Digital Compensation of Amplifier Nonlinearities in the Receiver of a Wireless System

Transcription

1 Dgtal Comensaton of Amlfer Nonlneartes n the Recever of a Wreless System Cedrc Dehos* and Tm C. W. Schen * CEA-LETI MINATEC, 7 rue des Martyrs, 8 05 Grenoble CEDEX 9, France, Tel , cedrc.dehos@cea.fr Phls Research, Hgh Tech Camus 7, WY5.0, 5656 AE Endhoven, the Netherlands, Tel , tm.schen@hls.com Abstract In order to enable effcent mlementaton of wreless systems wth hgh sgnal dynamcs, ths aer rooses an aroach to estmate and correct for the amltude and hase dstorton caused by nonlnear ower amlfers. The aroach comrses a reamble desgn, an estmaton method and a dgtal comensaton algorthm. Correcton for the nonlneartes s aled n the recever, whch requres no addtonal hardware. Performance evaluaton of a 60 GHz OFDM-based system shows that the roosed aroach can successfully be aled wth mnmal erformance degradaton comared to non-mared systems. Index Terms Amlfers, Nonlneartes, Comensaton, Dgtal sgnal rocessng, mm-wave communcatons. I. INTRODUCTION The defnton of the most sutable modulaton format for 60 GHz wreless ersonal area networ (WPAN) systems s currently under dscusson n several standardzaton bodes. Several roosals are based on orthogonal frequency dvson multlexng (OFDM), whch has the advantage of hgh achevable data rate and robustness to multath fadng. However, the hgh sectral effcency of OFDM comes at the cost of an ncrease n sgnal dynamcs. Ths roves to be an mortant erformance lmtng factor when effcent,.e., ntrnscally nonlnear, amlfers are used n system mlementaton. To mnmze the mact of these nonlneartes, sgnfcant ower bac-offs are requred when feedng the sgnal to the transmt amlfer, reducng ts oeratonal range and effcency. Snce oeratonal range s a ey ssue n 60 GHz systems, ths gves the advantage to constant envelo modulatons such as contnuous hase modulaton (CPM), mnmum shft eyng (MS) and OFDM hase modulaton (OFDM-PM) for alcaton n 60 GHz WPAN systems. Ths s because these modulaton formats allow for a ower-lmted ower amlfer to be oerated near ts saturaton level, as such, maxmzng the ower effcency. In ths aer we roose a set of technques whch also allow for effcent oeraton of transmt amlfers for modulaton formats wth hgh sgnal dynamcs, such as OFDM. Hence, ths mght change the dscusson about the most sutable modulaton for 60 GHz WPAN. To reduce the mact of nonlneartes, several dgtal sgnal rocessng based technques have been roosed to reduce the ea-to-average ower rato (PAPR) of the transmt sgnals for wreless systems, see e.g. [] and [] for CDMA and OFDM systems, resectvely. Moreover, technques have been roosed based on dgtal redstorton n amongst others [] and [], where the latter also treats memory effects. However, due to the addtonal rocessng requred for PAPR reducton and redstorton, they are generally not alcable for mlementaton at the moble devce sde n uln transmssons. Moreover, redstorton methods requre ower consumng broadband A/D converters, whch annhlate the acheved effcency mrovement. Also, the ntroduced delays due to feedbac loos n the redstorton mght not be accetable n ractcal mlementatons. Therefore, ths aer rooses to emloy the addtonally requred rocessng for reducton of the nfluence of nonlneartes n the recever base staton/access ont by means of ostdstorton. We roose a reamble desgn, whch enables searate estmaton of the channel state nformaton (CSI) and the nonlneartes. Comensaton for the nfluence of the transmtter nonlneartes s acheved before channel equalzaton n the recever and, as such, t reduces the comlexty of the total mlementaton. Moreover, we, dfferently from the majorty of revous lterature, treat the combned nfluence of both amltude (AM/AM) and hase (AM/PM) dstorton due to nonlneartes. Ths aer s organzed as follows. Frst Secton II rovdes an overvew of nonlnear ower amlfer models and shows that these models can be develoed nto olynomal seres exansons. It also ntroduces orthogonal olynomal bases to enable relable estmaton of, and correcton for, the nonlneartes. Secton III, subsequently, ntroduces the system model of a wreless system exerencng nonlneartes. A method to accurately estmate the nonlneartes n a olynomal bass s the roosed n Secton IV. Followng ths, a low comlexty comensaton algorthm s resented n Secton V. Smulaton results revealng the erformance of ths --70-/07/$ IEEE

