PAPR Constrained Power Allocation fo Carrier Transmission in Multiuser SI Communications. Author(s)Trevor, Valtteri; Tolli, Anti; Matsu

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1 JAIST Reposi Title PAPR Constrained Power Allocation fo Carrier Transmission in Mltiser SI Commnications Athors)Trevor, Valtteri; Tolli, Anti; Mats Citation IEEE Transactions on Wireless Commn 54): Isse Date Type Jornal Article Text version athor URL Rights This is the athor's version of the Copyright C) 05 IEEE. IEEE Transa Wireless Commnications, 54), DOI:0.09/TWC of this material is permitted. Permi IEEE mst be obtained for all other crrent or ftre media, inclding reprinting/repblishing this materia advertising or promotional prposes, collective works, for resale or redi servers or lists, or rese of any co component of this work in other work Description Japan Advanced Institte of Science and

2 PAPR Constrained Power Allocation for Mlti-Carrier Transmission in Mltiser SIMO Commnications Valtteri Tervo*, Member, IEEE, Antti Tölli, Senior Member, IEEE, Tad Matsmoto, Fellow, IEEE Abstract Peak-to-average power ratio PAPR) constrained power allocation for mlticarrier transmission in mltiser single-inpt mltiple-otpt SIMO) commnications is considered in this paper. Redcing the PAPR in any transmission system is beneficial becase it allows the se of inexpensive, energy-efficient power amplifiers. In this paper, we formlate a power allocation problem for single-carrier SC) freqency division mltiple access FDMA) and orthogonal FDMA OFDMA) transmission with instantaneos PAPR constraints. Moreover, a statistical approach is considered in which the power variance of the transmitted waveform is controlled. The constraints for the optimization problems are derived as a fnction of transmit power allocation and two sccessive convex approximations SCAs) are derived for each of the constraints based on a change of variables COV) and geometric programming GP). In addition, the optimization problem is constrained by a serspecific qality of service QoS) constraint. Hence, the proposed power allocation strategy jointly takes into accont the channel qality and the PAPR characteristics of the power amplifier. The nmerical reslts show that the proposed power allocation strategy can significantly improve the transmission efficiency of power-limited sers. Therefore, it is especially beneficial for improving the performance for cell edge sers. Index Terms Power minimization, soft interference cancellation, MMSE receiver, mltiser detection, PAPR redction I. INTRODUCTION Single-carrier SC) freqency division mltiple access FDMA) [] has been selected as the plink transmission scheme for the 3GPP long term evoltion LTE) standard and its advanced version LTE-A) [], de to its good peak-toaverage power ratio PAPR) properties. SC-FDMA can be viewed as a form of orthogonal FDMA OFDMA) [3] in which an additional discrete forier transform DFT) and an inverse DFT IDFT) are added at the transmitter TX) and receiver RX) ends, respectively. A DFT precoder [] spreads all the symbols across the whole freqency band, forming a virtal SC strctre which is known to lead to a redced PAPR. This work was spported by the Academy of Finland, Riitta and Jorma J. Takanen Fondation, Finnish Fondation for Technology Promotion, Walter Ahlström Fondation, Ulla Tominen fondation, KAUTE-fondation and Tano Tönning fondation. This work was also in part spported by the Japanese government fnding program, Grant-in-Aid for Scientific Research B), No V. Tervo and A. Tölli are with the Centre for Wireless Commnications, University of Ol, P.O. Box 4500, 9004 University of Ol, Finland, {valtteri.tervo, antti.tolli}@ee.ol.fi. T. Matsmoto is with the Centre for Wireless Commnications, University of Ol, and Japan Advanced Institte of Science and Technology, - Asahi-Dai, Nomi, Ishikawa, 93-9 Japan, tadashi.matsmoto@ee.ol.fi. It is well known that power allocation in mlti-carrier transmission provides significant improvement in terms of total power consmption [4]. In [5], [6], a power allocation techniqe taking into accont the convergence properties of an iterative RX was derived for SC-FDMA showing sbstantial improvement in terms of redcing the signal-to-noise ratio SNR) reqirements for the desired qality of service QoS) target. However, the se of freqency domain power allocation leads to an