Analytical Performance of Closed Loop Power Control Quantized under Fast Fading for CDMA Techniques

Transcription

1 Analytial Prforman of Closd Loop Powr Control Quantizd undr Fast Fading for CDMA Thniqus S.Nourizadh, T.Jans and R.Tafazolli CCSR Univrsity of Surry Guildford GU XH, UK Tl: (+).8.8. Fax: (+) Abstrat: Th prforman of SIRbasd Closd Loop powr ontrol (CLPC) is analytially analysd. Th valuation work has bn arrid out using th standard dviation of th powr ontrol rror (PCE) as th prforman mtri. A nonlinar ontrol thory mthod is applid to th fdbak systm undr fast fading. An analytial xprssion of th CLPC undr fast fading is also produd. Finally a quantizdstp siz powr ontrol algorithm, rplaing th hard limitr is onsidrd. Th proposd mthod is found to work onsidrably bttr for high spd MS as wll as bing a powrful tool to optimis th loop prforman. Introdution Thr has bn onsidrabl intrst latly in using Cod Division Multipl Ass (CDMA) thnology to improv th apaity of llular tlphon systms. Th apaity of th systm in suh a multipl ass shm dpnds havily on th Multipl Ass Intrfrn inhrnt in a CDMA basd llular strutur. Opn and Closd Loop Powr Control shms ar thrfor usd to nhan th apaity. Sin th fast fading is mitigatd via th losdloop shm, it is of primary intrst to b abl to valuat th prforman of th lattr, dpnding on paramtrs suh as th mobil spd and th hannl variations. Traditionally, th prforman of th losd loop is obtaind by simulation, a vry tim onsuming produr []. Th aim of this rsarh is to analytially valuat th prforman of th loop undr fast fading onditions, so that rsults an b obtaind by a losd formula. Sin it is found from both simulation and xprimntal rsults that th Powr Control Error (PCE) follows a normal distribution if xprssd in dbs [], th standard dviation (std) of th PCE is adoptd as th ritrion for th losd loop prforman. Th first stion of this papr xprsss th analytial xprssion of twofdbak mthods systm undr fading, proposd by []. Th sond stion applis and proposs th analytial xprssion of th PCE undr fast fading (Rayligh hannl). Th rsults (xprssd as th PCE std) obtaind from th analytial mthods ar ompard to thos obtaind from th simulation [] in th third stion. Finally a novl thniqu is introdud whih onsists of rplaing th hard limitr by a quantizr. Intrsting rsults about th prforman of th algorithm with rspt to th input rang of th quantizr; numbr of bits usd for quantization and th loop dlay ar obtaind. Mathmatial analysis of Closd Loop In CDMAbasd llular systms suh as UMTS/IMT, th CLPC prforman varis with diffrnt vhiular spds, propagation hannls and dlays. In rvrs link CLPC (Figur ), th BS masurs th rivd man SIR (Signal to Intrfrn Ratio) ovr th priod T P, and ompars it with th targt SIR. It thn transmits a powr ontrol ommand (in Figur notd +/ ) to th mobil station for a powr inras or dras. Tx MS P(t) G(t)+n(t) Propagation Channl P( tt ) P Z P (t) r bin P + P r Compar with Thrshold + Stp Sltion Figur : Convntional Closd Loop Powr Control Modl BS [] studid this problm undr slow fading suh as shadowing and ddud a modifid figur (Figur ) so as to apply nonlinar ontrol thory to th CLPC. B(z) + Y(z) K Z(z) z Figur : Modifid CLPC rprsntation N z X(z) Th prforman of th systm is analyzd through th Powr Control Error (PCE) and xprssd in db as: tar Y ( n) = Γ A( n) X ( n) + I Eq X ( n + ) = X ( n) + * sign[ Y ( n)] Eq whr I, A, X, and Γ tar rprsnt rsptivly, th intrfrn (assumd onstant), th fading, th transmit powr, th stp siz and th SIR targt. To failitat th analysis this squn is dfind: B tar ( n) = A( n ) + I + Γ Eq 89//$. IEEE. 9

