Suitted Spectrl Precoding for Out-of-nd Power Reduction under Condition Nuer Constrint in OFDM-Bsed Syste Leing Pn 1 No.50 Reserch Institute of Chin Electronic Technology Group Corportion, Shnghi 200311, Chin. forz@liyun.co Astrct: Due to the flexiility in spectru shping, orthogonl frequency division ultiplexing (OFDM) is proising technique for dynic spectru ccess. However, the out-of-nd (OOB) power rdition of OFDM introduces significnt interference to the djcent users. This prole is serious in cognitive rdio (CR) networks, which enles the secondry syste to ccess the instntneous spectru hole. Existing ethods either do not effectively reduce the OOB power lekge or introduce significnt iterror-rte (BER) perfornce deteriortion in the receiver. In this pper, joint spectrl precoding (JSP) schee is developed for OOB power reduction y the trix opertions of orthogonl projection nd singulr vlue decoposition (SVD). We lso propose n lgorith to design the precoding trix under receive perfornce constrint, which is converted to trix condition nuer constrint in prctice. This ethod well chieves the desirle spectru envelope nd receive perfornce y selecting zero-forcing frequencies. Siultion results show tht the OOB power decreses significntly y the proposed schee under condition nuer constrint. Keyword: Spectrl precoding, Out-of-nd, Orthogonl frequency division ultiplexing (OFDM), Sideloe suppression, Condition nuer constrint. 1. Introduction Dynic spectru ccess [1, 2] technology is extensively studied s n effective schee to chieve high spectrl efficiency, which is crucil step for cognitive rdio (CR) networks. Due to the flexile operility over non-continuous nds, orthogonl frequency division ultiplexing (OFDM) is considered s cndidte trnsission technology for CR syste [3]. However, due to the use of rectngulr pulse shping, the power ttenution of its sideloe is slow y the squre of the distnce to the in loe in the 1
Suitted frequency doin. Therefore, the out-of-nd (OOB) power rdition or sideloe lekge of OFDM cuses severe interference to the djcent users. Furtherore, this prole is serious in CR networks, which enles the secondry users to ccess the instntneous spectru hole. Therefore, these secondry users need to ensure tht the interference level of the power eission is cceptle for priry users. In prctice, typiclly of the order of 10% gurd-nd is needed for n OFDM signl in the long ter evolution (LTE) syste [4]. Therefore, the spectru efficiency is significntly reduced. The trditionl ethod of sideloe suppression is sed on windowing techniques [5], such s the rised cosine windowing [6], which is pplied to the tie-doin signl wve. However, this schee requires n extended gurd intervl to void signl distortion, nd the spectru efficiency is lso reduced for lrge gurd intervls. The cncelltion crriers (CCs) [7, 8] technique inserts few crriers t the edge of the spectru in order to cncel the sideloe of the dt crriers. However, this technique degrded the signlto-noise rtio (SNR) t the receiver. The sucrrier weighting ethod [9] is sed on weighting the individul sucrriers in wy tht the sideloe of the trnsission signl re iniized ccording to n optiiztion lgorith. However, the it-error-rte (BER) increses in the receiver, nd when the nuer of sucrriers is lrge, it is difficult to ipleent in rel-tie scenrio. The ultiple choice sequence ethod [10] ps the trnsitted syol into ultiple equivlent trnsit sequences. Therefore, the syste throughput is reduced when the size of sequences set grows. Constelltion djustent [11] nd constelltion expnsion [12] re difficult to ipleent when the order of qudrture plitude odultion (QAM) is high. Strikingly, the ethods [10, 11] require the trnsission of side infortion. The dptive syol trnsition [13] schee usully provides wek sideloe suppression in frequency rnges closely neighoring the secondry user occupied nd. However, the schees [7-14] depend hevily on the trnsitted dt syols. The precoding technology is widely used in OFDM syste to enhnce the perfornce of trnsission reliility over the wireless environent, in which there re ny precoding ethods 2