increased vale of the PAPR. Motivated by this fact, we have constrcted a framework in this paper by which the PAPR can be controlled via freqency domain power allocation. Redcing the PAPR in any transmission system is always desirable as it allows the se of more efficient and inexpensive power amplifiers PAs) at the TX. In order to maximize the PA efficiency PAE), the operating point of the PA shold be set as close to the satration as possible. However, in mlti-carrier systems, moving the operation point closer to the satration increases the probability that the amplified signal components appear in the nonlinear region of the PA. Frthermore, this probability is directly proportional to PAPR. Amplifying the signal components in PAs nonlinear region, introdces otof-band distortion. Ths, redcing the PAPR indces the following advantages: increased PAE with the same distortion or, decreased distortion with the same PAE. The problem of PAPR redction in mlti-carrier transmission has been an active research topic for several decades. In the past, the PAPR problem has been addressed in many papers and overview articles, e.g., [7] [9]. Existing techniqes, sch as selected mapping SLM) [0], partial transmit seqences PTS) [], [] and constellation shaping [3] [5] achieve a redced PAPR at the expense of a transmit signal power increase, bit error rate BER) increase, data rate loss, comptational complexity increase, etc. The most straightforward soltion for PAPR redction is clipping the amplitde of the OFDM signal. The drawback is that the clipping increases the noise level. Recent work on redcing the PAPR in SC-FDMA transmission can be fond in [6] [8], where the athors propose different precoding methods for PAPR redction. In [6], the idea is to se non-diagonal power allocation matrix in OFDMA transmission to distribte the symbols across the sbcarriers sch that the PAPR is minimized. The idea in the methods presented in [7], [8] is finding the optimal weights for the sbcarriers, i.e., power allocation matrix, sch that the power variance of the transmitted time domain signal is minimized.

3 However, the power allocation methods introdced in [6] [8] solely focs on the PAPR redction, while the method proposed in this paper considers joint PAPR redction and sm power minimization. Power allocation methods derived in this paper take also into accont a freqency selective channel and an iterative RX. The main objective in this paper is to improve the PAPR characteristics of a mlti-carrier transmission by introdcing a novel power allocation method taking into accont the properties of an iterative RX and the PAPR of the transmitted signal. The contribtions of this paper are smmarized as follows: Two approaches for a PAPR aware power allocation in mlti-carrier transmission are presented. The first approach optimally restricts the PAPR below a preset threshold vale while garanteeing the preset QoS target. The second approach controls the PAPR statistically by controlling the variance of the power of the transmitted signal. The instantaneos PAPR and the variance are derived for both SC-FDMA and OFDMA. The PAPR and power variance derivations presented in this paper apply in any normalized data modlation techniqe. The PAPR constraints are applied to the optimization framework presented in [6], where the objective is to minimize the sm power in plink transmission while garanteeing the convergence of an iterative eqalizer. Two sccessive convex approximations SCA) [9] commonly existing in the power control problems are derived for the PAPR constraints. Namely, sccessive convex approximation via change of variables SCACOV) [0] and sccessive convex approximation via geometric programming SCAGP) []. The athors have pblished the first reslts on PAPR constrained power allocation in [], [3] where SCACOV has been derived for instantaneos PAPR constraint and SCAGP has been derived for the power variance constraint. This paper provides a detailed derivation of the constraints and extends the concept to OFDMA. Frthermore, SCAGP is derived for the PAPR constraint and SCACOV is derived for the variance constraint. The rest of the paper is organized as follows: The system model assmed throghot the paper is presented in Section II. In Section II-A, the TX side of SC-FDMA and OFDMA plink transmission is described. In Section II-B, iterative eqalizers for