Time Delay Estimation in Cognitive Radio Systems

Transcription

1 Tme Delay Estmaton n Cogntve Rado Systems Invted Paper Fath Kocak, Hasar Celeb, Snan Gezc, Khald A. Qaraqe, Huseyn Arslan, and H. Vncent Poor Department of Electrcal and Electroncs Engneerng, Blkent Unversty, Blkent, Ankara 68, Turkey Department of Electrcal and Computer Engneerng, Texas A&M Unversty at Qatar, Doha, Qatar Department of Electrcal Engneerng, Unversty of South Florda, Tampa, FL, 3362, USA Department of Electrcal Engneerng, Prnceton Unversty, Prnceton, NJ 8544, USA arxv: v1 [cs.it] 24 Oct 29 Abstract In cogntve rado systems, secondary users can utlze multple dspersed bands that are not used by prmary users. In ths paper, tme delay estmaton of sgnals that occupy multple dspersed bands s studed. Frst, theoretcal lmts on tme delay estmaton are revewed. Then, two-step tme delay estmators that provde trade-offs between computatonal complexty and performance are nvestgated. In addton, asymptotc optmalty propertes of the two-step tme delay estmators are dscussed. Fnally, smulaton results are presented to explan the theoretcal results. Index Terms Cogntve rado, tme delay estmaton, maxmum-lkelhood, dversty, Cramer-Rao lower bound. I. INTRODUCTION Cogntve rado presents a promsng approach to mplement ntellgent wreless communcatons systems [1]-[4]. Cogntve rados can be regarded as more capable versons of software defned rados n the sense that they have sensng, awareness, learnng, adaptaton, goal drven autonomous operaton and reconfgurablty features [5], [6], whch facltate effcent use of rado resources such as power and bandwdth [1]. As the electromagnetc spectrum s a precous resource, t s mportant not to waste t. The recent spectrum measurement campagns n the Unted States [7] and Europe [8] ndcate that the spectrum s under-utlzed; hence, opportunstc use of unoccuped frequency bands s hghly desrable. Cogntve rado presents a soluton to neffcent spectrum utlzaton by opportunstcally usng the avalable spectrum of a legacy system wthout nterferng wth the prmary users of that spectrum [2], [3]. In order to facltate such opportunstc spectrum utlzaton, cogntve rado devces should be aware of ther locatons, and montor the envronment contnuously. Locaton awareness requres a cogntve rado devce to perform accurate estmaton of ts poston. Cogntve rado devces can obtan poston nformaton based on the estmaton of poston related parameters of sgnals travelng between them [9], [1]. Among varous poston related parameters, the tme delay parameter provdes accurate poston nformaton wth reasonable complexty [1]. The man focus of ths paper s tme delay estmaton n cogntve rado systems. The man dfference between tme delay estmaton n cogntve rado systems and that n conventonal systems s that a cogntve rado system can transmt and receve over multple dspersed bands. In other words, as a cogntve rado devce can use the spectral holes n a legacy system, t can have a spectrum that conssts of multple bands that are dspersed over a wde range of frequences (cf. Fg. 1). In [11], the Cramer- Rao lower bounds (CRLBs) for tme delay estmaton are obtaned for dspersed spectrum cogntve rado systems, and the effects of carrer frequency offset (CFO) and modulaton schemes of tranng sgnals on the accuracy of tme delay estmaton are quantfed. The CRLB expressons mply that Emal address for correspondence: gezc@ee.blkent.edu.tr frequency dversty can be utlzed n tme delay estmaton. Smlarly, the effects of spatal dversty on tme delay estmaton are nvestgated n [12] for sngle-nput multple-output systems. Also, the effects of multple antennas on tme delay estmaton and synchronzaton problems are studed n [13]. Ths paper studes tme delay estmaton for dspersed spectrum cogntve rado systems. Frst, the theoretcal lmts