Suitted proposed for OOB power reduction [15]. There re two in optiiztion schees with otining lrge suppression perfornce. One is to force the frequency response of soe frequency points to e zero y orthogonl projection [16-19]. These frequency points re regrded s zero-forcing frequencies. The other ethod [20, 21] is to iniize the power lekge in n optiiztion frequency region y dopting the qudrtic optiiztion ethod, using trix singulr vlue decoposition (SVD). The precoding schee in [16] is designed to stisfy the condition tht the first N derivtives of the signls re continuous t the edges of syols. However, this ethod introduces n error floor in error perfornce. The sideloe suppression with orthogonl projection (SSOP) ethod in [19] dopts one reserved sucrrier for recovering the distorted signl in the receiver. To intin the BER perfornce, dt cost is introduced in [17] y exploiting the redundnt infortion in the susequent OFDM syol. The sideloe suppression in [21] is sed on iniizing the OOB power y selecting soe frequencies in n optiized region. The suppression prole is first treted s trix Froenius nor iniiztion prole, nd the optil orthogonl precoding trix is designed sed on trix SVD. In [20], n pproch is proposed for ultiuser cognitive rdio syste. This ethod ensures user independence y constructing individul precoder to render selected spectru nulls. Unlike the ethods tht focus on iniizing or forcing the sideloe to zero, the sk coplint precoder in [22] forces the spectru elow the sk y solving n optiiztion prole. However, the lgorith leds to high coplexity. In this pper, spectrl precoding schee is proposed with trix orthogonl projection nd SVD for OOB reduction in OFDM-sed syste. The in ide of this schee is to reduce the OOB power under the receive qulity. The condition nuer of the precoding trix indictes the BER loss in the receiver. Therefore, we develop n itertion lgorith to design the precoding trix under trix condition nuer constrint. The proposed ethod hs n pproprite lnce ong suppression perfornce, spectrl efficiency nd receive qulity. Consequently, it is flexile for prcticl ipleenttion. 3
Suitted The rest of the pper is orgnized s follows. In Section 2, we introduce the syste odel of OOB power reduction y the spectrl precoding ethod. In Section 3, we present the proposed spectrl precoding pproch. Next, in Section 4, we provide n itertion lgorith to design the precoding trix ccording to the desirle spectru envelope, spectrl efficiency nd BER perfornce. Siultions re presented in Section 5 to deonstrte the perfornce of the proposed ethod, followed y sury in Section 6. d Precoding s IFFT Add Gurd DAC&UP Conversion Chnnel d Decoding s FFT Reove Gurd Down Conversion &ADC Fig. 1 Syste digr of spectrlly precoded OFDM. 2. Syste Model The lock digr of typicl OFDM syste using precoding technique is illustrted in Fig. 1. The nuer of totl crriers used in the trnsitter is M. The digitl spectrl precoding process efore inverse fst Fourier trnsfor (IFFT) opertion is expressed s s Pd, (1) where d is the originl OFDM syol of size N 1, nd s is the precoded vector. The size of the precoding trix P is M N ( M N ) nd the coding redundncy R M N is usully sll. The spectrl precoding ethod chieves etter suppression perfornce in zero-pdding (ZP) OFDM syste thn cyclic-prefix (CP) OFDM [19, 21]. Wht s ore, the ZP schee hs lredy een proposed s n lterntive to the CP in OFDM trnsissions [23] nd prticulrly for cognitive rdio [24]. The proposed ethod lso cn e directly pplied to CP systes with degrded perfornce on sideloe suppression. 4