SC-FDMA and OFDMA are presented. The optimization problem is introdced in Section III. The ser specific QoS constraints are presented in Section IV. The PAPR constraints with the SCACOV and SCAGP soltions are presented in Section V and Appendices A-F. The power variance constraints with the SCACOV and SCAGP soltions are presented in Section VI and Appendices G-L. The nmerical reslts are given in Section VII and the conclsions are drawn in Section VIII. Nomenclatre Following notations are sed throghot the paper: Vectors are denoted by lower boldface letters and matrices by ppercase boldface letters. The sperscripts H and T denote Hermitian and transposition of a complex vector or matrix, respectively. C, R, B denote the complex, real and binary nmber fields, respectively. I N denotes N N identity matrix. The operator avg{ } calclates the arithmetic mean of its argment, diag ) generates diagonal matrix of its Fig.. Block diagram of the TX side of the system model. argments, bdiag{ } generates the block diagonal matrix from its argment matrices, denotes the Kronecker prodct and is the Eclidean norm of its complex argment vector. E[ ] is the expectation operator. circ{ } constrcts a circlant matrix from its argment, in which each colmn of the matrix is a cyclically shifted version of its sccessive colmn. A list of the most relevant symbols sed in the paper is shown in Table I. II. SYSTEM MODEL In this section, the system model of plink transmission in a single-cell system with U single-antenna sers and a base station with N R antennas is presented. The channel state information CSI), inclding an instantaneos channel implse response and the second moment of additive thermal noise, is assmed to be perfectly known both at the TX and RX. A. Transmitter The TX side of the system model is depicted in Fig.. Each ser s data stream x B R c N Q,,,..., U, is encoded by a forward error correction FEC) code C with a code rate Rc. The symbol N Q denotes the nmber of bits per modlation symbol and is the nmber of freqency bins in DFT. The encoded bits c [c, c,..., c NQ ] T B N Q are bit-interleaved by mltiplying c by psedo-random permtation matrix Π B N Q N Q reslting a bit seqence c Π c. After the interleaving, the seqence c is mapped with a mapping fnction M ) onto a N Q -ary complex symbol b l C, l,,...,, reslting a complex data vector b [b, b,..., b ] T C. After the modlation, in SC- FDMA each ser s data stream is spread across the sbchannels by mltiplying b by a DFT matrix F C,,,..., U, where the elements of F are given by f m,l e iπm )l )/ ), m, l,,...,. In OFDMA, this spreading process is omitted. Each ser s data stream is mltiplied with its associated power allocation matrix P, where P diag[p,, P,,..., P, ] T ) R, with P,l being the power allocated to the lth freqency bin. Finally, before transmission, each ser s data stream is transformed into the time domain by the IDFT matrix F reslting in s [s, s,..., s ] T,. A cyclic prefix is appended to mitigate inter-block interference IBI) in SC- FDMA and inter-symbol interference ISI) in OFDMA.

4 3 TABLE I LIST OF SYMBOLS. b l soft estimate of the th ser s l th symbol b transmitted symbol vector of ser b soft estimate vector of ser ˆb time domain estimate vector of ser b l vector for the detected data stream of ser c encoded bit vector of ser with elements b l c interleaved encoded bit vector of ser H block circlant channel matrix of ser nmber of bins in discrete Forier transform N L length of the channel implse response N R nmber of receive antennas N Q nmber of bits per modlation symbol P diagonal power allocation matrix of ser r combination of the residal and the desired signal associated with ser r m received freqency domain signal vector ˆr m the otpt vector of soft cancelation associated with the m th freqency bin s transmitted symbol vector of ser after IDFT U nmber of sers v vector of noise samples x binary data stream of ser γ,m channel vector for the m th freqency bin of ser Γ freqency domain channel matrix for ser δ PAPR target for ser average residal interference of the soft symbol estimates of ser ζ the effective SINR of ser in SC-FDMA ζ the effective SINR of ser in OFDMA ξ,k axiliary constant describing the reqired ξ,k axiliary constant describing the reqired SINR for given MI target in SC-FDMA SINR for given MI target in OFDMA σv variance of noise σ power variance