on tme delay estmaton are revewed, and the concept of frequency dversty for tme delay estmaton s dscussed. Then, optmal and suboptmal tme delay estmaton technques are studed. Snce optmal maxmum lkelhood (ML) tme delay estmaton can have very hgh computatonal complexty for sgnals wth multple dspersed bands, two-step tme delay estmaton technques are nvestgated. The proposed two-step tme delay estmators frst extract unknown parameters related to sgnals n dfferent frequency bands, and then obtan the fnal tme delay estmate n the second step. In other words, multple observatons (sgnals at dfferent frequency bands) are processed effcently to provde a trade-off between computatonal complexty and estmaton performance. In addton, the optmalty propertes of the two-step estmators are nvestgated for hgh sgnal-to-nose ratos (SNRs), and smulaton results are presented to verfy the theoretcal analyss. II. SIGNAL MODEL Consder a scenaro n whch K dspersed frequency bands are avalable to the cogntve rado system, as shown n Fg. 1. The transmtter generates a sgnal that occupes all the K bands smultaneously, and sends t to the recever. Then, the recever s to estmate the tme delay of the ncomng sgnal. Snce the avalable bands can be qute dspersed, the use of orthogonal frequency dvson multplexng (OFDM) approach [14] can requre processng of very large bandwdths. Therefore, processng of the receved sgnal n multple branches s consdered n ths study, as n Fg. 2 [11]. For the recever n Fg. 2, the baseband representaton of the receved sgnal n the th branch s gven by r (t) = α e jωt s (t τ) + n (t), (1) for = 1,...,K, where τ s the tme delay of the sgnal, α = a e jφ and ω represent, respectvely, the channel coeffcent and the CFO for the sgnal n the th branch, s (t) s the baseband representaton of the transmtted sgnal n the th band, and n (t) s complex whte Gaussan nose wth ndependent components, each havng spectral densty σ 2. It s assumed that the sgnal n each branch can be modeled as a narrowband sgnal; hence, a sngle complex channel coeffcent s used to represent the fadng of each sgnal. Also, t should be noted that the effects of CFO are consdered n the sgnal model n (1) snce multple down-converson unts are employed n the recever, as shown n Fg. 2.

2 Fg. 1 ILLUSTRATION OF DISPERSED SPECTRUM UTILIZATION IN THE COGNITIVE RADIO SYSTEM, WHERE THE WHITE SPACES REPRESENT THE AVAILABLE BANDS. Fg. 3 THE BLOCK DIAGRAM OF THE TWO-STEP TIME DELAY ESTIMATION APPROACH. THE SIGNALS r 1 (t),..., r K (t) ARE AS SHOWN IN FIG. 2. Fg. 2 BLOCK DIAGRAM OF THE FRONT-END OF THE COGNITIVE RADIO RECEIVER. III. THEORETICAL LIMITS ON TIME DELAY ESTIMATION Estmaton of the tme delay parameter τ based on K receved sgnals n (1) nvolves (3K + 1) unknown parameters snce the channel coeffcents and CFOs are also unknown. In other words, the vector of unknown parameters, θ, can be expressed as θ = [τ a 1 a K φ 1 φ K ω 1 ω K ]. For an observaton nterval of [, T], the log-lkelhood functon for θ s expressed as 1 [15] Λ(θ) = c K 1 2σ 2 r (t) α e jωt s (t τ) 2 dt, (2) where c s a constant that s ndependent of θ. Then, the Fsher nformaton matrx (FIM) [15] can be obtaned from (2) as n [11], and the nverse of the FIM can be used to obtan the CRLB on mean-squared errors (MSEs) of unbased tme delay estmators. In ts most generc form, the CRLB can be expressed as [11] E{(ˆτ τ) 2 a 2 σ 2 (Ẽ (ÊR )2 /E ) ξ) 1, (3) where E = s (t τ) 2 dt s the sgnal energy, Ẽ = s (t τ) 2 dt, wth s (t) representng the frst dervatve of s(t), and ÊR = R{s (t τ )s (t τ )dt, wth R denotng the operator that selects the real-part of ts argument. In addton, ξ represents a term that depends on the spectral propertes of sgnals s (t) for = 1,...,K [11]. The CRLB expresson n (3) reveals that the accuracy of tme delay estmaton