Suitted Conventionlly, power spectrl density (PSD) nlysis for ulticrrier systes is sed on n nlog odel with sinc kernel function [25]. The PSD converges to the sinc function with the spling rte incresing. In ddition, the overspling constrint presented in [26] ensures the desirle power spectrl sideloe envelope property fter precoding for DFT-sed OFDM. In generl OFDM syste, the tie-doin signl cn e defined y rectngulr function (send-equivlent) s 1, 0 t T gt () (2) 0, elsewhere where T is the syol durtion. The frequency doin representtion of -th sucrrier is written s j T/2 sin(( ) T / 2) G ( ) e, ( ) / 2 (3) where is the center frequency of -th sucrrier. Therefore, the gnitude envelope in the OOB region ( ) is expressed y G ( ) sin(( ) T / 2) ( ) / 2 2. (4) Then we define the function C ( ) to indicte the gnitude envelope y 1 C ( ). (5) The expression (5) is otined fro (4) y ignoring constnt fctor. This opertion does not ffect the theticl nlysis for the design of the precoding trix [19]. The coplex coputing y (3) is converted to rel opertion, which reduce the coplexity of spectrl precoding in the following. Bsed on (5), using the superposition of M crriers, the trget function indicting the PSD t OOB frequency is defined y 5
Suitted M 1 A( ) s C ( ), (6) 0 2 where s is the -th eleent in the precoded OFDM vector s. In OFDM syste, the power spectru of its sideloe decys slowly s ( ) 2, where is the frequency distnce to the inloe. Fro (1) nd (6), the PSD trget function of the OFDM signl is expressed in the trix for s 1 1 P( ) ( ) c Pd, (7) T T s T A 2 where s c C0( ), C1 ( ),... C 1( ) T M, (.) T nd denote trnspose opertion nd expecttion respectively. P( ) nd C( ) re not the PSD presenttion of the OFDM signl nd the frequency response respectively, ut indicte their envelope chrcter. The gol of the precoding ethod is to design P to reduce the eission in OOB region. Siultneously, the process is irrespective of the vlue of vector d. 3. The Proposed Joint Spectrl Precoding (JSP) Method In this pper, joint spectrl precoding (JSP) ethod is proposed with two ties orthogonl projection nd one tie SVD. The process is given y three steps: Inner orthogonl projection SVD Outside orthogonl projection. As presented in the previous ppers [19, 21], in ech step, the in ide of orthogonl projection is to force the power of zero-forcing frequency points to e zero, nd the SVD opertion is to iniize the power in optiized region. However, the precoding trix derived fro the finl step does not chieve the gol in ech step gin. Unlike the ethods tht focus on iniizing or forcing the sideloe to zero, we reduce the OOB power under the receive qulity constrint y selecting the frequency points in ech step. Therefore, the proposed ethod hs n pproprite lnce etween the suppression perfornce nd the receive qulity. In ddition, when the nuer of the reserved crriers is sll, such s only one, then the nuer of zero-forcing frequencies in orthogonl projection ethod is one. But in the JSP, we set the zero-forcing frequency in inner nd outside orthogonl projection to e different nd etter suppression perfornce is otined. 6
Suitted reserved crriers dt crriers zero-forcing frequency optiized region Fig. 2 The spectru digr for OOB power reduction. Generlly, the sucrriers of OFDM-sed syste re considered to e continuous. The proposed ethod lso cn e used for non-continuous ulti-crriers syste. The nuer of eployed crriers is M nd the index is fro 0 to M 1. Fig. 2 illustrtes the frequency region of. The R reserved crriers t the upper nd lower spectrl edges, which re used to chieve sideloe suppression nd intin the receive qulity. Two groups of zero-forcing frequency points projection respectively. The corresponding nuer is nd N nd re selected in inner nd outside orthogonl N ( N R, N R). K frequency points in the optiized region re chosen to reduce the OOB power y SVD opertion. We first give the process of JSP in this section, nd the detil of how to select the preters is presented in the next. In the first step, the group of zero-forcing frequency points ( [,...,... ], 0 or 1 i N i i M 1 ) re chosen to chieve C ˆ Pd 0, (8) Where ˆd is vector of size M, derived fro the originl dt syol d N1 y dding zero in the loction of the reserved crriers, i.e., ˆ T T d[0,0,... d,...0,0]. C is the gnitude response trix of size N M, whose eleent C ( i, j) C ( ) coputed y (5). j is the index of the crriers. The solution of (8) is j i equivlent to p ˆd to the nullspce of C, In [27], the orthogonl projector P pping vector ontoc ( C is the nullspce of C ) long C is given y 7
