Gaussian and Flat Rayleigh Fading Channel Influences on PAPR Distribution in MIMO-OFDM Systems

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1 Gaussian and Flat Rayligh Fading Channl Influncs on PAPR Distribution in MIMO-OFDM Systms Basl Rihawi and Yvs Louët IETR/Supélc - Campus d Rnns Avnu d la Boulai, BP 87, 355 Csson-Sévigné, Franc Tlphon: + ( , Fax: +( basl.rihawi,yvs.lout@rnns.suplc.fr Abstract Multipl-input multipl-output orthogonal frquncy division multiplxing systms (MIMO-OFDM has bcam a promising candidat for high prformanc 4G broadband wirlss communications. Howvr, lik OFDM, on main disadvantag of MIMO-OFDM is that th signals transmittd on diffrnt antnnas may xhibit a prohibitivly larg pakto-avrag powr ratio (PAPR. In this papr, th influnc of gaussian and flat Rayligh fading channl on th PAPR of MIMO-OFDM passband signals at th rcivr-sid hav bn analyzd. A mathmatical xprssion for complmntary cumulativ distribution function (CCDF of th PAPR has bn drivd which dpnds on th numbr of transmitting antnnas and th paramtrs of channl. I. INTRODUCTION Multipl-input multipl output (MIMO radio systms hav attractd considrabl intrst and hav bn studid intnsivly sinc Tlatar [] du to its potntial of achiving high data rats in multipath channl. It is shown that whn th multipl transmit and rciv antnnas ar usd to form a MIMO systm, th systm capacity can b improvd by a factor of th minimum numbr of th transmit and rciv antnnas compard to a singl-input singl-output (SISO systm with flat Rayligh fading or narrowband channls [], [3]. For high data rat wirlss widband applications, MIMO combind with orthogonal frquncy division multiplxing (OFDM is bing considrd in a larg numbr of currnt tchnology applications. Important challngs, howvr, rmain in th fficint implmntation of ths kind of systms. In this papr, on of th inhrnt disadvantags of using OFDM is considrd: th tim-domain signal, consisting of a numbr of indpndntly modulatd subcarrirs, xhibits a high pak-to-avrag powr ratio (PAPR [4], which can rsult in clipping and nonlinar distortion in th powr amplifir (PA and low nois amplifir (. Svral studis hav analyzd th PAPR at th transmittr-sid (at th input of th PA. In [5], th PAPR in MIMO singl-carrir systms has bn considrd. Th objctiv of this papr is to study th influnc of th channl (gaussian and flat Rayligh fading on this PAPR at th rcivr-sid (at th input of th. This papr is organizd as follows: th scond sction prsnts som dfinitions rlatd to PAPR. In th third sction, MIMO-OFDM systm usd in this study is dscribd. Th main ida of this papr is discussd in fourth and fifth sctions, whr th influnc of gaussian channl and flat Rayligh fading channl on th CCDF of th PAPR of MIMO-OFDM passband signals at th rcivr-sid is analyzd. Som simulation rsults ar prsntd. Finally, a conclusion will b mad. II. OFDM SIGNAL AND PAPR DEFINITIONS Bfor procding furthr, lt us dfin th notations usd throughout th papr. Frquncy, basband tim and passband tim domain vctors ar dnotd by capital, small (. and small bold lttrs rspctivly. Scalars ar dnotd by small normal lttrs. In OFDM modulation, a block of N data symbols S k whr k = 0,,..., N, gathrd in a S vctor, will b transmittd in paralll such that ach on modulats a diffrnt subcarrir from a st {f k, k = 0,,..., N }. Th N subcarrirs ar orthogonal, with f k = k f, whr f = /T and T is th original symbol priod. Th rsulting analog OFDM complx nvlop s(t ovr on symbol of duration T can b xprssd as s(t = N S k jπfkt, t [0, T ]. ( N k=0 Th ral transmittd signal s(t is thn xprssd as s(t = R[ s(t jπfct ]. ( whr f c is th RF (radio-frquncy carrir frquncy and R(. rprsnts th ral part. Th basband modulation is don in th digital domain using an ovrsampld vrsion of s(t givn by s(n/l = N S k jπkn NL, t [0, NL ], (3 LN k=0 whr L is th ovrsampling factor. Notic that whn L =, s(n is th Nyquist-sampld vrsion of s(t. In practic, S is transformd into a discrt-tim vctor s of componnts s k whr k = 0,,..., N, via an invrs