2 [] applid th statistial linarisation mthod to th Eq., rplaing th nonlinar lmnt with th quivalnt gain K, to th output of whih is addd an unorrlatd signal (rprsntd by N in Figur ). Th rlation btwn th PCE and th gain K is givn in Eq.: Eq Y = K π From th Figur, [] proposd two mthods to alulat th PCE, th Sptrum Intgration Mthod (SIM) and th Lyapunov Equation Mthod (LEM), whih onsist of solving simultanously Eq. and th quation alulatd from ah mthod. SIM assums that th sptrum of th hannl P B is known and ddus th transfr funtions H YB and H YN (Figur ) Eq Y = HYB( ) PB + HYN( ) PN π π P N rprsnts th sptrum of th nois N. Th LEM, on th ontrary, oprats in tim domain and thrfor th autoorrlation funtion of th hannl B is rquird. Th standard dviation of th PCE y is thn alulatd via Eq. ( a b ) Eq y = x + N K a ( ) Th paramtrs a and b ar usd only for simplifiation, α is a paramtr that ontrols th spatial dorrlation of th fading and N is a zro man Gaussian random pross. a = α(k) b = K+α Th radr is rfrrd to [] on obtaining Eq.. For both mthods, th solution y will b th intrstion btwn th two urvs dfind by th Eq. & Eq. (or Eq.), as K taks valus from zro to two. [] solvs this problm for slow fading suh as Shadowing by assuming th variabl A, thrfor B (Eq.), to hav a lognormal distribution as xprssd in []. CLPC undr Rayligh fading In [], th as whr th hannl follows a Rayligh distribution was don smianalytially. To xprss a full analytial xprssion of th CLPC undr a fast fading hannl has bn provn to b not so simpl []. On rason is that in th proposd modl it is vry hard to inlud th autoorrlation of th Rayligh hannl known as th Jaks modl, xprssd in Eq.. Rµ ( t) = J o(π * fmt) Eq Instad w hav onntratd our fforts in th Gaussian powr sptral dnsity givn by [] f µ S µ ( f ) = Eq 8 π whr µ is th varian of th Rayligh variabl, f is th maximum dopplr frquny and is a paramtr that is rlatd to th db utoff frquny f of S µ (f) aording to = ln Eq 9 f Th invrs Fourir transform of Eq.8 givs th orrsponding autoorrlation funtion π R t = t Eq () ( ) µ µ Th lattr modl orrsponds to a wav inidn snario whr th signal nrgy is onntratd in two distint angular rgions, whr th angl sprad orrsponds to th varian of th gaussian shap and th man angl orrsponds to th offst of th Gaussian shap []. Lyapunov Mthod: As it was mntiond bfor, th Lyapunov mthod is a tim domain pross and for that rason th tim autoorrlation funtion of th Rayligh hannl is rquird. By applying Eq. to our losd loop modl, whr in th Eq. A now rprsnts th fast fading and, its autoorrlation is givn by Eq.. π R n = nt Eq ( ) ( ) A A is th standard dviation of th Rayligh signal. T is th sampling priod, whih is th lngth of a PCG (Powr Control Group, []) for th systm and is a paramtr xprssd as: f f v Eq = = ln ln From Eq., w an propos Eq., whr th paramtr α an b idntifid asily bfor bing insrtd to th Eq. via th paramtrs a and b. n R( n) = α Eq A Sptrum Intgration Mthod: From th ztransform of th autoovarian of R(n) (Eq.), an quivalnt xprssion for th Rayligh powr sptrum is obtaind: P B ( ω ) = A ( α ) + α α osω Eq By inluding th powr sptrum P B (Eq.) into Eq., th Sptrum Intgration Mthod will also propos a solution for fast fading. As in th Shadowing