PAPR REDUCTIO TECHIQUES I OFDM SYSTEMS USIG DCT AD IDCT 1 S. SUJATHA P. DAAJAYA 1 Rsarch Scholar, Dpartmnt ECE, Pondichrry Enginring Collg, Pondichrry, India Profssor, Dpartmnt of ECE, Pondichrry Enginring Collg, Pondichrry, India E-mail: 1 sujatha.s@cmrit.ac.in, pdananjayan@pc.com ABSTRACT Orthogonal frquncy division multiplxing (OFDM) has gaind much popularity in th fild of wirlss communication bcaus of its ability to transfr th data at highr rat, high bandwidth fficincy and its robustnss to multipath. Sinc OFDM has high Pak to Avrag Powr Ratio (PAPR) as major limitation, many tchniqus wr proposd to rduc it. In this papr, a DCT basd modifid slctiv mapping (SLM) tchniqu is proposd to rduc th PAPR of th transmittd signal and its PAPR rduction prformanc compard with that of IFFT basd modifid SLM tchniqu, IDCT basd modifid SLM tchniqu and IDST basd modifid SLM tchniqu. Th proposd tchniqu for PAPR rduction grants an improvmnt ovr th xisting IFFT. Kywords: OFDM, PAPR, Slctiv Lvl Mapping, Discrt Cosin Transform, IDCT, IDST. 1. ITRODUCTIO Orthogonal Frquncy Division Multiplxing (OFDM) is a multi-carrir modulation tchniqu for high data rat wirlss communication bcaus of its robustnss to frquncy slctiv fading, high spctral fficincy and lss computational complxity [1, ]. OFDM sparats th availabl spctrum into a numbr of ovrlapping but orthogonal narrowband sub-channls as shown in Figur 1, and hnc th OFDM convrts a frquncy slctiv channl into a non-frquncy slctiv channl. Th prim advantag of OFDM lis in its ability to cop with svr frquncyslctiv fading du to multi-path without complx qualization filtrs. On of th major drawbacks of OFDM transmission is high Pak to Avrag Powr Ratio (PAPR). Figur: 1 OFDM modulation High PAPR signals ar usually undsirabl for it usually strains th analog circuitry. A larg PAPR brings disadvantag lik an incrasd complxity of th analog to digital convrsion and digital to analog convrsion and rducs th fficincy of RF Powr amplifir PAPR of OFDM signal incrass with incras in th numbr of subcarrirs which causs poor fficincy and dgrads th systm prformanc of powr amplifir [3]. To ovrcom such ffct, many PAPR rduction tchniqu such as coding distortion and distortion lss tchniqus hav bn proposd for OFDM signals [4]. Distortion lss tchniqus such as Slctiv Lvl Mapping (SLM) [5] and Partial Transmit Squnc (PTS) [6] can improv th PAPR statistics with only a small data rat loss but has no inhrnt rror control. In th SLM, th input data bit squncs ar multiplid by ach of th phas squncs to gnrat altrnativ input symbol squncs. Each of ths altrnativ input data squncs is mad th Invrs Fast Fourir Transform (IFFT) opration, and thn th on with th last PAPR is slctd for transmission [7]. Although som tchniqus of PAPR rduction hav bn summarizd in [8], an ffctiv PAPR rduction tchniqu should b givn th bst trad-off btwn th capacity of PAPR rduction and transmission powr. In OFDM systm if th subcarrirs ar of larg valu th ral and imaginary valu of complx bas band signal bcom Gaussian distribution and th amplitud of OFDM signal has Rayligh distribution and xhibits strong fluctuations. Thrfor, th rsultant PAPR can b r high. In th worst cas, PAPR of an OFDM systm 597