2 fast Fourir transform (IFFT as s = IF F T (S. (4 Thn w can obtain th signal s(t by using a digital-toanalog convrtr (. Complx tim-domain sampls of th OFDM signal ar approximatly Gaussian distributd du to th statistical indpndnc of carrirs (cntral-limit thorm. This mans that thr can b som vry high paks prsnt in th signal thos can b quantifid by th PAPR (for signals RF which is dfind by P AP R{s(t} = max t [0,T ] R[ s(tjπfct ] E{ R[ s(t jπfct ] }, (5 whr E{u} dnots th xpctd valu of u. In this papr, to valuat th PAPR accuratly from a statistical point of viw, th complmntary cumulativ distribution function (CCDF of th OFDM PAPR signal is usd to quantify th probability of xcding a givn thrshold P R 0. It is dfind by CCDF P AP R (P R 0 = P r[p AP R > P R 0 ]. (6 In a classical OFDM contxt, th CCDF of PAPR of th sampld basband signal is approximatly givn by P r[p AP R{ s(n} > P R 0 ] ( N. (7 This last quation is a vry good approximation of th PAPR distribution of s(n, but diffrs as much as on db from th PAPR distribution of s(t. On of th attmpts to dtrmin th PAPR distribution of s(t cam from [6], whr it is statd that P r[p AP R{ s(t} > P R 0 ] (.8N. (8 Th.8 valu has bn obtaind by simulations, whr th PAPR of s(n/l approachs th PAPR of s(t for larg L. In practic L = 4 is nough to dtct th prsnc of continuoustim paks with satisfactory prcision [7]. III. SYSTEM MODEL Th MIMO-OFDM systm usd in this study is illustratd in Fig., whr N OFDM subcarrirs ar considrd.,, and rprsnt digital-to-analog convrtr, high powr amplifir, and low nois amplifir, rspctivly. Each of th subcarrirs is indpndntly modulatd using a quadratur amplitud modulation (QAM. data mapping spac tim ncodr OFDM Modulator OFDM Modulator MIMO channl In this work, th computation of th CCDF of th PAPR at th rcivr is considrd and is approximatd using a discrt-tim CCDF, which is obtaind from th sampls of th rcivd signal. Th sampling rat is th Nyquist rat or a multipl (i.. ovrsampling. Th rasons for this discrttim approach for CCDF computation ar : i th simulation tools ar implmntd in discrt-tim, ii computation in contiuous-tim is too complx as closd-form xprssion, and iii whn an ovrsampling rat of four tims th Nyquist rat is usd, it closly approximats continuous-tim CCDF of th PAPR. Thrfor, this approach is to mploy discrttim approximation of continuous-tim CCDF by using a sampling rat that is four tims th Nyquist rat. Th discrttim computation of CCDF is thn approachd by sampling th nvlop of th rcivd signal y(t (s Fig., which is quivalnt to sampling th complx modulatd signal. A Nyquist rat of N sampls ovr th OFDM symbol intrval is takn. To simplify th notations in th rst of this papr, w dnot th continuous-tim signals by small bold lttrs and thir discrt-tim sampls by small normal lttrs. Th signal s at on of th transmitting antnna is s (t = R[ s (t jπfct ] = s I (t cos(πf c t s Q (t sin(πf c t. (9 whr s is th OFDM basband signal, f c is th carrir frquncy of passband signal and I, Q dnots th ral and imaginary parts of th basband signal rspctivly. Each discrt-tim sampl of th transmittd signals s and s (Fig. follows a gaussian law with zro man and varianc s. Thir probability dnsity function (pdf is of th form f s (s = f s (s = s s. (0 πs whr s = E{s } = s I = s Q. In this work, idal powr amplifirs with gain G = ar considrd at th transmittr. IV. PAPR ANALYSIS IN MIMO-OFDM-AWGN CHANNEL Lt us analyz th CCDF of th PAPR of th rcivd signal y (t at on rciving antnna in th cas of AWGN channl (s Fig.. This signal can b xprssd as y = s + s + b, ( whr b is an additiv whit Gaussian nois (AWGN. By considring z = y, th pdf of any sampl of z is thn givn by [9] f z (z = z f y ( z U(z = z πy z y U(z, ( Fig.. Structur of th two-antnna MIMO-OFDM systm whr y is qual to s + s + b, f y follows a gaussian law as xprssd in (0 and U is th Havisid function.

