Maximum A Posteriori Approach to Time-of-Arrival ... - Semantic Scholar
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Fig. 7. BER versus equivalent SNR performance of both perfect and imperfect relaying Ir-DST and noncooperative IRCC-URC-G2 schemes for a frame length of 250 000 bits, where the cooperative network is configured with Gsr = 8, Grd = 2, and Ls = Lr , and the noncooperative system is equipped with two transmit and four receive antennas.
VI. C ONCLUSION In this contribution, we have proposed an Ir-DST coding scheme for near-capacity cooperative communications. We have derived the CCMC capacity and the constrained information-rate bounds of Alamouti’s G2 scheme for the half-duplex relay channel. The proposed joint source-and-relay mode design procedure is capable of finding the optimal Ir-DST coding scheme, which performs close to the capacity limit and achieves a maximum effective throughput. Furthermore, the code-design procedure is not limited to a specific networking scenario; it is applicable under virtually any network configuration.
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[12] M. Tüchler and J. Hagenauer, “EXIT charts of irregular codes,” in Proc. 36th Annu. Conf. Inf. Sci. Syst., Princeton, NJ, Mar. 2002. [CD-ROM]. [13] M. Tüchler, “Design of serially concatenated systems depending on the block length,” IEEE Trans. Commun., vol. 52, no. 2, pp. 209–218, Feb. 2004. [14] H. Ochiai, P. Mitran, and V. Tarokh, “Design and analysis of collaborative diversity protocols for wireless sensor networks,” in Proc. VTC—Fall, Los Angeles, CA, Sep. 26–29, 2004, pp. 4645–4649. [15] A. Host-Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 2020–2040, Jun. 2005. [16] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1522–1544, Apr. 2006. [17] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [18] D. Divsalar, S. Dolinar, and F. Pollara, “Serial turbo trellis coded modulation with rate-1 inner code,” in Proc. ISIT, Sorrento, Italy, Jun. 25–30, 2000, p. 194. [19] L. Hanzo, T. H. Liew, and B. L. Yeap, Turbo Coding, Turbo Equalisation and Space Time Coding for Transmission Over Wireless channels. New York: Wiley-IEEE Press, 2002. [20] T. M. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572–584, Sep. 1979. [21] S. X. Ng, S. Das, J. Wang, and L. Hanzo, “Near-capacity iteratively decoded space-time block coding,” in Proc. IEEE VTC—Spring, Marina Bay, Singapore, May 11–14, 2008, pp. 590–594. [22] S. X. Ng, J. Wang, and L. Hanzo, “Unveiling near-capacity code design: The realization of Shannon’s communication theory for MIMO channels,” in Proc. ICC, Beijing, China, May 19–23, 2008, pp. 1415–1419. [23] L. Kong, S. X. Ng, and L. Hanzo, “Near-capacity three-stage downlink iteratively decoded generalized layered space-time coding with low complexity,” in Proc. GLOBECOM, New Orleans, LA, Nov. 30–Dec. 4, 2008, pp. 1–6.
Maximum A Posteriori Approach to Time-of-Arrival-Based Localization in Non-Line-of-Sight Environment Kenneth W. K. Lui, H. C. So, and W.-K. Ma
R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity: Part I and II,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1948, Nov. 2003. [2] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] J. Laneman and G. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [4] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversitymultiplexing tradeoff in half-duplex cooperative channels,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4152–4172, Dec. 2005. [5] B. Zhao and M. Valenti, “Distributed turbo coded diversity for relay channel,” Electron. Lett., vol. 39, no. 10, pp. 786–787, May 2003. [6] S. X. Ng, Y. Li, and L. Hanzo, “Distributed turbo trellis coded modulation for cooperative communications,” in Proc. ICC, Dresden, Germany, Jun. 14–18, 2009, pp. 1–5. [7] S. ten Brink, “Convergence behaviour of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [8] S. ten Brink, “Designing iterative decoding schemes with the extrinsic information transfer chart,” AEU Int. J. Electron. Commun., vol. 54, no. 6, pp. 389–398, Nov. 2000. [9] Z. Zhang and T. Duman, “Capacity-approaching turbo coding for halfduplex relaying,” IEEE Trans. Commun., vol. 55, no. 10, pp. 1895–1906, Oct. 2007. [10] J. G. Proakis, Digital Communications., 4th ed. New York: McGrawHill, 2001. [11] S. X. Ng and L. Hanzo, “On the MIMO channel capacity of multidimensional signal sets,” IEEE Trans. Veh. Technol., vol. 55, no. 2, pp. 528–536, Mar. 2006.
Abstract—A conventional approach to mobile positioning is to utilize the time-of-arrival (TOA) measurements between the mobile station (MS) and several receiving base stations (BSs). The TOA information defines a set of circular equations from which the MS position can be calculated with the known BS geometry. However, when the TOA measurements are obtained from the non-line-of-sight (NLOS) paths, the position estimation performance can be very unreliable. Assuming that the NLOS probability and distribution are known and the NLOS-induced error dominates the corresponding TOA measurement, two maximum a posteriori probability (MAP) algorithms for NLOS detection and MS localization are derived in this paper. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. It is shown that the former achieves more accurate estimation performance at the expense of higher computational cost. Index Terms—Mobile positioning, non–line of slight (NLOS), time of arrival (TOA).
Manuscript received May 14, 2009; revised September 16, 2009. First published December 31, 2009; current version published March 19, 2010. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project CityU 119606. The review of this paper was coordinated by Prof. Y. Ma. K. W. K. Lui and H. C. So are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. W.-K. Ma is with the Department of Electronic Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2039762
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I. I NTRODUCTION The problem of locating a mobile station (MS) has been receiving considerable interest since the mandate of the wireless Enhanced 911 rules in the U.S. [1]. A typical mobile positioning approach utilizes the time-of-arrival (TOA) measurements, which are the signal propagation times between the MS and several base stations (BSs) [2]. The measured TOAs are easily converted to distance estimates between them by multiplying the former by the speed of light. For 2-D positioning, each TOA provides a circle centered at the BS on which the MS must lie, and at least three BSs are needed for unique localization. Numerous methods [2]–[5] have been derived for TOA-based mobile positioning when there are direct propagation paths between the MS and BSs, which is referred to as the line-of-sight (LOS) condition. However, non-LOS (NLOS) errors [2], [6], [7], which may occur in urban environments and are characterized by excess propagation distances due to reflection and diffraction, will lead to unreliable localization if its effects are not taken into account. Roughly speaking, there are three approaches for mobile positioning in the presence of NLOS propagation. The first approach [8], [9] uses propagation scattering models that allow NLOS error modeling to derive the MS location. Its difficulties lie in obtaining an accurate model, and the model may change with the seasons and addition or removal of building structures. The second approach [10]–[14] utilizes all NLOS and LOS measurements but provides weighting or scaling to minimize the effects of the NLOS contributions in producing the MS position estimate. Nevertheless, the weighting schemes may provide unreliable results because the NLOS measurements are always involved in the localization process. The last approach [15]–[19] aims to identify the LOS/NLOS paths and perform localization with the detected LOS measurements only. In case of perfect identification, accurate positioning performance can be attained, although there is always the possibility of wrong detection. Inspired by [19], which exploits the NLOS error prior probability and distribution for hybrid time-difference-of-arrival/angle-of-arrival location, we develop a maximum a posteriori probability (MAP) approach for joint LOS/NLOS detection and TOA-based mobile positioning. The rest of this paper is organized as follows. In Section II, the problem formulation and assumptions of the TOA measurement models are presented. Given the NLOS probability and distribution, two MAP algorithms for NLOS detection and MS localization are derived in Section III. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. An iterative procedure is also suggested for the latter to boost its estimation performance. Simulation results for performance evaluation are provided in Section IV, and conclusions are drawn in Section V.