with sub channls may rach up to a valu of. High PAPR valu has som disadvantags for OFDM modulation. High PAPR rsults from th natur of th modulation whr multipl subcarrirs and sinusoids ar addd togthr to th signal to b transmittd. Whn sinusoids add, th pak magnitud would hav a valu of, whr th (rms) avrag might b quit low du to dstructiv intrfrnc btwn th sinusoids. High PAPR signals would rquir a larg rang of dynamic linarity from th analog circuits, which usually rsults in xpnsiv dvics, and highr powr consumption/lowr fficincy for xampl, powr amplifir has to oprat with largr back off to maintain linarity and nds to b rducd to an accptabl lvl. In th litratur various mthods ar proposd for th purpos of PAPR rduction. It should b notd that most of th mthod ar basd on th sam ida of slcting th tim domain signal to b transmittd from a st of diffrnt rprsntations with th constraint of minimization of PAPR which would dgrad th prformanc of systm. This tchniqu can b catgorisd into Signal scrambling tchniqus and Signal distortion tchniqus. Th fundamntal principl of tchniqus is to scrambl ach OFDM signal with diffrnt scrambling squncs and slct on which has th smallst PAPR valu for transmission. This typ of approach includs Slctiv Mapping (SLM) and Partial Transmit Squncs (PTS). SLM mthod applis scrambling rotation to all sub-carrirs indpndntly whil PTS mthod only taks scrambling to part of th subcarrirs. Ppin Magnangana Zoko Goyoro, t al proposd a [3] schm which consists of DCT matrix transform followd by th SLM using Rimann matrix for th phas squnc to rduc PAPR. Howvr to rduc PAPR and computational complxity furthr DCT basd modifid SLM tchniqu is proposd in this papr.. DCT basd modifid slctiv mapping tchniqu is usd to rduc th PAPR. This mthod combins Discrt Cosin Transform (DCT) with modifid Slctiv Lvl Mapping (SLM). In DCT basd modifid SLM Invrs Discrt Sin Transform (IDST) and Invrs Discrt Cosin Transform (IDCT) ar also usd along with IFFT Th papr is organizd as follows. Sction brifly introducs PAPR of OFDM signals, Sction3 givs an ovrviw of SLM tchniqu is dscribd. Th dtaild dscription of DCT basd OFDM tchniqu is shown in Sction 4 and 5. Th discussions and simulation rsults ar prsntd in Sction 6. Finally, conclusions ar givn in Sction7.. PAPR OF OFDM SIGALS Th PAPR of transmittd OFDM is dfind as th ratio btwn th maximum instantanous powr and th avrag powr, dfind by: ( ) m ax x t PAPR = (1) E x ( t ) Whr x (t) dnots an OFDM signals aftr IFFT, and E [.] dnots xpctation. Lt us dnot th data block of lngth as a vctor Y= [Y 0, Y 1, Y Y -1 ] T whr is qual to th numbr of subcarrirs and (.) T dnots transpos. Th duration of a data symbol in X modulats on of a st of subcarrirs, {f m, m =0, 1,., -1}.Th subcarrirs ar chosn to b orthogonal, that is f m= m f, whr f=1/t and T is th duration of an OFDM data block.th complx nvlop of th transmittd OFDM signal is givn by 1-1 jπkδft x ( t ) = Y k () k=1 whr Y k is th data symbol carrid by th k th subcarrir. According to th limit cntral thorm, whn is larg, both ral and imaginary part of s(t) is Gaussian distributd. Th Cumulativ Distributd Function (CDF) of th signal is F (z) = 1 -z (3) If thr ar subcarrirs in an OFDM systm, and all th sampling valus ar complt indpndnt, th CDF of th systm is givn by th quation: ( ) ( ) 1 xp ( ) ( ) P PAPR z = F z = z (4) So in cas of no ovr sampling, th Complmntary Cumulativ Distribution Function (CCDF), this is usually usd as an important paramtr to dscrib th PAPR of an OFDM signal which is writtn as follows 598
