Asymptotic Probability Bounds on the Peak Distribution of Complex Multicarrier Signals Without Gaussian Assumption*

Transcription

1 Asymptotic Probability Bouds o the Peak Distributio of Complex Multicarrier Sigals Without Gaussia Assumptio* Masoud Sharif ad Babak Hassibi Departmet of Electrical Egieerig Califoria Istitute of Techology Pasadea, CA, 9 25 Abstract Multicarrier sigals exhibit a large peak to mea evelope power ratio (PMEPR). I this paper; we derive lower ad upper pmbabilitj bouds for the PMEPR distributio whe etries of the codeword, C, are chose idepedetly from a symmetric q-arj PSK or QAM costellatio, C E Q:, or C is chose jmm a complex dimesioal sphere, 0, whe the umber of subcarriers,, is large ad wifhout ay Gaussia assumptio o either the joit distributio or ay sample of the multicarrier sigal. Eve though the worst case PMEPR is of the order oj, the mai result is that the PMEPR of a radom codeword C chose from Q: or R, is log with pmbability oe, asymptotical.v. We the obfai a Varshaiwv-Gilbert (VG) style boud for the achievable rate ad miimum Hammig distace of codes chosefrom Q:. with PMEPR of less tha log. It is proved that asymptotically. the VG boud remais the same for codes chose from e: with PMEPR of less tha log. Itroductio ratio (PMEPR) of multicaier sigals is of the order of where is the umber of suhcarriers. O the other had, the umerical evaluatio of the distributio of PMEPR shows that ecouterig as the worst case is highly ulikely [7]. This i fact motivates the problem of fidig the PMEPR distributio to quatify how severe is that. I [7], by assumig that the multicarrier sigal is a Gaussia process, a expressio for the probability distributio of PMEPR is derived. This is a vety strog assumptio which is mathematically ot valid whe the modulatig iformatio is chose from QWSK costellatio or from a complex dimesioal sphere. O the other had, by oly assumig that each sample of the multicarrier sigal has a Gaussia distributio, a upper houd for the complemetary cumulative distributio fuctio of PMEPR has bee derived i [SI. I this paper for large, we derive upper ad lower houds for the PMEPR distributio whe the modulatig codewords are chose from symmetric QAMPSK costellatios or from the dimesioal complex sphere without ay Gaussia assumptio either o each sample or o the whole process. The bouds are a geeralizatio of the result of Halasz for Littlewood trigoometric polyomials to ay symmetric complex coste~~atio ad also to spherical codes for polyomials over the uit circle [g], we the use Multicaier modulatio has bee proposed i differet wireless ad wirelie applicatios such as wireless local area etworks (") ad lie (DsL). these houds to determie the achievable rate of codes with Eve though multicaier modulatio has a very good performace i a multipath fadig eviromet, it suffers from high amplitude variatio which is ufavorable from a practical poit of view. Differet schemes have hee proposed to reduce the peakto mea evelope power ratio (PMEPR) such as codig methods, clippig, dummy caiers, ad selective mappig (SLM) [l, 2, 3, 4, 5,6]. Clearly, the worst case peak to mea evelope power *?his work was supported i pa by the Natioal Siece Foudatio uder grat o. CCR-03388, by the office of Naval Research uder grat o. NW , ad by Caltech's Lee Ceter for Advaced Networkig PMFPR To summarize the results, we show that with probability oe, ay codeword either with etries chose idepedetly from a symmetric QAMPSK costellatio or chose uiformly from a complex sphere has PMEPR of log i O(logog) for a large umber of suhcaiers. Based o this result, we determie the achievable rate of codes with bouded PMEPR. We move that there exist codes with high rate ad large miimum Hammig distace as log as the PMEPR is less tha log for sufficietly large. The rest of the paper is outlied as follows. Sectio 2 i /02$ IEEE 7