as [], th systms dfind by (Eq. & Eq. or Eq.), an now b solvd numrially for th Rayligh hannl. For a partiular powr ontrol stp siz and MS spd, th solution to ah systm will b givn by th intrstion of th two urvs orrsponding to a partiular valu of K. Analytial Prforman of CLPC In ordr to vrify th rsults from both thniqus undr fast fading, w hav also simulatd th onvntional CLPC (shown in th Figur ) basd on th UMTS standard, whr th powr ontrol priod (or PCG) hosn is.ms and th stp siz is fixd to db []. By stting th lattr paramtrs, aording to th UMTS standard, th Lyapunov and th Sptrum Intgration Mthod rsults ar also produd. As mntiond bfor, th prforman of th systm is analyzd through th standard dviation of th PCE y. Th simulation assumptions and rsults ar idntial to []. Th rsults of all thr mthods ar rprsntd in th Figur. Rsults from th two analytial mthods 89//$. IEEE. 9

3 (Lyapunov & Sptrum Intgration) follow thos obtaind by simulations basd on th UMTS standards valus. B(z) + Y(z) Q(Y) Z(z) z X(z) z n Figur : Standard Dviation of th Rivd PCE for ConvCLPC Th thr urvs (or thir PCE standard dviations) prsnt th ommon haratristis to onvrg towards th standard dviation valu of th Rayligh hannl, ( A =.db) for high mobil vloity. This xplains that for high dopplr frquny th CLPC annot trak ffiintly th fast fading []. Noti, sam typ of onvrgn was obsrvd for shadowing []. Th thr plots math ah othr losly but th imprision btwn th analytial mthods and th simulation mthod is du to th assumptions mad by th analytial mthods as wll as xat auray of th simulation mthod. Th analytial mthods produ ths rsults muh fastr than th simulation []. Thrfor, to analyz th CLPC for diffrnt paramtrs, suh as stp siz, dlay fft or othrs vry rapidly, th analytial mthod boms thrfor a powrful tool ompard to th lngthy simulations. Quantizd stp siz powr ontrol So far only th prforman of th fixdstp siz powr ontrol algorithm was valuatd. It has bn provd that th proposd analytial modl prforman is rlativly los to th simulation prforman. W an thrfor optimis th xisting algorithm (onvntional CLPC) via th analytial modl. W hav implmntd and analysd th pross of quantization in th powr ontrol algorithm. Th powr ontrol rror is now quantizd, so that th stp siz an adapt to an xat valu. Th quantizr onsists of b lvls, whr b is th numbr of bits usd for a powr ontrol ommand. Th dgradation in prforman originats from th dlay introdud in th loop, sin now th powr ontrol ommand onsists of mor than on bit; a dlay is introdud in th loop. In gnral, tim dlays ar primarily of two kinds. First thr is a dlay du to th tim to masur and rport th masurmnts to th algorithm and sondly thr is a tim dlay du to th tim it taks bfor th omputd powr lvl is atually usd in th transmittr. Hr w onsidr mainly th sond on. Figur : Blok diagram for th Quantizd stp siz powr ontrol algorithm. W now assum that a quantizr rplas th hard limitr and th nw stat quation (driving from Eq.) is: X ( n + ) = X ( n) + Q( Y ( n k)) Eq, whr Q(.) dnots th opration of quantization pross. Th PCE dlayd by k powr ontrol ommands (or slots in UMTS standard) is xprssd as Y( n k) = B( n k) X ( n k) Eq For th analysis of th losd loop prsntd in Figur, th statistial approah prsntd in [8] is adoptd. Th blok diagram of Figur is th orrsponding mathmatial modl. It is assumd that th quantization rror is random in natur and that it is addd to th original signal