to distance by multiplying c. In practice, the distance measurements, which are denoted by di , are subject to disturbance and are modeled as [15], [19]
di =
x − xi 2 + εi ,
φi = 0
x − xi 2 + εi + ηi ,
φi = 1
,
i = 1, 2, . . . , M
(2)
where εi is the measurement error, ηi is the NLOS-induced error, and φi is the NLOS existence variable at the ith BS. The εi is a zeromean white Gaussian variable with known variance σi2 . The distance measurement corresponds to LOS and NLOS propagation when φi = 0 and φi = 1, respectively. We assume that ηi is a positive random variable with a known probability density function (pdf), which is denoted by p(di ), and its value is much larger than doi and |εi |. It is also assumed that {εi }, {φi }, and {ηi } are independent. Assuming that doi and εi are negligible, the observed measurements of (2) can then be well approximated by
di =
x − xi 2 + εi , φi = 0 ηi ,
,
i = 1, 2, . . . , M
(3)
φi = 1
and will be based on (3) for NLOS detection and mobile positioning in the next section. In doing so, the detection/estimation procedure is transformed into a simple threshold comparison problem, although the unknown MS position is involved. Let the probability of NLOS propagation at the ith BS be p(φi = 1) = qi , and thus, we have p(φi = 0) = 1 − qi . We assume that {qi } are known a priori, and their values are sufficiently small such that it is very likely that there are at least three LOS measurements. That is, the NLOS measurements will not be involved for position estimation in our methodology. It is worthy to point out that {σi2 }, {qi }, and the NLOS distribution can be acquired from field tests. In the field tests, the MS location is known, which means that its true distances between the BSs are available. For each path measurement between the MS and a BS, we can easily distinguish whether it corresponds to LOS or NLOS propagation. That is, we detect it as a LOS path if the corresponding derived distance is comparable with the true value, assuming that the disturbance in the LOS scenario is sufficiently small. Otherwise, the path is identified as an NLOS measurement because it largely deviates from the ideal LOS path. Based on the LOS/NLOS identification results from a large number of trials, we can obtain the empirical probabilities for {qi }, while the NLOS distribution is deduced from the identified NLOS measurements. Moreover, {σi2 } can be estimated with the use of the identified LOS measurements and their corresponding noise-free distances. Given the distance measurement vector d = [d1 , d2 , . . . , dM ]T , as well as the NLOS probability and pdf, the task is to find φ = [φ1 , φ2 , . . . , φM ]T and x.
II. P ROBLEM F ORMULATION AND A SSUMPTIONS In this study, we consider 2-D positioning with M BSs in the possibly NLOS condition. Let x be the position of the MS to be determined and xi , i = 1, 2, . . . , M be the known position of the ith BS. In the absence of measurement noise and NLOS propagation, the TOA at the ith BS, which is denoted by τi , is τi =
doi , c
i = 1, 2, . . . , M
(1)
where doi = x − xi 2 is the distance between the MS and the ith BS, with · 2 representing the Euclidean norm, and c is the known signal propagation speed. Obviously, the TOA can easily be converted
III. P ROPOSED A LGORITHMS In this section, two MAP-based algorithms will be developed for NLOS detection and mobile positioning. The first algorithm is a standard MAP estimator based on the approximate signal model of (3), whereas the second is a simplified version of this standard MAP algorithm with the use of geometric constraints. Utilizing the pdf of d conditioned on φ and x, namely, p(d|φ, x), and the prior pdf of p(φ), a MAP estimator for φ and x, which is ˆ and x ˆ , respectively, is then [20] denoted by φ ˆ x ˆ ) = arg (φ,
max
φ∈{0,1}M ,x∈R2
[ln p(d|φ, x) + ln p(φ)]
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(4)
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TABLE I S TANDARD MAP A LGORITHM
Fig. 1. Two overlapped regions (along the line passing through xi and xj ).
Fig. 2. Nonoverlapped case (along the line passing through xi and xj ).
(di − x − xi 2 )2 , and Ti , are nonnegative. This implies that when F (φ, x) is at its minimum value, we have
φi =
1, 0,
if (di − x − xi 2 )2 > Ti if (di − x − xi 2 )2 ≤ Ti .
(10)
That is, the solution of φ in terms of the unknown x is given in (10). We will now utilize (10) and geometric constraints to find φ without knowing x. When di corresponds to a LOS measurement, we have
where
p(d|φ, x) =
M
(di − xi − x2 )2 ≤ Ti .
pφi (di )
i=1
M
×
√
i=1
1 1 exp − 2 (di −x−xi 2 )2 2σi 2πσi
We take the square root on both sides of (11) to yield
1−φi
di −
(5) p(φ) =
M
qiφi
i=1
M
(1−qi )1−φi .
(6)
i=1
In the Appendix, we show that the MAP estimator of (4) is equivalent to ˆ x ˆ ) = arg (φ,
min
φ∈{0,1}M ,x∈R2
F (φ, x)
(11)
(7)
Ti ≤ xi − x2 ≤ di +
Ti .
(12)
Considering x as the variable, its admissible position will simply be bounded √ by two circles with a common center at xi and radii of di ± Ti . It can be proved that Ti = 0 when σi → 0 or noise is absent while Ti → ∞ if qi → 0 or there is no NLOS propagation, which means that the feasible regions can be anywhere, and these agree with our common sense because di = xi − x2 in the noise-free case, and there is no restriction √ on x for the LOS √ scenario. For simplicity, we define Ui = di + Ti and Li = di − Ti as the radii of the outer and inner circles in (12), respectively. Here, we only exploit the scenario when the outer circles of the ith and jth BSs are overlapped, and the corresponding necessary and sufficient condition is xi − xj 2 < Ui + Uj
(13)
where F (φ, x) =
M 1 − φi i=1
2σi2
Ti = 2σi2 ln
√
(di − x − xi 2 )2 − Ti
1 − qi 2πσi qi p(di )
(8)
which is illustrated in Fig. 1. However, we have to exclude the case when the ith outer circle is unable to touch the jth inner circle, which is geometrically illustrated in Fig. 2. The corresponding inequality is Uj + xi − xj 2 ≤ Li ⇒ xi − xj 2 ≤ Li − Uj .
.
(9)
Since each φi only has a binary value of 0 or 1, our first approach, which is referred to as the standard MAP algorithm, is to solve (7) by considering it as a combinational problem, and its estimation procedure is given in Table I. It is obvious that a major drawback of the standard MAP algorithm is that its computational load will drastically grow as the number of measurements increases. In the following, we will devise a computationally attractive solution for (7) by first finding φ and then x. From (8) and (9), we see that all components in F (φ, x), namely, 1 − φi , σi2 ,
(14)
Swapping the indices i and j, we also need to exclude xi − xj 2 ≤ Lj − Ui .
(15)
Negating (14) or (15) yields (xi − xj 2 > Lj − Ui )
(xi − xj 2 > Li − Uj )
⇒ xi − xj 2 > max(Lj − Ui , Li − Ui ) (16)
where is the intersection operator, and max(a, b) = a if a > b. Substituting di and Ti for Ui and Li and noting that the Euclidean
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norm is always nonnegative, (16) becomes
xi − xj 2 > max |di − dj | − (
Ti +
Tj ), 0 .
(17)
Similarly, (13) is identical to
xi − xj 2 < (di + dj ) + (
Ti +
Tj ).
(18)
Now, we have relaxed (12) to (17) and (18), which will be utilized to determine φ as follows. First, we construct an M × M matrix that assesses all pairs of distance measurements according to (17) and (18), which is denoted by Φ
[Φ]ij =
0, 1,
if the pair (di , dj ) satisfies (17) and (18) otherwise
(19)
where [Φ]ij denotes the (i, j) entry of Φ. That is, [Φ]ij = 0 means that the two distances di and dj agree with each other, and they are possibly LOS measurements. On the other hand, [Φ]ij = 1 indicates that di is inconsistent with dj , and thus, at least one of them corresponds to NLOS propagation. Assuming that {qi } are sufficiently small such that LOS paths contribute to the majority, the majority rule [21] is then ˆ via the removal of the employed in Φ to produce M vectors of φ contradicting distance measurements. We follow the same estimation procedure as in the standard MAP method by exploring all admissible M ˆ ˆ with φ < M − 3 to determine φ and x, which vectors of φ i=1 i is referred to as the simplified MAP algorithm. Since at most M vectors are involved in the simplified scheme, there will be a significant computational reduction over the standard method, which considers 2M combinations, particularly for a large value of M . In the case when the estimate of x is sufficiently accurate and some of the LOS distances are detected as NLOS measurements, the estimation performance will be improved by the following iterative steps. ˆ. 1) Estimate φ using (10) and x ˆ using d with φˆi = 0 by the two-step weighted least 2) Compute x square (TSWLS) method [3]. 3) Repeat steps 1 and 2 until parameter convergence. We refer to this enhancement as the iterative simplified MAP algorithm.
Fig. 3. LOS/NLOS detection probability with a larger NLOS error using the model of (3).