depends on the SNR at each branch (va the a 2 /σ2 terms), as well as on the propertes of sgnals s (t) n (1). In addton, the summaton term n (3) ndcates that the accuracy can be mproved as more bands are employed, whch mples that frequency dversty can be utlzed n tme delay estmaton. For nstance, when one of the bands s n a deep fade (that s, small a 2 ), some other bands can stll be n 1 The unknown parameters are assumed to be constant for t [, T]. good condton to facltate accurate tme delay estmaton. In order to nvestgate the effects of sgnal desgn on the tme delay estmaton accuracy, suppose that the baseband representaton of the sgnals n dfferent branches are of the form s (t) = l d,lp (t lt ), where d,l denotes the complex tranng data and p (t) s a pulse wth duraton T. Then, the ξ term n (3) becomes zero, whch results n a CRLB expresson that would be obtaned n the absence of CFOs [11]. In other words, the effects of CFOs can be mtgated va approprate sgnal desgn. In the specal case of d,l = d l and p (t) satsfyng p () = p (T ) for = 1,...,K, (3) reduces to [11] ) 1 E{(ˆτ τ) 2 Ẽ a 2 σ 2. (4) Hence, for lnearly modulated sgnals wth constant envelopes, mproved tme delay estmaton accuracy can be acheved. IV. TWO-STEP TIME DELAY ESTIMATION The ML estmate of θ can be obtaned from (2) as KX Z 1 T n o ˆθ ML = arg max R α θ σ 2 e jωt r (t)s (t τ) dt Ea2 2σ 2 (5) whch requres an optmzaton over a (3K + 1)-dmensonal space, hence s qute mpractcal n general. Therefore, a twostep tme delay estmaton approach s consdered n ths study, as shown n Fg. 3. In the frst step, each branch of the recever performs estmaton of the tme delay, the channel coeffcent and the CFO related to the sgnal n that branch. Then, n the second step, the estmates from all the branches are used to obtan the fnal tme delay estmate. A. Frst Step: Parameter Estmaton at Dfferent Branches In the frst step, the unknown parameters of each receved sgnal are estmated at the correspondng recever branch accordng to the ML crteron (cf. Fg. 3). Based on the sgnal model n (1), the log-lkelhood functon at branch becomes Λ (θ ) = c 1 2σ 2 r (t) α e jωt s (t τ) 2 dt, (6) for = 1,...,K, where θ = [τ a φ ω ] denotes the vector of unknown parameters related to the sgnal at the th branch, r (t), and c s a constant that s ndependent of θ. From (6), the ML estmator at branch s expressed as ˆθ = argmn θ r (t) α e jωt s (t τ) 2 dt, (7)

3 where ˆθ = [ˆτ â ˆφ ˆω ] s the vector of estmates at the th branch. After some manpulaton, (7) yelds ] { [ˆτ ˆφ ˆω = arg max R r (t)e j(ωt+φ) s (t τ ) dt φ,ω,τ (8) â = 1 { R r (t)e j(ˆωt+ˆφ ) s E (t ˆτ ) dt. (9) In other words, at each branch, optmzaton over a threedmensonal space s performed to obtan the unknown parameters, whch s sgnfcantly less complex than the ML estmaton n (5) that requres optmzaton over (3K + 1) varables. B. Second Step: Combnng Estmates from Dfferent Branches After obtanng tme delay estmates ˆτ 1,..., ˆτ K n (8), the second step combnes those estmates accordng to one of the crtera below and makes the fnal tme delay estmate. 1) Optmal Combnng: Accordng to the optmal combnng 2 crteron, the tme delay estmate s obtaned as ˆτ = κ ˆτ κ, (1) where ˆτ s the tme delay estmate of the th branch, whch s obtaned from (8), and κ = â 2 Ẽ/σ 2. In other words, the optmal combnng approach estmates the tme delay as a weghted average of the tme delays at dfferent branches, where the weghts are chosen as proportonal to the multplcaton of the SNR estmate, E â 2 /σ2, and Ẽ/E. As Ẽ s defned as the energy of the frst dervatve of s (t), Ẽ /E can be expressed, usng Parseval s relaton, as Ẽ/E = 4π 2 β 2, where β s the effectve bandwdth of s (t), whch s defned as β 2 = 1 E f2 S (f) 2 df wth S (f) denotng the Fourer transform of s (t) [15]. Therefore, t s observed that the optmal combnng technque assgns a weght to the tme delay estmate of a gven branch n proporton to the product of the SNR estmate and the effectve bandwdth related to that branch. The ntuton behnd ths combnng approach s that sgnals wth larger effectve bandwdths and/or larger SNRs facltate more accurate tme delay estmaton [15]; hence, ther weghts are larger n the combnng process. Ths ntuton wll be verfed n Secton IV-C theoretcally. 