Blind Low Complexity Time-Of-Arrival Estimation Algorithm for UWB Signals

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by IEEE Signal Processing Letters on 2005/06, available at: https://doi.org/ 10.1109/LSP.2015.2450999 ©2005 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” SIGNAL PROCESSING LETTERS, VOL. , NO. , 1 Blind Low Complexity Time-Of-Arrival Estimation Algorithm for UWB Signals Eva Arias-de-Reyna, Member, IEEE, Juan Jos´ e Murillo-Fuentes, Senior Member, IEEE, and Rafael Boloix-Tortosa, Member, IEEE Abstract—This paper presents a novel time-of-arrival (TOA) blind estimation technique for ultra wideband energy detection receivers with reduced complexity. The proposed method is blind in the sense that it does not exploit any information about channel or noise power. This new approach is based on a set of approximations of the exact likelihood function (ELF) of the observed energy. Even though these approximations achieve an important reduction of complexity, the shape of the new approximated function is accurate enough compared to the ELF. Application of a threshold to the differential of the approximated log-likelihood function completes the procedure. Simulations show that the performance of the proposed method in terms of the cumulative distribution function of the estimation error approaches that of a method based on the ELF and a genie-aided algorithm with perfect knowledge of the optimal threshold. Index Terms—Time-of-arrival, UWB, ranging, localization. I. INTRODUCTION HIGHLY accurate location information is required in a large variety of emerging applications such as inventory tracking, security, monitoring or navigation aids [1], [2]. In indoor environments the global positioning system (GPS) typically fails, and one needs to resort to an alternative technology. Among the possibilities, ultra wideband impulse radio (UWB-IR) [3], [4] is especially promising: thanks to the ultrashort duration of the transmitted pulses it exhibits very accurate estimation of the time-of-arrival (TOA) of the pulse corresponding to the direct path between nodes [5]. Further characteristics that make UWB attractive for localization applications are its obstacle penetration and multipath resolution capabilities. Also, it favors the design of low power and low cost devices, especially with non-coherent architectures and sub-Nyquist sampling rates [6]. These systems have become very popular for low data rate applications such as wireless sensor networks. In this paper we focus on TOA estimation for these devices, with UWB energy detection receivers [6] that provide energy measurements over a sequence of time slots [5]. A number of methods have been proposed for this problem, see for example [7], [8], [9], [10], [11], [5] and references therein, where one of the challenges is detection when the first Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. The authors are with the Department of Signal Theory and Communications, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain (e-mail: [email protected]; [email protected]; [email protected]) This work has been supported by the Spanish government (Ministerio de Educaci´ on y Ciencia, TEC2012-38800-C03-02), European Union (FEDER) and Junta de Andaluc´ ıa (TIC-155). arriving path is not the strongest one. Giorgetti and Chiani [8] have recently proposed a TOA estimation method based on model selection that, in spite of lacking any knowledge about channel or noise power, achieves the performance of a genie-aided threshold crossing technique with knowledge of the optimal threshold. To this end, the exact likelihood function (ELF) of the observed energy samples is computed. However, this estimation exhibits a high computational complexity that suggests replacing it by approximations [8]. We propose a new TOA estimation technique based on some simplifications to the ELF of the observed data, followed by the application of a threshold to the differential of the approximated