Báo cáo hóa học: " Research Article LCMV Beamforming for a Novel Wireless Local Positioning System: Nonstationarity and Cyclostationarity Analysis"

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Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2007, Article ID 98243, 12 pages doi:10.1155/2007/98243 Research Article LCMV Beamforming for a Novel Wireless Local Positioning System: Nonstationarity and Cyclostationarity Analysis Hui Tong, Jafar Pourrostam, and Seyed A. Zekavat Department of Electrical and Computer Engineering, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA Received 24 June 2006; Revised 29 January 2007; Accepted 21 May 2007 Recommended by Kostas Berberidis This paper investigates the implementation of a novel wireless local positioning system (WLPS). WLPS main components are: (a) a dynamic base station (DBS) and (b) a transponder, both mounted on mobiles. The DBS periodically transmits ID request signals. As soon as the transponder detects the ID request signal, it sends its ID (a signal with a limited duration) back to the DBS. Hence, the DBS receives noncontinuous signals periodically transmitted by the transponder. The noncontinuous nature of the WLPS leads to nonstationary received signals at the DBS receiver, while the periodic signal structure leads to the fact that the DBS received signal is also cyclostationary. This work discusses the implementation of linear constrained minimum variance (LCMV) beamforming at the DBS receiver. We demonstrate that the nonstationarity of the received signal causes the sample covariance to be an inaccurate estimate of the true signal covariance. The errors in this covariance estimate limit the applicability of LCMV beamforming. A modiﬁed covariance matrix estimator, which exploits the cyclostationarity property of WLPS system is introduced to solve the nonstationarity problem. The cyclostationarity property is discussed in detail theoretically and via simulations. It is shown that the modiﬁed covariance matrix estimator signiﬁcantly improves the DBS performance. The proposed technique can be applied to periodic-sense signaling structures such as the WLPS, RFID, and reactive sensor networks. Copyright © 2007 Hui Tong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION In WLPS, a single unit (the DBS) is capable of positioning transponders located in its coverage area. In systems such as cell phone positioning [3] and radio frequency ID [4], multi- This paper investigates how to implement optimal beam- ple units should cooperate in the process of positioning. Ac- forming for a novel wireless local positioning system cordingly, the WLPS has many civilian and military appli- (WLPS). We focus on how to estimate covariance matrix for cations. For example, in vehicle collision avoidance applica- optimal beamforming, because the speciﬁc signaling scheme tions, each vehicle (car) may carry a DBS and each pedestrian in this WLPS, that is, cyclostationarity, enables a novel co- may carry a transponder. Then, each vehicle is able to posi- variance matrix estimator. tion (and identify) pedestrians. Another possible application The WLPS consists of two main components [1]: a dy- of the WLPS is airport security, where security guards may namic base station (DBS) and a transponder (or possibly a carry DBSs and passengers may carry transponders. number of transponders), all mounted on mobiles. The DBS The WLPS can be considered as a merger of positioning periodically transmits ID request signals (a short burst of en- and communication systems. The TOA/DOA estimation is ergy). Each time a transponder detects the ID request signal, the primary procedure for positioning, while the ID detec- it sends its unique ID (a signal with a limited duration) back tion process is supported by communications. This paper in- to the DBS. In the WLPS, the DBS detects and tracks the vestigates the ID detection performance, that is, the commu- positions and IDs of the transponders in its coverage area. nication aspect of the WLPS, while the TOA/DOA estimation The position of a transponder is determined by the com- process is discussed in [5, 6]. bination of time-of-arrival (TOA) and direction-of-arrival (DOA). TOA is estimated via the time diﬀerence between the As depicted in [7], the main source of error in the ID de- tection process is the interference from other transponders. transmission of ID request signal and the reception of the To reduce this interference, direct sequence code division corresponding ID. DOA estimation would be possible if an multiple access (DS-CDMA) and beamforming techniques antenna array is installed at the DBS receiver [2].
2 EURASIP Journal on Advances in Signal Processing are adopted in the WLPS. The conventional beamforming methods (delay and sum) in the WLPS have been discussed Periodic ID request signal in [7]. In general, linear constrained minimum variance (LCMV) beamforming outperforms conventional beam- ID of transponder number 1 forming in terms of interference suppression [8]. Therefore, it is natural to extend our study from conventional beam- ID of transponder number 2 forming to LCMV beamforming. An important step to perform LCMV beamforming is the ID of transponder number 3 estimation of the covariance matrix of the received signal. DBS Transponders Considering stationary signals, sample covariance accurately estimates the true signal covariance [9]. However, in the Figure 1: WLPS basic structure. WLPS, the received signal at the DBS receiver is not station- ary, because the DBS transmits ID request signals noncontin- uously. The nonstationarity of the received signal causes the its coverage area. Once a transponder detects the ID request sample covariance to be an inaccurate estimate of the true signal, it sends its unique ID (a signal with limited dura- signal covariance. The errors in this covariance estimate limit tion) back to the DBS, as shown in Figure 1. The DBS is the applicability of LCMV beamforming in the WLPS. equipped with multiple antennas to support DOA estimation In this work, a modiﬁed covariance matrix estimator is and beamforming. proposed. The transponders transmit signals noncontinu- In the WLPS, a DBS communicates with multiple trans- ously and repetitively. Accordingly, the DBS received signal ponders simultaneously. This is the same as standard cellu- is nonstationary and cyclostationary. The proposed modiﬁed lar communication systems. However, diﬀerent from cellular covariance matrix estimator exploits the cyclostationarity to systems, the DBS received signal in the WLPS is not station- counter the nonstationarity problem. A detailed theoretical ary. analysis shows that, in most practical situations, the cyclo- As shown in Figure 1, the signal transmitted by a trans- stationarity duration is suﬃciently long to ensure an accurate ponders do not span over the whole time domain. This fea- estimate. Finally, the WLPS ID detection performance is nu- ture leads to a new performance measure metric: probability- merically simulated. The numerical results conﬁrm that the of-overlapping, povl , which is deﬁned as the probability that modiﬁed covariance matrix estimator improves the WLPS the desired ID is overlapped with the ID signals from other performance signiﬁcantly. It should be further noted that the transponders. In