PAPR0 ( 0 ) 1 ( 1 ) P PAPR > PAPR = (5) whr PAPR 0 is th clipping lvl. This quation is rad as th probability that th PAPR of a symbol block xcds som clip lvl PAPR 0. SLM is on of th probabilistic tchniqus adoptd to rduc th PAPR of th OFDM signal. Hnc it can achiv PAPR rduction without distorting th signal and will not caus any loss of data. Th main disadvantag of SLM is that th complxity is high. ow thr ar many xtnsion schms for rducing th complxity of SLM [13-16]. Slctiv Lvl Mapping As shown in Figur, In SLM tchniqu th input data is partitiond into Y data block of lngth.thn th OFDM data block is multiplid lmnt by lmnt with phas squnc E (u) =[ u,0, u,1,. u,-1 ] T whr u=1,, U, U phas rotatd OFDM data block Y u is obtaind. 3.1 DCT Transform Th pak valu of th auto corrlation is th avrag powr of input squnc. DCT concptually xtnds th original -point data squnc to -point squnc by doing mirror xtnsion of th -point data squnc. Sinc th both nd of data is always continuous in th DCT, th lowr ordr of componnts will b dominatd in th transform domain signal aftr convrtd by DCT. Th DCT is a Fourir-lik transform, which was first proposd by Ahmd t al. [17] [0]. Th ida to us th DCT transform is to rduc th autocorrlation of th input squnc to rduc th pak to avrag powr problm and th transmittd signal dos not rquir any sid information at th rcivr. In th sction, w brifly rviw DCT transform. Th 1D discrt cosin transform (1D DCT) A[k] of a squnc a[n] of lngth is dfind as: 1 π ( n+ 1) k A k = a k n= 0 a[ n] cos (7) For k = 0,1... - 1, th invrs DCT is dfind as ( n+ 1) 1 π k a n = n 0 a[ k] A k = cos (8) n= 0, 1-1 whr a[k] is dfind as: 1 fo r k = 0 a [ k ] = fo r k = 1,,... 1 (9) Figur. Slctiv lvl mapping schm All phas rotatd OFDM data blocks rprsnt th sam information as th unmodifid OFDM data block providd that th phas squnc is known. 1 j π k ft k m ( ) Y k Y t = (6) Among th phas rotatd OFDM data blocks on with th lowst PAPR is slctd and transmittd. Th information about th slctd phas squnc should b transmittd to th rcivr as sid information. At th rcivr, rvrs opration should b prformd to rcovr th unmodifid OFDM data block. In ordinary SLM tchniqu, thr is no rstriction on th construction of phas squnc s. Howvr, w st a structural limitation on th phas squnc for modifid SLM. Th basis squncs of th 1D DCT ar ral, discrt-tim sinusoids dfind by: C ( n 1) π + k n, k = co s (10) Th DCT basis consists of th following ral squncs. C n,0, C n,1,... C n n, 1 (11) Th quation (7) is xprssd in matrix A = C a (1) Whr A and a ar both th vctor with x1 and C is a DCT transform matrix with x. Th row (or column) of th DCT matrix C ar orthogonal matrix vctors. Thn w can us this proprty of th DCT matrix and rduc th pak powr of OFDM signals.dct can rduc th autocorrlation btwn th ach componnt of 599
