Spectral Density of Random Time -Hopping Spread-Spectrum UWB Signals with Uniform Timing Jitter

Transcription

1 Spectra Density o Rom ime -Hopping Spread-Spectrum UWB Signas with Uniorm iming Jitter Moe Z. Win Wireess Systems Research Department, A& Labs - Research 100 Schuz Drive, Red Bank, J USA e: Fax: e-mai: win@research.att.com Abstract We derive the power spectra density o timehopping H spread-spectrum signas in the presence o rom timing jitter, which is characterized typicay by a discrete-time stationary process independent o the H sequences. Detaied descriptions o dierent H schemes, empoying a rom H sequence, is given spectra anaysis o such H signas in the presence o uniorm timing jitter is carried out using a systematic tractabe technique. I. Introduction RECELY, there has been an interest in utra - wide bwidth UWB time-hopping H spreadspectrum SS mutipe access techniques or both commercia miitary appications 1], ], 3], 4], 5]. he UWB H-SS radio communicates with baseb puses o short duration, thereby spreading the energy o the radio signa very thiny rom near d.c. to severa GHz. he techniques or generating UWB signas have been around or more than three decades 6]. A comprehensive reerence a tota o 8 reerences o eary work in this area can be ound in a tutoria survey paper 7]. he key motivations or using UWB H-SS radio are the abiity to highy resove mutipath, as we as the avaiabiity o the technoogy to impement generate UWB signas with reativey ow compexity. he ine time resoution properties make UWB H-SS radio a viabe cidate or communications in dense mutipath environments such as short-range or indoor wireess communications 8]. UWB H-SS radios, operating with ow transmission power in an extremey arge transmission bwidth, have aso been under consideration or uture miitary networks because they inherenty provide a covertness he research described in this paper was supported in part by the Joint Services Eectronics Program under contract F , in part by the Integrated Media Systems Center, a ationa Science Foundation Engineering Research Center with additiona support rom the Annenberg Center or Communication at the University o Southern Caiornia the Caiornia rade Commerce Agency. Moe Win was with the Communication Sciences Institute, University o Southern Caiornia, Los Angees, CA. He is now with the Wireess Systems Research Department, ewman Springs Laboratory, A& Labs - Research, 100 Schuz Drive, Red Bank, J USA. property with ow probabiity o detection interception LPD/LPI capabiities 4], 5]. Since gigahertz bwidth aocations are not avaiabe at the proposed requencies or commercia appications, these radios must operate under Part 15 FCC reguations are treated as spurious intererence to a other systems. In addition, UWB H-SS radios operating over the highy popuated requency range beow a ew gigahertz must contend with a variety o interering signas must aso insure that they do not interere with narrowb radio systems operating in these dedicated bs. For miitary UWB H-SS radios, the major objective is to minimize the probabiity o detection interception by nonintended users. his requires transmitting H-SS signas with extremey ow spectra content. On the other h, they must maintain the average power eve required or reiabe communications. A o this impies that the spectra occupancy composition o a chosen H-SS signaing technique pays an important considerations in the design o a UWB H-SS mutipe access communication systems. As an exampe, the abiity o the receiver to reject narrowb intererence, the abiity o the transmitter to avoid interering with other radio systems, the probabiity o detection/interception by a non-intended receiver can be estimated rom the power spectra density PSD o the time-hopped puse trains. Evauation o the PSD or idea synchronous data puse streams 1 based upon stochastic theory is we documented in the iterature 9], 10], 11], 1]. More recent work on the time-series approach to spectra anaysis, using nonstochastic theory based upon time averages, provides an aternative technique to anayzing the spectra properties o digita signas 13]. 