standard wireless systems, povl is always proposed estimator is not restricted to this particular WLPS unity for multiple transponders. In the DBS receiver, the system: it is possible to apply this estimator to any system probability of overlapping is less than unity and corresponds that exhibits repetitive structures. Hence, the proposed co- to: variance matrix estimator has a wide range of applications. Beamforming [10] and cyclostationarity [11] have been K −1 povl = 1 − 1 − dc , (1) studied separately for more than ﬁfty years. In recent decades, a joint consideration of beamforming and cyclosta- where K denotes the number of transponders and dc repre- tionarity (i.e., beamforming for cyclostationary signals) at- sents duty cycle, which is deﬁned as: tracted certain attention [12, 13]. In those studies, the sig- τ nals are both stationary and cyclostationary. In other words, dc = . (2) IRTmin continuous signals with repetitive structures are considered. In our work, we study noncontinuous signals with repetitive Here, τ is the duration of the ID of a transponder, and IRTmin structures. Therefore, this paper exploits cyclostationarity to is the time diﬀerence between the ﬁrst responding transpon- counter the nonstationarity problem in optimal beamform- der and the last responding transponder. A comprehensive ing. results for IRTmin have been introduced in [1]; here, roughly, The rest of the paper is organized as follows: Section 2 in- Rmax troduces the fundamentals of the WLPS structure; Section 3 IRTmin = , (3) discusses the implementation of WLPS system and the non- 2c stationarity problem; Section 4 demonstrates how to exploit where Rmax is the maximum coverage distance of the DBS, cyclostationarity to counter the nonstationarity problem; and c denotes the speed of light. For vehicle collision avoid- Section 5 presents numerical results, and Section 6 concludes ance applications, typically Rmax should not exceed 1 km. The the paper. exact value of Rmax may vary with diﬀerent environments, for example, urban or highways. 2. WLPS BASIC STRUCTURE In general, through this preliminary study, the noncon- tinuous nature of the WLPS seems alleviate the interference The WLPS comprises of a set of DBS and transponders. In problem: the undesired signals from other transponders may the scope of this paper, we consider the communication be- or may not interfere with the desired signal. In contrast, in tween one DBS and multiple transponders. The DBS trans- standard communication systems, the undesired signals al- mits ID request signals periodically to all transponders in ways overlap with the desired signal.
Hui Tong et al. 3 1 3. SYSTEM IMPLEMENTATION AND NONSTATIONARITY ANALYSIS 0.9 0.8 Once a transponder detects the ID request signal, it would Probability of overlapping 0.7 transmit its unique ID back to the DBS. To suppress interfer- ence from other transponders, the bits in the ID are spread by 0.6 DS-CDMA techniques. Hence, the transponders would peri- 0.5 odically transmit DS-CDMA signals that are with a limited duration. In a multipath (urban) environments, the received 0.4 signal at the DBS receiver would be the summation of DS- 0.3 CDMA signals from multiple transponders through multiple 0.2 paths. Finally, in the DBS receiver, it is possible to apply DS- CDMA despreading and beamforming techniques to extract 0.1 the ID of the desired transponder, as explained in Section 3.1. 0 10 20 30 40 50 60 In this work, the DOA estimation for the paths of the Number of transmitters (TRX or DBS) desired transponder is assumed to be perfect. Although the nonstationarity nature does have eﬀect on DOA estimation, Duty cycle = 0.1 Duty cycle = 0.001 the eﬀect turns out to be minimal, and the DOA estimation Duty cycle = 0.01 Duty cycle = 0.000015 is accurate enough for most practical applications [5]. Since the only required information for LCMV beamforming is the Figure 2: The probability of overlapping. directions of the paths of the desired transponder and the es- timation of covariance matrix, a good estimation of the co- variance matrix would ensure a good ID detection perfor- mance, as depicted in Section 3.2. However, it is noted that the noncontinuous nature of In Section 3.3, it is shown that the standard sample co- the WLPS is not suﬃcient in terms of rejecting interference. variance matrix estimator does not lead to a good quality of As shown in Figure 2, the probability of overlapping is very covariance matrix estimation. The reason is that due to the high when dc = 0.1 with a moderate number of transpon- nonstationary nature of the WLPS, diﬀerent bits of the ID ders (N = 10). In many applications, for example, vehicle experience diﬀerence interference. Hence, averaging covari- collision avoidance, the duty cycle might be even larger than ance matrix over each bit does not lead to a consistent esti- 0.1. Therefore, one cannot expect to suppress interference re- mator, that is, increasing the number of averaged data does liably through the noncontinuous nature of the WLPS. not reduce the mean square error (MSE) of the estimation. To reduce interference power, DS-CDMA and beam- The consistent covariance matrix estimator, which exploits forming techniques are necessary in the WLPS. A detailed the cyclostationarity of the WLPS, would be introduced in analysis for conventional beamforming and DS-CDMA tech- Section 4. niques has been presented in [7]. In general, optimal beam- formers perform better than the conventional beamformer. 3.1. Signal model Hence, it is natural to extend our study from conventional beamformer to optimal beamformers. The transmitted DS-CDMA signal by the kth transponder Optimal beamformers generate a statistically optimum corresponds to estimation of the desired signal through applying a weight vector to the observed data. This weight vector is computed sk ( t ) = g τ ( t ) via optimizing a certain cost function. Examples of these cost functions include total power, SINR, entropy, mean square N −1 bk [n] · gTb t − nTb · ak t − nTb · cos 2π fc t , · error, or nonGaussianity [14, 15]. Here, LCMV beamformer n=0 is selected because: (1) it is particularly good at rejecting in- (4) terference and (2) it only requires the observations of the re- ceived signals and the direction of the desired signal. The for- where N denotes the number of bits per ID code (that rep- mer is easy to obtain and the latter has been available via the resents the maximum capacity of the WLPS, which is in the DOA estimation process, which is prior to the beamforming order of 2N ), bk [n] denotes the nth bit of transponder k’s ID, process. Tb = τ/N represents the transponder bit duration, gτ (t ), and The basic structure of the WLPS has been introduced gTb (t ) are rectangular pulses with the duration of τ and Tb , in this section. In the next section, we introduce the signal respectively. Here, ak (t ) denotes the spreading code for the model of the WLPS and describe the beamforming imple- kth transponder, that is, mentation in a mathematical form. It is emphasized that di- rectly applying LCMV beamforming in the WLPS is not ap- G−1 propriate due to its nonstationary nature. In Section 4, cyclo- ak (t ) = k k Cg gTc t − gTb , Cg ∈ {−1, 1}, (5) stationarity would be exploited to solve the nonstationarity g =0 problem.
