Level Crossing Rate in Terms of the Characteristic Function: A New Approach for Calculating the Fading Rate in Diversity Systems

Transcription

1 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in Diversity Systes Ali Adi Meer IEEE and Mostafa Kaveh Fellow IEEE Dept. of Elec. and Cop. Eng. New Jersey Institute of echnology Newark NJ 7 Dept. of Elec. and Cop. Eng. University of Minnesota Minneapolis MN Eails: ali.adi@njit.edu kaveh@ece.un.edu Astract he level crossing rate LCR of a rando process conveys useful inforation aout e underlying process and is of interest in diverse engineering fields. In wireless counications it is related to e syste characteristics such as handoff outage proaility fading rate average duration of fades velocity or axiu Doppler shift of e oile and e effect of diversity on fading. he LCR forula was originally derived y Rice in ters of e joint proaility density function PDF of e underlying process and its tie derivative. In is paper we express e LCR in ters of e joint characteristic function CF. his new forula is useful for any cases where e joint CF is sipler to derive an e associated joint PDF. As an application and for a DS-CDMA syste e fading rate at e output of a RAKE receiver wi eier axial ratio coiner or postdetection eual gain coiner operating over a freuency-selective fading channel wi different pa statistics is easily calculated using e new CF-ased LCR forula. Keywords: Characteristic function DS-CDMA systes Fading channels Level-crossing proles RAKE receivers. his work was supported in part y e National Science Foundation under e Wireless Initiative Progra Grant # Page of 8

2 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh I. INRODUCION Wireless counications systes are suject to e fading and diversity-ased techniues are powerful tools for itigating e ipact of fading []. he asic idea of all diversity eods is to coine several independent copies of e signal at e receiver. Application of ultiple antennas at e receiver is an efficient and coon diversity reception. Utilization of e RAKE receiver for wideand systes which are ipaired y freuency-selective fading is anoer type of diversity reception []. In e real world e pas over which e signal has propagated are not independent and identically distriuted i.i.d. [] [3] [4] [5]. So e replicas of e transitted signal at e receiver have different fading distriutions and are utually correlated. Direct seuence-code division ultiple access DS-CDMA systes operating over freuency-selective fading channels are typical exaples where we encounter non-i.i.d. fading aong e pas. Fading rate in a diversity syste defined as e level crossing rate LCR of e total instantaneous signal-to-noise ratio SNR per syol at e output of e diversity coiner is a useful easure for evaluating e dynaic perforance of at diversity syste. he coon approach for calculating e LCR of a rando process is to eploy Rice s forula which expresses e LCR in ters of e joint proaility density function PDF of e process and its derivative. However for diversity systes operating over non-i.i.d. fading channels it is very hard if not ipossile to derive an expression for e joint PDF of instantaneous SNR and its derivative at e output of e diversity coiner. On e oer hand for cases of interest such as Rayleigh and Rice fading channels e associated joint characteristic function CF can e derived in closed for. herefore Rice s LCR forula for a rando process should e rewritten in ters of e joint CF of e process and its derivative. In what follows such a forula has een derived for an aritrary rando process and its utility has een deonstrated y calculating e fading rate at e output of a DS- CDMA RAKE receiver wi eier axial ratio coining MRC or postdetection eual gain coining EGC operating over a Rice-Rayleigh freuency-selective fading channel. II. HE LCR FORMULA IN ERMS OF HE CHARACERISIC FUNCION For a given level and according to e celerated Rice s forula e LCR of a stationary rando process Y t is given y [6]: { Y t }] y p y dy E[ N Y Y where E is e aeatical expectation N { Y t} represents e nuer of ties per unit tie Page of 8