2 aroach n a 60 GHz OFDM system are reorted n Secton VI. Fnally, we conclude n Secton VII. II. NONLINEARITY MODELS AND APPROXIMATIONS A. Nonlnearty modelng The resonse of broadband ower amlfers can have recarous memory effects. For wreless devces wth lmted bandwdth comared to the aled carrer frequency, these effects can be effectvely mnmzed by careful desgn []. Moreover, these amlfer nonlneartes wth memory effect can generally be reduced to a Wener or Hammersten model [5],.e., the concatenaton of a lnear flter and nonlnear memoryless functon. For the Hammersten model the lnear flter functon can effectvely be regarded as beng art of the roagaton channel. Consequently, we wll n the followng assume the memory effects to be neglgble. The nfluence of a memoryless nonlnearty f( ) can be decomosed nto an amltude dstorton g G ( ) and a hase dstorton g φ ( ), whch are both functons of the amltude of the nut sgnal. The comlex sgnal s ( at the outut of the nonlnearty can, consequently, be wrtten as s ( = f ( s( ) = g( s( ) s( () = g G ( s( ) ex( jgϕ ( s( )) s(, where g( ) denotes the nonlnear dstorton functon, denotes the absolute value and s( s the comlex sgnal nut to the nonlnearty. In the lterature two models of nonlneartes are wdely used for wreless systems,.e., [7] and [8]. Both were orgnally roosed for ower amlfers. - The sold-state amlfer (SSA) model, as roosed by Ra n [7], for whch the nonlnearty s descrbed by s( g( s( ) = +, () A max where A max s the outut level at saturaton and determnes the smoothness of the transfer and s a ostve nteger. - The travelng wave tube (TWT) amlfer model, as roosed by Saleh n [8], for whch the nonlnearty s descrbed by α ( ) α s( G ϕ g s( = ex j, () + β s( G + βϕ s( where α G, β G are the arameters descrbng the amltude nonlnearty and α φ, β φ are the hase dslacement arameters. It has been shown revously that by a Taylor seres exanson around zero, these models can be brought bac to odd-order olynomals [9]. If we defne x= s( to be the amltude of the transmt sgnal, we can rewrte the dstorton functon as n ( n+ ) ( = ( + A x + A5 x A n+ x + ο ( x ) 5 n+ n+ ex j( Φ x + Φ x Φ x + ο( x ) g () ( 5 n+ ), where A n and Φ n denote the nth order olynomal coeffcents for g G ( ) and g φ ( ),.e., the AM/AM and AM/PM nonlnear functon, resectvely. We note that the coeffcents of the AM/AM characterstc can be related to the amltude of the ower amlfer nut nterceton onts, the common measures to characterze nonlnear amlfers. For the thrd and ffth order coeffcents ths yelds 8 A = and A 5 =, (5) AIIP 5AIIP5 resectvely. Here A IIP and A IIP5 denote the thrd and ffth order nut nterceton onts, resectvely. B. Orthonormal olynomal bass In fact, the revous olynomal aroxmaton corresonds to the rojecton of g G ( ) and g φ ( ) onto the canoncal bass {e 0, e, e, e, e, e 5, } = {, x, x, x, x, x 5,...}. (6) It s well-nown, however, that dentfcaton of nonlneartes s dffcult n ths bass, snce t wll easly ncur numercal errors [0]. Therefore, we wll derve a more sutable orthonormal olynomal bass, usng the Gram-Schmdt orthonormalsaton method. For ths orthonormalsaton, we wll aly the nner roduct u, v = E[ u( v( ] t [ 0, T ] over the observaton erod T. For a system samled wth a samle tme T s = T/N, the nner N roduct can be rewrtten as u, v = = u( ) v( ). Ths blnear 0 form s symmetrc and ostve-defnte. Now, let us defne χ n = E[x n ] to be the nth moment of the transmtted sgnal amltude. The teratve stes of the Gram- Schmdt orthonormalzaton rocess result nto the terms of the orthonormal olynomal bass { 0 (, (, (,...}, whch are gven by e + = 0 = + e + e e = 0 + +,,, (7) where e s the th element of the canoncal bass as defned n (6). The frst four orders of the develoed orthonormal olynomal bass, as functon of the moment χ n, are gven by x ( =, 0 ( =, χ χ ( ) = x (8) χ x χ x x, and ( =. χ χ χ χ χ χ We note that accordng to () the orthonormalzaton rocess can be reduced to the odd-terms of the canoncal bass. The nterest of the whole olynomal bass s, however, stressed n [0]. The dentfcaton of the nonlnearty arameters, as we wll resent n Secton IV, s facltated by the use of ths orthonormal bass. C. Polynomal nterolaton We also consder the use of nterolaton olynomals, such as Lagrange or Newton, snce they are good canddates for low comlexty aroxmaton of contnuous nonlnear functons. 6