Suitted P I C C C C (9) T 1 T M ( ), where I M is unit trix of size M M. This projector hs property tht the precoded dt vector Pd, ˆ in the nullspce of C, is closest to ˆd s in ˆ ˆ. ˆ P d d P dc (10) Oviously, P is non-orthogonl. Therefore, it will cuse significnt BER perfornce degrdtion in the receiver. After the step of the inner orthogonl projection, the precoded dt is given y s P d ˆ. (11) Then in the second step, the gnitude response trix of K frequency points o in the optiized region is written s Cop CoP, where C o is gnitude response trix of the K optiized frequency points. The eleents in C o is coputed y (5). We then use the SVD opertion to iniize the power lekge in the optiized region. The prole is deterined y P rg in C P. (12) o P op By decoposing the trix Cop into C U Σ V using SVD, the optil precoding trix Po to T op c c c chieve (12) is derived fro P V V V (13) R R1 M 1 o [ c, c,... c ], where i V c is the i-th colun of V c. P o is n orthogonl trix tht V c. The trix Po of size M T o o N N is coposed of the lst N coluns in P P I, where I N is unit trix of size N N. The originl dt d is pped to Pod, while the length is extended fro N to M. The verge power of ech syol of efore nd fter precoding re M 1 2 s i nd i0 P P N 1 s i0 2 i, where i is the i-th digonl eleent in Σ c nd P s 8
Suitted is the verge power of ech syol d. The second step is to ndon the lrgest R singulr vlues in then the OOB power is reduced fter precoding y Pod. Σ c, In the third step, the group of zero-forcing frequency points ( [,...,... ], 0 or 1 i N i i M 1 ) re chosen. The orthogonl projection trix P is otined siilr to (9) y P I C C C C (14) T 1 T M ( ), where the trix C of size N the precoding trix is otined y o. M, is the gnitude response trix of the frequency points. Finlly, P P P (15) After precoding, the iniu error prole of decoding in the receiver, is given y in PPd ˆ -d, (16) where ˆP is the decoding trix. The orthogonl projector P p the vector onto C, so rnk( P ) M N N, where rnk (.) denotes the rnk of trix. Due to rnk( P ) N o nd the vlue of N is sll, the cse of tht P is full colun rnk, is esy to e chieved in prctice. Then the pseudoinverse trix of P is the optil solution for (16) to chieve PP ˆ I, which is given y N ˆ ( ) T 1 T P P P P. This JSP schee is unlike the ethods tht iniize the OOB power or forcing the sideloe to zero. In the third step, we p Pdto o the nullspce of C rther thn the nullspce of CP. o If the nullspce of CP is selected, the JSP ethod turns to the trditionl orthogonl projection ethod, which introduces o lrge deteriortion in the receiver. Furtherore, fter the opertions in the lst two steps, the precoding trix P does not chieve the gol in the first step of pping d to the nullspce of C. In ddition, fter the opertion in the third step, P lso does not chieve the gol of iniizing the power lekge in the optiized region in the second step. 9
Suitted As presented in the SVD opertion [21] or orthogonl projection ethod [19], the receive qulity nd suppression perfornce hve not een well lnced. We lso hd soe tests to exine other cointions tht only eploying the first two steps or the lst two steps in the JSP ethod. The results indicte tht the suppression perfornce is siilr to only using orthogonl projection or SVD opertion. The reson is tht the etter OOB power reduction perfornce y the JSP ethod is otined y two ties orthogonl projection. In ddition, the BER perfornce is iproved y using reserved crriers in the second step. Wht s ore, the desirle spectru envelope lso cn e chieved y selecting the frequency points or optiized region in the three steps independently. 4. Design of The Precoding Mtrix Under Condition Nuer Constrint In this section, we develop n lgorith to design the precoding trix, otining the desirle spectru envelope under receive perfornce constrint. As illustrted in Fig.1, the dt fter FFT process in the receiver is expressed s s Qd +n, (17) where Q = HP, H is coplex digonl trix with the chnnel frequency response of M sucrriers nd n is coplex dditive white Gussin noise (AWGN) vector with zero en. Q = P for AWGN chnnel. s is the received signl fro noise esureent. The trix P is full colun ut T P P I N. Thus, this non-orthogonl precoding trix will introduce BER loss in the receiver. If soe singulr vlues of P re too sll, copred to the other vlues, low dditive noise will result of lrge errors. Therefore, the condition nuer of P is introduced, which is to esure the sensitivity of the solution of liner equtions to errors in the dt [28]. is given y x Con( P )=, (18) in 10