4 EURASIP Journal on Advances in Signal Processing where G (G ≤ 2N )1 is the processing gain (code length), Tc = Finally, the CDMA despreading is applied and the detected Tb /G = τ/ (N · G) represents the chip duration, and gTc (t ) is bit is given as a rectangular pulse with the duration of Tc . G With an antenna array mounted on the DBS receiver, the j z j [ n] = z j [ n , g ] Cg . (11) received signal at the DBS (see Figure 3), which is the sum- g =1 mation of signals from multiple transponders through mul- tiple paths, corresponds to The above description has included all necessary steps of WLPS ID detection process, except the calculation of the K Lk −1 N −1 j αk V θlk bk [n]gTb t − τlk − nTb gτ t − τlk r (t ) = weight vector W (θq ) in (9), which is the kernel part of l j k=1 l=0 n=0 this work. Here, we discuss how to determine W (θq ) in · ak t − τlk − nTb cos 2π fc t + φlk + n(t ), Section 3.2. (6) 3.2. Weight vector calculation where K denotes the total number of transponders, Lk is the number of paths for the transponder k, and αk , τlk , φlk denote The conventional beamforming weight vector simply corre- l the fading factor, time delay, and random phase shift for kth sponds to transponder’s lth path, respectively. Here, for simplicity of j j presentation, we assume that Lk = L, for all k. V (θlk ) denotes W f θq = V θq . (12) the array response vector that corresponds to j Noting that V (θq ) is a predeﬁned linear phase ﬁlter, which V θlk = 1 exp −i · 2πd cos θlk / λ · · · coincides with the deﬁnition of discrete Fourier transform, (7) T it is said that the conventional beamforming is equivalent to exp −i · 2(M − 1)πd cos θlk / λ . discrete Fourier transform [16]. Here, i denotes the imaginary unit, d is the spacing between The LCMV beamforming, which minimizes the total antenna elements, M is the total number of antennas, (·)T output power, while keeping the desired signal power con- denotes transpose, λ denotes the carrier wavelength, and θlk stant, corresponds to the solution of the following optimiza- is the direction of kth transponder’s lth path. Basically, in (7), tion problem [8]: we assume half wavelength spacing between antennas and the j j j j j precise knowledge of array manifold at the DBS receiver. min WcH θq Rq Wc θq s.t. WcH θq V θq = 1. (13) j After demodulation, the g th chip of the nth bit output for Wc (θq ) the j th transponder’s, the qth path would correspond to Using Lagrange multiplier, the solution of the above equa- j τq +(n+1)Tb +(g +1)Tc tion, that is, LCMV BF, is given by [17]: j = y q [ n, g ] r (t ) j τq +nTb +gTc j −1 j Rq W f θq j j j × cos 2π fc t + φq g t − τq − nTb − gTc d t. Wc θ q = , (14) j −1 j j W H θq Rq W f θq (8) f The g th chip of the nth bit output of the beamformer for j th j where Rq is the covariance matrix of j th transponder’s qth transponder’s qth path is given as jH j j path’s observed signal, that is, Rq = E[ yq · yq ]. j j j zq [n, g ] = W H θq · yq [n, g ], j (9) In this work, precise knowledge of the DOA θq and ar- j ray manifold is assumed, that is, W f (θq ) is perfectly known. j j where the weight vector W (θq ) and yq [n, q] are both 1 × M Then, the only left important implementation issue of the column vectors, and H denotes Hermitian transpose. j LCMV beamforming is the estimation of Rq . In general, the The receiver in Figure 3 and (9) resembles a spatial sample covariance matrix estimator corresponds to RAKE-like structure. Here, each RAKE corresponds to one path. Each path is received from a speciﬁc direction. Hence, Γ−1 1 jH beamforming on each RAKE is applied to capture the energy j j Rq = yq [n] yq [n], (15) Γ n=0 from the associated direction. After beamforming, the signals from diﬀerent paths are where Γ, (Γ ∈ {1, 2, 3 · · · N }), denotes the selected data combined via maximal Ratio combining: j j length for Rq estimation. If yq [n] is a stationary and ergodic L process, the sample average equals time average, and the sam- jj z j [ n, g ] = αl zl [n, g ]. (10) ple covariance matrix estimator leads to an accurate estimate l=1 j of Rq . In another word, the sample covariance matrix estima- tor would be consistent, and increasing the number of data samples reduces the error variance of the sample covariance Note that 2N is the maximum number of transponders that the system 1 matrix estimator. can accommodate.