log-likelihood function (LLF). As a result, we have a significant reduction in complexity with respect to the methods in [8]. II. SIGNAL MODEL AND PROBLEM STATEMENT The transmitted signal structure corresponds to the synchronization preamble of the IEEE 802.15.4a standard, used for acquisition, synchronization and ranging purposes. It consists of NSY NC identical symbols of duration TSY NC [12], [7]. The mathematical model for this preamble is s(t) = NSY NC −1 X m=0 Ncs−1 X n=0 dnp(t−nTf−mTSY NC),(1) where p(t)is an ultrashort pulse of duration of the order of the nanosecond, Tfis the pulse repetition period (PRP), usually referred to as frame period in the UWB literature, and {dn}Ncs−1 n=0 is a perfectly balanced code sequence prescribed by the 802.15.4a standard with elements ∈ D ={−1,0,+1} [12], [7]. Tfis usually larger than the channel delay spread. At the receiver we consider an energy detection architecture [5], [7], [8], [13]. The first stage is a bandpass filter (BPF) with center frequency f0and bandwidth B. The received signal at the output of the BPF is r(t) = NSY NC −1 X m=0 Ncs−1 X n=0 dnw(t−nTf−mTSY NC)+n(t)(2) where w(t)is the received waveform for each transmitted pulse p(t). Its shape is determined by both the attenuation coefficients and delays τ1< τ2<· · · τLof the Lpaths of the fading channel considered [14], and the bandpass filtering effect of the channel and BPF. Besides, n(t)is the result of filtering thermal noise with two-sided power spectral density N0/2with the BPF. The BPF is followed by an energy detection scheme, namely a square-law device plus an integrate-and-dump device [5]. 2 SIGNAL PROCESSING LETTERS, VOL. , NO. , Observe that for a small noise power the squared signal is very similar for all the PRPs with either dn= 1 or dn=−1. On the other hand, the PRPs with dn= 0 contain only noise. The output of the integrate-and-dump device is sampled each Tint seconds, obtaining an energy sample. The number of time slots (bins) for each PRP is M=Tf/Tint and the total number of energy samples for the preamble is NSY NC ×Ncs ×M. Without loss of generality we assume that Tfis a multiple of Tint. As in [8], we store these samples in a matrix ε={εi,j}, where εi,j is the energy sample corresponding to the jth bin of the ith PRP, hence εis an NSY NCNcs ×Mmatrix. We assume a two-step procedure for TOA estimation [5], [8]. After the first stage, alignment of the code sequence has been achieved, although the instant of beginning of the signal waveform (TOA) for each PRP with dn= 1 or dn=−1is not known yet. Here we focus on the second stage, which consists in a fine estimation of the TOA, i.e., the delay of the first path, τ1, within an uncertainty region of duration Tf. Determination of the beginning of this region is achieved through an initial detection of the highest energy bin and jumping backward by a given duration Tback to a bin prior to the beginning of the channel impulse response (CIR) [7]. Suitable values of Tback to ensure that the resulting first bin falls within the noise only region depend on the characteristics of UWB channels, but need not be optimized for individual CIRs [7]. Since the values of PRP envisioned by the IEEE standard 802.15.4a may cause the backward jump to fall within the tail of the previous CIR, we adopt the procedure in [7], exploiting just the signal-plusnoise PRPs preceded by a noise only PRP. Let Isel be the set of indices of the selected PRPs preceded by a noise only PRP. The input to our TOA estimation algorithm can be expressed as {εi,j}i∈ Isel, j = 0,1,· · · , M −1.