Suitted where x nd in is the lrgest nd the sllest singulr vlue of P. Con(.) denotes the condition nuer of trix. The vlue of indictes the BER loss in the precoding. [1, ) nd lrger vlue of leds to worse BER perfornce. Fig. 3 The condition nuer of trnsition trix Q in AWGN chnnel nd Ryleigh fding chnnel. The trnsition trix Q significntly influence the receive qulity. We exine the verge condition nuer of Q through different chnnels in Fig. 3. The results of through Ryleigh chnnel re verged over 2000 reliztions. Two Ryleigh chnnel is selected: 3GPP extended pedestrin A (EPA) odel [29], whose excess tp dely = (0 30 70 90 110 190 410)ns with the reltive power = (0-1 -2-3 -8-17.2-20.8)dB, nd ten-tp Ryleigh lock fding chnnel with exponentilly decying powers set s 2 l/3 9 j/3 E( h ( i) ) e e, j 0,1,...,9. As illustrted in Fig. 3, the receive perfornce is etter l j0 through the AWGN chnnel thn the Ryleigh fding chnnel. Copred to the trnsission without precoding, the condition nuer of Q linerly increses with condition nuer constrint 0 in AWGN chnnel, ut decrese in the Ryleigh fding chnnel when the vlue of 0 is sll. The zero-forcing equlizer is used for AWGN chnnel nd fding chnnel respectively y 11
Suitted T 1 T (P P) P s, d= (Q H Q) 1 Q H s, AWGN chnnel Fding chnnel (19) where (.) H nd d denotes conjugte trnspose nd the decoded dt respectively. In this prt, we develop n lgorith to otin P under condition nuer constrint y selecting the frequency points. In order to keep the receive perfornce decreses slightly, the nuer of zeroforcing frequency points in the first orthogonl projection is chosen s sll s possile. In prctice, we select N 1 for single-side suppression or N 2 for doule-side suppression to keep the spectru syetric. If the specil cse of R 1 is selected, then we fix N 1. The envelope of the precoded PSD curve is inly deterined y the outside orthogonl projection. Thus, N R is selected for high suppression perfornce. The two group zero-forcing frequencies nd re rrnged in the different OOB region, fr fro or close to the inloe. The one fr fro the inloe is fixed, for the plitude response is wek y (5). The one closed to the inloe is used to intin the receive qulity y djusting its loction. In the SVD opertion, in order to effectively ndon the lrgest R singulr vlues in Σ c, K should lrger or equl to R. We select o for siple ipleenttion nd K R. Then the vriles for otining P re only the group closed to the inloe. Thus, we chnge these frequency points to chieve P xcon( P) st.. Con( P ), (20) P 0 where 0 is the given condition nuer ccording to the BER constrint. The signl estition perfornce under condition nuer constrint in couniction syste is illustrted in [28, 30]. In ddition, s presented in [16, 19], the zero-forcing frequency point close to inloe leds to quicker power reduction on the edge of the inloe, ut the power fr fro the inloe is lrger. Inversely, if these points re fr fro the inloe, the power decreses slowly on the edge, while the eission fro the inloe decreses lrgely. 12
Suitted Tle 1 The correltion etween OFDM perfornce nd the preters in the proposed ethod. Cse 0 R N BER OOB power reduction Cse A: Quicker power reduction on the edge of the inloe Cse B: Lower power lekge fr fro the inloe : Negtive correltion; : Positive correltion; : Wek or no effect. N=1: Single-side suppression N=2: Doule-side suppression With surizing the properties nlyzed ove, we first give the correltion etween the preters nd the perfornce of OFDM trnsceiver in Tle 1, which is lso presented in the siultion section. In the following, we develop n lgorith to otin the precoding trix P y selecting the frequencies nd the initiliztion process step y step s (.) The vlue of, ccording to the sury in Tle 1. The in spectru envelope is decided in N nd Cse is selected ccording to the power spectru envelope property. (.) R is chosen ccording to sideloe suppression perfornce nd spectrl efficiency. (c.) 0 is the given condition nuer ccording to the BER qulity constrint. If Cse A is required, we fix given in the lgorith. 