3 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh where Y t prie denotes differentiation wi respect to tie t and y y is e joint PDF of p Y Y e rando variales Y Y t and Y Y t wi t as an aritrary instant of tie. Now we want to express E[ N { Y t }] in ters of Φ ω ω E[exp jω Y + jω ] e joint CF of Y and Y Y Y Y raer an y y note at j. Using e generalized Parseval s eore for e right-hand p Y Y side of noting at y y sgn y wi sgn. as e signu function we otain: E[ N { } I I Y t ] [sgn y ]{ [ y py Y y ]} dω π π d I I[ y sgn y ]{ [ py Y y ]} ω where I [.] is e Fourier transfor and is e coplex conjugate operator. Based on: ω YY YY I [sgn y ] jω I [ p y ] π Φ ω ω e j dω and e properties of e Fourier transfor we finally otain: d ΦYY ω ω Φ YY Y [ { }] j ω ω Φ ω ω jω E N Y t e dω dω e dω dω. π ω ω π ω d For cases where e derivation of Φ Y Y ω ω is easier an p Y y y plays an iportant Y role. An exaple of such a case is reported in [7] where an exact and siple solution for e expected nuer of axia of e envelope of a spherically invariant rando process is derived. Based on at siple solution which could not e otained using Rice s LCR forula in a new velocity estiator for cellular systes which is roust against e variations of SNR and channel characteristics is proposed in [8]. In e seuel we will see how euation allows us to solve an LCR prole which sees to e intractale using Rice s forula given in. III. FADING RAE A HE OUPU OF MRC AND POSDEECION EGC Assue at e diversity coiner has ranches fingers in e RAKE receiver. In e presence of additive white Gaussian noise which is independent of fading e instantaneous SNR per syol of e ranch is given y envelope of e ranch wi e average power γ t R t N B. In is forula R t is e signal Page 3 of 8 Ω E[ R t] N is e one-sided power spectral density of e noise and B is e andwid of e receive filter in each ranch. Clearly e average SNR per syol of e ranch can e written as γ Ω N B. he total instantaneous SNR per syol at e output of o MRC and postdetection EGC is given y []: γ γ t. 3 t he total average SNR per syol at e output of e coiners can e written as

4 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh γ Ω. Fro now on let us assue NB to siplify e notation wiout γ NB loss of generality. We define FR γ e fading rate wi respect to e given reshold γ at e output of e diversity coiner as e γ -crossing rate of γ t i.e. FRγ E[ Nγ { γ t }]. In is paper we assue at separate resolvale propagation pas are independent. However e pas can e distriuted according to a coon fading distriution wi different paraeter values [9] or even different failies of fading distriutions []. Note at such a general setting is not just of eoretical interest ecause wideand easureents of freuency-selective fading channels have shown non-identically distriuted pas [] [3] [4] [5]. In is paper we consider a diversity coiner wi e first ranches having a Rice-distriuted envelope and e second ranches having a Rayleigh-distriuted envelope. his odel corresponds to a typical freuency-selective fading channel in a ixed uran-suuran region [3]. In order to calculate E N { t } assuing R t γ t has a Rice distriution []: [ γ γ ] using we first derive an expression for ω ω Φ γ γ r r exp + a a r p R r I r 4 where I. is e odified Bessel function of order zero and a and are nonnegative nuers descried in e seuel. Based on e inphase and uadrature representation for R t we have + R t U t V t where U t is a Gaussian process wi ean a and variance while V t is a zero-ean Gaussian process wi e sae variance independent of U t. herefore: E j j E j U V j U U VV Φ γ γ ω ω [exp ω γ + ω γ ] [exp{ ω + + ω + }]. 5 Let us define e following vectors and atrices: [ U V V U ] [ a ] W ω D ω ω ω ω ω 6 wi as e transpose operator. Since differentiation is a linear operation e processes U t and V t are also Gaussian wi zero ean and e sae variance. So W is a Gaussian vector wi e ean-vector and e covariance-atrix given in 6. Let us also define e lowpass euivalent coplex envelope of e ranch as R t R t exp[ jθ t] where Θ t tan [ V t U t ] is Page 4 of 8