(3) The energy average for each bin over different PRPs is typically used as an intermediate variable in TOA estimation methods. In this case it yields ¯εj=1 Nsel X i∈Isel εi,j, j = 0,1,· · · , M −1(4) where Nsel is the cardinality of Isel. Let us call jmax the index of the highest energy bin mentioned above, jmax = arg maxj{¯εj}. Hence, as an output of the energy detector, for each of the Nsel PRPs of the signal preceded by a noise only PRP we have a set of Menergy samples. The first bins of the PRP belong to the noise only region, while the bins from the one with index j=nT OA onwards contain signal plus noise (although some of these bins might eventually contain only noise for some channel realizations). Our goal is the estimation of nT OA, denoted as ˆnT OA, as an intermediate step for the estimation of the TOA, determined as ˆnT OATint +Tint/2[5], [8]. We need to choose between M−1hypotheses, the kth hypothesis corresponding to ˆnT OA =k, that is, a duration of the initial noise only region of kbins, k∈ {1,2,· · · , M −1}. III. PROPOSED METHOD As in the TEMOS-E variant in [8], we assume that the signal plus noise region does not begin later than at the highest energy bin (with index jmax), thus we replace the number of hypotheses M−1by Mmax −1 = jmax. For the kth hypothesis, there are M−k+ 1 parameters to estimate, namely the energies λjof the received signal in the bins of indices from j=kto j=M−1plus the value of N0 [8]. Accordingly, we arrange these parameters in a vector θ(k)= [λk, λk+1,· · · , λM−1, N0], where the noise variance is σ2=N0/2. The ELF of the observed data, f(ε;θ(k)), depends on this vector and can be approximated [8] as the product of the probability density functions (PDFs) of the individual energy samples εi,j i∈ Isel, j = 0,1,· · · , M −1: f(ε;θ(k)) = Y i∈Isel M−1 Y j=0 g(εi,j;λj, N0, ν)(5) where g(εi,j;λj, N0, ν)is the PDF of a noncentral chi-square random variable with noncentrality parameter λjif λj6= 0 (signal plus noise bins), while it is the PDF of a central chi-square random variable if λj= 0 (noise only bins) [8]. For both distributions, the number of degrees of freedom is ν= 2BTint. For simplicity we are assuming that BTint is an integer, so that νis an even integer. The method in [8] involves the computation of f(ε;ˆ θ(k)), where ˆ θ(k)is the estimate of the unknown parameters for the kth hypothesis. In particular the estimate of N0for the kth hypothesis is computed from the observed data at the noise only bins corresponding to this hypothesis, ˆ N(k) 0=2 νPk−1 j=0 ¯εj k, while the estimates for the signal energies are computed as ˆ λ(k) j=¯εj−νˆ N(k) 0 2+ j=k, k + 1,· · · , M −1, where the notation (x)+indicates the maximum between 0 and x. It is interesting to point out that obtaining the value of f(ε;ˆ θ(k))involves quite complex computations. We propose an approximation that leads to a less complex estimation algorithm. The validity of the proposed transformations for the purposes of this paper will be assessed by simulation with the channel models standardized by IEEE for UWB [14]. A. Approximation of the log-likelihood function Let us first rewrite (5) as if all factors were PDFs of noncentral chi-square variables, f(ε;ˆ θ(k)) = Y i∈Isel M−1 Y j=0    ε ν 2−1 i,j ˆ N(k) 0ν 2exp −εi,j +ˆ λ(k) j ˆ N(k) 0! · Iν 2−1 qεi,j ˆ λ(k) j ˆ N(k) 0/2! 1 2ν 2 −1 qεi,j ˆ λ(k) j ˆ N(k) 0/2!ν 2−1        (6) where ˆ λ(k) j=     0j= 0,1,· · · , k −1 ¯εj−νˆ N(k) 0 2+ j=k, k + 1,· · · , M −1 The subsequent approximations will eliminate the indeterminacy that arises when ˆ λ(k) j= 0. In particular, we propose to ARIAS-DE-REYNA et al.: BLIND LOW COMPLEXITY TIME-OF-ARRIVAL ESTIMATION ALGORITHM FOR UWB SIGNALS 3 replace function Iν 2−1(z)with the first two terms of its series expansion In(z) z 2n≈1 n!1 + z2 4(n+1) . Therefore, f(ε;ˆ θ(k))≈Y i∈Isel M−1 Y j=0    ε ν 2−1 i,j ˆ N(k) 0ν 2· exp −εi,j +ˆ λ(k) j ˆ N(k) 0!1 ν 2−1! 1+2 εi,j ˆ λ(k) j νˆ N(k) 02!!.(7) Taking the logarithm yields an approximation of the LLF. The terms that do not depend on kare irrelevant for our purpose, which is to choose the most likely hypothesis. Hence, an approximation of the exact LLF up to a constant is ln f(ε;ˆ θ(k))≈X i∈Isel M−1 X j=0 −ν 2ln ˆ N(k) 0−εi,j +ˆ λ(k) j ˆ N(k) 0 + ln   1+2 εi,j ˆ λ(k) j νˆ N(k) 02    .(8) Next we propose to use a low signal to noise ratio (SNR) approximation, in particular ln 1+2 εi,j ˆ λ(k) j νˆ N(k) 02!