2) Selecton Combnng (SC): Another approach to obtan the fnal tme delay estmate s to determne the best branch and to use ts estmate as the fnal tme delay estmate. Accordng to SC, the best branch s defned as the one wth the maxmum value of κ = â 2 Ẽ/σ 2 for = 1,...,K. That s, the branch wth the maxmum multplcaton of the SNR estmate and the effectve bandwdth s selected as the best branch and ts estmate s used as the fnal one. In other words, { ˆτ = ˆτ m, m = arg max â 2 Ẽ/σ 2, (11) {1,...,K where ˆτ m represents the tme delay estmate at the mth branch. 3) Equal Combnng: The equal combnng approach assgns equal weghts to the estmates from dfferent branches and obtans the tme delay estmate as ˆτ = 1 K ˆτ. Consderng the combnng technques above, t s observed that they are counterparts of the dversty combnng technques employed n communcatons systems [16]. However, the man dstncton s that the am s to maxmze the SNR or to reduce the probablty of symbol error n communcatons systems [16], whereas, n the current problem, t s to reduce the MSE of the tme delay estmaton. In other words, ths study focuses on dversty combnng for tme delay estmaton, where the dversty s due to the dspersed spectrum utlzaton of the cogntve rado system. C. Optmalty Propertes of Two-Step Tme Delay Estmaton In ths secton, the asymptotc optmalty propertes of the two-step tme delay estmators are nvestgated n the absence of CFO. In order to analyze the performance of the estmators at hgh SNRs, the result n [12] for tme-delay estmaton at multple receve antennas s consdered frst. Lemma 1 [12]: Assume that s (t τ)s (t τ)dt = for = 1,...,K. Then, for the sgnal model n (1), the delay estmate n (8) and the channel ampltude estmate n (9) can be modeled, at hgh SNR, as ˆτ = τ + ν and â = a + η, (12) for = 1,..., K, where ν and η are ndependent zero mean Gaussan random varables wth varances σ 2/(Ẽ a 2 ) and σ 2/E, respectvely. In addton, ν and ν j (η and η j ) are ndependent for j. From Lemma 1, t s obtaned that E{ˆτ = τ for = 1,..., K. In other words, the tme delay estmates of all branches are asymptotcally unbased. Snce the combnng technques n the prevous secton consders only one, or a lnear combnatons of the tme delay estmates at dfferent branches, the two-step tme delay estmaton technques have an asymptotc unbasedness property. Regardng the varance of the estmators, t s frst shown that the optmal combnng technque has a varance that s approxmately equal to the CRLB at hgh SNRs. 3 To that am, the condtonal varance of ˆτ n (1) gven â 1,...,â K s expressed as follows: Var{ˆτ â 1,..., â K = κ2 Var{ˆτ â 1,...,â K ) 2, (13) κ where the ndependence of the tme delay estmates s used to obtan the result (cf. Lemma 1). Snce Var{ˆτ â 1,...,â K = Var{ˆτ â = σ 2/(Ẽ a 2 ) from Lemma 1 and κ = â 2 Ẽ/σ 2, (13) can be manpulated to obtan Var{ˆτ â 1,...,â K = K â 4 Ẽ a 2 σ2 â 2 Ẽ σ 2 ) 2. (14) Lemma 1 states that a s dstrbuted as a Gaussan random varable wth mean a and varance σ 2/E at hgh SNRs. Hence, for suffcently large values of E,..., EK, (14) can σ 2 σk ( 2 K ) Ẽ be approxmated by a 1, 2 σ whch s equal to the 2 CRLB expresson n (4). Therefore, the optmal combnng technque n (1) yelds an approxmately optmal estmator at hgh SNRs. 2 The optmalty property s nvestgated n Secton IV-C. 3 Ths s the man reason why ths combnng technque s called optmal.