5 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh e signal phase of e ranch. We represent e autocovariance function of R t y τ φ R R and e associated power spectru y e Fourier transfor S f I [ φ τ ]. he paraeter R R R R n n in 6 is e n spectral oent of R t defined y [6]: n n π f S f df. n RR Using e properties of e Fourier transfor it is easy to show at: n n n j d φ τ dτ. 7 n RR τ Based on e definitions in 6 5 can e written as: Φ ω ω E[exp jw DW ]. 8 γ γ Since W in 6 is a Gaussian vector wi e ean-vector and e covariance-atrix scalar rando variale e W DW is a uadratic for of Gaussian variales and its characteristic function E[exp jω W DW ] can e evaluated according to []: E[exp jωw exp DW ] [ I I det I jω jω D D ] 9 where I is e 4 4 unit atrix and det. denotes deterinant. Now Φ γ γ ω ω in 8 can e easily derived y replacing ω in 9 wi. After soe algeraic anipulations we otain: exp + 4 j Φ ω ω γ γ a ω jω ω ω + 4 ω j ω For a Rayleigh distriution a significantly siplifies to: [ 4 ] j Φ γ γ ω ω + ω ω.. Based on e independence of γ t s and e definition of γ t in 3 we have: Φ ω ω Φ γ γ ω ω. γγ Sustitution of into gives e fading rate E N { t } [ γ γ ]. Now suppose e diversity coiner is ounted on a oile receiver. If e scattering over e pa to e oile receiver is nonisotropic en n for all n can e different fro zero []. However for isotropic scattering we have φ R R τ J πf τ where J. is e Bessel function of order zero and according to f is e axiu Doppler freuency [] f is related to e speed of e oile v v f λ c wi c λ as e waveleng of e carrier. So e spectral oents are: Page 5 of 8

6 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh π f. As a result reduces to: a 4 π f ω jω exp + 8π f ω j ω + 8π f ω j ω Φ ω ω γ γ. 3 It is coon to express a and in ters of Rice factor K and e average power K a and + a Ω K into 3 we otain: Φ ω ω γ γ K exp Page 6 of 8 Ω defined y Ω + []. By sustituting a K Ω K + and + K f j K ΩK[ π f Ωω j K + ω] + + π Ωω Ω + ω K + + π f Ωω jω K + ω For e Rayleigh distriution K 4 drastically siplifies to: f j Φ γ γ ω ω + π Ω ω Ω ω.. 4 o see e difference etween e fading rates of coiners wi non-i.i.d. and i.i.d. ranches we consider two nuerical exaples: a non-i.i.d. case and an i.i.d. case. In e non-i.i.d. scenario we have 3 and. his eans Rice-distriuted envelope in e first ranch and Rayleigh-distriuted envelopes in e second and ird ranches. herefore K K. Aong e proposed odels for 3 e power delay profile we consider e coon eually-spaced exponential profile [3]: ρ Ω Ω e ρ... 5 where e paraeter ρ reflects e rate at which e average power decays. By sustituting 4 into and en into FR γ for e non-i.i.d. case is plotted in Fig. using e Maeatica software for ese paraeter values: K db Ω db ρ. and f 5Hz which corresponds to e oile speed non iid Ω Ω [ + exp ρ + exp ρ] Ω v.5/s if e carrier freuency f c is GHz. Notice at db. For e i.i.d. scenario 3 and. his iplies Rayleigh-distriuted envelope for all e ranches. Hence K K K 3. We also assue ρ so Ω Ω Ω3. For is case we take Ω db which yields iid non iid Ω Ω 3 Ω db Ω. he corresponding FR γ for e i.i.d. case is also plotted in Fig.. Visual coparison of ese two plots exhiits e significant difference etween e non-i.i.d. and i.i.d. cases aout which a syste designer should always e aware. For e case of i.i.d. ranches descried aove γ t in 3 is a chi-suare process wi 6 degrees