as nois. Th rror { q (n)} is a stationary whit nois squn uniformly distributd in [ /, /], whr is th quantizr stp siz. E B(z) + Y(z) Yq(z) X(z) z Figur : Mathmatial modl of Quantizd stp siz powr ontrol algorithm In othr words th rror sampls ar unorrlatd and th rror squn { q (n)} is unorrlatd with th signal squn. Th nois powr is thn givn by = = Eq Pn E p( ) d =, whr R = and R is th rang of th quantizr. b Sin th systm dpitd in Figur is linar, th varian of th powr ontrol rror an b found in th sam way usd in Eq.: π π Eq 8 Y = HYB( ) PB + E HYE( ) π π π π and th nw transfr funtions ar: j ( ) = Eq 9 ω H YB ( ) + n n Eq H YE ( ) = n + z n 89//$. IEEE. 9

4 Th sptrum of th Rayligh hannl is givn by P B (ω) (Eq.) and onsidrd for ths hannl valus ( A =.db, f =GHz, T=.ms). Som intrsting rsults rgarding th fft of th quantizr rang, numbr of bits usd and dlays in th prforman of th CLPC hav bn obtaind. standard dviation of PC rror (db) 8 Quantizd stp siz Rang=PCG Dlay= bit usd bit usd bit usd bit usd 8 9 mobil spd (kmph) Figur : Efft of numbr of bits usd Figur shows th fft of th numbr of bits usd in th quantization pross on th standard dviation of th powr ontrol rror. Figur and Figur 8 illustrat th fft of th rang and loop dlay rsptivly. standard dviation of PC rror (db) Quantizd stp siz Numbr of bits usd=pcg Dlay= Rang= Rang= standard dviation of PC rror (db) 8 Quantizd stp siz Rang Numbr of bits usdpcg Dlay PCG dlay PCG dlay 8 9 mobil spd (kmph) Figur 8: Efft of dlay As th numbr of bits b usd inras, th powr ontrol algorithm shows bttr prforman. This is xptd sin it is provd from Eq that th nois powr is invrsly proportional to b. Howvr thr is no point in using a lot of quantization lvls to improv prforman. Thr bits ar nough as shown in Figur. From this point, th inras in prforman dos not justify th dlay imposd. Sin mor than on bit ar now usd for th powr ontrol ommand transmission, thr is an unavoidabl dlay in th updat of th mobil powr. It should b xptd that as th dlay inrass, th prforman of th powr ontrol shm gts wors (Figur 8), sin th hannl gain is badly trakd. Bfor to prod furthr, w hav fixd som of ths paramtrs to analys th rdibility of this mthod vrsus th simulation. Compar STD of rivd powr Quantizd SS Analytial vs Simulation 8 9 mobil spd (kmph) Figur : Efft of Quantizr input rang Th obsrvation of th Figurs, & 8 hlps th optimisation of th Quantizr paramtrs, namly rang and numbr of lvlsbits usd, as wll as in th visualization of th loop dlay fft, whih is unavoidabl sin now mor than on bits pr powr ontrol ommand ar transmittd. As th input rang R inrass, th powr ontrol algorithm shows wors prforman. This is xptd sin it is provd from Eq that th nois powr is proportional to R. PCE Sigma = STD QSS simul QSS math 8 8 Spd km/h Figur 9: Quantizd CLPC Analytial vs Simulation mthod From th Figur 9 it an b sn that th rsults obtaind analytially ar vrifid by thos obtaind via simulation. In this lattr ah powr ontrol ommand onsists of thr bits (thrfor th stp siz an hav ight diffrnt valus) and th rang is fixd to db. Th PDF of th rivd powr at th BS (rprsntd by QSS simul in Figur 9) for diffrnt MS vloity is also drawn in th Figur. 89//$. IEEE. 9