IV. S IMULATION R ESULTS
Fig. 4. Mean square error performance with a larger NLOS error using the model of (3).
Computer simulations have been conducted to evaluate the performance of the proposed algorithms for LOS/NLOS detection and mobile positioning. We also include the results of the TSWLS, clairvoyant, Riba–Urruela [18], and Cong–Zhuang [19] estimators for comparison. The TSWLS method uses all distance measurements, including those from NLOS propagation, for position determination, while the clairvoyant estimator employs the actual x to find {φi } in (10) and then computes the position estimate with the detected LOS measurements using the TSWLS algorithm. On the other hand, both [18] and [19] calculate the MS positions for all LOS/NLOS distance measurement combinations, and the one with the lowest error is selected as the estimate. Note that the former utilizes NLOS probability and maximum distance error information, while the latter does not. The BSs are located at (1000, 1000) m, (−1000, −1000) m, (1000, −1000) m, (−1000, 1000) m, (−500, 0) m, (0, −500) m, (500, 0) m, and (0, 500) m, and the MS position is (300, 400) m. The standard deviation of each measurement noise is σi = 0.1 × doi , while ηi is uniformly distributed in the range of [0, α × max(doi )], where α is a scaling factor. For simplicity, the NLOS probabilities {qi } are set to have identical M values. The LOS/NLOS detection rate, which is defined as E{ i=1 (1 − |φi − φˆi |)}, and the mean square position
ˆ 22 }, are employed as the performance error (MSPE), that is, E{x − x measures. Note that the former has a value of one only when all LOS/NLOS measurements are correctly identified and is zero otherwise. All simulation results are averages of 1000 independent runs, where each run contains at least three LOS measurements to ensure that the MS is localizable. For digital signal processing implementation with field-programmable gate arrays, see [22]. Figs. 3 and 4 show the LOS/NLOS detection and MSPE performance versus qi based on the approximate model of (3). We set α = 20, which corresponds to a larger NLOS error scenario. In Fig. 3, it is observed that all estimators, except the TSWLS and Riba–Urruela methods, have comparable LOS/NLOS detection performance when qi is sufficiently small. For a larger value of qi , we see that the clairvoyant and Cong–Zhuang estimators perform the best, followed by the standard MAP, iterative simplified MAP, and simplified MAP algorithms, although their differences are not significant. As expected, the TSWLS method performs poorly in the whole range because NLOS measurements are employed in the positioning process. On the other hand, [18] provides a detection rate that is close to zero. The MSPE results in Fig. 4 also show good agreement with the
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Fig. 5. LOS/NLOS detection probability with a larger NLOS error using the model of (2).
Fig. 7. LOS/NLOS detection probability with a smaller NLOS error using model of (2).
Fig. 6. Mean square error performance with a larger NLOS error using the model of (2).
Fig. 8. Mean square error performance with a smaller NLOS error using the model of (2).
LOS/NLOS detection rates, except for the Riba–Urruela algorithm. This indicates that the poor detection probability of [18] is mainly due to incorrect identification of LOS paths. We see that the MSPE difference between the clairvoyant and simplified MAP estimators is less than 6.5 dB for qi ∈ (0, 0.2). Furthermore, the complexities of the algorithms are studied according to the average computational time per trial. The measured times for the standard MAP, simplified MAP, iterative simplified MAP, TSWLS, clairvoyant, Riba–Urruela, and Cong–Zhuang estimators are 5.6 × 10−2 , 1.0 × 10−3 , 1.4 × 10−3 , 5.1 × 10−4 , 3.2 × 10−4 , 6.5 × 10−2 , and 1.5 × 10−1 s, respectively. For the standard MAP method, we observe that it combines the advantages of relatively low complexity of [18] and high performance of [19], while significant computational reduction of the two simplified MAP schemes over the standard one is observed, and their complexity is comparable with the TSWLS and infeasible clairvoyant estimators. The aforementioned test is repeated using the standard model of (2). The results, which are shown in Figs. 5 and 6, are comparable with those in Figs. 3 and 4. In particular, the proposed MAP approach performs similarly in both models of (2) and (3), although our math-
ematical development is founded on the latter. This demonstrates the robustness of the MAP schemes. Finally, we study the effect of a smaller NLOS error condition. We repeat the aforementioned experiment with α = 2 and the model of (2). The results are plotted in Figs. 7 and 8, and similar findings are obtained, except that now, the detection rate is smaller. V. C ONCLUSION Assuming that the NLOS probability and distribution is known, two MAP algorithms for joint NLOS detection and mobile positioning have been developed. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. An iterative procedure has also been suggested for the latter to boost its estimation performance. In terms of estimation performance, the standard MAP algorithm is superior, while the simplified versions have the advantage of being computationally attractive over the former. A possible extension of this paper is to study the robustness of the proposed approach for imperfect knowledge of the NLOS probability and distribution, which occurs in real environments.
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A PPENDIX A In this Appendix, we will prove that (4) is equivalent to (7). Substituting (5) and (6) into (4) yields
As the second term of (A.1) takes no effect on the optimization, the MAP estimator is then ˆ x ˆ ) = arg (φ,
ln p(d|φ, x) + ln p(φ)
M
= ln
M 1
√
pφi(di)·
i=1
i=1
+ ln p(φ) =
M
2πσi
qiφi
i=1
=
M
φi ln(p(di )) +
+
M
2σi2
i=1
=−
+
M
2σi2
M 1 − φi i=1
+
M
2σi2
+
(1 − φi ) ln
(1 − φi ) ln
i=1
=−
M 1 − φi i=1
+
M
2σi2
M
(1 − φi ) ln(1 − qi )
1 − qi √ 2πσi
1 qi p(di ) 1 − qi √ 2πσi
(1 − φi ) ln
M
φi ln(qi p(di ))
+
M
ln (qi p(di ))
i=1
1 − qi √ 2πσi qi p(di )
ln (qi p(di ))
i=1
=−
M 1 − φi i=1
+
M i=1
2σi2
(di − x − xi 2 )2 − Ti
ln (qi p(di )) .
× (di − x − xi 2 )2 − Ti
(A.2)
R EFERENCES
(di − x − xi 2 )2
i=1
+
1 2πσi
(di − x − xi 2 )2
i=1 M
√
i=1
i=1
=−
(1 − φi ) ln
(di − x − xi 2 )2 +
(1 − φi ) ln
2σi2
The authors would like to thank the anonymous reviewers for their useful comments and suggestions.
i=1
M 1 − φi i=1
M
i=1
ACKNOWLEDGMENT
1−φi
(di − x − xi 2 )2
φi ln(qi ) +
φ∈{0,1}M ,x∈R2
which is (7).
