Moment-Based Bound on Peak-to-Average Power Ratio and Reduction with Unitary Matrix

Transcription

1 Mome-Based Boud o Peak-o-Average Power Raio ad Reducio wih Uiary Marix Hirofumi Tsuda Deparme of Applied Mahemaics ad Physics Graduae School of Iformaics Kyoo Uiversiy Kyoo Japa suda.hirofumi.38u@s.kyoo-u.ac.jp arxiv: v1 [cs.it] 18 Feb 019 Absrac Reducig Peak-o-Average Power Raio (PAPR is a sigifica ask i FDM sysems. To evaluae he efficiecy of PAPR-reducig mehods he complemeary cumulaive disribuio fucio (CCDF of PAPR is ofe used. I he siuaio where he ceral limi heorem ca be applied a approximae form of he CCDF has bee obaied. he oher had i geeral siuaios he boud of he CCDF has bee obaied uder some assumpios. I his paper we derive he boud of he CCDF wih o assumpio abou modulaio schemes. Therefore our boud ca be applied wih ay codewords ad ha our boud is wrie wih fourh momes of codewords. Furher we propose a mehod o reduce he boud wih uiary marices. Wih his mehod i is show ha our boud is closely relaed o he CCDF of PAPR. Idex Terms rhogoal Frequecy Divisio Muliplexig Peak-o-Average Power Raio Mome Boud Uiary Marix I. INTRDUCTIN Peak-o-Average Power Raio (PAPR is he raio of he squared maximum ampliude o he average power. I is kow ha i-bad disorio ad ou-of-bad disorio are caused by a large ipu power sice he oupu power is o-liear wih respec o a ipu power for a large power regime [1]. I is kow ha hese disorios reduce Sigal-o-Noise Raio (SNR ad he chael capaciy []. Sice sigals wih large PAPR ed o be disored i is demaded o reduce PAPR. Thus may mehods o reduce PAPR have bee proposed [3]- [6]. By defiiio PAPR depeds o a give codeword. However PAPR is ofe regarded as a radom variable sice a codeword ca also be regarded as a radom variable. To ivesigae performaces of PAPR-reducig mehods he complemeary cumulaive disribuio fucio (CCDF of PAPR is ofe evaluaed [7]-[11]. Therefore i is demaded o obai he form of he CCDF. Whe each codeword is radomly ad idepedely chose from a give disribuio ad he ceral limi heorem ca be applied approximae forms of he CCDF have bee obaied [] [1]. These resuls are based o ha a FDM sigal ca be regarded as a Gaussia process. Furhermore i has bee prove ha usual coded-fdm sigals ca be regarded as Gaussia processes [13]. he oher had i he case where he ceral limi heorem cao be applied a approximae form has o bee obaied. I he case where he ceral limi heorem cao be applied he upper bouds of he CCDF of PAPR have bee obaied [14]-[16]. I is expeced o achieve lower PAPR as he upper boud decreases. Furher classes of error correcio codes achievig low PAPR have bee obaied [14]. To obai upper bouds some assumpios are ofe required. e of usual assumpios is abou modulaio schemes. Thus wih a give modulaio scheme mehods o reduce PAPR have bee discussed. However here is a case where codewords do o belog o a popular modulaio scheme. For example afer applyig a ieraive clippig ad filerig mehod [10] [17] i is uclear wha modulaio scheme each symbol i codewords belogs o. I such a siuaio kow PAPR bouds could o be valid. Therefore i is demaded o obai a more geeralized boud uder o assumpio abou a modulaio scheme. I his paper we derive a upper boud of he CCDF of PAPR wih o assumpio abou a modulaio scheme. ur boud is wrie i erms of fourh momes of codewords. As a similar boud i has bee prove ha here is a boud which is wrie i erms of momes i BPSK sysems [15]. Therefore our resul ca be regarded as a geeralizaio of such a exisig resul. To reduce PAPR we apply he echique which