Hui Tong et al. 5 Beamforming Demodulation for the 1st path of transponder j Beamforming Demodulation for the 2nd path of transponder j Despreading Beamforming for the 3rd path . of transponder j . . Path . . diversity . combining Beamforming for the last path Demodulation Decision of transponder j rule Figure 3: DBS receiver implementation via antenna arrays and DS-CDMA systems. jH j since there is no interference at all, E[ yq [n] yq [n]] = jH j E[ yq [n + 1] yq [n + 1]] and the sample covariance ma- Interfereing signal 2 trix estimator leads to an accurate estimation. How- ever, the main advantage of LCMV beamforming is Desired signal interference suppression, and in this situation, LCMV will not provide better performance than conventional j beamforming even with accurate estimation of Rq . Interfering signal 1 (ii) Large values of dc in dense transponder environment leads to povl → 1. In this case, the sum of interfer- From transponders to DBS ences would approximately be white noise, and the re- Interference from diﬀerent directions ceived signal statistically tends to be stationary, that is, for diﬀerent bits jH jH j j E[ yq [n] yq [n]] E[ yq [n + 1] yq [n + 1]]. In this case, the covariance matrix would be an identity matrix and Figure 4: Diﬀerent chips experience diﬀerent interference. LCMV beamforming becomes equivalent to conven- tional beamforming. (iii) Medium dc values and moderate transponder density 3.3. Nonstationarity analysis lead to a spatial structure for the interference, that is, several interfering signals are received in diﬀerent di- Standard wireless communication systems are stationary be- rections. In this case, the received data samples would cause of transmission of very long sequences from a large be nonstationary, large selection of Γ does not improve number of users. In other words, in these systems, diﬀerent the quality of covariance estimation, and the sample chips of the desired signal would experience the same inter- covariance matrix estimator is not consistent. ference. However, because the WLPS transponder transmit- ted signal is a short burst signal, the interfering signal may Figure 5 represents the mean square error (MSE) be- only interfere with some, but not all chips of the desired tween the true value and the estimated values of covariance signal (see Figure 4). Hence, the interference changes within matrix as a measure of nonstationarity, assuming a ﬂat fading each bit of the desired signal. This is especially the case for channel. The MSE corresponds to medium probability-of-overlapping, povl , values. Therefore, j in WLPS, Rq varies for diﬀerent chips and large selection of M M j j 2 MSE = Rq (m, u) − Rq (m, u) , (16) Γ does not necessarily lead to a high quality of the covariance m=1 u=1 matrix estimation. To have a better understanding when the received signal is not stationary, we have the following dis- j j where M is the number of antenna array elements, Rq and Rq cussion. denote the true and estimated covariance matrixes via sample (i) Small values of dc in (1) leads to low povl (see Figure 2). covariance matrix, respectively. The direction and distance In an extreme situation, povl → 0. In this case, of the transponders are assumed to be uniformly distributed
6 EURASIP Journal on Advances in Signal Processing remain the same for a number of periods, same chips of transponder ID in diﬀerent period experience the same inter- 0.5 ference (See Figure 6). Here, the period of transponder trans- mission is called ID request time (IRT). The repetition property of transponder transmission is 0.4 also known as cyclostationarity: although diﬀerent chips in MSE the same period does not experience same interferences, 0.3 same chips in diﬀerent periods experience same interfer- ences. Hence, it is possible to apply beamforming to each 0.2 chip, if the covariance matrix for each chip can be estimated. As shown in Figure 6, the covariance matrix estimation via 0.1 cyclostationarity for the g th chip of the nth bit corresponds to 0 10 20 30 40 50 60 Ω Number of users 1 jH j j R q [ n, g ] = yq [n, g , ω] yq [n, g , ω], (17) Ω ω=1 dc = 0.1 dc = 0.01 dc = 0.001 where Ω denotes the number of period within which the cy- clostationarity holds. Using (17), consequently (8) and (9) Figure 5: Simulation results: the mean square error of estimated would correspond to covariance matrix by standard estimation method. j y q [ n, g , ω ] in [0, π ] and [0, Rmax ]. The estimated covariance matrix is j τq +(n+1)Tb +(g +1)Tc +(ω−1)IRT j = normalized before comparing it with true covariance matrix. r (t ) cos 2π fc t + φq j τq +nTb +gTc +(ω−1)IRT It is seen that when dc is small (= 0.001) and the num- j ω ∈ {1, 2, . . . , Ω}, ber of transponder is small (60), the MSE is small as well. This corresponds to j j j W H θq · yq [n, g , ω], z q [ n, g ] = (19) Ω ω=1 the second case discussed. A large number of interferences lead to a spatially white structure. In other words, every chip is interfered by signals in many directions. Hence, the inter- respectively. Equation (19) reﬂects both beamforming and ference over diﬀerent chips would be similar, which leads to equal gain time diversity combining processes. Because each a stationary process. frame experiences independent fading, we also achieve time When dc is moderate (= 0.01), the MSE is large, that is, diversity beneﬁts via combining the chips from diﬀerent the nonstationarity problem is severe. IRT. The receiver structure via cyclostationarity is shown in The high MSE shown in Figure 5 leads to low probability Figure 7. Here, a separate block is considered for the covari- of detection. As a result, directly applying LCMV beamform- ance matrix estimator via cyclostationarity, since the new es- ing does not improve the system performance compared to a timator requires a temporary storage of the received signals. conventional beamforming. This point is veriﬁed by ID de- It should be noted that the proposed consistent co- tection simulations in Figure 11 (see Section 5). variance matrix estimator may not be restricted to LCMV beamforming, various optimal [18] or robust beamforming 4. ESTIMATOR BASED ON THE CYCLOSTATIONARITY [19, 20] methods may also use this estimator. In this paper, the application of LCMV beamforming in the WLPS is in- Section 3.3 introduced the nonstationarity problem in the troduced. The proposed concept may be easily extended to WLPS. This section proposes a modiﬁed covariance matrix any signal processing algorithm that requires an estimation estimator to solve the nonstationarity problem, which ex- of covariance matrix, as long as the system exhibits a repeti- ploits the cyclostationarity property of the WLPS. tive nature. 4.1. New estimator via cyclostationarity 4.2. Cyclostationarity duration The nonstationarity is mainly generated by the noncontin- uous transmission of transponders. However, it should be An important issue of the new estimator is the maximum possible value of Ω, that is, the number of periods that the noted that, in addition to the noncontinuousness, the trans- cyclostationarity holds. A larger value of Ω leads to better mission is also periodical. In every period, a transponder re- estimation, while a small value of Ω (e.g., 1 or 2) will render transmits the same ID bits with the same spreading code. Now, assuming all transponders’ directions and distances the estimator via cyclostationarity improper.