5 probability kmh kmh kmh kmh PDF of rivd powr for Quantizd Figur : PDF of QuantCLPC spd in km/h Anothr important onlusion oms from th omparison of th fixdstp powr ontrol algorithm against th proposd quantizd algorithm. For a bttr prforman omparison of both algorithms, w hav gathrd th rsults of Figur & 9 in Figur. PCE Sigma = STD Compar STD of rivd powr Analytial vs Simulation FSS simul QSS simul FSS math QSS math 8 8 Spd km/h Figur : Fixdstp vs Quantizdstp algorithms In almost all ass th quantizdstp shm (QSS) outprforms th fixdstp shm (FSS). Sin th quantizd stp siz an tak a valu losr to th aurat valu rquird by th BS to hang th MS transmit powr, it is obvious that this algorithm prforms bttr than th onvntional mthod. Not, that in this prforman th dlay produd du to th xtra numbr of bits usd, is not affting th gain obtaind. Howvr, as th mobil spd inrass (from km/h to km/h) and th fading boms fastr, th fixd stpsiz is not nough to ompnsat for th fading sin biggr stps ar ndd []. In this as th quantizd stp siz prforms bttr sin it an tak biggr valus as wll. Nvrthlss, abov km/h, th rat of th hang in th fast fading is too high, and th rror rivd at th BS is vry los to th rror du to th Rayligh fading std. But at this spd, thniqus suh as intrlaving ar xptd to improv th systm prforman. Conlusion This papr has analytially valuatd th prforman of th CLPC usd in a CDMA systm. Sin th valuation is usually don by lngthy simulations, th analytial mthod provids fastr rsults. W hav shown th wllknown diffiulty to valuat this xprssion for th as of fast fading. W hav proposd a fully analytial mthod undr fast fading, whih provids los rsults ompard to th simulatd ons. This mthod allowd us to analys th quantizd stp siz algorithm and its prforman was also analytially valuatd vrsus th simulation. W hav shown that th quantizd mthod is an intrsting tool to optimis th prforman. For an optimum prforman, a ompromis must b mad btwn th numbr of quantization lvls and th dlay imposd. An improvmnt to this work would b th us of diffrntial quantization in ordr to rdu adaptivly th dynami rang and onsquntly th numbr of bits to transmit th powr ontrol ommand. Also thr hav bn many analysis on th CLPC basd on adaptiv stp siz, but any fast hangs in th transmit powr ould afft th intrfrn stabilisation (or onvrgn) at th systm lvl. Th proposd modl is appliabl to any CDMA systms rgarding an appropriat hoi of th hannl, suh as Riian hannl for Satllit Communiations basd on CDMA systms. Rfrns [] Sim M.L., Gunawan E., Soong BH., Soh CB., Prforman Study of ClosLoop Powr Control Algorithms for a Cllular CDMA Systm, IEEE Trans., VT8, No., 99, 999 [] Vitrbi, A. and Padovani, R. Impliation of Mobil Cllular CDMA, IEEE Communiations Magazin, pp. 8, Dmbr 99 [] L. Song, N. Mandayam, and Z. Gaji, " Analysis of an Up/Down Powr Control Algorithm for th CDMA Rvrs Link undr Fading ", IEEE JSAC Wirlss Sris, vol. 9, No., pp. 8, Fbruary [] S.Nourizadh, P.Taaghol, R.Tafazolli, A Novl ClosdLoop Powr Control for UMTS, G Mobil Communiation Thnologis Conf. IEE, pp9, Marh. [] Gudmundson, M Corrlation Modl for Shadow Fading in Mobil Radio Systms, Eltronis Lttrs, vol., pp., Novmbr 99 [] M. Patzold, A Dtrministi Mthod for th Drivation of a Disrt WSSUS Multipath Fading Channl Modl, submittd to Europan Transations on Tlommuniations and Rlatd Thnologis, 99. [] A. A. Huttr, A Simpl Smart Antnna Approah to Rdu th Error Floor for Mobil OFDM Systms, Intrnational onfrn on Tlommuniations ICT. [8] Proakis, J.G and Manolakis, D.G Digital Signal Prossing, Prinipls, Algorithms and Appliations, Prnti Hall Intrnational, 99 89//$. IEEE. 9