1 Idea synchronous data puse streams reers to a baseb data waveorm with a ixed transmission interva ixed transmission epochs where the underying data sequence has known statistica properties the transmitted waveorm is a singe known puse shape. he duaity between the two compementary theories, stochastic theory using ensembe averages nonstochastic theory using time averages, can be ound in 14], where the raction-o-time distribution concept is introduced as a means o obtaining probabiistic interpretation o time averages as expected vaues or ensembe av /99/$10.00 c 1999 IEEE

2 Evauation o the PSD or nonidea data puse streams, due or instance to impementation imperections, is aso o interest. One such exampe is data asymmetry where the rising aing transition time instants are oset by ixed amounts reative to the nomina ones 15], 16]. Another source o imperection is rom timing jitter typicay characterized by a discrete-time stationary rom process with known statistica properties. Genera expressions or the PSD o M-ary digita puse streams in the presence o timing jitter were derived using stochastic theory in 17]. he empirica spectra anaysis o a time-jittered puse-ampitude moduated signa was carried out using the nonstochastic time-series approach in 13]. Previous work in the iterature or evauation o the PSD o SS signas have been ocused mainy on idea directsequence SS signas 18], 19], 0], 1], ]. he PSD o idea requency-hopped SS signas was derived in 19]. However, the eect o timing jitter was not incuded in these anayses. he purpose o this paper is to characterize the PSD o H-SS signaing schemes in the presence o rom timing jitter, using a stochastic approach simiar to 17], 9], 10], 11]. Detaied descriptions o dierent H schemes, empoying a rom H sequence stochastic mode, is given spectra anaysis o such H signas in the presence o uniorm timing jitter is carried out using a systematic tractabe technique. he stochastic mode is o particuar interest in appications such as rom signa detection /or interception. Furthermore, the eect o changes in various H parameters on PSD o transmitted signa can be estimated, without requiring the detaied H sequence design procedure. II. Signa Mode Preiminary Deinitions A rom H-SS signa in the presence o timing jitter can be modeed as st = a n wt n b n ɛ n 1 where a n } b n } are arbitrary stochastic romy chosen sequences, ɛ n } represents the rom timing jitter. It is assumed that a n }, b n }, ɛ n } are stationary mutuay independent o each other. he covariance unction o st is derived in 3] as K s t; τ =E st st] st + τ st + τ] } = K β n; m n, y, z m= y z W yw ze jπyn1 e +jπzm1 e +jπy zt e jπzτ dydz, where K β n;,, can be interpreted as the covariance unction o the equivaent stationary sequence β n deined by β n y a n e +jπybn e +jπyɛn. 