(1 − qi )
M
M 1 − φi
M
i=1
M 1 − φi i=1
1 (di −x−xi 2)2 2σi2
×exp −
i=1
i=1
−
1−φi
min
(A.1)
[1] Revision of the Commissions Rules to Ensure Compatibility With Enhanced 911 Emergency Calling Systems, RM-8143, CC Docket no. 94102, Jul. 26, 1996. [2] J. J. Caffery, Jr., Wireless Location in CDMA Cellular Radio Systems. Norwell, MA: Kluwer, 1999. [3] Y. T. Chan and K. C. Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 1905–1915, Aug. 1994. [4] M. A. Spirito, “On the accuracy of cellular mobile station location estimation,” IEEE Trans. Veh. Technol., vol. 50, no. 3, pp. 674–685, May 2001. [5] K. W. Cheung, H. C. So, W.-K. Ma, and Y. T. Chan, “Least squares algorithms for time-of-arrival-based mobile location,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1121–1130, Apr. 2004. [6] B. Denis, J.-B. Pierrot, and C. Abou-Rjeily, “Joint distributed synchronization and positioning in UWB ad hoc networks using TOA,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1896–1911, Jun. 2006. [7] D. Harmer, M. Russell, E. Frazer, T. Bauge, S. Ingram, N. Schmidt, B. Kull, A. Yarovoy, A. Nezirovic, L. Xia, V. Dizdarevic, and K. Witrisal, “EUROPCOM: Emergency ultrawideband radio for positioning and communications,” in Proc. IEEE Int. Conf. Ultra-Wideband, Hannover, Germany, Sep. 2008, vol. 3, pp. 85–88. [8] J. Takada, J. Fu, H. Zhu, and T. Kobayashi, “Spatio-temporal channel characterization in a suburban non line-of-sight microcellular environment,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 532–538, Apr. 2002. [9] S. Al-Jazzar, J. Caffery, Jr., and H.-R. You, “Scattering-model-based methods for TOA location in NLOS environments,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 583–593, Mar. 2007. [10] P. C. Chen, “A non-line-of-sight error mitigation algorithm in location estimation,” in Proc. IEEE Wireless Commun. Netw. Conf., New Orleans, LA, Sep. 1999, vol. 1, pp. 316–320. [11] W. Wei, X. J. Yu, and Z. Z. Liang, “A new NLOS error mitigation algorithm in localization estimation,” IEEE Trans. Veh. Technol., vol. 54, no. 6, pp. 2048–2053, Nov. 2005. [12] R. Casas, A. Marco, J. J. Guerrero, and J. Falco, “Robust estimator for non-line-of-sight error mitigation in indoor localization,” EURASIP J. Appl. Signal Process., vol. 2006, pp. 1–8, 2006, Article ID 43429. [13] C. Morelli, M. Nicoli, V. Rampa, and U. Spagnolini, “Hidden Markov models for radio localization in mixed LOS/NLOS conditions,” IEEE Trans. Signal Process., vol. 55, no. 4, pp. 1525–1542, Apr. 2007. [14] W.-Y. Chiu and B.-S. Chen, “Mobile location estimation in urban areas using mixed Manhattan/Euclidean norm and convex optimization,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 414–423, Jan. 2009. [15] M. P. Wylie and J. Holtzman, “The non-line of sight problem in mobile location estimation,” in Proc. IEEE Int. Conf. Universal Pers. Commun., Sep. 29–Oct. 2, 1996, vol. 2, pp. 827–831. [16] S. Gezici, H. Kobayashi, and H. V. Poor, “Non parametric nonline of sight identification,” in Proc. IEEE 58th Veh. Technol. Conf., Oct. 6–9, 2003, vol. 4, pp. 2544–2548. [17] Y. T. Chan, W. Y. Tsui, H. C. So, and P. C. Ching, “Time-of-arrival based localization under NLOS conditions,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 17–24, Jan. 2006.
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[18] J. Riba and A. Urruela, “A non-line-of-sight mitigation technique based on ML-detection,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Montreal, QC, Canada, May 2004, vol. 2, pp. 153–156. [19] L. Cong and W. Zhuang, “Non-line-of-sight error mitigation in mobile location,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 560–573, Mar. 2005. [20] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [21] M. Fey, “May’s Theorem with an infinite population,” Soc. Choice Welf., vol. 23, no. 2, pp. 275–293, Oct. 2004. [22] U. Meyer-Baese, Digital Signal Processing With Field Programmable Gate Arrays. Berlin, Germany: Springer-Verlag, 2001.
Error Rate Analysis of Band-Limited BPSK With Nakagami/Nakagami ACI Considering Nonlinear Amplifier and Diversity M. A. Rahman, Member, IEEE, C. S. Sum, Member, IEEE, R. Funada, Member, IEEE, T. Baykas, Member, IEEE, J. Wang, Member, IEEE, S. Sasaki, Member, IEEE, H. Harada, Member, IEEE, and S. Kato, Fellow, IEEE
Abstract—An exact expression of the average error rate is developed for a band-limited binary phase-shift keying (BPSK) system in the presence of adjacent channel interference (ACI) considering a Nakagami fading channel. The system employs a root-raised cosine (RRC) filter both in the transmitter and receiver sides and a transmitter power amplifier (PA) that may have nonlinear input/output characteristics. These practical considerations are accurately taken into account. By utilizing the characteristic function (CF) method of error-rate analysis, we interestingly blend the concepts of time and frequency domains and develop an error-rate expression that is simple and general for several system parameters, including adjacent channel separation, RRC filter roll-off factor, and PA output back-off (OBO). Hence, the error rate in the presence of cochannel interference (CCI) is obtained as a special case. The developed results for a single-branch receiver are then extended for diversity receivers, and several interesting observations are made. Index Terms—Adjacent channel interference (ACI), cochannel interference (CCI), error-rate analysis, Nakagami fading, power amplifier (PA) nonlinearity and diversity receivers.
I. I NTRODUCTION The frequency spectrum is a valuable resource. For its proper utilization, the available spectrum is usually subdivided into several frequency bands. This facilitates simultaneous operation of several systems without interfering with each other. However, in many practical cases, it may not be possible to perfectly confine a desired signal within a desired band. Due to imperfection of the transmit filter and nonlinearity of the power amplifier (PA), a portion of the transmitted Manuscript received June 10, 2009; revised October 6, 2009. First published December 4, 2009; current version published March 19, 2010. The review of this paper was coordinated by Prof. H.-C. Wu. M. A. Rahman, C. S. Sum, R. Funada, T. Baykas, J. Wang, H. Harada, and S. Kato are with the National Institute of Information and Communications Technology, Yokosuka 239-0847, Japan (e-mail: [email protected]; aziz.jp@ ieee.org; [email protected]; [email protected]; [email protected]; junyi. [email protected]; [email protected]; [email protected]). S. Sasaki is with the Department of Electrical and Electronic Engineering, Niigata University, Niigata 950-2181, Japan (e-mail: [email protected]. ac.jp). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2037821
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signal spectrum may spill over adjacent bands and cause unwanted interference. Due to imperfection of the receive filter, a receiver may also receive a portion of signals transmitted in adjacent bands. Such interferences are usually known as adjacent channel interference (ACI). As opposed to ACI, there may be systems simultaneously operating within the same frequency band of the desired system. Interferences from such sources are commonly known as cochannel interference (CCI). The development of an accurate expression of the error rate for a conventional band-limited binary phase-shift keying (BPSK) system in the presence of CCI has been a topic of several recent publications [1]–[6]. Among those, [4] presented a characteristic function (CF)based exact error-rate analysis for band-limited BPSK in a Nakagami fading environment considering Rayleigh-faded CCI. A complete version of the study was presented in [3] considering Nakagami-faded CCI. The results presented in [3] are precisely accurate, although not exact. However, these works were later extended for diversity in [2], although the error rate was given based on rather inaccurate Gaussian approximation of CCI. Other authors also addressed CCI analysis in Rician fading [5]. Recently, Smadi et al. [6] have presented an exact CCI analysis for partially coherent modulations considering diversity. However, despite several papers being published on CCI analysis, papers that contributed on ACI analysis are quite limited [7]–[9]. A basic difference of the ACI signal from its CCI counterpart is that it has a center frequency different from, but usually somewhat close to, the desired signal. As a result, works on CCI cannot readily be extended for ACI. In addition, there is an important concern that, for an ACI analysis to be useful, transmitter PA nonlinearity should accurately be taken into consideration. PA nonlinearity gives rise to spectrum regrowth, which itself is a major reason to create ACI. Because the equivalent time-domain pulse shape of a band-limited signal extends infinitely, developing a tractable error rate analysis for such signal taking into account PA nonlinearity is a challenging task. A basic work on analyzing and comparing performance of various digital modulation schemes in the presence of ACI can be found in [7]. Later, [8]–[10] addressed ACI analysis for offset phase-shift keying, differential phase-shift keying, and minimum shift keying, respectively. Reference [11] investigated the performance of a neuralnetwork-based receiver in the presence of ACI. However, none of the works considered the effect of nonlinear amplification. The effect of nonlinear amplification on ACI potential was considered in [12]–[15]; however, [12]–[14] were based on simulations, whereas [15] only considered a fully saturated PA. In addition, [16] and [17], which considered the effect of nonlinearity on system performance, were on a very specific application. There are also a few other works (see [18]–[20] and the references therein) that investigated the impact of PA nonlinearity on spectral regrowth, signal-to-noise ratio degradation, etc. Nevertheless, what all the aforementioned papers have in common is that none presented an analytical work on developing an exact error rate of a band-limited signal considering ACI in a fading environment, either for a single-branch receiver or for diversity combining. Due to immense practical importance, there is a current demand in the literature for such a general analysis considering nonlinear amplification in the transmitter, fading in the channel, and diversity in the receiver. In this paper, we consider a practical band-limited BPSK system in the presence of ACI in a Nakagami fading environment. Because Nakagami fading is a general representation of fading that includes Rayleigh and Rician fading as special cases, developing an error rate in Nakagami fading is of practical importance. By employing the CF-based method, a simple exact expression of the error rate is developed that is valid for ACI with an arbitrary center frequency
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Fig. 7. BER versus equivalent SNR performance of both perfect and imperfect relaying Ir-DST and noncooperative IRCC-URC-G2 schemes for a frame length of 250 000 bits, where the cooperative network is configured with Gsr = 8, Grd = 2, and Ls = Lr , and the noncooperative system is equipped with two transmit and four receive antennas.
