environments, assuming a serial search scheme based on correlation detection. Discussion also includes the effect of code-division multiple-access (CDMA) schemes on the ranging performance. Finally, the overall TOA estimation performance is evaluated by simulation using the UWB channel model. II. PROBLEM STATEMENT The performance of a TOA estimation would be dependent on the algorithm employed for the estimation and various factors embedded in it, e.g., the location of the initial lock point, the length of the search region, and threshold [3]. In this paper, we suppose a conventional serial search method based on correlation. To obtain a general perspective into the error performance, we define our problem in the following way. The serial , while the search is performed over the region direct path signal is assumed to be located at , as shown in Fig. 1. We select the first crossing point of the correlator output at given threshold levels as the estimate of TOA of the direct path signal. This scenario can be regarded as a special case of the ranging algorithm presented in [11], where the initial lock and the length of the search region is point is located at . equal to When a UWB pulse is transmitted into a multipath channel, can be represented by the general received signal

(2) where . Assuming that the first arriving path is the direct path, the TOA and the amplitude of the direct and path signal are and , respectively. The parameters denote the TOA and amplitude of th multipath component. is unknown a priori. The The number of multipath signals waveform represents the canonical single-path signal with seconds and energy of . The thermal noise a width of

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006

Fig. 1. Problem setup for TOA estimation of the direct path signal. The true arrival time of the direct path signal is assumed to be zero while it is searched over a time period [  ;  ].

0

is assumed to be AWGN with a two-sided spectral density of , denotes multiuser interference (MUI). The and the signal can be output of the correlator with a template waveform written as

(3) where denotes the autocorrelation function of the signal ” and is the cross-correlation function between sig“ ” and “ ,” i.e., nals “

Fig. 2. Level-crossing problem with two-sided barrier. Parameters  and  denote the first passage time in one-sided and two-sided barrier problems, respectively.

search region. The occurrence of the large error is partitioned into these three events and its mean square (MS) can be evaluated by

(7) where , , and mean the events of occurrences of early false-alarm error, late error, and missing error, respectively. We in the occurrence of missing error, which assume that means that the TOA of the initial lock point is regarded as that of the direct path. A. Early False-Alarm Error

(4)

(5) The estimate of the TOA of the direct path signal, namely determined by

, is

(6) where

is a threshold.

1) Single-User Environments: The value of has a critical effect on the error performance. If it is too small, then occurrences of early false alarms remain a significant possibility. Conversely, with a large , probabilities of late error and missing error will be significant. Ignoring interference, one way to determine the threshold is to select it relative to the noise floor, i.e.,

(8) is a constant. where In single-user environments, the probability of an early false, can be written as alarm error, namely

III. ERROR ANALYSIS Since the true TOA of the direct path signal is assumed to be 0, the estimation error would be equal to . In this paper, we define a large error as an estimation error whose magnitude is greater than the signal correlation time. Then, the error can be separated into small and large error regions and , respectively. We can classify large errors into three catagories [12]: early false-alarm error, late error, and missing error. An early false-alarm error occurs when a false detection in the noise and interference-only portion of the signal is regarded as that of the direct path signal. A late error occurs when the actual direct path signal is missed and a multipath signal is falsely declared to be the direct path signal. A missing error occurs when no signal component is detected in the search region, i.e., the magnitude of the correlator output does not exceed the threshold level at any time instant in the

(9) . The probability defined in Here, we assume (9) is recognized as a level-crossing probability of a continuous at threshold levels in a time period random process , in particular, a two-sided barrier problem [13] is a zero-mean (see Fig. 2). The random process wide-sense stationary Gaussian process. Some results of evaluations of level-crossing probabilities for certain classes of random processes are available, and a zero-mean stationary Gaussian process is one of them. Let us first consider the one-sided barrier problem in which only the positive threshold

LEE AND YOO: LARGE ERROR PERFORMANCE OF UWB RANGING IN MULTIPATH AND MULTIUSER ENVIRONMENTS

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is considered. The level-crossing probability for this case, , is defined as namely

(10) is the length of the time that it takes for to where reach a threshold level , namely, the first passage time. The distribution of is approximated by [14]

(11) where denotes the time between a down-crossing and the next at a level . The expected value adjacent up-crossing of of is given by [15]

Fig. 3. Simulated distribution of  in single-user environments with K = 3:5. Upper and lower bounds for it are shown together.

(12) where

(13) is the expected number of zero-crossings of the process is defined as The cross-spectral density

.