≈2εi,j ˆ λ(k) j νˆ N(k) 02. Applying this approximation, performing the summations wherever possible and taking into account (4) and λ(k) j= 0∀j < k, it follows that ln f(ε;ˆ θ(k))≈Nsel −νM 2ln ˆ N(k) 0−PM−1 j=0 ¯εj ˆ N(k) 0 −PM−1 j=kˆ λ(k) j ˆ N(k) 0 + 2PM−1 j=k¯εjˆ λ(k) j νˆ N(k) 02  .(9) As a final step in order to simplify the expressions we propose to remove the operator (x)+in the estimates of signal energies ˆ λ(k) j. The rationale behind this choice is that while kbelongs to the noise only region, the estimation of N(k) 0 involves noise only bins and ¯εj−νˆ N(k) 0/2is expected to be either positive (jth bin contains signal plus noise) or close to zero (jth bin contains only noise). As kgets into the signal plus noise region, the estimate ˆ λ(k) jloses quality, with or without the operator (x)+. Thus, we approximate M−1 X j=k ˆ λ(k) j≈ M−1 X j=k ¯εj−ν(M−k)ˆ N(k) 0 2(10) M−1 X j=k ¯εjˆ λ(k) j≈ M−1 X j=k (¯εj)2−νˆ N(k) 0 2 M−1 X j=k ¯εj,(11) which allows us to approximate the LLF, again up to irrelevant constants and a factor Nsel, as ln f(ε;ˆ θ(k))≈ −νM 2ln ˆ N(k) 0−PM−1 j=0 ¯εj ˆ N(k) 0 −2PM−1 j=k¯εj ˆ N(k) 0 +2 νPM−1 j=k(¯εj)2 ˆ N(k) 02−νk 2.(12) 0 10 20 30 40 50 60 70 0 0.02 0.04 0.06 εj j 0 10 20 30 40 50 60 70 −432 −430 −428 −426 −424 −Eq. (12) k 0 10 20 30 40 50 60 70 −0.5 0 0.5 1 d(k) k 0 10 20 30 40 50 60 70 −2 0 2 4 c(k) k (a) (b) (c) (d) nTOA (true TOA) jmax transition region correct hypothesis estimate of nTOA max(c(k)) j0 Tf/Tint Tback/Tint jini c(jini)<0 Te/Tint Fig. 1. (a) A realization of ¯εj. (b) Corresponding realization of Eq. (12) (the figure shows its opposite). (c) Corresponding realization of d(k). (d) Convolution of d(k)with a rectangular function. B. Estimation of nT OA from the approximation of the LLF Eq. (12) allows detection of the correct hypothesis with an accuracy comparable to the method in [8] or a genie-aided optimal thresholding (OT) technique in many cases, as will be shown in Section IV. This is due to the shape of the expression in (12) as a function of k, which resembles the shape of the exact LLF. Fig. 1(b) illustrates the approximation of −ln f(ε;ˆ θ(k))(opposite of Eq. (12)), for a given realization of ¯εj(Fig. 1(a)). The hypotheses from k= 1 to k=nT OA have a similar likelihood. Then, −ln f(ε;ˆ θ(k)) typically exhibits an increase starting at k=nT OA + 1, due to the low likelihood that the noise only region includes the first signal plus noise bin, and subsequent ones. Thus, our estimation problem boils down to detecting the increase at k=nT OA + 1, and the proposed procedure is as follows. First, we compute the differential of (12) as d(k) = −ln f(ε;ˆ θ(k+1))−(−ln f(ε;ˆ θ(k))), k = 1,· · · , Mmax −1 (see Fig. 1(c)). This new function has small values, either positive or negative, except for a region with larger positive values: the transition region starting from the beginning of the signal plus noise region until the likelihood stabilizes. The ripple of the noise only region of d(k)decreases with k, due to a better estimate of N0as more bins are involved in the estimation. We propose to choose the value of Tback large enough to ensure that the noise only region of d(k) spans at least a few (let us say, j0) bins. This allows us to discard these first bins in the estimation of nT OA and avoid the corresponding poor estimates of N0, thus reducing the uncertainty region of kto the range [j0, Mmax −1]. A further reduction of the uncertainty region is achieved by computing c(k), the convolution of d(k)with a rectangular function of duration denoted by Tconv. The maximum of c(k)is expected to fall within the transition region, thus allowing to locate this region. From this maximum we search backwards until we find a negative value of c(k), as shown in Fig. 1(d), or reach j0. We assume that the transition region is not likely to begin 4 SIGNAL PROCESSING LETTERS, VOL. , NO. , before this position, denoted as jini, thus we consider only the bins starting from jini, one by