VI. C ONCLUSION In this contribution, we have proposed an Ir-DST coding scheme for near-capacity cooperative communications. We have derived the CCMC capacity and the constrained information-rate bounds of Alamouti’s G2 scheme for the half-duplex relay channel. The proposed joint source-and-relay mode design procedure is capable of finding the optimal Ir-DST coding scheme, which performs close to the capacity limit and achieves a maximum effective throughput. Furthermore, the code-design procedure is not limited to a specific networking scenario; it is applicable under virtually any network configuration.
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[12] M. Tüchler and J. Hagenauer, “EXIT charts of irregular codes,” in Proc. 36th Annu. Conf. Inf. Sci. Syst., Princeton, NJ, Mar. 2002. [CD-ROM]. [13] M. Tüchler, “Design of serially concatenated systems depending on the block length,” IEEE Trans. Commun., vol. 52, no. 2, pp. 209–218, Feb. 2004. [14] H. Ochiai, P. Mitran, and V. Tarokh, “Design and analysis of collaborative diversity protocols for wireless sensor networks,” in Proc. VTC—Fall, Los Angeles, CA, Sep. 26–29, 2004, pp. 4645–4649. [15] A. Host-Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 2020–2040, Jun. 2005. [16] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1522–1544, Apr. 2006. [17] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [18] D. Divsalar, S. Dolinar, and F. Pollara, “Serial turbo trellis coded modulation with rate-1 inner code,” in Proc. ISIT, Sorrento, Italy, Jun. 25–30, 2000, p. 194. [19] L. Hanzo, T. H. Liew, and B. L. Yeap, Turbo Coding, Turbo Equalisation and Space Time Coding for Transmission Over Wireless channels. New York: Wiley-IEEE Press, 2002. [20] T. M. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572–584, Sep. 1979. [21] S. X. Ng, S. Das, J. Wang, and L. Hanzo, “Near-capacity iteratively decoded space-time block coding,” in Proc. IEEE VTC—Spring, Marina Bay, Singapore, May 11–14, 2008, pp. 590–594. [22] S. X. Ng, J. Wang, and L. Hanzo, “Unveiling near-capacity code design: The realization of Shannon’s communication theory for MIMO channels,” in Proc. ICC, Beijing, China, May 19–23, 2008, pp. 1415–1419. [23] L. Kong, S. X. Ng, and L. Hanzo, “Near-capacity three-stage downlink iteratively decoded generalized layered space-time coding with low complexity,” in Proc. GLOBECOM, New Orleans, LA, Nov. 30–Dec. 4, 2008, pp. 1–6.
Maximum A Posteriori Approach to Time-of-Arrival-Based Localization in Non-Line-of-Sight Environment Kenneth W. K. Lui, H. C. So, and W.-K. Ma
R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity: Part I and II,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1948, Nov. 2003. [2] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] J. Laneman and G. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [4] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversitymultiplexing tradeoff in half-duplex cooperative channels,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4152–4172, Dec. 2005. [5] B. Zhao and M. Valenti, “Distributed turbo coded diversity for relay channel,” Electron. Lett., vol. 39, no. 10, pp. 786–787, May 2003. [6] S. X. Ng, Y. Li, and L. Hanzo, “Distributed turbo trellis coded modulation for cooperative communications,” in Proc. ICC, Dresden, Germany, Jun. 14–18, 2009, pp. 1–5. [7] S. ten Brink, “Convergence behaviour of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [8] S. ten Brink, “Designing iterative decoding schemes with the extrinsic information transfer chart,” AEU Int. J. Electron. Commun., vol. 54, no. 6, pp. 389–398, Nov. 2000. [9] Z. Zhang and T. Duman, “Capacity-approaching turbo coding for halfduplex relaying,” IEEE Trans. Commun., vol. 55, no. 10, pp. 1895–1906, Oct. 2007. [10] J. G. Proakis, Digital Communications., 4th ed. New York: McGrawHill, 2001. [11] S. X. Ng and L. Hanzo, “On the MIMO channel capacity of multidimensional signal sets,” IEEE Trans. Veh. Technol., vol. 55, no. 2, pp. 528–536, Mar. 2006.
Abstract—A conventional approach to mobile positioning is to utilize the time-of-arrival (TOA) measurements between the mobile station (MS) and several receiving base stations (BSs). The TOA information defines a set of circular equations from which the MS position can be calculated with the known BS geometry. However, when the TOA measurements are obtained from the non-line-of-sight (NLOS) paths, the position estimation performance can be very unreliable. Assuming that the NLOS probability and distribution are known and the NLOS-induced error dominates the corresponding TOA measurement, two maximum a posteriori probability (MAP) algorithms for NLOS detection and MS localization are derived in this paper. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. It is shown that the former achieves more accurate estimation performance at the expense of higher computational cost. Index Terms—Mobile positioning, non–line of slight (NLOS), time of arrival (TOA).
Manuscript received May 14, 2009; revised September 16, 2009. First published December 31, 2009; current version published March 19, 2010. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project CityU 119606. The review of this paper was coordinated by Prof. Y. Ma. K. W. K. Lui and H. C. So are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. W.-K. Ma is with the Department of Electronic Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2039762
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I. I NTRODUCTION The problem of locating a mobile station (MS) has been receiving considerable interest since the mandate of the wireless Enhanced 911 rules in the U.S. [1]. A typical mobile positioning approach utilizes the time-of-arrival (TOA) measurements, which are the signal propagation times between the MS and several base stations (BSs) [2]. The measured TOAs are easily converted to distance estimates between them by multiplying the former by the speed of light. For 2-D positioning, each TOA provides a circle centered at the BS on which the MS must lie, and at least three BSs are needed for unique localization. Numerous methods [2]–[5] have been derived for TOA-based mobile positioning when there are direct propagation paths between the MS and BSs, which is referred to as the line-of-sight (LOS) condition. However, non-LOS (NLOS) errors [2], [6], [7], which may occur in urban environments and are characterized by excess propagation distances due to reflection and diffraction, will lead to unreliable localization if its effects are not taken into account. Roughly speaking, there are three approaches for mobile positioning in the presence of NLOS propagation. The first approach [8], [9] uses propagation scattering models that allow NLOS error modeling to derive the MS location. Its difficulties lie in obtaining an accurate model, and the model may change with the seasons and addition or removal of building structures. The second approach [10]–[14] utilizes all NLOS and LOS measurements but provides weighting or scaling to minimize the effects of the NLOS contributions in producing the MS position estimate. Nevertheless, the weighting schemes may provide unreliable results because the NLOS measurements are always involved in the localization process. The last approach [15]–[19] aims to identify the LOS/NLOS paths and perform localization with the detected LOS measurements only. In case of perfect identification, accurate positioning performance can be attained, although there is always the possibility of wrong detection. Inspired by [19], which exploits the NLOS error prior probability and distribution for hybrid time-difference-of-arrival/angle-of-arrival location, we develop a maximum a posteriori probability (MAP) approach for joint LOS/NLOS detection and TOA-based mobile positioning. The rest of this paper is organized as follows. In Section II, the problem formulation and assumptions of the TOA measurement models are presented. Given the NLOS probability and distribution, two MAP algorithms for NLOS detection and MS localization are derived in Section III. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. An iterative procedure is also suggested for the latter to boost its estimation performance. Simulation results for performance evaluation are provided in Section IV, and conclusions are drawn in Section V.