3 Hence, we roose ther alcaton n Secton V for the nonlnearty comensaton. Let {a,b } be the onts for the corresondng nterolaton olynomal L{a,b}(, where generally b = g(a). The frst ont of the olynomal,.e., {a 0,b 0 } should be zero, snce the amlfers exhbt a lnear behavor for low nut amltudes. The quadratc error n the aroxmaton of the dstorton usng ths olynomal can then be exressed as: Amax ε = ( L{ a, b}( g( ) P ( dx, (9) 0 where P s( ( s the robablty densty functon of the enveloe of the transmt sgnal. For an OFDM sgnal P s( ( s well aroxmated by a Raylegh dstrbuton. Although one could choose the abscssa onts a to be lnearly dstrbuted, t seems more otmal to ut more onts close to zero. Ths s esecally true for systems alyng sgnals wth a hgh dynamc range, such as an OFDM system wth a hgh number of carrers, snce the hgh nut values, near A max, have a low robablty of occurrence. Let us now defne γ to be the lnear dstrbuton of the abscssa onts, wth a mnmum and maxmum value of 0 and, resectvely. When we then focus on owers of γ as ossble dstrbutons of the abscssa onts, numercal evaluaton of the quadratc error n the aroxmaton, as defned n (9), can be used to fnd the otmal abscssa onts for the combnaton of nut sgnal and nonlnearty. Ths s llustrated by Table for a TWT amlfer wth α G =, β G =0.5, α φ =0.07 and β φ =0. and an OFDM sgnal wth dfferent varances for the amltude of the nut sgnal. Varance of the nut sgnals enveloe dstrbuton s( Best abscssa onts γ 7 γ γ γ.5 Table : Otmal nterolaton abscssa onts for dfferent dstrbutons of the amltude of the nut sgnal. For unnown nut sgnal dstrbuton and/or ower amlfer attrbutes, the squared lnear bass wth M onts M a = 0,,.., Amax and b = g( a), M M (0) leads to a good aroxmaton for a wde range of nut sgnal dstrbutons and amlfer characterstcs. We note that another ossble method could be the use of Chebyshev nodes, whch are the roots of the Chebyshev olynomal of the frst nd, snce they can also be used to mnmze the roblem of unwanted oscllatons n the nterolaton olynomal rocess, also nown as Runge s henomenon []. III. SYSTEM MODEL Let us now consder a baseband equvalent model for our transmsson scheme, as schematcally dected n Fg.. Fg. : Schematc model of the regarded wreless system. The receved baseband sgnal after transmsson through the wreless channel can be wrtten as y ( = g( s( ) s( h( + n(, () where h( and n( denote the comlex channel mulse resonse and addtve whte Gaussan nose, resectvely. Here * denotes convoluton. We can conclude from () that the receved sgnal s nfluenced by a channel transfer functon, a nonlnear functon and a random nose rocess. In the followng we wll focus on a secfc system,.e., an (uncoded) OFDM system, to mae our aroach more ntellgble. It s noted, however, that the roosed technques are more generally alcable. We regard a system whch transmts a reamble wth tranng sequence q n front of the actual data acet. Also, a cyclc refx s added to every OFDM symbol n the tme doman to create robustness aganst nter-symbol nterference. The rocessng we roose n order to comensate for both the nonlneartes and the channel n such an OFDM system s summarzed n Fg. and wll be further detaled n Sectons IV and V. It conssts of dervng an estmate of the (wreless) channel and the nonlnearty and subsequently correctng for both. For otmum comensaton of both effects, comensaton for the nonlneartes should theoretcally occur after the channel equalzaton. However, to yeld an mlementaton wth lmted comlexty, nonlnearty comensaton n ths wor wll be done before the FFT oeraton. In Secton VI we show ths yelds a good bt-error rate (BER) erformance. Fg. : Strategy for the estmaton and comensaton of nonlneartes n the recever of an OFDM system.