has bee developed i Idepede Compoe Aalysis (ICA [18]. The mai idea of ICA is o fid a suiable uiary marix o reduce he kurosis which is a saisical quaiy wrie i erms of fourh momes. From his idea used i ICA i is expeced ha our boud ca be reduced wih uiary marices sice our boud is also wrie i erms of fourh momes of codewords. The kow mehods a Parial Trasmi Sequece (PTS echique ad a Selecive Mappig (SLM mehod are o modulae he phase of each symbol o reduce PAPR [11] [7]. Therefore hese kow mehods are o rasform codewords wih diagoal-uiary marices ad our mehod ca be regarded as a geeralizaio of hese mehods. II. FDM SYSTEM AND PAPR I his secio we show he FDM sysem model ad he defiiio of PAPR. Firs a complex basebad FDM sigal

2 is wrie as [19] s(= ( A k exp π j k 1 T 0 < T (1 where A k is a rasmied symbol K is he umber of symbols j is he ui imagiary umber ad T is a duraio of symbols. Wih Eq. (1 a radio frequecy (RF FDM sigal is wrie as ζ(=re{s( exp(π j f c { ( ( k 1 = Re A k exp π j T + f c where Re{z is he real par of z ad f c is a carrier frequecy. Wih RF sigals PAPR is defied as [14] [0] PAPR(c { Re = max 0 <T ( ( k 1 A k exp π j + f c T P av where c=(a 1 A... A K C is a codeword x is he raspose of x C is he se of codewords P av correspods o he average power of sigals P av = K E{ A k ad E{X is he average of X. he oher had wih basebad sigals Peak-o-Mea Evelope Power Raio (PMEPR is defied as [14] [0] PMEPR(c = max 0 <T ( A k exp ( (3 π j k 1 T. (4 P av As see i Eqs (3 ad (4 PAPR ad PMEPR are deermied by he codeword c ad i is clear ha PAPR(c PMEPR(c for ay codeword c. I [1] i has bee prove ha he followig relaio is esablished uder some codiios described below ( 1 π K r PMEPR(c PAPR(c PMEPR(c (5 where r is a ieger such ha f c = r/t. The codiios ha Eq. (5 holds are K r ad exp(π jk/r 1. I addiio o hese aoher relaio has bee show i [16]. From Eq. (5 PAPR is approximaely equivale o PMEPR for sufficiely large f c. Throughou his paper we assume ha he carrier frequecy f c is sufficiely large ad we cosider PMEPR isead of PAPR. Noe ha his assumpio is ofe used []. III. BUND F PEAK-T-AVERAGE PWER RATI I his secio we show he boud of a CCDF of PAPR. As see i Secio II PAPR ad PMEPR deped o a give codeword. Sice codewords are regarded as radom variables PAPR ad PMEPR are also regarded as radom variables. I wha follows we merely wrie PAPR i formulas whe PAPR is a radom variable. Firs we make he followig assumpios he probabiliy desiy of c p(c is give ad fixed. he carrier frequecy f c is sufficiely large. For 1 k l m K he saisical quaiy E{A k A l A m A exiss where z is he cojugae of z. The secod assumpio abou a carrier frequecy is ofe used []. As see i Secio II PAPR is approximaely equivale o PMEPR if he carrier frequecy f c is sufficiely large. Thus we cosider PMEPR isead of PAPR. The las assumpio has bee used i [16]. We call he quaiy E{A k A l A m A he fourh mome of A k A l A m ad A. For deails abou complex mulivariae disribuios ad momes we refer he reader o [] [3]. From he Cauchy-Schwarz iequaliy ad his assumpio i ca be prove ha he average power P av exiss ha is P av <. Le us cosider he PAPR wih a give codeword c = (A 1 A... A K. I [4] he followig relaio has bee prove where max s( ρ(0+ ρ(i= K i ρ(i (6 A k A k+i. (7 We leρ(k be 0. Noe ha he quaiyρ(0 is he power of a codeword ad ha he ime does o appear i he righ had side (r.h.s of Eq. (6. I is o sraighforward o aalyze Eq. (6 sice he absolue-value erms appear i Eq. (6. To overcome his obsacle we obai he upper boud of r.h.s of Eq. (6. From he Cauchy-Schwarz iequaliy we obai he followig relaio max s( ρ(0+ The above boud is rewrie as { max s( 4 (K 1 ρ(i K 1 ρ(0 + ρ(i. ρ(0 + The r.h.s of Eq. (9 is rewrie as { (K 1 ρ(0 + ρ(k =(K 1 = K 1 K 1 + { { K 1 K 1 ρ(i ρ(k + ρ(k k ρ(k+ρ(k k ρ(k ρ(k k. (8. (9 (10