to distance by multiplying c. In practice, the distance measurements, which are denoted by di , are subject to disturbance and are modeled as [15], [19]
di =
x − xi 2 + εi ,
φi = 0
x − xi 2 + εi + ηi ,
φi = 1
,
i = 1, 2, . . . , M
(2)
where εi is the measurement error, ηi is the NLOS-induced error, and φi is the NLOS existence variable at the ith BS. The εi is a zeromean white Gaussian variable with known variance σi2 . The distance measurement corresponds to LOS and NLOS propagation when φi = 0 and φi = 1, respectively. We assume that ηi is a positive random variable with a known probability density function (pdf), which is denoted by p(di ), and its value is much larger than doi and |εi |. It is also assumed that {εi }, {φi }, and {ηi } are independent. Assuming that doi and εi are negligible, the observed measurements of (2) can then be well approximated by
di =
x − xi 2 + εi , φi = 0 ηi ,
,
i = 1, 2, . . . , M
(3)
φi = 1
and will be based on (3) for NLOS detection and mobile positioning in the next section. In doing so, the detection/estimation procedure is transformed into a simple threshold comparison problem, although the unknown MS position is involved. Let the probability of NLOS propagation at the ith BS be p(φi = 1) = qi , and thus, we have p(φi = 0) = 1 − qi . We assume that {qi } are known a priori, and their values are sufficiently small such that it is very likely that there are at least three LOS measurements. That is, the NLOS measurements will not be involved for position estimation in our methodology. It is worthy to point out that {σi2 }, {qi }, and the NLOS distribution can be acquired from field tests. In the field tests, the MS location is known, which means that its true distances between the BSs are available. For each path measurement between the MS and a BS, we can easily distinguish whether it corresponds to LOS or NLOS propagation. That is, we detect it as a LOS path if the corresponding derived distance is comparable with the true value, assuming that the disturbance in the LOS scenario is sufficiently small. Otherwise, the path is identified as an NLOS measurement because it largely deviates from the ideal LOS path. Based on the LOS/NLOS identification results from a large number of trials, we can obtain the empirical probabilities for {qi }, while the NLOS distribution is deduced from the identified NLOS measurements. Moreover, {σi2 } can be estimated with the use of the identified LOS measurements and their corresponding noise-free distances. Given the distance measurement vector d = [d1 , d2 , . . . , dM ]T , as well as the NLOS probability and pdf, the task is to find φ = [φ1 , φ2 , . . . , φM ]T and x.
II. P ROBLEM F ORMULATION AND A SSUMPTIONS In this study, we consider 2-D positioning with M BSs in the possibly NLOS condition. Let x be the position of the MS to be determined and xi , i = 1, 2, . . . , M be the known position of the ith BS. In the absence of measurement noise and NLOS propagation, the TOA at the ith BS, which is denoted by τi , is τi =
doi , c
i = 1, 2, . . . , M
(1)
where doi = x − xi 2 is the distance between the MS and the ith BS, with · 2 representing the Euclidean norm, and c is the known signal propagation speed. Obviously, the TOA can easily be converted
III. P ROPOSED A LGORITHMS In this section, two MAP-based algorithms will be developed for NLOS detection and mobile positioning. The first algorithm is a standard MAP estimator based on the approximate signal model of (3), whereas the second is a simplified version of this standard MAP algorithm with the use of geometric constraints. Utilizing the pdf of d conditioned on φ and x, namely, p(d|φ, x), and the prior pdf of p(φ), a MAP estimator for φ and x, which is ˆ and x ˆ , respectively, is then [20] denoted by φ ˆ x ˆ ) = arg (φ,
max
φ∈{0,1}M ,x∈R2
[ln p(d|φ, x) + ln p(φ)]
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TABLE I S TANDARD MAP A LGORITHM
Fig. 1. Two overlapped regions (along the line passing through xi and xj ).
Fig. 2. Nonoverlapped case (along the line passing through xi and xj ).
(di − x − xi 2 )2 , and Ti , are nonnegative. This implies that when F (φ, x) is at its minimum value, we have
φi =
1, 0,
if (di − x − xi 2 )2 > Ti if (di − x − xi 2 )2 ≤ Ti .
(10)
That is, the solution of φ in terms of the unknown x is given in (10). We will now utilize (10) and geometric constraints to find φ without knowing x. When di corresponds to a LOS measurement, we have
where
p(d|φ, x) =
M
(di − xi − x2 )2 ≤ Ti .
pφi (di )
i=1
M
×
√
i=1
1 1 exp − 2 (di −x−xi 2 )2 2σi 2πσi
We take the square root on both sides of (11) to yield
1−φi
di −
(5) p(φ) =
M
qiφi
i=1
M
(1−qi )1−φi .
(6)
i=1
In the Appendix, we show that the MAP estimator of (4) is equivalent to ˆ x ˆ ) = arg (φ,
min
φ∈{0,1}M ,x∈R2
F (φ, x)
(11)
(7)
Ti ≤ xi − x2 ≤ di +
Ti .
(12)
Considering x as the variable, its admissible position will simply be bounded √ by two circles with a common center at xi and radii of di ± Ti . It can be proved that Ti = 0 when σi → 0 or noise is absent while Ti → ∞ if qi → 0 or there is no NLOS propagation, which means that the feasible regions can be anywhere, and these agree with our common sense because di = xi − x2 in the noise-free case, and there is no restriction √ on x for the LOS √ scenario. For simplicity, we define Ui = di + Ti and Li = di − Ti as the radii of the outer and inner circles in (12), respectively. Here, we only exploit the scenario when the outer circles of the ith and jth BSs are overlapped, and the corresponding necessary and sufficient condition is xi − xj 2 < Ui + Uj
(13)
where F (φ, x) =
M 1 − φi i=1
2σi2
Ti = 2σi2 ln
√
(di − x − xi 2 )2 − Ti
1 − qi 2πσi qi p(di )
(8)
which is illustrated in Fig. 1. However, we have to exclude the case when the ith outer circle is unable to touch the jth inner circle, which is geometrically illustrated in Fig. 2. The corresponding inequality is Uj + xi − xj 2 ≤ Li ⇒ xi − xj 2 ≤ Li − Uj .
.
(9)
Since each φi only has a binary value of 0 or 1, our first approach, which is referred to as the standard MAP algorithm, is to solve (7) by considering it as a combinational problem, and its estimation procedure is given in Table I. It is obvious that a major drawback of the standard MAP algorithm is that its computational load will drastically grow as the number of measurements increases. In the following, we will devise a computationally attractive solution for (7) by first finding φ and then x. From (8) and (9), we see that all components in F (φ, x), namely, 1 − φi , σi2 ,
(14)
Swapping the indices i and j, we also need to exclude xi − xj 2 ≤ Lj − Ui .
(15)
Negating (14) or (15) yields (xi − xj 2 > Lj − Ui )
(xi − xj 2 > Li − Uj )
⇒ xi − xj 2 > max(Lj − Ui , Li − Ui ) (16)
where is the intersection operator, and max(a, b) = a if a > b. Substituting di and Ti for Ui and Li and noting that the Euclidean
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norm is always nonnegative, (16) becomes
xi − xj 2 > max |di − dj | − (
Ti +
Tj ), 0 .
(17)
Similarly, (13) is identical to
xi − xj 2 < (di + dj ) + (
Ti +
Tj ).
(18)
Now, we have relaxed (12) to (17) and (18), which will be utilized to determine φ as follows. First, we construct an M × M matrix that assesses all pairs of distance measurements according to (17) and (18), which is denoted by Φ
[Φ]ij =
0, 1,
if the pair (di , dj ) satisfies (17) and (18) otherwise
(19)
where [Φ]ij denotes the (i, j) entry of Φ. That is, [Φ]ij = 0 means that the two distances di and dj agree with each other, and they are possibly LOS measurements. On the other hand, [Φ]ij = 1 indicates that di is inconsistent with dj , and thus, at least one of them corresponds to NLOS propagation. Assuming that {qi } are sufficiently small such that LOS paths contribute to the majority, the majority rule [21] is then ˆ via the removal of the employed in Φ to produce M vectors of φ contradicting distance measurements. We follow the same estimation procedure as in the standard MAP method by exploring all admissible M ˆ ˆ with φ < M − 3 to determine φ and x, which vectors of φ i=1 i is referred to as the simplified MAP algorithm. Since at most M vectors are involved in the simplified scheme, there will be a significant computational reduction over the standard method, which considers 2M combinations, particularly for a large value of M . In the case when the estimate of x is sufficiently accurate and some of the LOS distances are detected as NLOS measurements, the estimation performance will be improved by the following iterative steps. ˆ. 1) Estimate φ using (10) and x ˆ using d with φˆi = 0 by the two-step weighted least 2) Compute x square (TSWLS) method [3]. 3) Repeat steps 1 and 2 until parameter convergence. We refer to this enhancement as the iterative simplified MAP algorithm.
Fig. 3. LOS/NLOS detection probability with a larger NLOS error using the model of (3).
IV. S IMULATION R ESULTS
Fig. 4. Mean square error performance with a larger NLOS error using the model of (3).