one, until the beginning of the signal plus noise region is detected. To this end, a criterion as simple as the following leads to the good results reported in Section IV for a variety of channel models: 1) Estimate the variance of d(k)in the noise only region, σ2 d, from a few bins -spanning a duration denoted by Teprior to jini. 2) Set a threshold u= 5σd. 3) Estimate ˆnT OA =kas the lowest k≥jini such that d(k)2> u2. The proposed method is not very sensitive to the exact value of u, for example u= 4σdor u= 6σdlead to similar results in the simulations reported in Section IV. As in [8], we have used a genie-aided OT method as a benchmark. For each SNR, the selected optimal threshold is the value leading to minimum RMSE, averaged over all the channel realizations. A practical approximation to this ideal scheme would rely on previous knowledge or determination of the optimal threshold-to-noise ratio and estimation of noise power, leading to performance degradations [8]. The version of the method in [8] used for comparison is the TEMOS-E method with the ELF in Eq. (5) and penalty function based on efficient detection criteria (EDC). The computational advantage of the proposed method with respect to this one is twofold. On one hand, the proposed method avoids computation of Bessel functions and square roots, alleviating the memory requirements, which is a desirable feature for low cost sensors. On the other hand, the dominant term of the complexity of the method based on ELF is O(MmaxMNsel), where the operation to be repeated is the computation of a chisquare PDF, while for the proposed method it is O(Mmax), where the operation to be repeated is Eq. (12). IV. RESULTS In our simulations the transmitted signal is compliant with the IEEE 802.15.4a standard [12], for the shortest preamble and shortest code sequence envisioned: NSY NC = 16 and Ncs = 31, which corresponds to 8 PRPs preceded by a noise only PRP. This is the most challenging situation and leads to Nsel = 128. The pulse p(t)is a root raised cosine bandpass pulse with central frequency 4 GHz, roll-off factor 0.6 and duration parameter 0.75 ns as in [8]. The PRP is Tf= 128 ns, TSY NC = 3968 ns and the BPF is ideal with central frequency 4 GHz and bandwidth 1.5 GHz. We have conducted simulations for three of the channel models standardized by IEEE [14]: CM1, CM3 and CM4. We have taken Tback = 120 ns, which ensures that the first 40 ns contain only noise. This leads to j0= 40 ns/Tint. Some suitable values for the remaining parameters are Tconv = 20 ns, Te= 16 ns. Fig. 2 shows the root mean square error (RMSE) of the TOA estimation for 1500 channel realizations for CM3 and Tint = 2 ns, as a function of SNR, defined as Ew/N0where Ew=R∞ −∞ w2(t)dt. The algorithms tested are: the proposed one, the Maximum Energy Selection (MES) [5] technique, the genie-aided OT algorithm and the method in [8] as described in the previous section. Fig. 3 shows the 0 10 20 30 40 50 60 10−10 10−9 10−8 10−7 TOA RMSE (s) SNR (dB) Proposed method, u=5σd Proposed method, u=4σd Method based on ELF MES benchmark: genie−aided OT Fig. 2. RMSE of TOA estimation for channel model CM3. 0 0.5 1 1.5 2 x 10−8 0 0.2 0.4 0.6 0.8 1 P(|error|< e) e (s) CM3 Proposed method, u=5σd Proposed method, u=4σd Method based on ELF MES benchmark: genie−aided OT 0 0.5 1 1.5 2 x 10−8 0 0.2 0.4 0.6 0.8 1 P(|error|< e) e (s) CM4 Proposed method, u=5σd Proposed method, u=4σd Method based on ELF MES benchmark: genie−aided OT 0 0.5 1 1.5 2 x 10−8 0 0.2 0.4 0.6 0.8 1 P(|error|< e) e (s) CM1 Proposed method, u=5σd Proposed method, u=4σd Method based on ELF MES benchmark: genie−aided OT Fig. 3. CDF of absolute value of TOA estimation error for SNR=10dB (solid lines) and 20 dB (dotted lines). cumulative distribution function (CDF) of the absolute value of the TOA estimation error for three channel models, Tint = 2 ns and two values of SNR, 10 and 20 dB. The proposed method is observed to perform close to the method in [8] and the benchmark