2 troduces the otatio, multicaier sigals, ad the PMEPR of a codeword. The lower ad upper probability bouds for the PMEPR distributio are derived i Sectio 3. I Sectio 4 we discuss the cosequeces of the bouds ad we obtai a Varsharmov-Gilbert type boud for the achievable rate of codes with bouded PMEPR ad with give miimum Hammig distace. Fially, Sectio 5 cocludes the paper with further results ad ope problems. 2 Defiitios The ormalized complex evelope of a multicarrier sigal with subcarriers may be writte as where C = (cl,..., c,) is the complex modulatig vector with etries from a give complex costellatio &. The admissible modulatig vector is called codeword ad the esemble of all possible codewords costitute the code C. The peak to mea evelope power ratio (PMEPR) of each codeword C is defied as, Similarly, PMEPh is defied as the maximum of (2) over all codewords i C. I this paper, we will cosider two classes of codes, amely, complex symmetric q-ary codes i which each coordiate is chose from a complex symmetric costellatio icludig QAM ad PSK with q alphabets, C E Qr, ad spherical codes. By symmetry, we mea that the probability distributio fuctio for the costellatio is eve, equivaletly, each poit A has the same probability as -A i the costellatio. O the other had, codewords of a spherical code are poits o a complex dimesioal sphere defied as,,={(c,,..., C) EC":Clc<j2=}. (3) It is worth otig that E{llCll*} = E,, for radom q-ary codes where E,, is the average eergy of the costellatio Qq. Also, for spherical codes chose from R,, E{llC2} = sice all the codewords have costat orm. It is clear from the defiitio of PMEPR that if all the carriers add up coheretly, PMEPR ca be of the order of. However umerical results suggest that this barely occurs. The mai goal of this paper is to get a better isight to the distributio of PMEPR. Throughout the paper we use the followig otatios: C ad C represet a code family ad a codeword, ci deotes i=l the i'th coordiate of the codeword C, ad log is the atural logarithm. We use cy ad B as arbitrary costats ad f() = O(g()) deotes that lilimm\sl I ]a(. 3 Bouds for Cumulative Distributio Fuctio of PMEPR I this sectio, we cosider codewords are either chose fmm Q; or from R,. It is a commoly assumed i the literature that whe the cj's are idepedetly chose, sc(6) ca be approximated as a Gaussia process [7]. However, this is ot mathematically rigorous for spherical codewords ad codewords with etries from Qq. I other words, by assumig that the c;'s are idepedetly ad idetically distributed, eve though it is coceivable that ay fiite samples of sc(0) is joitly Gaussia for large values of, this statemet is ot valid for samples of sc(0). I this sectio without usig ay Gaussia assumptio, we derive upper ad lower bouds for PMEPR distributio for differet schemes. Derivatio of the bouds are the geeralizatio of the result of Halasz, for the asymptotic distributio of the maximum of E:=, a;cos (is)[ whe ai's are chose from {+I, -) with equal probability, to polyomials over the uit circle with coefficiet chose from Qq as well as spherical codes. 3. Upper Boud I the followig theorem, we fid a upper boud for the probability of havig a PMEPR slightly larger tha log ad we show that asymptotically this probability goes to zero. Theorem is for the symmetric QAh4 costellatios ad it ca be geeralized to symmetric PSK ad spherical codes as well [lo]. Theorem (Upper Boud-QAM Case). Let sc(0) = E:=, c,ejzaei where q = a; + jbi ad the ai's ad b3.s each have i.i.d ad eve disrributios. Also, let be the esemble of all the admissible codewords. The, for large values of, ad for ay radom codeword C chose from CfAM. Pr { PMEPR,QAM (C)>log+4loglog}< 0 Proof: Let's defie the real fuctio SC((Y, 6) as, 4) sc(a,o) = Re{ej"sc(0)} (5) As poited ot i [9], this defiitio eables us to deal with the maximum of a real fuctio ad geeralize the result of Halasz. Let's first defie K = max)sc(cy,o)l = maxlsc(0)l (6) a,@ E 72