(14)

From (10) and (11),

is evaluated by

(15) For the case of a two-sided barrier problem, the first passage time is defined as

(16) where denotes the first passage time at the level distribution of is given by

Fig. 4. Correlator template waveform modeled as the fifth derivative of Gaussian with 10-dB points of 2 and 5.34 GHz.

given by (18) Fig. 3 shows upper and lower bounds for . The template waveform used and simulation results for in the simulation is the fifth derivative of a Gaussian model, which is represented by [17]

. The

(19) (17) which is not easy to compute without a complete statistical deand . However, assuming that and are scription of identically distributed, it would satisfy [16]

(18)

where (15), and (18),

and is plotted in Fig. 4. From (9), (11), would satisfy the following inequalities: (20) (21)

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TABLE I RMS BANDWIDTH OF GAUSSIAN PULSE

Fig. 5. RMS of an early false-alarm error that was simulated with  Upper and lower bounds for it are shown together.

= 20 ns.

The MS of an early false-alarm error can be evaluated by

(22)

Fig. 6. Distribution of  which was simulated with K = 3:5 and varying  .

and, from (18) and (22), it satisfies

(23)

(24) The rms early false-alarm errors which were simulated with different values of are shown in Fig. 5. One interesting observation for the probability of an early false-alarm error is that, according to (11)–(13), it is dependent on the rms bandwidth of the signal. One possible way to ex, varying the signal amine this dependence is to evaluate bandwidth, while keeping its energy the same, so that the output SNR is calibrated. Table I shows the rms bandwidth of the signal represented by (19) with different values of . Figs. 6 and 7 and rms of early false-alarm errors with show simulated different signal bandwidths, respectively. We can observe that the rms error decreases with decreasing signal bandwidth. 2) Multi-User Environments: When more than one UWB link is present in the multiple-access system, the effect of MUI

Fig. 7. RMS of an early false-alarm error that was simulated with  and varying  .

= 20 ns

can be critical to accurate detection of the direct path signal. In this paper, we assume perfect power control conditions where all users are assumed to have the same power. We further assume that the frame time is larger than the delay spread of the channel. Under these assumptions, the average power of the inin the presence of UWB interferers can terference signal be approximated by (25)

LEE AND YOO: LARGE ERROR PERFORMANCE OF UWB RANGING IN MULTIPATH AND MULTIUSER ENVIRONMENTS

where

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is given as the input to the users are asynchronous. When correlator, the output can be expressed as (26)

and is the pulse repetition time. Equation (26) can be approximated by (33) (27) where The total energy contained in defined as

terms, namely

, is

(34)

(28)

If the receiver takes averages over pulse transmissions to obtain a processing gain, the averaged response, namely , can be written as

and is assumed to be the same for all users under perfect power control conditions. Under this assumption, is approximated by

(29) (35) Considering the presence of MUI, the threshold can be determined relative to noise and interference level, i.e.,

From (32) and (35), it reduces to

(30) where denotes the number of integrations performed to obtain a processing gain. The MUI signal may have some different statistical properties according to CDMA schemes, and its effect on ranging performance can also be different. The typical CDMA schemes employed by impulse radio are time hopping (TH) and direct sequence (DS). The MUI signal under a TH scheme, namely , in the presence of other users can be represented by [18]

(36) Similarly, the MUI signal in the DS system, namely can be expressed as

,

(37) (31) where (32) The superscript indicates that the quantities are dependent on the signal transmitted by the th user while the transmitter of our interest is the zeroth one. The delay pattern of the th user signal is determined by the TH sequence . Parameter denotes the chip time, and the delay parameter is assumed to , which indicates that the simultaneous be uniform over

where represents the pseudorandom (PN) sequence of the th user and satisfies . The correlator output is given by

(38)

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Fig. 8. MUI at the output of correlator in TH (upper) and DS (lower) systems under indoor multipath channel model CM 4. It was assumed that N = 8, N = 16, and T = 200 ns. Signals before and after taking integrations are shown together in each plot.

The correlator output after taking averages equal to

times will be

(39) In both TH and DS cases, the averaged power of the correlator output with the MUI signal being given as input is reduced by taking integrations, but through different mechanisms. This is because of the fact that randomness lies at the time-shift pattern of the signal in a TH system, while it lies at polarity in a DS system, as shown in (36) and (39). Fig. 8 shows a typical example of these signals which were generated using the multipath channel model recommended by the IEEE 802.15.3a standards task group [19]. The TH sequence employed here was ns, and assumed to be uniform over 128 time bins with the PN sequence for DS-CDMA was also uniformly generated. In this figure, we can observe the averaged correlator output is more flat over the time window in the TH case than for the DS case. In other words, the averaged MUI of the DS system has a greater peak-to-average-power ratio (PAR) than that of the TH system. Intuitively, we can anticipate that a larger PAR of the

( ) and Fig. 9. Distributions of the first passage time of R ( ) with K = 4, N = 8, and T = 200 ns under indoor R multipath channel model CM 4. Thermal noise was assumed to be absent.