3 erages. he spectra density obtained by using the nonstochastic time-series approach is reerred to as empirica spectra density. It can be shown that ] ] } K β n;,, =E β n β n β n+ β n+ = R an; Φ bn,bn+, Φ ɛn,ɛn+, 4 ] a na n+ Φ b n Φ ɛn Φ bn+ Φ ɛn+ where Φ bn,b m y, z =E e jπybn+zbm} Φ ɛn,ɛ m y, z =E e jπyɛn+zɛm} Φ bn y =E e jπybn} Φ ɛn y =E e jπyɛn}. he unction β n y E β n y} is the mean o β n y, a n E a n } is the mean R a n; E a n a n+} is the correation unction o the sequence a n }. III. PSD o Rom H-SS Signas with Uniorm Jitter In genera, the PSD S s o the rom H-SS signa st consists o continuous as we as discrete components 9], namey Ss c Ss d, respectivey. Irrespective o the properties o the sequences a n }, b n }, ɛ n }, the rom H-SS signa st is itse wide-sense cycostationary since K s t; τ is, in addition to being a unction o τ, a periodic unction o t. hereore, the continuous spectrum o st is S c s =F τ K s t; τ t } 5 where < > t denotes time average, F τ } denotes the Fourier transorm operator with respect to variabe τ. For the stationary stochastic sequences a n }, b n }, ɛ n }, S c s is cacuated in 3] as where S c s =S w S β 6 S w = 1 W 7 is the normaized energy spectra density o the puse shape wt, S β = he discrete PSD is ound rom K β n;,, e jπ1. 8 S d s =F τ sts t + τ t }. 9 For the stationary stochastic sequences a n }, b n }, ɛ n }, S d s is cacuated in 3] as S d s = 1 1 W Gβ δd /99/$10.00 c 1999 IEEE

3 where G β =E a n } Φ bn Φ ɛn. 11 he resuts obtained above are genera appy to arbitrary stationary timing jitter statistics. In the oowing, the expressions or the PSD are evauated or the case o practica interest, namey uniorm distributed timing jitter UDJ. Speciicay, ɛ n } is modeed as a sequence o independent identicay distributed i.i.d. uniorm rom variabes with probabiity density unction 1 ɛn x = 1 <x< 1 0 otherwise, where = 1. ote that, in genera, ɛ n has non-zero mean, thus this mode incudes the asymmetry o the timing jitter. It is easy to show that Φ ɛn = sinπ π Φ ɛn,ɛ n+, = hereore, rom 4 ] e sin π π + 1 +jπ 1 =0 ] ] R an;0 a n Φ bn sinπ =0 π K β n;,, = R an; Φ bn,bn+, ] ] a na n+ Φ b n sinπ 0, π 15 rom 11 G β =a n Φ bn ] sinπ e π + 1 +jπ. 16 Finay, substituting into 8 10 respectivey, using 6 7 o Section III, the continuous discrete PSD components o a rom H-SS signa in the presence o i.i.d. UDJ asymmetric or symmetric over the interva become Ss c = 1 ] } sinπ W R an;0 a n Φ bn π + 1 W R an; Φ bn,b n+, } ] a na n+ Φ b n ]e jπ sinπ π S d s = 1 W sinπ an Φ bn π ] δ D 18 respectivey. ote that are independent o the jitter asymmetry. he imiting case o 1 or 0 impies constant cock deay with no timing jitter. hereore, the resuts o the PSD o an idea synchronous rom H-SS signa can be obtained rom by setting 1 or 0. IV. Appications A. Cocked H by a Rom Sequence Consider a digitay controed H-SS signa which produces rom transmissions at mutipes o the basic cock period c. his signaing technique, known as cocked H by a rom sequence CHRS, can be modeed as s CH t = a n wt n c ɛ n, 19 where a n } is an unbaanced binary i.i.d. rom sequence with p an =1 Pr a n } = 0 1 p a n =0. ote that s CH t can be obtained rom the rom H-SS signa st o 1 by setting = c =0 or b n =0. It can be shown that a n } has the correation unction p =0 R a = p 1 0, mean vaue a = p. For an independent stationary stochastic sequence ɛ n }, it can be shown using 4 11 that p p K β n;,, = Φ ɛn =0 0 0 G β =pφ ɛn. 3 Finay, substituting 3 into 8 10 respectivey, using 6 7 o Section III, gives the desired resuts S c CH = 1 c W p p Φ ɛn } 4 S d CH = p c W Φɛn δd. 5 he above two equations are the continuous discrete PSD components o CHRS signa in the presence o arbitrary stationary timing jitter statistics. It can be seen that /99/$10.00 c 1999 IEEE