3 We the defie a fuctio 0 < u(z) 5 as, where E,, = E{lc;*}, A = -, ad M = JE,, log + O(log log ). We also assume U(.) is a fuctio which is 0 times differetiable such that u")(z) = O(A-') for 5 T 5 0. Cosequetly, we defie the radom variable q as, where u(f) is the Fourier trasform of ~(2). O the other had, usig Bemstie iequality [4, ad the defiitio of q, we ca deduce that if q < &&, this implies that K 5 A4 + 2A [9],[]. Therefore, By computig the mea of the radom variable q, we ca the use the Markov ieaualitv to fid a lower boud for I _ Eq. (0). This is derived i [IO] ad by lettig M = JE,, log + O(loglog), it is show that theorem follows from (0) ad (). It is show i [IO] that the asymptotic aalysis i Theorem boils dow to the computatio of the characteristic fuctio of sc(b). For the PSK costellatio, we ca use a similar argumet to that of the QAM case sice the c;'s are idepedet. However, for spherical codes, the cj's are ot idepedet aymore, ad we istead use the followig lemma that derives the characteristic fuctio of sc(b) whe Cis uiformly distributed over,. The same aalysis goes through for computig the characteristic fuctio of sc(a, theta). Lemma. Let C = (cl,..., c,,) be a radom complex vector uiform.y distributed over R, ad s,(s) = Cy==, c;ejei. The Pmof: The proof relies OD the fact that the vector (CI,..., c) isotropically distributed ad so its distributio remais the same uder rotatio or multiplicatio by ui- tary matrices. Therefore, p(sc(b)) = p(< (cl,...,c,), (eis,...,ejs) >) = P(<Li(Cl,..., ctl)>(,0 I..., 0) >) = P(fiC) (3) sice we ca premultiply the Vadermode vector by a uitary matrix, U, ad rotate the vector s.t. U(eje,...,e'"') = (fi,o,...,0) without chagig the distributio of C. It ca be further show that by lettig ci = r,ej+<, the r ad are idepedet with the followig distributio, 2 2 -I P ( d = -r(-r), P($Jl) = 2;; The lemma follows by usig (3) ad (4). We ca ow use the asymptotic expasio of Bessel fuctios to write (I?) as, qejks,(m,e)} = e-(k2/2+ut + D(ke))(l + O(l/)) which allows us to geeralize the proof for the QAM case to spherical codes as well. 3.2 Lower Boud Theorem 2 obtais the probability of havig PMEPR slightly less tha log for QAM, PSK, ad spherical codes ad shows that this probability goes to zero as teds to ifiity. Theorem 2 (Lower Boud-QAM Case). Let sc(b) = Cy=, c;ej2ei where y = a; + jbi ad the a; s ad bi s be the esemble of all the admissible codewords. The. for large values of, ad for ay radom codeword C chose from C,QAM, I each have i.id ad eve distributios. Also, let CyAf Pr {PMEPR,PAM(C)< log-6.5loglog (3 Proof: The proof is similar to the upper boud without usig the parameter a. Sice we are lookig for a lower boud for Pr{PMEPRc(C) > A}, istead of cosiderig the maximum of sc(b) over all B, we may cosider the maximum of sg(b) = R{sc(B)} over its samples B, = x(2i + l)/ fori =,...,. We also agai defie U(.) 73

4 as i Theorem. The the radom variable q is defied as, 4 Varsharmov-Gilbert Boud for Codes with Bouded PMEPR I this sectio, we use the result of Sectio 3 to derive a Varsharmov-Gilbe type boud o the rate of code with give miimum distace ad PMEPR of less tha log. Here, we use the Hammig distace betwee two codewords defied as the umber of coordiates i which the two codewords are differet, dmi,(c',c2) = I{i : C: # c:, i =,...,}l, (22) ad miimum distace of C is the miimum of (22) over all codewords C' # Cz. The rate of the code family, C, is also defied as, R = ;log, (CI (23) where (CI is the cardiality of the set C. Therefore, the evaluatio of the lower boud boils dow to the asymptotic aalysis of the first ad secod momets of q. It is show i [lo] for M = JE,,log - 6.5loglog, E{q} O(log6), ad moreover,.;: 5 o(eiq} log2 + log5 ) (20) By replacig the mea ad secod order of q i (9), the theorem follows immediately. This theorem ca be also geeralized to symmetric PSK costellatios as well as spherical codes by usig their characteristic fuctio which is discussed i Eq. (2) [lo]. 3.3 Review of The Bouds To get better isight ito the above results, let Ct correspods to or C, defied as radom codes chose from symmetric QAWSK costellatios or radom spherical codes. Usig the iequality, Pr(A) + Pr(B) - 5 Pr(A B) 5 Pr(A) together with Theorems ad 2, we ca state the followig corollary: I Corollary. Let C be a radom vector whose etries are chose idepeder.v fmm a q-a? symmetric costellatio icludig PSK ad QAM or let C be radomly chose from Q, adct E {Cy',CrSK, C,} be the correspodig esemble of radom code words^ The, with pmbability oe, Theorem 3. Let Q, be a carplex q-ary symmetric PSK or QAM cosreliario, R > 0, ad _< 9, if we have R 5 - ffq(6) - O(;)> (24) the, asymptotically, there exists a code C of legth, with etries chose from Q,. rate R, miimum Hammig distaced,<, = L hj, adpmepb < logf4loglog. Proof: For the costructio