target for ser σ,k the variance of the LLRs at the inpt of Σˆr,m interference covariance matrix of the the decoder of the th ser at the k th MI index m th freqency bin of ser in SC-FDMA Σˆr covariance matrix of the otpt of the soft cancelation ω,m receive beamforming vector for the m th freqency bin of ser in SC-FDMA ω,m receive beamforming vector for the m th Ω filtering matrix of ser freqency bin of ser in OFDMA B. Receiver In this section, freqency domain soft cancelation minimm mean sqared error MMSE) receiver is derived for SC-FDMA and OFDMA. ) SC-FDMA: For SC-FDMA, the RX presented in [6] is sed. After the soft cancelation, the residal and estimated received signal of ser are smmed in r C N R as U r Γ l P l Fb l b l ) + Γ P F b + F NR v, ) l where b C is a soft symbol estimate vector composed by b [ b, b,..., b ] T with b n being the soft symbol estimate of b n given in [6, Eq. 6)]. A matrix Γ bdiag{γ,, Γ,,..., Γ, } C N R is the spacefreqency channel matrix for ser expressed as Γ F NR H F. The block diagonal DFT matrix F NR is expressed as F NR I NR F C N R N R, and Γ,m C N R N R is the diagonal channel matrix for the m th freqency bin of the th ser. A matrix H [H, H,..., H N R ] T C N R is the space-time channel matrix for ser and H r circ{[h r,, h r,,..., h r,n L, 0 N L ] T } C is the time domain circlant channel matrix for ser at receive antenna r. The operator circ{} constrcts a circlant matrix from its argment vector, N L denotes the length of the channel implse response, and h r,l, l,,..., N L, r,,..., N R, is the fading factor of mltipath channel. A vector v C N R in ) denotes white additive Gassian noise vector with variance σv. The time domain otpt of the receive filter for the th ser can be written as ˆb F ΩH r, where Ω [ Ω, Ω,..., Ω N R ] T C N R is the filtering matrix for the th ser and Ω r C is the filtering matrix for the r th receive antenna of the th ser. The effective signal to interference pls noise power ratio SINR) of the prior symbol estimates for the th ser can be expressed as ζ m P,m ω H,mγ,m γ H,mω,m ω H,mΣˆr,m ω,m, ) where γ,m C N R consists of the diagonal elements of Γ,m, i.e., γ,m is the channel vector for the m th freqency bin of ser. The receive beamforming vector for the [ m th freqency bin of ser is denoted as ω,m [ Ω ] [m,m], [ Ω ] [m,m]..., [ Ω ] T N R ] [m,m] C N R, and Σˆr,m C N R N R is the interference covariance matrix of the m th freqency bin given by U Σˆr,m P l,m γ l,m γl,m H l + σvi NR. 3) l The average residal interference of the soft symbol estimates is denoted as l avg{ b l }, where b l [ b l, b l,..., b l ] T C. Solving the optimal RX via MMSE criterion yields [4] Ω avg{ b }ζ + Σ ˆr Γ P, 4) where Σˆr C N R N R is the covariance matrix of the otpt of the soft cancelation given by U Σˆr Γ l P l l P l Γ H l + σvi NR, 5) and l l I. l

5 4 ) OFDMA: The received signal at the m th sbcarrier is r m U γ,m P,m b m + ṽ m C N R, 6) where ṽ m C N R denotes white additive Gassian noise vector with variance σ v. The freqency domain signal after soft cancelation is expressed as ˆr m r m U γ,m P,m b m. 7) The filtered signal can be expressed as ˆb m ω H,m r,m, 8) where r,m ˆr + γ,m P,m b m, and ω,m C N R is the receive filter of the th ser at the m th sbcarrier which can be fond by solving minimize ω,m E b m,ṽ m [b m ˆb m)b m ˆb m) H ]. 9) Sbstitting the soltion of 9) to 8) gives the MMSE estimate of the transmitted symbol as [ U ˆb m γ,m γ,mp H,m + γ l,m γl,mp H l,m b l,m ) l l + σ vi NR ) γ,m P,m ] H r,m. 0) Similarly to the case of SC-FDMA, the interference cancelation term b l,m can be approximated by l avg{ b l } leading to a more compact notation γ,m ˆb m [γ,m γ,mp H,m + Σr,m) ṷ P,m ] H r,m, where Σ ṷ r,m ) U γ l,m γl,mp H l,m l + σvi NR. ) l l The effective SINR after the MMSE filter is given by ζ,m Ul l P,m γ H,m ω,m γ H l,m ω,m P l,m b l,m ) + σ v ω,m. 3) Ths, the fndamental differences to SC-FMDA are that the received signal decoples sch that all the operations can be performed per sbcarrier and there is no self interference in the SINR eqation nlike in ). In fact, l is an essential approximation in order to se higher order modlations where the power of a symbol is not eqal to one. In order to se the approximation l, the expectation of a symbol power has to be one and the length of a block needs to be large enogh. ˆP 0) : Initialize ˆP : repeat 3: Calclate the optimal Ω from 4). 