4 CHRS removes a portion o the signa s power, namey the amount given by p W Φ ɛn d, converts it into the set o spectra ines indicated by the deta unctions which occur at mutipes o the cock requency 1 Ṫhe resuts or UDJ can now be obtained easiy by substituting 13 into 4 5 as SCH c = 1 ] } W sinπ p p 6 c π S d CH = p c W sinπ π ] δ D. 7 ote that, the UDJ mode incudes the asymmetry o the timing jitter, however PSD resuts or CHRS signa are independent o the jitter asymmetry. As pointed out in Section III, the PSD resuts o an idea CHRS signa in the absence o timing jitter can be obtained by setting 1 or 0as S d CH = p c S c CH = 1 c W p p } 8 W δd. 9 Consider a purey periodic transmission with one puse every c seconds p = 1. It can be seen rom the above two equations that, when the timing jitter is absent, the continuous spectrum disappears the spectrum contains ony spectra ines spaced at 1 as expected. When the timing jitter is present, however, the continuous spectrum sti exists in 4 6. hereore timing jitter heps rom the viewpoint o smoothing the spectrum. B. Framed H by a Rom Sequence Some H systems may require somewhat reguar spacing between puses, in addition to cocked time ocations. his signaing technique is reerred to as ramed H by a rom sequence FHRS. he FHRS signa s FH t can be written as s FH t = wt n b n c ɛ n, 30 where is a rame time or average puse repetition time, b n } is an integer-vaued i.i.d. rom H sequence, with pm, 0 m< Pr b n = m} = h, 31 0, otherwise. he vaue o the sequence eements are in the range 0 b n < h h c. ote that the number h o possibe hop times can be any number rom 1 a reguar puse train with puses spaced apart, to / c one puse at any romy seected cock time within each rame o seconds. he signa s FH t can be obtained rom the rom H-SS signa st o 1 by setting It is easy to veriy that Φ bn = a n =1 = = c. p m e +jπm 3 1 =0 Φ bn,b n+, = p me +jπm For an independent stationary stochastic sequences b n } ɛ n }, it can be shown using 4 11 that K β n;,, = 1 pme+jπm Φɛn = G β = p m e +jπm Φ ɛn. 35 Substituting into 8 10 respectivey, using 6 7 o Section III, the continuous discrete PSD components o a FHRS signa in the presence o arbitrary stationary timing jitter statistics become SFH c = 1 W 1 p me +jπm Φ ɛn 36 S d FH = 1 W p m e +jπ m Φ ɛn δd. 37 ote that FHRS, characterized by p m }, removes the power rom the continuous spectrum by the actor p me +jπm Φɛn. his actor, however, reappears in the discrete spectrum as expected. Roughy speaking, the addition o the raming constraint transers the energy in the continuous PSD to spectra ines 1 apart in the discrete PSD /99/$10.00 c 1999 IEEE

5 he resuts or UDJ can now be obtained by substituting 13 into as SFH c = 1 W 1 p me +jπm ] sinπ π 38 S d FH = 1 W sinπ π p m e +jπ m ] δ D. 39 he PSD resuts o an idea FHRS signa in the absence o timing jitter becomes SFH c = 1 W 1 p m e +jπm S d FH = 1 W p m e +jπ m δ D. V. Concusions Expressions or the PSD o rom H-SS signas in the presence o uniorm timing jitter are derived using stochastic theory. he resuts show that the asymmetry o the timing jitter does not aect the PSD, that timing jitter heps in smoothing the ine spectrum. he PSD o idea synchronous H-SS signas are shown to be a specia case o the resuts obtained in this paper, can be obtained as a imiting case when the timing jitter goes to zero i.e. 1 or 0. he PSD expressions obtained in this paper, based on a rom H sequence mode, are o particuar interest in appications such as rom signa detection /or interception. Furthermore, our resuts can be used to estimate the eect o changes in various H parameters on PSD o transmitted signa, without going through the detaied H sequence design procedure. Acknowedgments he author wishes to thank R. A. Schotz, L. R. Wech, G. Chrisikos, J.