4 IV. ESTIMATION OF THE NONLINEARITIES A. Preamble desgn To enable the searate estmaton of the nonlneartes and the channel resonse, we roose a new reamble format q consstng of two sequences: - A constant modulus tranng sequence q for (lnear) channel estmaton. q should be frequency whte to enable accurate estmaton of frequency selectve channels. - A hgh PAPR tranng sequence q for nonlnearty estmaton. q should cover the same dynamc range as the data sgnal and have the same mean ower. B. Lnear channel estmaton The receved sgnal corresondng to the transmtted tranng sequence q s gven by z( = g( q( ) q( h( + n( () = q( Qh ( + n(, where Q =g( q ( ) s tme-ndeendent due to the constant modulus roerty of q. An estmate of the channel state nformaton s derved from the receved reamble symbol z, usng a classcal least-squares or mnmum mean squared error estmaton algorthm []. Note that a scalng of Q wll be resent n ths estmate, whch can be smly removed after estmaton of the nonlnearty. C. Nonlnearty estmaton Wth the nowledge of the moments of the tranng sequence q, a dedcated orthonormal bass s comuted usng (7), to revent numercal errors n the dentfcaton of the nonlneartes. The nonlnear functon can be exressed n ths olynomal bass { 0 (, (, (,...} as g( = g ( ) ex ( ) x j ϕ x. () = = A sutable aroxmaton of g( ) s the th order seres gˆ( = g ( )ex ( ) x j ϕ x. () = = The rojecton of the receved sgnal z, corresondng to the transmtted tranng sequence q, onto the bass of () gves a relable aroxmaton of the AM/AM g and AM/PM φ nonlnearty characterstcs. Alternatvely, one could use a smlar method to drectly estmate the nverse of the nonlnearty, as was llustrated by the authors n [] for a MIMO OFDM system. D. On aroxmaton accuracy The accuracy of the aroxmated functon deends on: - The seres length comared to the effectve order of the ower amlfer nonlnearty. - The length of the tranng sequence relatve to the sgnalto-nose rato. From our exerence we conclude that a ffth order olynomal can cover a wde range of ower amlfer classes and technologes. We aly (0) to be the bass for the olynomal nterolaton wth b = gˆ ( a). V. COMPENSATION FOR THE NONLINEARITIES Postdstorton would be most straghtforwardly carred out by alyng the estmate of the nverse nonlnear functon to the receved sgnal y(. The nverson of the estmated nonlnearty functon ĝ s, however, often dffcult to calculate. However, the nverson of abscses and ordnates of the nterolaton onts s smle. As revously ntroduced n Secton III.C, the nterolaton olynomal L{a,b} can gve an accurate aroxmaton of g. Hence, the nterolaton olynomal L{b,a} can be used to acheve smle and accurate aroxmaton of g -. In fact, thans to the nterolaton olynomals the estmaton of g or g - corresonds to the same rocedure. A. Polynomal nterolaton We frst consder the Lagrange olynomal nterolaton, whch can be exressed as x b L Lagrange{ b, a}( = a. (6) = b b The second s the Newton form nterolaton, whch can be exressed as L { b, a}( = c x b, (7) Newton = = where c denotes the dvded dfference as defned teratvely n [] and s gven by a c =. = b b l= l Note that the obtaned olynomals are the same, snce there s only one th order olynomal whch nterolates + onts, for the Lagrange and Newton methods. B. Comensaton aroach Comensaton for the nonlneartes s now carred out n the tme doman usng the revously defned nverse nterolaton olynomal. The nonlnearty olynomal nterolaton functon s aled on the amltude of the receved sgnal, yeldng w ( = L{ b, a} ( y( ), where w( reresents the sgnal corrected for AM/AM dstorton. Subsequently, AM/PM dstorton can be evaluated ether n the orthonormal bass wth () or wth the nterolaton olynomal, yeldng l gˆ( w( ) = ϕ ( w( ), (8) = where ( ) yelds the hase of ts argument. Eventually, the receved sgnal s corrected for both the amltude and hase dstorton, yeldng w( = w( ex j y( gˆ( w( ). (9) ( ( )