as compared with MES, without the need to optimize any parameter for the channel model or SNR value. An increase in SNR leads to a closer approximation of the proposed technique to the method in [8] and the benchmark. V. CONCLUSION We have proposed a blind TOA estimation method for an UWB energy detection receiver. The proposed technique is based on certain approximations of the exact LLF of the measured energy samples, which avoid computation of modified Bessel functions and allow combination of some terms. Thus, a computational advantage is obtained with respect to methods based on the exact LLF. Through the simulations the shape of the resulting function resembles the shape of the exact LLF with enough accuracy for a simple criterion -thresholding of the differential of the approximated LLFto allow estimation of the TOA for a variety of channel models and SNR values. Performance of the proposed method in terms of CDF of the estimation error is close to that of a method based on the ELF and a genie-aided OT method. ARIAS-DE-REYNA et al.: BLIND LOW COMPLEXITY TIME-OF-ARRIVAL ESTIMATION ALGORITHM FOR UWB SIGNALS 5 REFERENCES [1] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero III, R. L. Moses, and N. S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” IEEE Signal Processing Magazine, vol. 22, no. 4, pp. 54–69, July 2005. [2] D. Dardari, E. Falletti, and M. Luise, Satellite and Terrestrial Radio Positioning Techniques. A Signal Processing perspective. Elsevier, 2011. [3] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molish, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: a look at positioning aspects for future sensor networks,” IEEE Signal Processing Magazine, vol. 22, no. 4, pp. 70–84, July 2005. [4] L. Yang and G. B. Giannakis, “Ultra-wideband communications: An idea whose time has come,” IEEE Signal Processing Magazine, vol. 21, no. 6, pp. 26–54, November 2004. [5] D. Dardari, A. Conti, U. Ferner, A. Giorgetti, and M. Z. Win, “Ranging with ultrawide bandwidth signals in multipath environments,” Proceedings of the IEEE, vol. 97, no. 2, pp. 404–426, February 2009. [6] K. Witrisal, G. Leus, G. J. M. Janssen, M. Pausini, F. Troesch, T. Zasowski, and J. Romme, “Noncoherent ultra-wideband systems,” IEEE Signal Processing Magazine, vol. 26, no. 4, pp. 48–66, July 2009. [7] A. A. D’Amico, U. Mengali, and L. Taponecco, “TOA estimation with the IEEE 802.15.4a standard,” IEEE Transactions on Wireless Communications, vol. 9, no. 7, pp. 2238–2247, July 2010. [8] A. Giorgetti and M. Chiani, “Time-of-arrival estimation based on information theoretic criteria,” IEEE Transactions on Signal Processing, vol. 61, no. 8, pp. 1869–1879, April 2013. [9] W. Liu, H. Ding, X. Huang, and Z. Liu, “TOA estimation in IR UWB ranging with energy detection receiver using received signal characteristics,” IEEE Communications Letters, vol. 16, no. 5, pp. 738–741, May 2012. [10] W. Liu, X. Huang, T. Jin, Z. Zhou, and X. Li, “Entropy based TOA estimation in IR UWB ranging with energy detection receiver under dense multipath environment,” in Proc. IEEE International Conference on Ultra Wideband (ICUWB), 2013, pp. 31–36. [11] H. Cai, T. Lv, H. Gao, A. Hu, and Y. Lu, “An optimized first path detector for UWB ranging using error characteristics,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC), 2014, pp. 1230 – 1235. [12] “IEEE standard wireless medium access control (MAC) and physical layer (PHY) specifications for low-rate wireless personal area networks (WPANs), IEEE Standard 802.15.4a-2007 (amendment to IEEE Standard 802.15.4-2006),” IEEE, Tech. Rep. [13] A. A. D’Amico, U. Mengali, and E. Arias-de-Reyna, “Energy-Detection UWB Receivers with Multiple Energy Measurements,” IEEE Transactions on Wireless Communications, vol. 6, no. 7, pp. 2652–2659, July 2007. [14] A. F. Molisch et al., “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Transactions on Antennas and Propagation, vol. 54, no. 11, pp. 3151–3166, November 2006.