3 Thn, w can obtain th cumulativ distribution function of z by intgrating ( as F z (z = πy z z 0 v v y dv = rf(, (3 y whr rf(. dnots th rror function, dfind as rf(x = x t dt. π Thn, th CCDF of th PAPR of th continuous rcivd signal y is approximatly givn by P r[p AP R{y } > P R 0 ] (rf(.8n. (4 This rsult can b gnralizd : in a gaussian channl contxt and for MIMO-OFDM systms, th transmittd and rcivd signals at ach of transmitting and rciving antnnas rspctivly all follow gaussian laws. Th consqunc is that th gaussian channl dos not influnc th PAPR distribution of th rcivd signals, whatvr th signal to nois ratio valu is. Simulation rsults ar dpictd on Fig.3. V. PAPR ANALYSIS IN FLAT RAYLEIGH FADING CHANNEL A. Th flat Rayligh fading channl modl Th flat Rayligh fading channl is a much-usd modl for narrowband transmissions ovr wirlss and mobil communication channls. This modl has bn adoptd in this papr. On a flat Rayligh fading channl, all signal frquncis ar attnuatd by th sam factor. For any discrt-tim sampl, th rcivd signal y can b writtn as 0 y = αs + b. (5 whr α is th fading amplitud and s is th transmittd signal sampl. Th fading amplitud α can b dscribd as a random variabl, whos distribution follows a Rayligh law [8]. B. SISO-OFDM cas First of all, w will sarch th CCDF of th PAPR of th rcivd signal in SISO-OFDM systm sktchd in Fig.. Th pdf of x = αs, (whr s is an OFDM signal following a gaussian law with zro man and varianc s is [9] f x (x = + Aftr dvlopmnts, w find f α ( x s f s(s ds. (7 s f x (x = x αs. (8 α s Th distribution function of y is thn xprssd as F y (y = P r[x + b y] = = 4 [ + rf( y b x= y x b= b +αsy + α s rfc( b + α s y α s b f b (bf x (xdbdx b αsy α s rfc( b α s y ]. (9 α s b Considring z = y, thn w obtain F z (z = F y ( z F y ( z = z [rf( b b +αsy + α s rfc( b + α s y α s b b αsy α s rfc( b α s y ], (0 α s b whr rf c(. dnots th complmntary rror function dfind as rfc(. = rf(.. Thanks to th indpndnc of th N sampls, th probability that non of thm xcds P R 0 can b simplifid as P r[ max 0 n<n z(n P R 0] [ rf( b + b +αs α s rfc( b + α s α s b b αs α s rfc( b α s N ]. ( α s b data mapping OFDM modulator LO Rayligh Channl Thn, th CCDF of th PAPR of th continuous rcivd signal can b approximatd by Pm. P r[p AP R{y} > P R 0 ] [rf( b Fig.. Structur of th SISO-OFDM systm From th abov figur, w can writ y = αs + b = x + b, (6 whr α is a Rayligh random variabl, whos pdf is givn by f α (α = α α α α ; α 0. + b +αs Pm. α s rfc( b + α s Pm. α s b b αs Pm. α s rfc( b α s Pm. ].8N, α s b whr P m is th man powr of y givn by P m = α s + b. ( Fig. 3 dpicts th CCDFs of th PAPR of rcivd signal in two cass (AWGN and flat Rayligh fading channl. Ths

4 curvs ar obtaind by th simulation of 0 5 symbols with N = 04 subcarrirs modulatd with a 4-QAM modulation. Th signal-to-nois ratio is qual to 0dB. W st th ovrsampling factor L = 4. It can b shown from this figur that for a probability of 0, th flat Rayligh fading channl incrass th PAPR about 5dB compard to th gaussian channl. It can also b sn that th similitud btwn th thory and simulation is vry good. Pr(PAPR>PR rayligh, (simulation rayligh, (thorical AWGN, (simulation AWGN, (thorical PR0(dB Fig. 3. CCDFs of th PAPR of th rcivd signal in a SISO-OFDM (AWGN and flat Rayligh fading channl C. MIMO-OFDM cas Now, w analyz th CCDF of th PAPR of th signal y in a two-antnna MIMO-OFDM systm rprsntd in th Fig.. From this figur, a rcivd sampl of th signal can b xprssd