1. where Algorith. 1: Initiliztion: 0, R, 1. i : i i+1. i-th itertion. fr fro the inloe nd djust N, itertion increent, 0. The process to solve (20) is,, 0, i 0. 0 2. Design P i y Section 3 nd copute the condition nuer Con( P i ). 3. if ( 0) :, go ck to Step 1. else Stop. Output: P P i1. 0 is the initilizing frequency points close to the inloe, nd fro the inloe. If the nuer of reserved crriers of R is lrge, little djusting of 0 is the fixed frequencies fr will led to 13
Suitted lrge chnge to. Therefore, the itertion increent the frequency points in should not e too sll. In prctice, we select R 4. lso should e sll, nd the distnce etween If Cse B is required, we fix fr fro the inloe nd djust : in the Step 3. The distnce etween the frequency points in Otherwise, the trix e inccurte in (13) y SVD in prctice. lso should not e too sll or too fr fro the inloe. C op y e close to singulr or dly scled. Tht y led to tht the results y 5. Siultion Results nd Discussions In this section, soe nuericl results re presented to deonstrte the perfornce of the proposed ethod. An instnce in LTE is selected tht the sucrrier spcing is 15 khz. The nuer of the sucrriers is N 300 in 5 MHz ndwidth [4]. The frequency xis is norlized to the spcing 2 /T. The index of dt sucrriers is fro -150 to 150, while the direct current crrier 0 is not eployed. To illustrte the OOB power reduction effect, the PSD is otined y coputing the power of DFT coefficients of tie-doin OFDM signl over tie spn nd verging over thousnds of syols. The frequency-doin overspling rte is eight nd the QPSK odultion is eployed. The siulted re inly ZP-OFDM syste, unless noted otherwise. A. Sideloe Suppression Perfornce In the first experient, the OOB reduction perfornce coprison is presented using different precoding technologies. The nuer of reserved sucrriers is selected s R 2. The SSOP [19], the orthogonl projection (OP) [16], the optil orthogonl precoding (OOP) [21] nd CC [8] ethods re selected s coprison. The zero-forcing frequencies re fixed t 180 for OP nd SSOP ethod. The optiized region for OOP nd CC schee is lso selected nery the frequencies 180. N 2 for doule-side suppression to keep the spectru syetric in the proposed JSP ethod. The spectrl 14
Suitted coding rte is 300/302, 4000 nd 180. The condition nuer of the precoding trix is 3.2279. Fig. 4 PSD of the OFDM signl with different precoding techniques. As the PSD curves illustrted in Fig. 4, ll the trnsitted signls re ZP-OFDM, esides CP-OFDM signl is presented y the proposed JSP ethod. It is oviously tht the OOB power using the JSP ethod is lower thn other pproches. The OOP/OP/SSOP ethods hve siilr suppression perfornce. The CC, which is not spectrl precoding ethod, chieves less OOB power reduction. In ddition, the results show the perfornce degrdtion of sideloe suppression in CP systes. Fig. 5 Quicker power reduction on the edge of the inloe (Cse A) with different condition nuer constrint. 15
Suitted Fig. 6 Lower power lekge fr fro the inloe (Cse B) with different condition nuer constrint. In Fig. 5, the precoding trix is designed under condition nuer constrint of 0 { 1.5,3,5,1 0,20} respectively. The PSD curves of Cse A is presented with N 2. is fixed fr fro the inloe nd 4000, R 2 nd 0.25. Oviously, the suppression perfornce is iproved y relxing the condition nuer constrint while the distnce etween nd inloe incresing. In Fig. 6, the Cse B is presented with N 2. We fix fr fro the inloe nd 4000, nd the other preters re se with in Fig. 5. The suppression perfornce is lso iproved y relxing 0. The OOB power in the region fr fro the inloe is lower thn Cse A. However, the power on the edge of inloe decreses slowly. choosing Fig. 7 Single-side suppression with different nuer of revered crriers, N =1. Fig. 8 PSD of Multi-nd OFDM signl. In Fig. 7, the Cse B is selected, N 1 nd 0 1.5. The right-side suppression is presented y nd in the right side of the inloe. The distnce etween two djcent frequencies in is five when R 1. As the syetry presented in (5), (9) nd (14), the difference etween dopting the OOB frequency nd to design the precoding trix is only in the second step of SVD. Therefore, the power on the left side OOB region is lso reduced, while the power eission is higher thn 16