3 From he above equaios he r.h.s of Eq. (9 is wrie wih periodic correlaio erms ad odd periodic correlaio erms. These erms are wrie as ρ(k+ρ(k k=c B (k 11 c ρ(k ρ(k k=c B (k 11 c (11 where z is he cojugae raspose of z he marices B (k 11 ad B (k 11 are B (k 11 = ( I K k I k B (k 11 = ( I K k I k. (1 Sice hese marices are regular hey ca be rasformed o diagoal marices. From his geeral discussio hese marices are decomposed wih he eigevalue decomposiio as [5] B (k 11 = V D (k V B (k 11 = ˆV ˆD (k ˆV (13 where V ad ˆV are uiary marices whose (m -h elemes are V m = 1 ( exp π j m K K ˆV m = 1 ( ( m exp π j K K + 1 (14 K ad D (k ad ˆD (k are diagoal marices whose -h diagoal elemes are ( D (k = exp π jk ( K( ˆD (k = exp π jk K + 1 (15. K Wih hese expressios Eq. (9 is wrie as { max s( 4 K(K 1 α k 4 + β k 4 (16 whereα k adβ k are he k-h eleme of α ad β wrie as α= Vc ad β= ˆVc respecively. Wih he codeword c he above iequaliy is wrie as max s( 4 K(K 1 { ( (c V G k Vc + c ˆV G k ˆVc (17 where G k is a marix whose (k k-h eleme is uiy ad he oher elemes are zero. Noe ha G k G k= G k. For he laer coveiece we se C k = V G k V ad Ĉ k = ˆV G k ˆV respecively. Noe ha he marices C k ad Ĉ k are posiive semidefiie Hermiia marices sice C k ad Ĉ k are he Gram marices. From Eq. (17 wih a give codeword c he boud of he squared PAPR is obaied as PAPR(c max s( 4 P av K(K 1 P av { ( (c C k c + c Ĉ k c. (18 I he above relaios he firs iequaliy is obaied from he resul ha PAPR(c PMEPR(c. From he above discussios we have arrived a he boud of PAPR wih a give codeword c. From his boud we ca obai he boud of he CCDF of PAPR as follows. Le Pr(PAPR>γ be he CCDF of PAPR whereγ is posiive. The he followig relaios are obaied Pr(PAPR>γ = Pr(PAPR >γ Pr(max s( 4 > P avγ E{ max s( 4 P avγ K(K 1 Pavγ { ( E (c C k c + c Ĉ k c. (19 I he course of derivig Eq. (19 he firs equaio has bee obaied from he fac ha PAPR is posiive. The firs iequaliy has bee obaied from Eq. (4 ad he fac ha PAPR(c PMEPR(c for ay codeword c (see Secio II. The secod iequaliy has bee obaied wih he Markov iequaliy [6]. The las iequaliy has bee obaied from Eq. (17. As see i Eq. (19 he boud of he CCDF is wrie i erms of he fourh momes of codewords ad he boud does o deped o a modulaio scheme. Furher if each codeword c is radomly ad uiformly chose from he se of codewords C ad he umber of codewords is fiie he Eq. (19 is wrie as [16] Pr(PAPR>γ 1 K(K 1 C Pavγ c C { ( (c C k c + c Ĉ k c where C is he umber of compoes i C. IV. REDUCING PAPR WITH UNITARY MATRIX (0 I Secio III we have obaied he boud of he CCDF of PAPR. From Eq. (19 he boud is wrie i erms of he fourh momes of codewords. I is expeced ha PAPR decreases as he boud decreases. I his secio we propose a mehod o reduce he boud wih uiary marices. ur echique ca be see i ICA [18] [7] sice he mai idea of ICA is o reduce he kurosis which is wrie i erms of he fourh mome. I kow mehods i has bee proposed o modulae he phase of each symbol o reduce PAPR ad hese mehods are o rasform a codeword wih a diagoal-uiary marix. Thus our echique ca be regarded as a exesio of hese mehods. I addiio o he assumpios made i Secio III we make he followig assumpios o iroduce a echique o reduce our boud he umber of compoes i codewords C is fiie ha is C =M<.