of the code, we use the Varsharmov-Gilbert argumet as follows. Clearly, we have q" umber of codewords i which ba = gpr{pmeprc9 (C) > log ( log4 ]} umber of them have PMEPR of greater tha log ( log4 ). Moreover we defie V,(, t) as the maximum umber of q-ary dimesioal codewords withi Hammig distace t of a codeword which is equal to E:=, (;)(q - I)' [2J. Now we costrct the code C by first elimiatig the ba codewords with udesired PMEPR from the q codewords, ad the we select a codeword from the remai- ig codewords ad elimiate all codewords withi distace &i = 6 which is at most V,(, 6~~). Therefore, as. log as we have the followig iequality qrvqh PJ) +A < q", (25) we ca choose qr codewords with miimum distace at least 6 ad PMEPR of at most log ( log4 ). We ca rewrite Eq (25) as, Now takig the logarithm of both sides of (26), ad usig 74

5 lim log, V,(, l6l) = &(6) [2, we get, +m R 5 - H,(6) + -logq Pr{PMEPR < log(logy a)}. (27) Usig Theorem to houd the third term i (27), ad usig - log ( - z) = O(z) we ca deduce the Theorem. Accordig to Theorem 3, it follows that ot oly do there exist spherical high rate codes with PMEPR of 8log, hut there also exist codes chose from usual costellatios Lie QAM ad PSK with the same asymptotic. Similarly, accordig to the VG boud, we ca see that there is o pealty i terms of rate ad miimum distace of codes with bouded PMEPR up to log if the codewords are chose from symmetric PSK or QAM costellatios. 5 Future Work ad Coclusio As metioed before, multicarrier sigals may exhibit PMEPR of order. However, the result of this paper shows that ecouterig this worst case is highly ulikely. More precisely, asymptotically ad i the probability sese, the PMEPR is log for a large class of costellatios icludig QAWSK symmetric ad spherical codes. Now it is ot surprisig at all to have either spherical codes or codes chose from QAMPSK costellatios with high rate ad large miimum distace ad PMEPR of less tha log [5. O the other had, this result does ot cotradict the fact that there might be expoetially may codewords with costat PMEPR. However the ratio of the umber of these codewords to q has to ted to zero asymptotically. So there still remais a ope problem of what is the rate of codes with costat PMEPR? Aother issue would be to cosider the trade off betwee rate ad miimum distace of codes with bouded PMEPR. We kow that the Varsharmov-Gilbert houd remais the same by restrictig codewords to PMEPR less tha log. However, what will happe to the rate ad miimum distace of a code by reducig the PMEPR to less tha log, e.g. costat? Refereces [I] X. Li ad L. J. Cimii, Effects of clippig ad filterig o the performace of OFDM: i Pmc. IEEE Veh. Tech. Co$, May 997, pp [?I S. H. Muller ad J. B. Huber, A compariso of peak power reductio schemes for OFDM, i Pmc. IEEE Glob. Comm. Co$, 997, pp J. Tellado ad J. M. Cioffi, Efficiet algorithms for reducig PAR i multicarrier system, ipmc. IEEE Iter: Symp. Ifo., August 998, p. 9. [4 M. Sharif, M. Gharavi-Alkhasari, ad B. H. Khalaj, O the peak to average power of OFDM sigals based o oversamplig, to appear i IEEE Tras. Comm., [5 K.G. Paterso ad V. Tarokh, O the existece ad costructio of good codes with low peak to average power ratios: IEEE Tras. Iform., vol. 46, o. 6, pp , Sep (6 J.A. Davis ad J. Jedwah, peak to mea power cotrol i OFDM, Golay complemetary sequeces, ad Reed-Muller codes, IEEE Tras. Ifo., vol. 45, o. 7, pp , Nov [7 H. Ochiai ad H. Imai, O the distributio of the peak to average power ratio i OFDM sigals, IEEE Tras. Comm, vol. 49. o. 2, pp , Feb [8] M. Sharif ad B.H. Khalaj, Peak to mea evelope power ratio of multicarrier sigals: a aalytical approach: i Pmc. ofleeeicc, 200, pp [9] G. Halasz, O the result of Salem ad Zygmud cocerig radom polyomials, Studia Scie. Math. Hug., pp , 973. [lo] M. Shaif ad B. Hassibi, Asymptotic probability houds o the peak distributio of complex multicarrier sigals ad their implicatios, to be submitted to IEEE Tras. Iform., [Ill A. Gersho. B. Gopiath, ad A. M. Odlyrko, Coefficiet iaccuracy i trasversal filterig, The Bel/ Systems Techical Joural, vol. 58, o. 0, pp , Dec S. B. Wicker, Ermr cotrol systems for digital commuicatio ad storage, Pretice Hall: NJ, Eglewood cliffs,