Suitted on the right side. The suppression perfornce is iproved y incresing the nuer of revered crriers. Although the siultion is not illustrted, this property is lso presented in the cse of N 2 y djusting R. In cognitive rdio syste, the secondry users ccess the spectru hole dyniclly. In ddition, crrier ggregtion (CA) [31] is criticl technology in LTE syste, which enle user to eploy noncontiguous sunds. Therefore, in Fig. 8, the PSD of two non-contiguous sunds occupied y one user or two is presented. The condition nuer is 0 10 for ll the cses. The nuer of the crriers of ech sund is 150. The design of the precoding trix for two users is independent, while the dt trnsitted y one user through two sunds is dependent. The results indicte tht the proposed schee is lso suitle for ulti-users trnsitting through non-contiguous sunds. B. BER perfornce Fig. 9 BER perfornce with different spectrl precoding techniques through AWGN chnnel. Fig. 10 BER perfornce with se condition nuer constrint through AWGN chnnel, N =1. A coprison of the BER perfornce using spectrl precoding techniques is shown in Fig.9. The condition nuer constrint is 0 2 for the JSP. The optiized region or zero-forcing frequencies for OOP, SSOP nd OP ethod is se with the JSP ethod. As illustrted in Fig. 4, the sideloe suppression 17
Suitted perfornce y OOP, SSOP nd OP ethod is siilr. The difference is tht OOP schee introduces no qulity loss in the receiver nd the OP ethod leds to lrge error s presented in Fig. 9. The SSOP ethod iproves the receive qulity y dopting reserved sucrrier to decode the distorted signl. The BER qulity loss y the JSP ethod is not notle. In Fig. 10, the condition nuer constrint is fixed s 0 10, s well s single-side suppression is selected s N 1. The results illustrted tht the receive qulity is pproxite identicl when the condition nuer nd different. This property indictes tht if N re fixed, lthough oth of the nuer of reserved crriers R nd the Cse re N hs een selected ccording to single-side suppression or doule-side suppression, the BER perfornce constrint cn e converted to the condition nuer constrint in prctice. Fig. 11 BER perfornce with difference condition nuer constrint nd N through AWGN chnnel, R=2. In Fig. 11, the results deonstrte tht the BER perfornce of N 1 is etter thn N 2 under se constrint. This is why we select the vlue of N s sll s possile to intin the receive qulity. 18
Suitted Fig. 12 BER perfornce with difference condition nuer constrint through AWGN chnnel. Fig. 13 BER perfornce with difference condition nuer constrint through 3GPP EPA fding chnnel. In Fig. 12, the receive qulity is presented with N 2. The nuer of revered crriers is R 2. The results show tht the BER perfornce decreses y relxing 0, ut the suppression perfornce is iproved shown in Fig. 5-6. For exple, the power in the OOB region cn e reduced y nerly 40dB when 0 10, while the SNR loss is less thn 1dB t BER=10 7. When the condition nuer of precoding trix is sll, such s 0 1.5 in Fig. 12, the BER perfornce is slightly etter thn tht of without precoding. Becuse the nuer of the odulted crriers of ech syol is extended fro N to M, the receive qulity is iproved y frequency diversity. However, this property is not notle when the vlue of 0 is lrge. In Fig. 13, the preters re set se with in Fig.12. Copring to without precoding, the BER perfornce through the fding chnnel is siilr to AWGN chnnel, when SNR is sll. The receive qulity decreses y relxing 0. But with SNR incresing, the BER line converges to without precoding. 6. Conclusions In this pper, joint spectrl precoding (JSP) ethod is proposed to reduce the OOB power eission in OFDM-sed syste, s well s n itertion lgorith is given to otin the precoding trix. We lso 19
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