Computer simulations have been conducted to evaluate the performance of the proposed algorithms for LOS/NLOS detection and mobile positioning. We also include the results of the TSWLS, clairvoyant, Riba–Urruela [18], and Cong–Zhuang [19] estimators for comparison. The TSWLS method uses all distance measurements, including those from NLOS propagation, for position determination, while the clairvoyant estimator employs the actual x to find {φi } in (10) and then computes the position estimate with the detected LOS measurements using the TSWLS algorithm. On the other hand, both [18] and [19] calculate the MS positions for all LOS/NLOS distance measurement combinations, and the one with the lowest error is selected as the estimate. Note that the former utilizes NLOS probability and maximum distance error information, while the latter does not. The BSs are located at (1000, 1000) m, (−1000, −1000) m, (1000, −1000) m, (−1000, 1000) m, (−500, 0) m, (0, −500) m, (500, 0) m, and (0, 500) m, and the MS position is (300, 400) m. The standard deviation of each measurement noise is σi = 0.1 × doi , while ηi is uniformly distributed in the range of [0, α × max(doi )], where α is a scaling factor. For simplicity, the NLOS probabilities {qi } are set to have identical M values. The LOS/NLOS detection rate, which is defined as E{ i=1 (1 − |φi − φˆi |)}, and the mean square position
ˆ 22 }, are employed as the performance error (MSPE), that is, E{x − x measures. Note that the former has a value of one only when all LOS/NLOS measurements are correctly identified and is zero otherwise. All simulation results are averages of 1000 independent runs, where each run contains at least three LOS measurements to ensure that the MS is localizable. For digital signal processing implementation with field-programmable gate arrays, see [22]. Figs. 3 and 4 show the LOS/NLOS detection and MSPE performance versus qi based on the approximate model of (3). We set α = 20, which corresponds to a larger NLOS error scenario. In Fig. 3, it is observed that all estimators, except the TSWLS and Riba–Urruela methods, have comparable LOS/NLOS detection performance when qi is sufficiently small. For a larger value of qi , we see that the clairvoyant and Cong–Zhuang estimators perform the best, followed by the standard MAP, iterative simplified MAP, and simplified MAP algorithms, although their differences are not significant. As expected, the TSWLS method performs poorly in the whole range because NLOS measurements are employed in the positioning process. On the other hand, [18] provides a detection rate that is close to zero. The MSPE results in Fig. 4 also show good agreement with the
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Fig. 5. LOS/NLOS detection probability with a larger NLOS error using the model of (2).
Fig. 7. LOS/NLOS detection probability with a smaller NLOS error using model of (2).
Fig. 6. Mean square error performance with a larger NLOS error using the model of (2).
Fig. 8. Mean square error performance with a smaller NLOS error using the model of (2).
LOS/NLOS detection rates, except for the Riba–Urruela algorithm. This indicates that the poor detection probability of [18] is mainly due to incorrect identification of LOS paths. We see that the MSPE difference between the clairvoyant and simplified MAP estimators is less than 6.5 dB for qi ∈ (0, 0.2). Furthermore, the complexities of the algorithms are studied according to the average computational time per trial. The measured times for the standard MAP, simplified MAP, iterative simplified MAP, TSWLS, clairvoyant, Riba–Urruela, and Cong–Zhuang estimators are 5.6 × 10−2 , 1.0 × 10−3 , 1.4 × 10−3 , 5.1 × 10−4 , 3.2 × 10−4 , 6.5 × 10−2 , and 1.5 × 10−1 s, respectively. For the standard MAP method, we observe that it combines the advantages of relatively low complexity of [18] and high performance of [19], while significant computational reduction of the two simplified MAP schemes over the standard one is observed, and their complexity is comparable with the TSWLS and infeasible clairvoyant estimators. The aforementioned test is repeated using the standard model of (2). The results, which are shown in Figs. 5 and 6, are comparable with those in Figs. 3 and 4. In particular, the proposed MAP approach performs similarly in both models of (2) and (3), although our math-
ematical development is founded on the latter. This demonstrates the robustness of the MAP schemes. Finally, we study the effect of a smaller NLOS error condition. We repeat the aforementioned experiment with α = 2 and the model of (2). The results are plotted in Figs. 7 and 8, and similar findings are obtained, except that now, the detection rate is smaller. V. C ONCLUSION Assuming that the NLOS probability and distribution is known, two MAP algorithms for joint NLOS detection and mobile positioning have been developed. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. An iterative procedure has also been suggested for the latter to boost its estimation performance. In terms of estimation performance, the standard MAP algorithm is superior, while the simplified versions have the advantage of being computationally attractive over the former. A possible extension of this paper is to study the robustness of the proposed approach for imperfect knowledge of the NLOS probability and distribution, which occurs in real environments.
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 3, MARCH 2010
A PPENDIX A In this Appendix, we will prove that (4) is equivalent to (7). Substituting (5) and (6) into (4) yields
As the second term of (A.1) takes no effect on the optimization, the MAP estimator is then ˆ x ˆ ) = arg (φ,
ln p(d|φ, x) + ln p(φ)
M
= ln
M 1
√
pφi(di)·
i=1
i=1
+ ln p(φ) =
M
2πσi
qiφi
i=1
=
M
φi ln(p(di )) +
+
M
2σi2
i=1
=−
+
M
2σi2
M 1 − φi i=1
+
M
2σi2
+
(1 − φi ) ln
(1 − φi ) ln
i=1
=−
M 1 − φi i=1
+
M
2σi2
M
(1 − φi ) ln(1 − qi )
1 − qi √ 2πσi
1 qi p(di ) 1 − qi √ 2πσi
(1 − φi ) ln
M
φi ln(qi p(di ))
+
M
ln (qi p(di ))
i=1
1 − qi √ 2πσi qi p(di )
ln (qi p(di ))
i=1
=−
M 1 − φi i=1
+
M i=1
2σi2
(di − x − xi 2 )2 − Ti
ln (qi p(di )) .
× (di − x − xi 2 )2 − Ti
(A.2)
R EFERENCES
(di − x − xi 2 )2
i=1
+
1 2πσi
(di − x − xi 2 )2
i=1 M
√
i=1
i=1
=−
(1 − φi ) ln
(di − x − xi 2 )2 +
(1 − φi ) ln
2σi2
The authors would like to thank the anonymous reviewers for their useful comments and suggestions.
i=1
M 1 − φi i=1
M
i=1
ACKNOWLEDGMENT
1−φi
(di − x − xi 2 )2
φi ln(qi ) +
φ∈{0,1}M ,x∈R2
which is (7).