as y = α s + α s + b = x + x + b = x + b. (3 whr x = α s and x = α s ar supposd indpndnt, and x = x + x. Th pdf of x can b writtn as f x (x = u= x = β β = f x (uf x (x udu γ x γ (β γ ; β γ x δ (δ + x 4δ ; δ = β = γ, (4 whr β = α. s and γ = α. s. Th CCDF of th PAPR of th continuous signal y in th cas of β γ can b approximatd by P r[p AP R{y } > P R 0 ] [ (β γ ((β γ Pm. rf( b +β b +β Pm. β rfc( b + β P m. βb β b β Pm. β rfc( b β P m. βb γ b +γ Pm. γ rfc( b + γ P m. γb +γ b γ Pm. γ rfc( b γ P m. ].8N, γb whr P m is th man powr of y givn by P m = (β + γ + b. (5 Th CCDF of th PAPR of y in th cas of β = γ = δ can b approximatd by whr and P r[p AP R{y } > P R 0 ] Pm. [ 4δ (4δ rf( b +λ rfc( b + δ P m. δb λ rfc( b δ P m. δb ].8N, (6 λ = (δ b δ P m. b +δ Pm. δ, λ = (δ b + δ P m. b δ Pm. δ. In th last cas, P m is th man powr of y : P m = 4δ + b. (7 Fig. 4 shows th CCDF of th PAPR for 0 5 symbols, with N = 04 subcarrirs modulatd with a 4-QAM modulation. Th ovrsampling factor is L = 4 and th signal-to-nois ratio is qual to 0dB. It can b noticd that th PAPR dcrass as a function of numbr of transmitting antnnas. Fig. 5 compars th CCDFs of th PAPR for svral valus of SNR = Pm b. It is obvious that th CCDF of th PAPR b incrass as a function of SNR. VI. CONCLUSIONS In this papr, th influnc of AWGN and flat Rayligh fading channl on th PAPR distribution in MIMO-OFDM systms has bn analyzd. By giving th mathmatical formulations and simulations, it has bn dmonstratd that th rcivd signal at any rciving antnna of MIMO-OFDM systms hav a PAPR distribution qual to that of transmittd

5 Pr(PAPR>PR SISO (simulation SISO (thorical MIMO(, (simulation MIMO(, (thorical MIMO(4,4 (simulation PR0(dB Fig. 4. CCDFs of th PAPR of th rcivd signal in SISO, MIMO(, and (4,4 (flat Rayligh fading channl,snr = 0dB REFERENCES [] I. E. Tlatar, Capacity of multi-antnna Gaussian channls, AT T Bll Labs Intrnal Tch. Mmo., Jun 995. [] G. J. Foschini, Layrd Spac-Tim Architctur for Wirlss communication in a Fading Environmnt whn using Multi-Elmnt Antnnas, Bll Labs Tch. J., vol., no., Autumn 996, pp [3] G. J. Foschini and M. J. Gans, On limits of wirlss communications in a fading nvironmnt whn using multipl antnnas, Wirlss Prsonal Communication, vol. 6, pp , March 998. [4] R. Van N and R. Prasad, OFDM Wirlss Multimdia Communications, Artch Hous, 000. [5] B. Rihawi, Y. Louët Pak to Avrag Powr Ratio analysis in MIMO systms, ICCTA 06, Damas, Syria, May 06. [6] R. Van N and A. d Wild, Rducing th pak-to-avrag powr ratio of OFDM, Proc. IEEE Vhicular Tchnology Confrnc, vol. 3, pp , 998. [7] J. Tllado, Pak to Avrag Powr rduction for Multicarrir Modulation, Ph.D. Thsis Dissrtation, Stanford Univrsity, Sptmbr 999. [8] J. G. Proakis, Digital Communications, fourth dition, pp. 80-8, 00. [9] A. Papoulis, S. U. Pillai, Probability, Random Variabls and Stochastic Procsss, fourth dition, McGraw-Hill, Pr(PAPR>PR SNR=0dB (simulation SNR=0dB (thorical SNR=0dB (simulation SNR=0dB (thorical SNR=30dB (simulation SNR=30dB (thorical PR0(dB Fig. 5. Comparisons of th CCDFs of th PAPR (thorical and simulation of th rcivd signal in MIMO(, (flat Rayligh fading channl for svral valus of SNR signals in th cas of gaussian channl. Whras, in th flat Rayligh fading channl, th PAPR is mor than that of th gaussian channl. Morovr, th PAPR in MIMO-OFDM is lss than that in SISO-OFDM systms, and it dcrass with th numbr of transmitting antnnas in th cas of flat Rayligh fading channl.