OFDM signal this is th root caus to rduc PAPR. 3. DCT with Modifid Slctiv Lvl Mapping DCT along with Slctiv Lvl Mapping bcom an fficint PAPR rduction tchniqu by intgrating SLM and DCT matrix transform. PAPR is rducd by modifying th OFDM signal without any distortion, but still th complxity of SLM is high. For vry OFDM fram, SLM tchniqu rquirs n IFFT opration and this opration maks th systm complicatd. So as to prvail ovr systm complxity of SLM, modifid SLM is proposd. Th modifid SLM rducs IFFT block and also th PAPR. This tchniqu compriss of an IFFT block at th transmittr nd and th dcision of slcting data with lowst PAPR is accomplishd using a dcision algorithm bfor IFFT. Th algorithm for modifid SLM is givn as follows, Th block diagram of DCT with modifid SLM using IFFT is shown in Fig. and th corrsponding output signal is dscribd by assum, s = [ s, s s ] T 0 1 n 1 as a discrt tim OFDM signal vctor. Thn, th IFFT of vctor s taks th form as, s = QS (13) whr, Q symbolizs th IFFT matrix and it can b rprsntd as, [13] 1 1 L 1 Q = 1 M 1 jπ / n M jπ ( n 1)/ n L O L jπ ( n 1)/ n M jπ ( n 1)( n 1)/ n (14) Thus, as column matrix IFFT matrix Q can b xprssd as, [ Q Q Q ] Q L (15) = 0 1 n 1 In this proposd systm, DCT with modifid slctiv lvl mapping tchniqu is considrd to rduc th PAPR in OFDM systm. Th squnc of procss is givn blow. Figur. 3 Modifid SLM with IFFT Algorithm i. Lt X 1, X,, X n b th Binary information blocks. ii. Assum W as th ncoding cod word. iii. Evry block is ncodd into w using Hamming ncodr. iv. A control bit is appndd to w and xtndd hamming cod of 8-bits is calculatd. v. Th rror tabl and cost ladr is computd. vi. Vctors w+1, w+,, w+16 ar constructd for (vry cod word) Calculat A = X + Y + Z. vii. Cod word that has minimum A is chosn and transformd into OFDM signal through constllation mapping and IFFT. i. At th transmittr nd, th sourc data is forwardd to th linar block ncodr ii. To th outcom of ncodr DCT is applid and th transformd data is procssd by modifid SLM unit iii. Th modifid algorithm illustratd in algorithm- 1 is invokd iv. Th information with low PAPR is slctd and output is gnratd 3.3. LIEAR BLOCK CODES Whn th rror control coding and OFDM modulation procss work togthr such systm is calld COFDM. Th purpos of taking adjacnt bits in th sourc data is to sprad thm out across multipl subcarrirs. On or mor subcarrirs may b lost or impaird du to a frquncy null and this loss would caus a continuous stram of bit rror. Such an rror is a burst of rrors would typically b hard to corrct. Th main purpos of th modifid SLM tchniqu is to rduc PAPR and IFFT, IDCT block. Thr is only on IFFT and on IDCT block at transmittr.th squnc which is th lowst 600
PAPR can b obtaind out by a dcision algorithm bfor IFFT and IDCT. Considr an [n, k] Linar cod C with parity-chck matrix H, whr n is th lngth and k is th dimnsion of W. Sinc Hw t =0 for any codword w W, any vctor X +w has th sam syndrom as, that is [] t t t Hx = H( + w) = H (16) A binary information squnc is dividd into blocks of 4 bits. Each mssag block is ncodd into a codword W which is 7 bits by a [7, 4] hamming ncodr. Th paramtrs of th binary hamming cods ar typically xprssd as a function of a singl intgr u (for u=3, w hav a (7, 4) Hamming cod) not ncssarily prim, it is any positiv intgr. A hamming cod on GF () has cod lngth n= u -1, mssag lngth k= u -1-u, rdundancy n-k=u and rror conncting capability t=1bit. Hamming cods ar only singl rror corrcting. To improv th rror dtction and connction capability by adding parity chck digit. Th rsulting cod is calld th xtndd binary hamming cod. Suppos that w is a cod ovr th alphabt {0,1}. Lt w b th cod obtain