5 As schematcally dected n Fg., conventonal equalzaton can, subsequently, be aled to the ostdstorted sgnal w( to comensate for the wreless channel resonse. VI. SIMULATION RESULTS To evaluate the erformance of the roosed estmaton and comensaton scheme, smulatons were carred out for a 60 GHz OFDM-based system based on the Medum Rate Transmsson roosal (MRT) of the IEEE 80.5.c grou [5]. The relevant system arameters are gven n Table. Ths 60 GHz OFDM system s a tycal examle of a system wth hgh PAPR sgnals, whch uts severe constrants on the ower amlfcaton. Table : System smulaton arameters. Number of data sub-carrers 80 Number of sub-carrers 0 Channel bandwdth GHz Cyclc refx duraton 96 ns Modulaton 6-QAM 80.5.c channel CM., CM., CM. Codng Rate / PA model SSPA (=.0) IBO 0 db In order to study realstc nonlneartes, we consdered a CMOS SOI medum ower amlfer (MPA), a cture of whose layout s resented n Fg.. The ower amlfer acheved dbm outut ower and a db comresson ont of 7 dbm. Measurement results of the resonse of ths MPA were matched to those of dfferent amlfer models. It was found that the resonse of the SSA model wth =.0 was very close to that of the mlemented MPA. Hence ths model was used for smulatons. was relaced by the reamble ntroduced n Secton IV.A, consstng of q and q, to allow for estmaton of the channel resonse and nonlnearty, resectvely. For q a constant modulus sequence wth unform random hase of one OFDM symbol length s used. For q a random bt sequence modulated OFDM symbol s used, based on the MRT data sgnal. The nonlnear SSA ower amlfer model wth =.0 was smulated wth 0 db nut bac off from the db comresson ont,.e., a hghly nonlnear amlfer oeratng close to saturaton. For estmaton of both the AM/AM and AM/PM characterstcs of the nonlneartes, the receved tranng sequence was rojected onto a ffth order orthonormal bass, usng the method defned n Secton II.B. The correcton was acheved through a fve ont Lagrange olynomal nterolaton, wth a squared lnear nterolaton bass, as defned n (0). The accuracy of the nverse (Lagrange) nonlnearty estmaton and comensaton s studed n Fg.. The fgure dects the normalzed mean squared error (MSE) between the receved lnear sgnal not exerencng nonlneartes s(*h(+n( and the noseless undstorted sgnal s(*h( ( lnear ), the MSE between the nonlnear dstorted sgnal y( and the noseless undstorted sgnal s(*h( ( dstorted ), and the MSE between the comensated sgnal w( and the noseless undstorted sgnal s(*h( ( comensated ) lnear dstorted comensated MSE 0 - Fg. : 60 GHz cascode MPA layout. Note that, due to the guard subcarrers n the OFDM modulaton and the relatvely low emtted ower of ths CMOS amlfer, sectral regrowth remaned below the sectrum mas roosed by the IEEE 80.5.c grou. Due to the 0 subcarrers and 6-QAM modulaton, the OFDM sgnal exhbts an average PAPR of 8.8 db. The short tranng art of the roosed reamble n the MRT roosal SNR (db) Fg. : Normalzed mean squared error (MSE) for the dstorted and the comensated OFDM system. The observed MSE ndcated by lnear reveals the nfluence of the addtve nose. Consderable error n the sgnal s observed for the system exerencng the nonlnearty, ndcated by dstorted. When the roosed comensaton s aled, shown by the curve ndcated by comensated, only a mnor degradaton comared the system only exerencng nose s found. Hence, t can be concluded from ths fgure that