7 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh of freedo where e variance of all e six underlying Gaussian coponents is Ω. Based on e LCR of a chi process derived in [4] 5 FR γ for e aove i.i.d.-ranches exaple can e written as: FRγ π f γ Ω exp γ Ω. 6 he nuerical results otained y our approach coply very accurately wi 6 as a special case. IV. CONCLUSION In is paper we have introduced a characteristic function CF-ased approach for calculating e level crossing rate LCR of a rando process as a new alternative to e traditional proaility density function PDF-ased eod first proposed y Rice. As an application we have considered a DS-CDMA syste and calculated e fading rate at e output of a RAKE receiver operating over a freuency-selective fading channel wi different pa statistics. Such a prole defies even a nuerically tractale solution using Rice s LCR forula [5]. As e last word e LCR of an eual gain coiner is calculated in [6] where e joint CF is first derived en its Fourier transfor e joint PDF is coputed and plugged into Rice s LCR expression. However as we have shown here e LCR can e directly expressed in ters of e CF and ere is no need to calculate e associated PDF. REFERENCES [] G. L. Stuer Principles of Moile Counication. Boston MA: Kluwer 996. [] P. C. Fannin and A. Molina Analysis of oile radio channel sounding easureents in inner city Dulin at.88 Ghz IEE Proc. Coun. vol. 43 pp [3] K. A. Stewart G. P. Laedz and K. Sohrai Wideand channel easureents at 9 MHz in Proc. IEEE Vehic. echnol. Conf. Chicago IL 995 pp [4] W. R. Braun and U. Dersch A physical oile radio channel odel IEEE rans. Vehic. echnol. vol. 4 pp [5] H. Suzuki A statistical odel for uran radio propagation IEEE rans. Coun. vol. 5 pp [6] S. O. Rice Maeatical analysis of rando noise reprinted in Selected Papers on Noise and Stochastic Processes. N. Wax Ed. New York: Dover 954 pp [7] A. Adi and S. Nader-Esfahani Expected nuer of axia in e envelope of a spherically invariant rando process revised version suitted to IEEE rans. Infor. heory Jul.. [8] A. Adi and M. Kaveh A new velocity estiator for cellular systes ased on higher order crossings in Proc. Asiloar Conf. Signals Systes Coputers Pacific Grove CA 998 pp [9] G. P. Efyoglou V. A. Aalo and H. Helken Perforance analysis of coherent DS-CDMA systes in a Page 7 of 8

8 Level Crossing Rate in ers of e Characteristic Function: A New Approach for Calculating e Fading Rate in A. Adi & M Kaveh Nakagai fading channel wi aritrary paraeters IEEE rans. Vehic. echnol. vol. 46 pp [] R. D. J. van Nee H. S. Misser and R. Prasad Direct-seuence spread spectru in a shadowed Rician fading land-oile satellite channel IEEE J. Select. Areas Coun. vol. pp [] G. L. urin he characteristic function of Heritian uadratic fors in coplex noral variales Bioetrika vol. 47 pp [] A. Adi H. Allen Barger and M. Kaveh A paraetric odel for e distriution of e angle of arrival and e associated correlation function and power spectru at e oile station accepted for pulication in IEEE rans. Vehic. echnol.. [3] M. K. Sion and M. S. Alouini Digital Counication over Fading Channels: A Unified Approach to Perforance Analysis. New York: Wiley. [4] R. A. Silveran he fluctuation rate of e chi process IRE rans. Infor. heory vol. 4 pp [5] A. M. Hasofer he upcrossing rate of a class of stochastic processes in Studies in Proaility and Statistics. E. J. Willias Ed. New York: Nor-Holland 974 pp [6] W. C. Y. Lee Moile Counications Engineering. New York: McGraw-Hill 98. l og F adin g R at e log Fig.. Fading rate versus e SNR reshold postdetection EGC RAKE receiver. γ db at e output of a ree-ranch MRC or non-i.i.d. ranches Rice distriution for e first ranch and Rayleigh distriutions wi different powers for e second and e ird ranches i.i.d. ranches wi Rayleigh distriution Page 8 of 8