by adding a singl charactr to th nd of ach word in w in such a way that vry word in w. has vn wight. Th parity chck matrix of [8, 4] xtndd hamming cod w is H: (17) According to th formula S= H T, th syndroms which ar corrsponding to th nonrror and on rror pattrns could b obtaind. And othr svn two rrors pattrns could b obtaind from th othr syndroms. So th standard array of c. is constructd. Th standard array an [n, k] binary linar cod C is a X Y array and for xtndd array an [8, 4] for binary linar cod c. is also X Y array. whr X=u-K, Y=K. At last sixtn vctors ar constructd as w +1, w +... w+ 16, whr 1 =0 and 1,, ----- 16 ar proprly slctd as th cost ladrs of th standard array in trms of thir PAPR. Thn th Dcision critrion is usd to calculat th valu of A. Finally, th scrambld codword with th minimum A is slctd and thn transformd to an OFDM signal by constllation mapping and IFFT. Tabl 1 Standard array of [n, k] linar cod In this array thr ar M rows and ach row is a cost w dnots th codword and dnots th rror in transmission. This critrion is usd for ach codword to calculat th valu. Finally th codword with th minimum valu is slctd and thn transformd to an OFDM signal by constllation mapping and IFFT. At th rcivr, th rcivd signal is convrtd into r by FFT and constllation d-mapping. Th syndrom calculatd from r is usd for stimating th cost ladr chosn at th transmittr. Th codword c is obtaind by calculating w= +r and thn is convrtd into a mssag squnc of k bits. Th block diagram is shown in Figur. Instad of using th modifid SLM with IFFT, th proposd tchniqu xamins th prformanc nhancmnt of modifid SLM with IDCT. 3. MODIFIED SLM WITH IVERSE DISCRETE COSIE TRASFORM OFDM systm is mployd considring orthogonal basis of complx xponntial function st. But, OFDM can also b implmntd using a singl st of Cosinusoidal function as an orthogonal basis. This Cosinusoidal function is intgratd along with a DCT, and hnc this schm is trmd as DCT- OFDM and th output signal is givn as, [] n 1 iπ n x ( n ) = d s i D i co s n i = 0 T s (18) In quation (18) ds0, ds1, L ds n rprsnt th 1 indpndnt data symbols that ar attaind as of a (17) 601
modulation constllation. T s symboliz th sampling intrval. D i taks th valu as, 1 i = 0 D i = 1 i = 1,..... n 1 (19) Th systm that uss modifid SLM with Invrs Discrt Cosin Transform (IDCT) is picturizd in Fig. 3. In this work, th data stram is transformd by DCT. It can linarly transform data into th frquncy domain, whr th data can b rprsntd by a st of cofficints. Th advantag of DCT is that th nrgy of th original data may b concntratd in only a fw low frquncy componnts of DCT dpnding on th corrlation in th data.and thn th transformd data is procssd by th SLM unit whr ach data block is multiplid by C diffrnt numbr of phas squnc vctors. If th data passd by DCT matrix bfor IDCT, th autocorrlation cofficint of IDCT input squnc is rducd with low complxity and no sid information is rquird bcaus th matrics can b gnratd to rcovr th original data at th rcivr and th data is transmittd with lowst PAPR. Figur 3: Modifid SLM with IDCT 5. MODIFIED SLM WITH IVERSE DISCRETE SIE TRASFORM (IDST) Th OFDM systm can also b stablishd via a singl st of Sinusoidal functions as an orthogonal basis. This systm is incorporatd using a discrt Sin transform (DST). Thrfor, this schm is dscribd as DST-OFDM schm. According to this schm th output signal can b lucidatd as, [] n 1 iπ n x( n) = dsi Di sin n i= 0 Ts (0) Th OFDM systm that uss modifid SLM with Invrs Discrt Sin Transform (IDST) is illustratd in figur-4 6. SIMULATIO RESULTS AD DISCUSSIO Th analysis of th modifid SLM with DCT, IFFT, IDCT and IDST has bn carrid out using MATLAB softwar. Th ntir simulation paramtrs considrd for this analysis is summarizd in Tabl 1 as follows. Tabl : Simulation paramtrs Simulation paramtr Typ/valu umbr of subcarrirs() 18,56,51,104 Modulation Schm QAM Phas wighting factor() 16,64 Coding tchniqu Linar block coding In th OFDM Systm undr considration, modifid SLM tchniqu is applid to th uncodd information in th sub blocks, which is modulatd by QAM modulation. Th prformanc valuation is don in trms of CCDF. Fig. 5 shows th PAPR rduction prformanc using convntion SLM mthod. In SLM th numbr of blocks prsnt rprsnts as th sam information. As th numbr subblock (D) incrass it is sn from th Fig.5 that th PAPR of th signal dcrass from 9.6dB to 7.9dB. Figur 4: Modifid SLM with IDCT 60
10 0 CCDF (Pr[PAPR>PAPR0]) 10-1 Figur 5: CCDF of PAPR of convntional SLM systm Figurs 6, 7, 8 and 9 shows th prformanc of modifid SLM with IFFT, IDCT and IDST for diffrnt subcarrirs = 18,56,51 and 104. It givs th bttr PAPR rduction at =18 compard to all othr subcarrirs. Th Figur 6 shows th prformanc rsults of modifid SLM with Linar coding for subcarrir whn =18.th graph shows th comparison of PAPR using thr tchniqus IFFT= 4.1db, IDCT=.3db and IDST=1.6db. From th rsult th prformanc of IDCT and IDST is improvd whn compard to IFFT. As th IDCT and IDST matrix ar ral and orthogonal. Th symmtry of an orthogonal matrix indicats that algorithms for th forward and invrs transform computation will b th sam. Hnc it prforms fast multiplication and rducs th complxity. Th rsults of modifid SLM with IFFT, modifid SLM with IDCT and modifid SLM with IDST ar compard to find th bst tchniqu to obtain rducd PAPR OFDM systm. It is obsrvd that PAPR valu is rducd using th proposd tchniqu compard to xisting tchniqu. 10 0 DCT-IFFT DCT-IDCT DCT-IDST 10-0 0.5 1 1.5.5 3 3.5 4 4.5 PAPR0 [db] Figur 7: Comparison of CCDF of PAPR of OFDM systm with modifid SLM whn =56 CCDF (Pr[PAPR>PAPR0]) 10 0 10-1 DCT-IFFT DCT-IDCT 10 - DCT-IDST 0 0.5 1 1.5.5 3 3.5 4 4.5 PAPR0 [db] Figur 8: Comparison of PAPR of OFDM systm with modifid SLM whn =51 10 0 DCT-IFFT DCT-IDCT DCT-IDST CCDF (Pr[PAPR>PAPR0]) 10-1 CCDF (Pr[PAPR>PAPR0]) 10-1 DCT-IFFT DCT-IDCT 10 - DCT-IDST 0 0.5 1 1.5.5 3 3.5 4 4.5 PAPR0 [db] Figur 6: Comparison of CCDF of PAPR of OFDM systm with modifid SLM whn =18 10-0.5 1 1.5.5 3 3.5 4 4.5 5 PAPR0 [db] Figur 9: Comparison of PAPR of OFDM systm with modifid SLM whn =104 7. COCLUSIO In this papr, PAPR analysis in DCT basd OFDM is prformd. This papr considrd a Discrt 603