MUI signal will introduce a greater chance of occurrence of an early false-alarm error. Fig. 9 shows the simulated distribution of the first passage and in the presence of eight times of decreases with increasing interferers. Notice that the in the TH system, while it changes only a little in the DS system. This is probably because of the difference in PAR of MUI signals of the two. The resulting rms of an early false-alarm error is shown in Fig. 10.

LEE AND YOO: LARGE ERROR PERFORMANCE OF UWB RANGING IN MULTIPATH AND MULTIUSER ENVIRONMENTS

Fig. 10. RMS of an early false-alarm error with N = 8, T = 200 ns, and  = 20 ns under indoor multipath channel model CM 4. Thermal noise was assumed to be absent.

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Fig. 11. Normalized histogram of  under indoor multipath channel model CM 4 and its lognormal fit.

overlapping in the small error region probability can be approximated by

, then this

B. Late Error and Missing Error The probability of occurrence of late or missing errors can be expressed as (41) To evaluate this probability, the PDF of is required. However, since in our analysis total energy in all multipath com, the small-scale fading statistics ponents is normalized to described in the modified – model are not applicable here. Instead, we need the PDF of the strength of the direct path signal normalized by the square root of channel energy, namely , which is defined as

(42) (40)

where

Fig. 11 shows the normalized histogram of which was generated using CM 4 of modified – model. By curve-fitting, the PDF of is obtained as a lognormal function, which is given by

represents the event that (43) and where can be computed as

denotes the complement of event . Bounds for can be found from the analysis given in Section III-A. , comTo determine the conditional probability putation of the level-crossing probability of another kind of random process is required, which was not completed in this study. If we ignore the effect of noise, interference, and pulse

. From (41)–(43),

and

(44)

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Fig. 12. Normalized histograms of occurrence of event E [ E with  = 20 ns under indoor multipath channel model CM 4 when K = 0:15 (upper) and K = 0:20 (lower). Thermal noise was assumed to be absent.

Fig. 13. RMS error conditioned on the occurrence of late or missing errors with  = 20 ns under indoor multipath channel model CM 4. Thermal noise was assumed to be absent.

In Section III-A, the threshold was determined relative to noise and interference levels to limit the probability of early false-alarm errors. To control the probability and variance of late or missing errors, the threshold needs to be determined relative to the signal energy level, e.g., (45) where is a constant. Fig. 12 shows the relative frequency of when is equal to 0.15 and occurrence of an event 0.20. We can observe that the probability of a missing error is . The rms of these errors with considerable when different thresholds is shown in Fig. 13. C. Overall Error As indicated in previous sections, the threshold level is the most important factor when determining the accuracy of serial-search-type TOA estimation. In Sections III-A and III-B, we presented an approximate analysis on the effect of threshold on different kinds of errors. Finding the optimal threshold that minimizes overall error variance is a very difficult problem. Instead, we may use some empirical rules to determine based on the knowledge of noise floor, interference level, and averaged power of signal. We carried out a set of Monte Carlo simulations to examine the overall TOA estimation error with the following empirical thresholding: (46) The averaged SNR which is defined as

(47)

Fig. 14. RMS of overall estimation error with T = 200 and SNR = 9 dB under indoor multipath channel model CM 4.

was assumed to be 9 dB. The rms of overall error is plotted over the number of integrations in Fig. 14. Notice that, while the rms errors in TH and DS systems are similar when there is in no integration, it is reduced more rapidly with increasing TH systems than for DS systems. This may be related to the difference in statistical property of MUI signals in TH and DS formats, which was mentioned in Section III-A. IV. CONCLUSION An approximate evaluation of the performance of TOA estimation of the direct path signal based on correlation detection in multipath and multiuser environments were carried out. Upper and lower bounds for the MS of an early false-alarm error were found by modeling it as a two-sided level crossing event of a zero-mean Gaussian random process. The dependence of the probability of this error on the signal bandwidth may introduce