4 each codeword is chose wih equal probabiliy from C. Uder he above assumpios we propose a mehod o reduce PAPR wih uiary marices. The mai idea of our mehod is o fid uiary marices which make our boud small. Through our echique he average power P av ad SNR are preserved. To iroduce our mehod we defie subses of codewords. Firs from he above assumpio we ca divide he codewords C io N disjoi subses which saisfy N C= C C m C = for m. (1 =1 Sice he umber of compoes i C is fiie each umber of compoes i C is also fiie. For each subse C we defie a uiary marix W. The scheme of our mehod is described as follows. Firs le he rasmier ad he receiver kow he uiary marices {W =1 N. A he rasmier side each codeword c C i is modulaed o W i c wih he uiary marix W i. The he rasmier seds he umber i ad W i c. A he receiver side he symbol y ad he umber i are received. The he receiver esimaes he codeword ĉ as ĉ=wi y. I is clear ha ĉ=c if y=w i c. Wih he above scheme he boud i Eq. (0 is wrie as Pr(PAPR>γ K(K 1 1 M N P avγ =1 c C { (c W C kw c + (c WĈkW c. ( I kow mehods a PTS echique ad a SLM mehod oe diagoal uiary marix correspods o oe codeword. By coras i our mehods oe uiary marix correspods o oe se of codewords. This is he mai differece bewee our mehod ad he kow mehods. Le us cosider he case where he chael is a Gaussia chael ad he codeword c C i is se. I such a siuaio he received symbol y is wrie as y=w i c+ (3 where is a oise vecor whose compoes follow he complex Gaussia disribuio idepedely. The he esimaed codeword is wrie as ĉ=c+ Wi. (4 From he above equaio SNR is preserved hrough our mehod sice he marix W i is uiary. We have show he mai idea of our mehod. The remaied problem is how o fid W which achieves low PAPR for =1... N. I our mehod uiary marices W are give as he soluios which make our boud i Eq. ( small. To aalyze our boud we defie ( f {W N =1 N = =1 c C { (c W C kw c + (c WĈkW c. (5 Noe ha he variables of he fucio f is he uiary marices {W N =1 ad ha f is a real fucio. ( To fid he {W N =1 achievig low PAPR we miimize f {W =1 N uder he codiio ha {W =1 N is uiary. To miimize f is gradie is ecessary. However i geeral expressios ivolvig complex cojugae or cojugae raspose do o saisfy he Cauchy-Riema equaios [8]. Thus he fucio f may o be differeiable. To avoid his he geeralized complex gradie of f is defied as [9] f W = f Re{W + j f Im{W (6 where Re{Z ad Im{Z are he real par ad imagiary par of he marix Z respecively. Wih his defiiio he gradie of f wih respec o W is calculaed as f W =4 c C { (c W C kw c C k + ( c WĈkW c Ĉk W cc. (7 Wih he above equaio we propose he followig gradie desce algorihm a he l-h ieraio W (l+1 W (l ǫ f W (l (8 where W (l is he marix obaied a he l-h ieraio ad ǫ is a posiive parameer. Wih he above ieraio we ca obai he marix W (l+1 from W (l. Sice he marix W (l+1 oo he se is o always uiary we have o projec W (l+1 of uiary marices. e mehod is o use he Gram-Schmid process [30] [7] [31]. Firs we decompose he marix W (l+1 as W (l+1 = (w 1... w K ad updae w 1 w 1 / w 1 where z is he l orm of z. The he followig seps are ieraed for k=... K 1 w k w k k 1 w k w iw i. w k w k / w k. Fially he projeced marix is obaied as W (l+1 = (w 1... w K. Wih he above ieraios we ca obai a uiary marix. However i is uclear wha order o choose ad o ormalize vecors. To avoid his ambiguiy a symmeric decorrelaio echique has bee proposed [30] [7] [31]. A symmeric decorrelaio echique is o ormalize W (l+1 i as ( ( 1/ W (l+1 W (l+1 W (l+1 (l+1 W (9 where (Z Z 1/ is obaied from he eigevalue decomposiio of Z Z = FΛF as FΛ 1/ F wih F beig a uiary marix Λ ad Λ 1/ beig diagoal posiive marices wrie as Λ = diag(λ 1 λ...λ K ad Λ 1/ = diag(λ 1/ 1 λ 1/...λ 1/ K respecively. Wih he above projecio we ca obai he uiary marix W (l+1. The algorihm of our mehod is summarized i Algorihm 1.