Hui Tong et al. 7 Received signal in IRT period T + Ω − 1 Received signal in IRT period T Received signal in IRT period T + 1 Interference signal 1 Desired signal ··· Interference signal 2 − → y (n1 , g1 , Ω) − (n2 , g2 , Ω) → − → y (n1 , g1 , 1) − (n2 , g2 , 1) → − → y (n1 , g1 , 2) − (n2 , g2 , 2) → y y y Same interference Same interference Figure 6: Same chips in diﬀerent IRT periods have the same interference. where θB is the half power beam width, d denotes the dis- 4.2.1. Cyclostationarity duration for a single transponder tance between transponder and DBS, and v⊥ is depicted in Basically, Ω is determined by IRT and the duration within Figure 8. which the cyclostationarity remains available, and corre- Combining the above two conditions, the ﬁnal condition sponds to corresponds to θ ·d c Tcy ,B Tcy . min (23) Ω≤ , (20) B · v 2v⊥ IRT Note that the ﬁrst condition (TOA constraint) is independent where Tcy is the time within which cyclostationarity condi- of distance, while the second condition (DOA constraint) de- tion holds, and IRT denotes the repetition time of the ID pends on both velocity and distance. request signal. Two parameters impact the cyclostationar- Equivalent to (23), we have the conditions for cyclosta- ity: The direction and the distance of transponder. Hence, tionarity Doppler frequency, which corresponds to the Tcy is the time within which (a) the direction of the B · v 2v⊥ 1 transponder approximately remains constant and (b) the dis- fcy = . max , (24) θB · d Tcy c tance of the transponder approximately remains unchanged (see Figure 8). This means that the changing rate of cyclostationarity should Therefore, we consider the impact of the movement of be much larger than DOA/TOA changing rate. the transponder in two directions. The ﬁrst is in the direction Knowing v = v ·cos(ψ ) and v⊥ = v ·sin(ψ ) (see Figure 8) that is parallel to the line connecting transponder and an- and considering ψ a uniform random variable within 0 and tenna array. In this direction, the variation of the TOA within 2π , the cyclostationarity Doppler spread (Bcy,d ), which is the the duration of Tcy should be much smaller than the chip du- root-mean-square (RMS) value of cyclostationarity Doppler ration Tch , that is, TOA is relatively ﬁxed during Tcy , which frequency, corresponds to corresponds to ⎛ ⎞ 2 2 B·v ⎜ ⎟ 2v⊥ c = max ⎝ Æ ⎠, Bcy,d ,Æ (25) Tcy , (21) Bd · d c B·v where Æ(·) denotes expectation operation. where c is the speed of light, B = 1/Tch denotes the transpon- Applying simple mathematical manipulations (25) der signal bandwidth, and v represents the Doppler velocity would correspond to of the transponder; √ The second direction is the direction that is perpendicu- B·v 2v √, Bcy,d = max . (26) lar to the line connecting transponder and antenna array. In 2c θB · d this direction, the variation of DOA should be much smaller Then, using (26) and similar to the deﬁnition of channel co- than the antenna array half power beamwidth, that is, DOA herence time, we deﬁne the cyclostationarity coherence time is relatively ﬁxed during Tcy , which corresponds to as [21] ∼ 1. θB / 2 · d Tcy,c = (27) Tcy , (22) Bcy,d v⊥
8 EURASIP Journal on Advances in Signal Processing Beamforming Demodulation for the 1st path Time diversity of transponder j combining Beamforming for the 2nd path Time diversity Demodulation of transponder j combining Beamforming Time diversity for the 3rd path . combining of transponder j . . . . . Beamforming Time diversity for the last path combining Demodulation of transponder j Delay line and Despreading covariance matrix estimation Path diversity combining Decision rule Figure 7: Receiver structure with using cyclostationarity. TRX DBS various velocity and distance values has been computed in Figure 9. Here, we assume 300 MHz bandwidth and 27◦ half v power beamwidth (consistent with four antenna elements). ψ The ﬁrst area of interest in Figure 9 is low-velocity and short- v⊥ range area, which is mainly suitable for applications such as v indoor and airport security. Note that the cyclostationarity Doppler spread varies with distance in this area. Hence, we can conclude that for short range applications, DOA would be the dominant condition for cyclostationarity. The second area of interest is high-velocity, long-range area, which is mainly suitable for vehicle collision avoidance system. Note that the cyclostationarity Doppler spread is independent of Figure 8: Relationship between v, v⊥ , and v . distance in this area. We can conclude that for long-range applications, the main constraint is the rate of change of TOA. In order to guarantee cyclostationarity during Tcy , Tcy 4.2.2. Cyclostationarity duration for multiple transponders should be selected smaller than Tcy,c , or The cyclostationarity duration for a single transponder is Tcy < Tcy,c . (28) straightforward. However, in the WLPS system, multiple transponders may present. In this situation, the cyclosta- To demonstrate the eﬀects of the two conditions on tionarity duration computation is much more complicated. Here, we compute the probability (Pcs ) that the position of cyclostationarity, the cyclostationarity coherence time with