4 RMSE (sec.) Optmal Combnng Equal Combnng Selecton Combnng Theoretcal Lmt SNR (db) Fg. 4 RMSE VS. SNR FOR THE TWO-STEP ALGORITHMS, AND THE THEORETICAL LIMIT (CRLB). THE SIGNAL OCCUPIES THREE DISPERSED BANDS WITH B 1 = 2 KHZ, B 2 = 1 KHZ AND B 3 = 4 KHZ. RMSE (sec.) Optmal combnng Equal combnng Selecton combnng Theoretcal lmt Number of Bands Fg. 5 RMSE VS. THE NUMBER OF BANDS FOR THE TWO-STEP ALGORITHMS, AND THE THEORETICAL LIMIT (CRLB). EACH BAND IS 1 KHZ WIDE, AND σ 2 =.1. Regardng the selecton combnng approach n (11), the condtonal varance can be approxmated at hgh SNR as { Var{ˆτ â 1,..., â K mn σ 2 1 σ 2 K,..., Ẽ 1 a 2 1 Ẽ K a 2 K. (15) In general, the SC approach performs worse than the optmal combnng technque. However, when the estmate of a branch s sgnfcantly more accurate than the others, ts performance can get very close to that of the optmal combnng technque. Fnally, for the equal combnng technque n Secton IV-B.3, the varance can be calculated as Var{ˆτ = 1 K 2 σ 2. The equal combnng approach s expected Ẽ a 2 to have the worst performance snce t does not make use of any nformaton about the SNR or the sgnal bandwdths n the estmaton of the tme delay, as nvestgated next. V. SIMULATION RESULTS AND CONCLUSIONS In ths secton, smulatons are performed to evaluate the CRLBs and the performance of the tme delay estmators. Sgnal s (t) n (1) at branch s modeled by a unt-energy Gaussan doublet as n [11] wth bandwdth B. In all the smulatons, the spectral denstes of the nose at dfferent branches are assumed to be equal; that s, σ = σ for = 1,...,K. Also, the SNR of the system s defned wth respect to the total energy ( Pof the sgnals at dfferent branches,.e., K SNR = 1 log E 1 2 σ ). 2 In assessng the root-mean-squared errors (RMSEs) of the dfferent estmators, a Raylegh fadng channel s consdered. Namely, the channel coeffcent α = a e jφ n (1) s modeled as a beng a Raylegh dstrbuted random varable and φ beng unformly dstrbuted over [, 2π). In addton, the same average power s assumed for all the bands; that s, E{ α 2 = 1 s used. The tme delay, τ, n (1) s unformly dstrbuted over the observaton nterval, and t s assumed that there s no CFO n the system. Frst, the performance of the two-step estmators s evaluated wth respect to the SNR for a system wth K = 3, B 1 = 2 khz, B 2 = 1 khz and B 3 = 4 khz. The results n Fg. 4 ndcate that the optmal combnng technque has the best performance as expected from the theoretcal analyss, and SC, whch estmates the delay accordng to (11), has performance close to that of the optmal combnng technque. On the other hand, the equal combnng technque has sgnfcantly worse performance than the others, as t combnes all the delay estmates equally. Snce the delay estmates of some branches can have very large errors due to fadng, the RMSEs of equal combnng become qute sgnfcant. Fnally, t s observed that the performance of the optmal combnng technque gets qute close to the CRLB at hgh SNRs, n agreement wth the asymptotc arguments n Secton IV-C. Next, the RMSEs of the two-step estmators are plotted aganst the number of bands n Fg. 5, where each band s assumed to have 1 khz bandwdth. In addton, the spectral denstes are set to σ 2 = σ 2 =.1. From Fg. 5, t s observed that the optmal combnng has better performance than the selecton combnng and equal combnng technques. Also, as the number of bands ncreases, the amount of reducton n the RMSE per addtonal band decreases (.e., dmnshng return). In fact, the selecton combnng technque seems to converge to a constant value for large numbers of bands. Ths s ntutve as the selecton combnng technque always uses the estmate from one of the branches; hence, n the presence of a suffcently large number of bands, addtonal bands do not result n a sgnfcant ncrease n the dversty. On the other hand, the optmal combnng technque has a slope that s qute smlar to that of the CRLB; that s, t makes use of the frequency dversty effcently. REFERENCES [1] J. Mtola and G. Q. Magure, Cogntve rado: Makng software rados more personal, IEEE Personal Commun. Mag., vol. 6, no. 4, pp , Aug [2] S. Haykn, Cogntve rado: Bran-empowered wreless communcatons, IEEE J. Select Areas Commun., vol. 23, no. 2, pp , Feb. 25. [3] Z. Quan, S. Cu, H. V. Poor, and A. H. Sayed, Collaboratve wdeband sensng for cogntve rados, IEEE Sgnal Process. Mag., vol. 25, no. 6, pp. 6 73, Nov. 28. [4] Q. Zhao and B. Sadler, A survey of dynamc spectrum access, IEEE Sgnal Process. Mag., vol. 24, no. 3, pp , May 27. [5] J. O. Neel, Analyss and desgn of cogntve rado networks and dstrbuted rado resource management algorthms, Ph.D. dssertaton, Vrgna Polytechnc Inst. and State Unv., Blacksburg, VA, Sep. 26. [6] H. Celeb and H. Arslan, Enablng locaton and envronment awareness n cogntve rados, Elsever Computer Communcatons, vol. 31, no. 6, pp , Aprl 28. [7] Federal Communcatons Commsson (FCC), Facltatng opportuntes for flexble, effcent, and relable spectrum use employng cogntve rado technologes, ET Docket No. 3-18, Mar. 25.

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