Cosin Transform (DCT) basd modifid Slctiv Lvl Mapping (SLM) as an ffctiv tchniqu to rduc Pak to Avrag Powr Ratio (PAPR) in OFDM. In that, instad of using th modifid SLM with Invrs Fast Fourir transform (IFFT), this tchniqu xamins th prformanc nhancmnt of modifid SLM with Invrs Discrt Cosin Transform (IDCT) and Invrs Discrt Sin Transform (IDST). PAPR analysis is mad using thr tchniqus. Th tchniqu is simulatd in MAT lab. Modifid SLM tchniqu with DCT improvs th prformanc of OFDM signal with rspct to PAPR.As thr is an incrasing dmand for fficint frquncy spctrum utilization.ofdm provs invaluabl dmand to nxt gnration communication systm. REFERECES: [1]. S. H. Mullr and J. B. Hubr., A novl pak powr rduction schm for OFDM In procdings of IEEE intrnational Symposium on prsonal, Communication (PIMRC), 1090-1094, 1997. []. M.Paul and H.P.Kuchmbchr., On th rduction of th out of band radiation of OFDM signals.in Procdings of IEEE intrnational Confrnc on Communication (Jun 1998), 1304-1308 (1998). [3]. Ppin Magnangana Zoko Goyoro, Ibrahim Jams Moumouni., Sroy Abouty, SLM Using Rimann Squnc Combind with DCT Transform for PAPR Rduction in OFDM Communication Systms, World Acadmy of Scinc, Enginring and Tchnology 64 01 [4]. Osama A. Omr and Ahmd Abdl-Rhm., Adaptiv Hybrid PAPR Rduction in OFDM Systm, Intrnational Journal of Futur Computr and Communication, Vol., o. 3, Jun 013. [5]. M.Briling, S.H Mullr, and J.B Hubr., (001). SLM pak powr rduction without xplicit sid information, IEEE Communication Lttrs, (Jun 001), 6-18. [6]. A.D.S Jayalath and C.Tllambura., Adaptiv PTS approach for rduction of pak to avrag powr ratio of OFDM signals, Elctronics lttrs,(july 000),6-18,000. [7]. R. W. Bauml, R. F. H. Fishr, and J. B. Hubr, Rducing th Pak-to-Avrag Powr Ratio of Multicarrir Modulation by Slctd Mapping, IEE Elctronics Lttrs, vol. 3, no., pp. 056 057, Oct.1996.. [8]. Xiang, Wi, Julian Russll, and Yafng Wang., ICI rduction through shapd OFDM in codd MIMO-OFDM systms. Intrnational Journal on Advancs in Tlcommunications 3.3 and 4 (011): 194-05. [9]. Moussa Diallo, Rodrigu Rabinau, Laurnt Cariou and Marylin H lard, On improvd DCT basd Channl Estimation with vry low complxity for MIMO-OFDM systms, procdings of vhicular tchnology, 009 [10]. S. Litsyn and G.Wundr, Gnralizd bounds on th crst-factor distribution of OFDM signals with applications to cod dsign, IEEE Trans. Information Thory, vol. 5, no. 3, pp. 99 1006, Mar. 006. [11]. Arslan and Husyin, "Channl stimation for wirlss OFDM systms." IEEE Survys and Tutorials 9. (007): 18-48. [1]. MRA BAIG and VARU JEOTI, On th PAPR Rduction In OFDM Systms: A ovl ZCT Prcoding Basd SLM Tchniqu, Journal of Enginring Scinc and Tchnology Vol. 6, o. 3 (011) 357 368. [13]. Zhongpng Wang, Combind DCT and Companding for PAPR Rduction in OFDM Signals, Journal of Signal and Information Procssing, 011,, 100-104. [14]. P.Mukunthan and P. Dananjayan, Modifid PTS with FECs for PAPR rduction of OFDM signals, Intrnational Journal of Computr Applications, vol. 11, no. 3, pp. 38-43, Dcmbr 010. [15]. K,.Sathananthan and C.Tllambura,. ovl adaptiv modulation schm to rduc both PAR and ICI of an OFDM signal, Procdings in 6 th Intrnational symposium on DSP for Communication Systms,(Sydny, Australia),pp. 9-33, Jan, 00. [16]. Kyvan Kasiri, Iraj Hossini, Omid Tahri, Mohammad Javad Omidi, and P. Glnn Gulak, A Prprocssing Mthod for PAPR Rduction in OFDM systms by Modifying FFT and IFFT Matrics, Th 18th Annual IEEE Intrnational Symposium on Prsonal, Indoor and Mobil Radio Communications, (PIMRC'07), 007. [17]..T.Ahmd, atarajan and K.R.Rao, Discrt Cosin Transform, IEEE Transactions on Computr, vol.3, PP.90-93, 1974. 604
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