Hui Tong et al. 9 0.8 all transponders remain relatively ﬁxed in (Tcs ) seconds, that is, Pcs = prob The position of all transponder 0.6 Cyclocoherence time (s) Vehicle collision nodes remain unchanged within Tcs ≤ p, avoidance application Airport security (29) application 0.4 where p is generally selected close to unity. Assuming posi- tioning statistics of diﬀerent transponders are independent, 0.2 then M γ ( m) , Pcs = (30) 0 m=1 100 101 102 103 Distance (m) where γ(m) refers to the probability that the mth transponder v = 2 m/s v = 30 m/s node remains unchanged during Tcs . Based on the discus- v = 5 m/s v = 60 m/s sions of Section 4.2.1, γ(m) corresponds to v = 10 m/s (m v⊥ ) c θB Figure 9: Cyclostationarity coherence time for diﬀerent applica- γ(m) = prob v(m) . , ( m) (31) B · Tcs d 2Tcs tions, single transponder. Now the same movement statistics is assumed for all transponders: (i) the speed of each transponder node, vm Assuming all transponders have the same movement follows Rayleigh distribution with mean mv ; (ii) direction statistics, we would have γ(m) = γ and Pcs = γM . Incorpo- of each transponder node, ψ (m) , is uniformly distributed in rating (32), (33), and (34), Pcs in (29) would correspond to [0, 2π ); and (iii) all transponder nodes are uniformly dis- tributed in DBS coverage area, that is, R(m) is uniform in M α 11 + erf √ Pcs = (0, Rmax ], where Rmax is the maximum radius of the DBS cov- 22 2σ erage. Rmax β 1 1 Rmax erf √ Using assumptions (i) and (ii), v(m) = v(m) sin ψ (m) and · + 2 2Rmin 2σ (m v⊥ ) = v(m) cos ψ (m) would be two independent random √ M √2 variables with zero mean and variance σ = 2/πmv for 2σ 1 − e−Rmax β / 2σ 2 2 . + all m ∈ {1, 2, . . . , M } transponders. Let X (m) = v(m) and πβ (m Y (m) = v⊥ ) /d(m) , then (31) would correspond to (35) Note that β = sα (s = B · θB / 2c), then (35) would be a ﬁxed- γ(m) = prob X (m) < α, Y (m) < β , (32) point equation of α. Based on the deﬁnition of α in (32) where α = (1/ Ω)(c/B · Tcs ), β = (1/ Ω)(θB / 2Tcs ), B is intro- 1c Tcs = . (36) duced in (21), and Ω is a constant that satisﬁes Ω 1. ΩB·α Note that X (m) and Y (m) are two independent random Hence, α is a function of the DBS antenna array half power variables; hence, beamwidth and coverage range, the number of transponders, transponder speed, and transponder pulse duration. As a re- prob X (m) < α, Y (m) < β = FXm) (α) · FYm) (β). ( ( (33) sult, Tcs would be a function of those parameters. Solving (35) and ﬁnding an analytic solution for Tcs is FXm) (α) and FYm) (β) are cumulative distribution functions ( ( not trivial. Hence, in Figure 10, numerical results for Tcs (m) and Y (m) , respectively, that is, (CDF) of X are generated in terms of (a) the number of transponder and transponder average speed for a system with (uniform α 11 FXm) (α) = ( + erf √ linear array with 4 elements and half wavelength element for any m, 22 2σ spacing) and (transponder bandwidth of 8.33 MHz) and (b) Rmax β 1 1 transponder bandwidth and DBS antenna array half power FYm) = + ( Rmax erf √ (34) beamwidth for a system with M = 10 transponders with av- 2 2Rmin 2σ erage speed of mv = 5 m/s. In these simulations, other se- √2 2σ lected parameters are Rmax = 1000 m, p = 0.95 [see (29)], 1 − e−Rmax β / 2σ 2 2 + , πβ and Ω = 10. It is observed that Tcs decreases as the number of transponders, transponder average speed, and bandwidth √ x −t 2 where erf(x) = (2/ π ) 0e dt . increase. Moreover, Tcs decreases as half power beamwidth of
10 EURASIP Journal on Advances in Signal Processing (4) carrier frequency = 3 GHz, τTRX = 1.2 μs, and τDBS = 101 24 μs; (5) the antenna array is linear with 4 elements, and el- ement spacing d = λ/ 2 = 0.05 m (half power beamwidth = 27◦ ); (6) four multipaths lead to L = 4 fold path diversity; 100 (7) the transponder distance and angle are uniformly dis- tributed in [0 1] km and [0 π ], respectively; Tcs (s) (8) uniform multipath intensity proﬁle, that is, bit energy is distributed in each path identically; (9) binary phase shift keying (BPSK) modulation; 10−1 (10) perfect power control and DOA/TOA estimation. The above assumptions are particularly suitable for vehicle safety applications. Based on the assumed setup, transponder signal TOA is uniformly distributed in [Td Tmax ] at the DBS receiver. Assuming that Td Tmax , 10−2 0 10 20 30 40 50 approximately TOA of transponder signal is uniformly Number of TRXs distributed in [0 Tmax ], and the required bandwidth of a DS-CDMA transponder transmitter is 320 MHz. Using these = 2 m/s = 10 m/s mv mv parameters, IRTmin = 12 μs, then the duty cycle for DBS = 2 m/s (simul.) = 10 m/s (simul.) mv mv = 5 m/s = 30 m/s receivers would correspond to dc,DBS 0.1, which leads to a mv mv = 5 m/s (simul.) = 30 m/s (simul.) mv mv high probability of overlapping (see Figure 2). Assuming that the vehicle speed is 30 m/s, the cyclosta- tionarity coherence time (based on an average distance of Figure 10: Cyclostationarity coherence time for multiple transpon- ders. 500 m) would be 47.1 milliseconds, as shown in Figure 9. As we mentioned in Section 2, usually IRT is selected much larger than IRTmin in order to reduce interference power at transponder receiver. Here, we select IRT = 1.2 milliseconds. the antenna array decreases (e.g., using more elements in the Using (28), Tcy ∼ Tcy,c / 5 = 9.42 milliseconds, and using = array). The doted curves in Figure 10 represents the simula- (20), ﬁnally Ω ∼ 8. In other words, within 8 IRT frames, the = tion results generated using similar assumptions. The theo- conditions for cyclostationarity would well exist. It should be retical results have a good match with numerical results. mentioned that the conditions simulated in this paper lead The standard sample covariance estimator does not per- to a conservative selection of Ω, and in many applications, form for nonstationary signals. Hence, its MSE does not al- higher value than Ω = 8 is expected. ter with the number of temporal samples. In contrast, the The simulation results are shown in Figure 11. The mea- proposed cyclostationary-based covariance matrix estimator surement of ID detection performance is probability of miss improves the MSE as the number of samples increases. The detection (Pmd ), that is, the probability that the ID of the number of samples increases as the cyclostationary duration desired transponder is not detected correctly. Here, Pmd = increases. In Section 4, we substantially discussed that the cy- 1 − (1 − Pd )N , where N is the number of bits per ID and clostationarity duration is suﬃciently long in practical situ- Pd denotes the probability that one bit of the ID is detected ations. Hence, the MSE for the proposed estimator is small correctly. As discussed in Section 3.3, due to nonstationarity enough for most practical applications. Numerical results in nature of the WLPS, traditional sample covariance matrix es- Section 5 verify this claim. timator computation leads to a high probability of miss de- tection. It can also be seen that the performance of LCMV BF with the covariance matrix estimator via cyclostationary 5. NUMERICAL RESULTS property leads to a signiﬁcantly improved performance com- pared to the standard covariance matrix estimator. It is ob- In this section, we use MonteCarlo simulations to evaluate served that the proposed technique doubles the capacity of the ID detection performance of the WLPS system imple- this system at the Pmd = 10−3 (i.e., from 25 to 50). mented via LCMV beamforming with and without the newly The result not only beneﬁts from solving nonstationarity proposed covariance matrix estimator via the cyclostationar- problem, but also the time diversity attained over the 8 IRT ity property. Here, we consider a multitransponder, multi- periods, since the fading is assumed to be independent over path environment. For simulation purposes, we assume the chips in diﬀerent frames (IRT). This diversity improves the following: performance in conjunction with cyclostationarity. In order (1) the ID code has 6 bits (N = 6); to demonstrate the diﬀerent eﬀects of time diversity combin- (2) the DS-CDMA code has 64 chips (G = 64); ing and optimum beamforming, we also perform the opti- (3) channel delay spread for a typical street area is 27 mum beamforming without using time diversity combining. nanoseconds [22]; It can be seen that both of the two techniques contributes to
Hui Tong et al. 11 10−2 [2] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 5, pp. 720–741, 1989. 10−3 [3] M. Hellebrandt, R. Mathar, and M. Scheibenbogen, “Esti- mating position and velocity of mobiles in a cellular radio network,” IEEE Transactions on Vehicular Technology, vol. 46, 10−4 no. 1, pp. 65–71, 1997. Pmd [4] A. Juels, “RFID security and privacy: a research survey,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 2, pp. 10−5 381–394, 2006. [5] J. Pourrostam, S. A. Zekavat, and H. Tong, “Novel direction- of-arrival estimation techniques for periodic-sense local posi- 10−6 tioning systems,” in Proceedings of the IEEE Radar Conference (RADAR ’07), Waltham, Mass, USA, April 2007. [6] Z. Wang and S. A. Zekavat, “Manet localization via multi-node 10−7 TOA-DOA optimal fusion,” in Proceedings of the Military Com- 10 20 30 40 50 60 munications Conference (MILCOM ’06), pp. 1–7, Washington, Number of TRX DC, USA, October 2006. [7] H. Tong and S. A. Zekavat, “A novel wireless local posi- Conventional Fourier BF Standard LCMV BF tioning system via a merger of DS-CDMA and beamform- LCMV BF via cyclostationarity w/o time diversity ing: probability-of-detection performance analysis under ar- LCMV BF via cyclostationarity with time diversity ray perturbations,” IEEE Transactions on Vehicular Technology, vol. 56, no. 3, pp. 1307–1320, 2007. [8] O. L. Frost, “An algorithm for linearly constrained adaptive Figure 11: LCMV BF result with using cyclostationarity. array processing,” Proceedings of the IEEE, vol. 60, no. 8, pp. 926–935, 1972. [9] B. D. Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays,” IEEE Transactions on DBS receiver performance. It is observed that as the number Aerospace and Electronic Systems, vol. 24, no. 4, pp. 397–401, of transponders increases, the time diversity has a dominant 1988. impact on the performance improvement. [10] J. Capon, “High resolution frequency-wavenumber spectrum analysis,” Proceedings of the IEEE, vol. 57, no. 8, pp. 1408–1418, 6. CONCLUSION 1969. [11] W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationar- ity: half a century of research,” Signal Processing, vol. 86, no. 4, This paper proposes a novel covariance matrix estimator, pp. 639–697, 2006. which is the critical step for optimal beamforming imple- [12] Q. Wu and K. M. Wong, “Blind adaptive beamforming for cy- mentation, in a wireless local positioning system with a pe- clostationary signals,” IEEE Transactions on Signal Processing, riodic signaling structure. Diﬀerent from standard wireless vol. 44, no. 11, pp. 2757–2767, 1996. systems, the standard sample covariance matrix estimator is [13] J.