4: Set Ω Ω ) and solve problem 4) with variables P. SCA is employed here) 5: Update ˆP P ) 6: ntil Convergence Fig.. Alternating Optimization for SC-FDMA. III. OPTIMIZATION PROBLEM AND SOLVING METHOD The optimization problem considered in this paper is expressed as minimize P, Ω sbject to tr{p} z i P, Ω) 0, i,,..., N y k P) 0, k,,..., K, 4) where z i P, Ω) 0, i,,..., N, is a set of QoS constraints and y k P) 0, k,,..., K, is a set of constraints controlling the PAPR. Ω denotes the set of receive filters of all sers and all freqency bins. In this paper, we will derive y k P) in the form of y k P) ˆK n ρ k np qk n P qk n P qk n, ρ k n, q i mk R, 5) which can be split as y k P) y k P) + + y k P), where y k P) + ˆK n ρk n ρ k n ˆK n ρk+ n P qk n P qk n P qk n, and y k P) P qk n P qk n P qk n, with ρ k+ n max{0, ρ k n} and min{0, ρ k n}. Constraint y k P) 0 can be rewritten as y k P) + y k P), 6) where the fnctions y k P) + and y k P) are referred as posynomials. Similarly to [6], posynomials can be transformed into a convex form. However, the fnction y k P) on the RHS of 6) has to be approximated by a concave fnction to make the overall constraint convex. In this paper, we will show two type of approximations, which are garanteed to converge towards a local soltion. Note that the PAPR constraints depend only on the power allocation P. Hence, the PAPR constraints derived in this paper can be applied to any SC-FDMA or OFDMA optimization framework. The joint optimization of TX P and RX Ω can be performed via alternating optimization [6]. The alternating optimization in SC-FDMA is described in Fig., where P ) indicates a soltion to problem 4) for fixed Ω and Ω ) represents the optimal Ω for fixed P. The idea is to perform joint optimization by alternating between the TX and RX optimizations, where SCA is employed for TX optimization. Local convex approximations for non-convex constraints needed in SCA are described in forthcoming sections. After solving the approximated convex problem, the soltion is sed to pdate the approximation point. Then, the approximated problem is solved again sing the new approximation point. This procedre is repeated ntil convergence.

6 5 In the following three sections, we will derive QoS, PAPR and power variance constraints for SC-FDMA and OFDMA. De to the non-convexity of the constraints, two alternative SCAs are also presented. IV. QOS CONSTRAINTS The QoS constraint considered in this paper, is the convergence constraint for iterative RX derived in [6]. Convergence constrained power allocation CCPA) [6] is a power allocation method for an iterative RX that ses trbo eqalization. CCPA takes the convergence properties of the RX into accont by tilizing extrinsic information transfer EXIT) charts. More specifically, the idea is to sample the EXIT chart p to K points and check the convergence condition at each sampled MI vale. This approach leads to a problem with several SINR constraints, where each constraint is for a different vale of a priori information. In this section, the convergence constraint for SC-FDMA and OFDMA is briefly presented. A. Convergence Constraint for SC-FDMA The convergence constraint for SC-FDMA can be written as [6, Eq. 0] m P,m γ H,mω k,m U l γh l,m ωk,m P l,m k + σ v ω k,m ξ,k,,..., U, k,,..., K, 7) where ω k,m is the receive beamformer of the th ser at the m th freqency bin at the k th mtal information MI) index and k is the cancelation factor at the k th MI index. De to the additional DFT spreading at the SC-FDMA TX the symbol seqence is spread across the whole freqency band. Hence, the left hand side LHS) of 7) is the average SINR taken over all sbcarriers. The right hand side RHS) of 7) is a constant depending on the modlation coding scheme MCS), the reqired QoS and the amont of a priori information at the k th MI index. 