5 Algorihm 1: How o Fid Uiary Marix i ur Mehod 1 Se he iiial uiary marix W (1 for =1... N ad he ieraio cou l= 1. For =1... N calculae W (l+1 as 3 Projec W (l+1 W (l+1 f W (l W (l ǫ f ad obai he marix W (l oo he se of uiary marices for =1... N wih Gram-Schmid process or Eq. (9. 4 Le W be he orm of he marix W. If W (l+1 W (l 0 for =1... N or he ieraio cou l ges sufficiely large he sop. herwise se l l+ 1 ad go o sep. V. NUMERICAL RESULTS I his Secio we show he performace of our proposed mehod. As see i Secio II we assume ha he carrier frequecy f c i Eqs ( ad (3 is sufficiely large ad he PAPR is approximaely equivale o PMEPR. Thus we measure PMEPR isead of PAPR. We se he parameers as K = 18 ad M = 000. To measure PMEPR we choose oversamplig parameer J = 16 [1] [3]. The modulaio scheme is 16-QAM. All symbols are geeraed idepedely from he 16QAM se ad he we obai he se of codewords C={c 1... c M. I each N we se he subses of codewords as { C = c M N ( 1+1 c M N ( c M (30 N for = 1... N. Thus each subse of codewords is radomly obaied from he origial 16QAM se ad he umber of compoes i C is M/N for = 1... N. As iiial pois we se W (1 = E. where E is ideiy marix for =1... N. The gradie parameerǫ is se as ǫ= N/M 1/K. I Algorihm 1 we have used he symmeric decorrelaio echique described i Eq. (9. Figure 1 shows PAPR i our mehod wih he parameer N= 50. Each curve i he figure correspods o he ieraio imes. As see i his figure PAPR ges small as he ieraio ime icreases. This resul shows ha we ca obai he uiary marices which achieve lower PAPR as he ieraio ime icreases. Sice our mehod is o reduce he boud of PAPR i Eq. ( his resul implies ha decreasig he our boud may lead o decrease PAPR. We coclude ha our boud closely relaed o he CCDF of PAPR. Figure shows PAPR i our mehod wih he parameer N= 100. Similar o he resul wih he parameer N= 50 our mehod achieves lower PAPR as he ieraio ime icreases. However from Fig. 1 ad Fig. our mehod wih he parameer N = 100 achieves lower PAPR ha oe of our mehods wih N = 50 a each ieraio. The reaso may be explaied as follows. I our simulaios for c C codewords are radomly ad idepedely geeraed from 16QAM se. ad he average of c is 0. The he average of he quaiy cc is E{cc =E (31 where E is he ideiy marix. From he Cauchy-Schwarz iequaliy [33] he lower boud of he boud i Eq. (19 ca be wrie as K(K 1 P avγ K(K 1 Pavγ K(K 1 = Pavγ = K (K 1 P avγ { ( E (c C k c + c Ĉ k c Tr(E{C k cc + Tr (E { Ĉ k cc Tr(C k + Tr (Ĉk (3 where Tr(X is he race of X. I he above iequaliies we have used Tr(C k =Tr(Ĉ k =1ad Eq. (31. I is clear ha he above lower boud is ivaria uder he acio c Wc where W is a uiary marix. Le us cosider he siuaios of he simulaios wih N= 50 ad N= 100 ad defie he sample mea for each subse of codewords as g(c = 1 cc. (33 C c C Here C is he umber of compoes i he se C ad we have assumed ha c is radomly ad idepedely chose ad ha he quaiy he average of c equals o 0. The by he Law of Large Numbers he quaiy g(c may be closer o he ideiy marix as he umber of compoes i C icreases. From hese discussios if each subse C is radomly chose from C ad he umber of compoes i C icreases he he quaiy g(c is early equivale o he ideiy marix. I such a siuaio he lower boud i Eq. (3 may be igh. For hese reasos sice each umber of compoes i he subses wih N= 50 is larger ha oe wih N= 100 our mehod wih he parameer N= 100 achieves lower PAPR ha oe of our mehods wih N= 50 a each ieraio. VI. CNCLUSIN I his paper we have show he boud of CCDF of PAPR ad our proposed mehod o reduce PAPR. The mai idea of our mehod is o rasform each subse of codewords wih he uiary marix o reduce he boud of CCDF of PAPR. Furher he uiary marices are obaied wih he gradie mehod ad he projecig mehod. As see i Secio V i may o be sraighforward o reduce PAPR wih our mehod whe he quaiy g(c is early equivale o he ideiy marix. This obsacle may be overcome whe we choose efficiely he subses of codewords C. Therefore oe of remaied issues is o explore how o obai he subses of codewords C. Furher i is ecessary o explore oher mehods o reduce our boud.

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