-H. Lee and Y.-T. Lee, “Robust adaptive array beamforming not consistent in the WLPS system due to its nonstationar- for cyclostationary signals under cycle frequency error,” IEEE ity nature. A new consistent estimator, which exploits the Transactions on Antennas and Propagation, vol. 47, no. 2, pp. cyclostationarity property of the WLPS, is proposed. It is 233–241, 1999. demonstrated that in most applications, the cyclostationar- [14] L. C. Godara, “Application of antenna arrays to mobile ity duration is suﬃciently large for covariance matrix esti- communications—part II: beam-forming and direction-of- arrival considerations,” Proceedings of the IEEE, vol. 85, no. 8, mation. Numerical simulations verify that the new estimator pp. 1195–1245, 1997. improves system performance signiﬁcantly. [15] A. Hyvarinen, J. Karhunen, and E. Oja, Independent Compo- nent Analysis, John Wiley & Sons, New York, NY, USA, 2001. ACKNOWLEDGMENTS [16] P. Stoica and R. L. Moses, Introduction to Spectral Analysis, Prentice-Hall, Upper Saddle River, NJ, USA, 1997. [17] P. Stoica, Z. Wang, and J. Li, “Robust Capon beamforming,” This work was partially reported in [1]. This work is sup- IEEE Signal Processing Letters, vol. 10, no. 6, pp. 172–175, 2003. ported by the US NSF Grant ECS-0427430. WLPS US Patent [18] P. Xia and G. B. Giannakis, “Design and analysis of transmit- is Pending at Michigan Tech. University. beamforming based on limited-rate feedback,” IEEE Transac- tions on Signal Processing, vol. 54, no. 5, pp. 1853–1863, 2006. REFERENCES [19] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,” IEEE Transactions on Signal Processing, vol. 53, no. 5, pp. 1684–1696, 2005. [1] H. Tong and S. A. Zekavat, “LCMV beamforming for a novel [20] S. A. Vorobyov, A. B. Gershman, Z.-Q. Luo, and N. Ma, “Adap- wireless local positioning system: a stationarity analysis,” in tive beamforming with joint robustness against mismatched Sensors, and Command, Control, Communications, and Intelli- signal steering vector and interference nonstationarity,” IEEE gence (C3I) Technologies for Homeland Security and Homeland Signal Processing Letters, vol. 11, no. 2, part 1, pp. 108–111, Defense IV, vol. 5778 of Proceedings of SPIE, pp. 851–862, Or- 2004. lando, Fla, USA, March-April 2005.
12 EURASIP Journal on Advances in Signal Processing [21] T. S. Rappaport, Wireless Communications: Principles and Prac- tice, Prentice-Hall, Upper Saddle River, NJ, USA, 2nd edition, 2002. [22] A. A. Arowojolu, A. M. D. Turkmani, and J. D. Parsons, “Time dispersion measurements in urban microcellular envi- ronments,” in Proceedigs of the 44th IEEE Vehicular Technology Conference (VTC ’94), vol. 1, pp. 150–154, Stockholm, Sweden, June 1994. Hui Tong is currently pursuing his Ph.D. degree at Michigan Technological Univer- sity. His research interests span over the ar- eas of signal processing, information the- ory, and wireless communications. He has authored more than 15 papers on refer- eed international journals and conference proceedings. Recently, he focuses on multi- antenna channel modeling, channel capac- ity analysis, and signal processing. Jafar Pourrostam received the B.S. degree from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2000, and the M.S. degree from University of Tehran, Tehran, Iran, in 2003. He is cur- rently working toward the Ph.D. degree at the Department of Electrical and Computer Engineering, Michigan Technological Uni- versity, Houghton, MI, USA. His research interests are in digital signal processing, wireless communication systems, and array signal processing. Seyed A. Zekavat received his Ph.D. from Colorado State University, Fort Collins, Colorado in 2002 in telecommunications. He has published more than 50 journal and conference papers, and has coauthored the book Multi-Carrier Technologies for Wire- less Communications, published by Kluwer, an invited chapter in the book Adaptive An- tenna Arrays, published by Springer, and an invited paper in Journal of Communications, published by Academy Publisher. His research interests are in wire- less communications at the physical layer, dynamic spectrum allo- cation methods, radar theory, blind signal separation and MIMO and beamforming techniques, feature extraction, and neural net- working. He is the inventor of wireless local positioning systems (WLPS) with variety of military and civilian applications. He has been awarded by the US NSF Information Technology Research for National Priorities program to study and develop prototypes of WLPS. He is also an active technical program committee mem- ber for several IEEE international conferences. At Michigan Tech, he has founded two research laboratories on wireless systems, and is currently principal advisor for several Ph.D. students.