6 the roosed algorthms acheve accurate estmaton of the nonlnearty functon and ts nverse. The man contrbuton to the error can be attrbuted to the model msmatch because of lmted order of the Lagrange olynomal. The senstvty to nose can easly be decreased by consderng longer tranng sequences. Also, Monte Carlo smulatons were carred out wth CM., CM. and CM. channels wth nonlnearty comensaton through the Lagrange olynomal nterolaton. Fgure 5 shows the BER erformance for MRT and CM. wth a squared lnear bass for the nterolaton onts. BER lnear dstorted comensated Eb/No (db) Fg. 5: Bt-error rate (BER) erformance for the dstorted and the comensated OFDM system for channel CM.. The system wthout comensaton exhbts a sgnfcant BER floor n resence of PA nonlneartes for all smulated channels. For SNR values above 5 db, the estmaton and comensaton aroach successfully reduces the BER to close to the deal lnear sgnal erformance, thus allowng the use of the studed ower amlfer u to the db comresson ont. VII. CONCLUSIONS A novel aroach has been resented to estmate and correct for the n-band amltude and hase dstorton caused by nonlnear ower amlfers n wreless communcaton systems. The aroach s based on ostdstorton and olynomal nterolaton. Smulaton results for a 60 GHz based OFDM system show that the combned estmaton and comensaton aroach can consderably reduce the erformance mact of the nonlneartes. As such, the n-band nfluence of amlfer nonlneartes can be comensated for and wll not hnder the alcaton of OFDM n 60 GHz systems. Hence, the achevable PA effcency wll be manly drven by non-lnearty-caused sectral regrowth comared to the mosed emsson mas. ACNOWLEDGMENTS The wor of Cedrc Dehos was suorted by ST Mcroelectroncs Front-end Technology and Manufacturng, France. The wor of Tm Schen was sonsored n art by the B BroadBand Rado@Hand roject (BTS006) and n art by a travel grant of the Netherlands Organzaton for Scentfc Research (NWO). REFERENCES [] A. Srzycza, J.-P. Javaudn, and P. Sohan, Reducton of the Pea-to-Average Power Rato for the OFDM/OQAM Modulaton Proc. of IEEE VTC Srng 006, vol.,. 08-0, May 006. [] F. Nadal, S. Sezgner, and H. Sar, Pea-to-Average Power Rato Reducton n CDMA Systems Usng Constellaton Extenson Proc. of PIMRC 005,. 7-0, Set [] S. usuno et al., Power-Amlfer Module Wth Dgtal Adatve Predstorton for Cellular Phones, IEEE Trans. on Mcrowave Theory and Technques, vol. 50, no., Dec. 00 [] L. Dng, et al., A Robust Dgtal Baseband Predstorter Constructed Usng Memory Polynomals, IEEE Trans. on Comm., vol. 5, Jan. 00. [5] A. hanfar, N. Maslennov, and B. Vasslas, Bas crcut toologes for mnmzaton of RF Amlfer Memory Effects, Proc. of rd Euroean Mcrowave Conference, Munch, 00 [6] P. L. Glabert, G. Montoro, and E. Bertran, On the Wener and Hammersten Models for Power Amlfer Predstorton, Proc. of IEEE APMC 005, Dec [7] C. Ra, Effects of HPA-Nonlnearty on a -DPS/OFDM- Sgnal for a Dgtal Sound Broadcastng System, Proc. of nd Euroean Conf. on Satellte Commun.,. 79-8, Oct. 99. [8] A. A. M. Saleh, Frequency ndeendent and frequency deendent nonlnear models of TWT amlfers, IEEE Trans. Com., vol. 9, no.,.75-70, Nov. 98. [9] T. C. W. Schen, P. F. M. Smulders, and E. R. Fledderus, Imact of nonlneartes n multle-antenna OFDM transcevers, Proc. IEEE SCVT006,. 5-56, Nov [0] R. Rach and G. T. Zhou, Orthogonal olynomals for comlex Gaussan rocesses, IEEE Trans. on Sgn. Proc., vol. 5, , Oct. 00. [] L. Dng and G. T. Zhou, Effects of Even-Order Nonlnear Terms on Power Amlfer Modelng and Predstorton Lnearzaton, IEEE Transactons on Vehcular Technology, vol. 5, no.,. 56-6, Jan. 00. [] S. Coler, M. Ergen, A. Pur, and A. Baha, Channel Estmaton Technques Based on Plot Arrangement n OFDM Systems, IEEE Trans. on Broadcastng, vol. 8,. -9, Set 00. [] T. C. W. Schen, C. Dehos, D. Morche and E. R. Fledderus, Recever-based comensaton of transmtter-ncurred nonlnear dstorton n multle-antenna OFDM systems, Proc. IEEE VTC 007 Fall, Baltmore, US, Oct [] M. Abramowtz and I. A. Stegun, Handboo of Mathematcal Functons wth Formulas, Grahs, and Mathematcal Tables, 9th rntng. NY: Dover, 97. [5] IEEE 80.5 WPAN Mllmeter Wave Alternatve, PHY Tas Grou c (TGc).