-M. Cramer, M. K. Simon, S. M. Hinedi or hepu discussions. Reerences 1] Moe Z. Win Robert A. Schotz, Impuse radio: How it works, IEEE Commun. Lett., vo., no., pp , Feb ] Moe Z. Win Robert A. Schotz, Comparisons o anaog digita impuse radio or mutipe -access communications, in Proc. IEEE Int. Con. on Commun., June 1997, vo. 1, pp , Montréa, Canada. 3] J.. Conroy, Joseph L. LoCicero, Donad R. Ucci, Communication techniques using monopuse waveorms, in Submitted to MILCOM, Oct. 1999, Atantic City, J. 4] Santosh S. Koenchery, J. Keith ownsend, James A. Freebersyser, Gri Bibro, Perormance o oca power contro in peer-to-peer impuse radio networks with bursty traic, in Proc. IEEE Goba eecomm. Con., ov. 1997, vo., pp , Phoenix, AZ. 5] Santosh S. Koenchery, J. Keith ownsend, James A. Freebersyser, A nove impuse radio network or miitary communications, in Proc. Miitary Comm. Con., Oct. 1998, vo. 1, pp , Boston, MA. 6] Gerad F. Ross, he transient anaysis o certain EM mode our-post networks, IEEE rans. Microwave heory ech., vo. M-14, pp. 58, ov ] C. Leonard Bennett Gerad F. Ross, ime-domain eectromagnetics its appications, Proc. IEEE, vo. 66, no. 3, pp , Mar ] Moe Z. Win Robert A. Schotz, On the robustness o utra - wide bwidth signas in dense mutipath environments, IEEE Commun. Lett., vo., no., pp , Feb ] Marvin K. Simon, Sami M. Hinedi, Wiiam C. Lindsey, Digita Communication echniques: Signa Design Detection, Prentice Ha, Engewood Cis, ew Jersey 0763, irst edition, ] John G. Proakis, Digita Communications, McGraw-Hi, Inc., ew York, Y, 1000, third edition, ] Robert A. Schotz, How do you deine bwidth?, in Proc. Internationa eemetering Conerence, oct 197, pp , Los Angees, Caiornia, Invited Paper. 1] Robert C. itsworth Loyd R. Wech, Power spectra o signas moduated by rom pseudorom sequences, echnica Report 3-140, Jet Propusion Laboratory, oct ] Wiiam A. Gardner, Statistica Spectra Anaysis: A onprobabiistic heory, Prentice Ha, Engewood Cis, ew Jersey 0763, irst edition, ] Wiiam A. Gardner, Introduction to Rom Processes with Appications to Signas Systems, McGraw-Hi, Inc., second edition, ] Marvin K. Simon, Kwei u, Bartus H. Batson, Eects o data asymmetry on shutte ku-b communications ink perormance, IEEE rans. Commun., vo. COM-6, no. 11, pp , ov ] Marvin K. Simon Samson Miion, Power spectrum o unbaanced RZ biphase signas in the presence o data asymmetry, DA Progress Report 4-16, Jet Propusion Laboratory, Aug ] Moe Z. Win, On the power spectra density o digita puse streams generated by M-ary cycostationary sequence in the presence o stationary timing jitter, IEEE rans. Commun., vo. 46, no. 9, pp , Sept ] Rodger L. Peterson, Rodger E. Ziemer, David E. Borth, Introduction to Spread Spectrum Communications, Prentice Ha, Engewood Cis, ew Jersey 0763, irst edition, ] Marvin K. Simon, Jim K. Omura, Robert A. Schotz, Barry K. Levitt, Spread Spectrum Communications Hbook, McGraw-Hi, Inc., ew York, Y, 1000, revised edition, ] Jack K. Homes, Coherent Spread Spectrum Systems, Wiey- Interscience, ew York, irst edition, ] Savo G. Gisic, Power density spectrum o the product o two time-dispaced versions o a maximum ength binary pseudonoise signa, IEEE rans. Commun., vo. COM-31, no., pp , Feb ] ikos B. Pronios Andreas Poydoros, On the power spectra density o certain digitay moduated signas with appications to code despreading, IEEE J. Seect. Areas Commun., vo. SAC-8, no. 5, pp , June ] Moe Z. Win, Utra -Wide Bwidth Spread-Spectrum echniques or Wireess Mutipe -Access Communications, Ph.D. thesis, Communication Sciences Institute, Department o Eectrica Engineering, University o Southern Caiornia, Los Angees, CA, May /99/$10.00 c 1999 IEEE