(1 − qi )
M
M 1 − φi
M
i=1
M 1 − φi i=1
1 (di −x−xi 2)2 2σi2
×exp −
i=1
i=1
−
1−φi
min
(A.1)
[1] Revision of the Commissions Rules to Ensure Compatibility With Enhanced 911 Emergency Calling Systems, RM-8143, CC Docket no. 94102, Jul. 26, 1996. [2] J. J. Caffery, Jr., Wireless Location in CDMA Cellular Radio Systems. Norwell, MA: Kluwer, 1999. [3] Y. T. Chan and K. C. Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 1905–1915, Aug. 1994. [4] M. A. Spirito, “On the accuracy of cellular mobile station location estimation,” IEEE Trans. Veh. Technol., vol. 50, no. 3, pp. 674–685, May 2001. [5] K. W. Cheung, H. C. So, W.-K. Ma, and Y. T. Chan, “Least squares algorithms for time-of-arrival-based mobile location,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1121–1130, Apr. 2004. [6] B. Denis, J.-B. Pierrot, and C. Abou-Rjeily, “Joint distributed synchronization and positioning in UWB ad hoc networks using TOA,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1896–1911, Jun. 2006. [7] D. Harmer, M. Russell, E. Frazer, T. Bauge, S. Ingram, N. Schmidt, B. Kull, A. Yarovoy, A. Nezirovic, L. Xia, V. Dizdarevic, and K. Witrisal, “EUROPCOM: Emergency ultrawideband radio for positioning and communications,” in Proc. IEEE Int. Conf. Ultra-Wideband, Hannover, Germany, Sep. 2008, vol. 3, pp. 85–88. [8] J. Takada, J. Fu, H. Zhu, and T. Kobayashi, “Spatio-temporal channel characterization in a suburban non line-of-sight microcellular environment,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 532–538, Apr. 2002. [9] S. Al-Jazzar, J. Caffery, Jr., and H.-R. You, “Scattering-model-based methods for TOA location in NLOS environments,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 583–593, Mar. 2007. [10] P. C. Chen, “A non-line-of-sight error mitigation algorithm in location estimation,” in Proc. IEEE Wireless Commun. Netw. Conf., New Orleans, LA, Sep. 1999, vol. 1, pp. 316–320. [11] W. Wei, X. J. Yu, and Z. Z. Liang, “A new NLOS error mitigation algorithm in localization estimation,” IEEE Trans. Veh. Technol., vol. 54, no. 6, pp. 2048–2053, Nov. 2005. [12] R. Casas, A. Marco, J. J. Guerrero, and J. Falco, “Robust estimator for non-line-of-sight error mitigation in indoor localization,” EURASIP J. Appl. Signal Process., vol. 2006, pp. 1–8, 2006, Article ID 43429. [13] C. Morelli, M. Nicoli, V. Rampa, and U. Spagnolini, “Hidden Markov models for radio localization in mixed LOS/NLOS conditions,” IEEE Trans. Signal Process., vol. 55, no. 4, pp. 1525–1542, Apr. 2007. [14] W.-Y. Chiu and B.-S. Chen, “Mobile location estimation in urban areas using mixed Manhattan/Euclidean norm and convex optimization,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 414–423, Jan. 2009. [15] M. P. Wylie and J. Holtzman, “The non-line of sight problem in mobile location estimation,” in Proc. IEEE Int. Conf. Universal Pers. Commun., Sep. 29–Oct. 2, 1996, vol. 2, pp. 827–831. [16] S. Gezici, H. Kobayashi, and H. V. Poor, “Non parametric nonline of sight identification,” in Proc. IEEE 58th Veh. Technol. Conf., Oct. 6–9, 2003, vol. 4, pp. 2544–2548. [17] Y. T. Chan, W. Y. Tsui, H. C. So, and P. C. Ching, “Time-of-arrival based localization under NLOS conditions,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 17–24, Jan. 2006.
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[18] J. Riba and A. Urruela, “A non-line-of-sight mitigation technique based on ML-detection,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Montreal, QC, Canada, May 2004, vol. 2, pp. 153–156. [19] L. Cong and W. Zhuang, “Non-line-of-sight error mitigation in mobile location,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 560–573, Mar. 2005. [20] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [21] M. Fey, “May’s Theorem with an infinite population,” Soc. Choice Welf., vol. 23, no. 2, pp. 275–293, Oct. 2004. [22] U. Meyer-Baese, Digital Signal Processing With Field Programmable Gate Arrays. Berlin, Germany: Springer-Verlag, 2001.
Error Rate Analysis of Band-Limited BPSK With Nakagami/Nakagami ACI Considering Nonlinear Amplifier and Diversity M. A. Rahman, Member, IEEE, C. S. Sum, Member, IEEE, R. Funada, Member, IEEE, T. Baykas, Member, IEEE, J. Wang, Member, IEEE, S. Sasaki, Member, IEEE, H. Harada, Member, IEEE, and S. Kato, Fellow, IEEE
Abstract—An exact expression of the average error rate is developed for a band-limited binary phase-shift keying (BPSK) system in the presence of adjacent channel interference (ACI) considering a Nakagami fading channel. The system employs a root-raised cosine (RRC) filter both in the transmitter and receiver sides and a transmitter power amplifier (PA) that may have nonlinear input/output characteristics. These practical considerations are accurately taken into account. By utilizing the characteristic function (CF) method of error-rate analysis, we interestingly blend the concepts of time and frequency domains and develop an error-rate expression that is simple and general for several system parameters, including adjacent channel separation, RRC filter roll-off factor, and PA output back-off (OBO). Hence, the error rate in the presence of cochannel interference (CCI) is obtained as a special case. The developed results for a single-branch receiver are then extended for diversity receivers, and several interesting observations are made. Index Terms—Adjacent channel interference (ACI), cochannel interference (CCI), error-rate analysis, Nakagami fading, power amplifier (PA) nonlinearity and diversity receivers.
I. I NTRODUCTION The frequency spectrum is a valuable resource. For its proper utilization, the available spectrum is usually subdivided into several frequency bands. This facilitates simultaneous operation of several systems without interfering with each other. However, in many practical cases, it may not be possible to perfectly confine a desired signal within a desired band. Due to imperfection of the transmit filter and nonlinearity of the power amplifier (PA), a portion of the transmitted Manuscript received June 10, 2009; revised October 6, 2009. First published December 4, 2009; current version published March 19, 2010. The review of this paper was coordinated by Prof. H.-C. Wu. M. A. Rahman, C. S. Sum, R. Funada, T. Baykas, J. Wang, H. Harada, and S. Kato are with the National Institute of Information and Communications Technology, Yokosuka 239-0847, Japan (e-mail: [email protected]; aziz.jp@ ieee.org; [email protected]; [email protected]; [email protected]; junyi. [email protected]; [email protected]; [email protected]). S. Sasaki is with the Department of Electrical and Electronic Engineering, Niigata University, Niigata 950-2181, Japan (e-mail: [email protected]. ac.jp). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2009.2037821
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signal spectrum may spill over adjacent bands and cause unwanted interference. Due to imperfection of the receive filter, a receiver may also receive a portion of signals transmitted in adjacent bands. Such interferences are usually known as adjacent channel interference (ACI). As opposed to ACI, there may be systems simultaneously operating within the same frequency band of the desired system. Interferences from such sources are commonly known as cochannel interference (CCI). The development of an accurate expression of the error rate for a conventional band-limited binary phase-shift keying (BPSK) system in the presence of CCI has been a topic of several recent publications [1]–[6]. Among those, [4] presented a characteristic function (CF)based exact error-rate analysis for band-limited BPSK in a Nakagami fading environment considering Rayleigh-faded CCI. A complete version of the study was presented in [3] considering Nakagami-faded CCI. The results presented in [3] are precisely accurate, although not exact. However, these works were later extended for diversity in [2], although the error rate was given based on rather inaccurate Gaussian approximation of CCI. Other authors also addressed CCI analysis in Rician fading [5]. Recently, Smadi et al. [6] have presented an exact CCI analysis for partially coherent modulations considering diversity. However, despite several papers being published on CCI analysis, papers that contributed on ACI analysis are quite limited [7]–[9]. A basic difference of the ACI signal from its CCI counterpart is that it has a center frequency different from, but usually somewhat close to, the desired signal. As a result, works on CCI cannot readily be extended for ACI. In addition, there is an important concern that, for an ACI analysis to be useful, transmitter PA nonlinearity should accurately be taken into consideration. PA nonlinearity gives rise to spectrum regrowth, which itself is a major reason to create ACI. Because the equivalent time-domain pulse shape of a band-limited signal extends infinitely, developing a tractable error rate analysis for such signal taking into account PA nonlinearity is a challenging task. A basic work on analyzing and comparing performance of various digital modulation schemes in the presence of ACI can be found in [7]. Later, [8]–[10] addressed ACI analysis for offset phase-shift keying, differential phase-shift keying, and minimum shift keying, respectively. Reference [11] investigated the performance of a neuralnetwork-based receiver in the presence of ACI. However, none of the works considered the effect of nonlinear amplification. The effect of nonlinear amplification on ACI potential was considered in [12]–[15]; however, [12]–[14] were based on simulations, whereas [15] only considered a fully saturated PA. In addition, [16] and [17], which considered the effect of nonlinearity on system performance, were on a very specific application. There are also a few other works (see [18]–[20] and the references therein) that investigated the impact of PA nonlinearity on spectral regrowth, signal-to-noise ratio degradation, etc. Nevertheless, what all the aforementioned papers have in common is that none presented an analytical work on developing an exact error rate of a band-limited signal considering ACI in a fading environment, either for a single-branch receiver or for diversity combining. Due to immense practical importance, there is a current demand in the literature for such a general analysis considering nonlinear amplification in the transmitter, fading in the channel, and diversity in the receiver. In this paper, we consider a practical band-limited BPSK system in the presence of ACI in a Nakagami fading environment. Because Nakagami fading is a general representation of fading that includes Rayleigh and Rician fading as special cases, developing an error rate in Nakagami fading is of practical importance. By employing the CF-based method, a simple exact expression of the error rate is developed that is valid for ACI with an arbitrary center frequency
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