LEE AND YOO: LARGE ERROR PERFORMANCE OF UWB RANGING IN MULTIPATH AND MULTIUSER ENVIRONMENTS

a tradeoff between large and small error performances. Simulation results showed that the probability of an early false-alarm error is also dependent on the CDMA schemes in the presence of MUI. A study on the statistical properties of MUI and its effect on ranging performance would be of interest. Evaluation of variance of late error was not completed here, while its probability was computed semi-analytically. Exact computation of it would require the statistics of the first passage time of another class of random process. A further study on optimal thresholding to minimize the overall error variance remains as a future work. REFERENCES [1] Z. N. Low, J. H. Cheong, C. L. Law, W. T. Ng, and Y. J. Lee, “Pulse detection algorithm for line-of-sight (LOS) UWB ranging applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 63–67, 2002. [2] W. C. Chung and D. Ha, “An accurate ultra wideband (UWB) ranging for precision asset location,” in Proc. UWBST, Nov. 2003, pp. 389–393. [3] J.-Y. Lee and R. A. Scholtz, “Ranging in dense multipath environments using an UWB radio link,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1677–1683, Dec. 2002. [4] B. Denis, J. Keignart, and N. Daniele, “Impact of NLOS propagation upon ranging precision in UWB systems,” in Proc. UWBST, Nov. 2003, pp. 379–383. [5] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 70–84, Jul. 2005. [6] K. Yu and I. Oppermann, “UWB positioning for wireless embedded networks,” in Proc. RAWCON, Sep. 2004, pp. 459–462. [7] R. N. McDonough and A. D. Whalen, Detection of Signals in Noise. Burlington, MA: Academic, 1995. [8] C. H. Knapp and G. C. Carter, “The generalized correlation method for estimation fo time delay,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-24, no. 8, pp. 320–327, Aug. 1976. [9] J. P. Ianniello, “Time delay estimation via cross-correlation in the presence of large estimation errors,” IEEE Transactions on Electrmagn. Compat., vol. 31, no. 4, pp. 369–375, Dec. 1982. [10] J. Zhang, R. A. Kennedy, and T. D. Abhayapala, “Cramer–Rao lower bounds for the time delay estimation of UWB signals,” in Proc. ICC, Jun. 2004, vol. 6, pp. 3424–3428. [11] R. A. Scholtz and J.-Y. Lee, “Problems in modeling UWB channels,” in Proc. 36th Asilomar Conf. Signals, Syst. Computers, Nov. 2002, pp. 706–711. [12] J.-Y. Lee and S. Yoo, “Large error performance of UWB ranging,” in Proc. ICU, Sep. 2005, pp. 308–313. [13] I. F. Blake and W. C. Lindsey, “Level-crossing problems for random processes,” IEEE Trans. Inf. Theory, vol. IT-19, no. 3, pp. 295–315, May 1973.

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[14] J. R. Rice, “First-occurrence time of high-level crossings in a continuous random process,” J. Acoust. Soc. Amer., vol. 39, no. 2, pp. 323–335, 1966. [15] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 23, pp. 282–332, 1945. [16] H. A. David and H. N. Nagaraja, Order Statistics. Hoboken, NJ: Wiley, 2003. [17] H. Sheng, P. Orlik, A. M. Haimovich, L. J. Cimini, and J. Zhang, “On the spectral and power requirements for ultra-wideband transmission,” in Proc. ICC, May 2003, vol. 1, pp. 738–742. [18] R. A. Scholtz, “Multiple access with time-hopping impulse modulation,” in Proc. MILCOM, Jul. 1993, vol. 2, pp. 447–450. [19] J. Forester, Channel Modeling Sub-Committee Report (Final) Working Group for Wireless Personal Area Networks (WPAN’s), Tech. Rep. P802-15-02/368r5-SG3a, IEEE P802.12, Dec. 2002. Joon-Yong Lee (S’99–M’02) was born in Seoul, Korea, in 1970. He received the B.S. degree from Hong-Ik University, Seoul, Korea, in 1993, and the M.S. and Ph.D. degrees from the University of Southern California (USC), Los Angeles, CA, in 1997 and 2002, respectively, all in electrical engineering. From 1998 to 2002, he was a Research Assistant with the Ultra-Wideband Radio Laboratory (UltraLab), USC. At USC, he worked primarily on the design of the ultra-wideband (UWB) ranging system. In the summer of 2000, he worked as an intern with Time Domain Inc., developing a UWB precision location system. In 2002, he joined the faculty of Handong University, Pohang, Korea, where he is now an Assistant Professor of computer science and electrical engineering. He has consulted for the Electronics and Telecommunications Research Institute (ETRI) and the Ministry of Information and Communication Republic of Korea (MIC). His research interests include the design of UWB ranging and positioning systems and the characterization of UWB propagation.

Sungyul Yoo was born in Daejeon, Korea, in 1979. He received the B.S. degree in computer science and electrical engineering from Handong University, Pohang, Korea, in 2006, and is currently working toward the M.S. degree at the Department of Electronics Engineering, Kookmin University, Seoul, Korea. From 2004 to 2006, he was a Research Assistant with the Ultra-Wideband Communication Laboratory (UWBLAB), Handong University, where he was developing a simulator for UWB communication and ranging systems. His current research interests are in multidimensional image segmentation and MPEG-7.