7) is not convex in general and hence, convex approximations presented in [6, Secs. V.B. and V.C.] shold be sed. B. Convergence Constraint for OFDMA Similarly to SC-FDMA, the convergence constraint for OFDMA can be written as P,m γ,m H ωk,m Ul γl,m H ωk,m P l,m k + σv ω,m ξ,k, k l,..., U, m,...,, k,,..., K, 8) where ξ,k σ,k 4 is a constant depending on the variance of the a priori LLRs σ,k. Constraint 8) is clearly convex with respect to P. In OFDMA, the sbcarriers are decopled for fixed SINR and hence, the constraint 8) is per sbcarrier. In practise, one cold consider a rate constraint across the sbchannels reslting in a varying MCS and ths, bit and power loading algorithms shold be considered. However, in this paper we consider fixed SINR target and constant MCS across the sbcarriers. V. INSTANTANEOUS PAPR CONSTRAINT In this section, we derive instantaneos PAPR constraints for SC-FDMA and OFDMA. In addition, SCACOV and SCAGP are presented for non-convex constraints. A. PAPR constraint for SC-FDMA Let s m be the m th otpt of the transmitted waveform for the th ser after the IDFT. The PAPR constraint in general form is expressed as P AP Rs ) max m s m avg [E { s m }] δ, 9) where δ is a ser specific parameter controlling the PAPR. The max operator can be eliminated by reqiring s m avg[ s m ] δ, m,,...,. 0) Assming E{ b n },, n and E{b nb i } 0, n i, where b n denotes the complex conjgate of b n, the average can be calclated as avg[ s m ] m { E s m } m P,m. ) The assmption E{ b n } can be jstified for any modlation scheme with a proper normalization factor. After a lengthy derivation of s m, shown in Appendix A, the instantaneos PAPR constraint 9) for SC-FDMA can be expressed as l ) κ + d l P,l + δ P,l + n,n n >n + ˆη n n m P,n P,n ˆη n n m) P,n P,n, l n,n n >n m,,...,,,,..., U, ) where κ R,, d l R, l,, and ˆη n + n m, ˆη n n m R, n, n, m,. It can be seen that both sides of ) are posynomials and hence, the constraint is in the form of 6). The constraint ) is a non-convex constraint and many approximations can be applied. For example, the term P,n P,n existing on both sides of ) is actally a geometric mean and ths a concave fnction. While we cold directly apply SCA by approximating P,n P,n on the LHS of ), we present two SCAs so that the reformlated constraint can be incorporated to the optimization framework introdced in [6, Secs. V.B. and V.C.]. In the following we apply SCACOV and SCAGP for constraint ). ) SCACOV: For SCACOV, we reformlate the constraint ) sch that it has a convex and concave part. The concave part can be locally approximated by a linear fnction and similarly to [6, Algorithm ], a local soltion can be fond iteratively by pdating the approximation point.

7 6 : Set ˆα,m ˆα,m, 0), m. : repeat 3: Solve Eq. 4) with constraints 7) and 3). 4: Update ˆα,m α,m, ) k. 5: ntil Convergence. Fig. 3. Sccessive convex approximation via change of variables. Denoting P,l e α,l,,,..., U, l,,...,, constraint ) can be approximated at a point ˆα as l κ + d l )e α,l + n,n n >n + ˆη n n me α,n +α,n ) ˆT m α, ˆα ),,,..., U, m,,...,, 3) where ˆT m α, ˆα ) is a local linear approximation of the RHS of ) after the change of variables COV). Details are shown in Appendix B. The SCA algorithm starts by a feasible initialization ˆα,m ˆα,m, 0), m. After this, 4) is solved with constraints 7) and 3) and with appropriate change of variables to obtain a soltion α,m ) which is sed as a new point for the linear approximation. The procedre is repeated ntil convergence. The SCACOV is smmarized in Fig. 3. Becase the linear approximation ˆT m α, ˆα ) is always below the approximated convex fnction RHS of )), the points satisfying the approximated constraint 3) always satisfy the original constraint ). By projecting the optimal soltion from the approximated problem to the original convex fnction RHS in 38)) the constraint becomes loose and ths, the objective can always be redced. Hence, the objective vale of this type of iterative algorithm converges monotonically towards a local optimm of the original problem. ) SCAGP: In a standard form of geometric programming GP) [5] constraint, the LHS is a posynomial and RHS is a monomial. For SCAGP, the constraint needs to be in sch a form that it has a posynomial on both sides of the ineqality sign. Then, the RHS can be sccessively approximated by a monomial sing [6, Eq. 36)], and a local soltion can be fond iteratively by pdating the parameters in monomial approximation. Constraint ) can be approximated, m sing l κ + d l )P,l + n,n n >n + ˆη n n m P,n P,n ÂmP, b ), 4) where ÂmP, b ) is a monomial approximation of the RHS of ). Detailed derivation of SCAGP for PAPR constraint in SC-FDMA is shown in Appendix C. The LHS is a posynomial and RHS is a monomial and hence, 4) is a valid GP constraint. Similarly to SCACOV, 4) can be sed pdating the approximation point iteratively. Becase the monomial approximation is never above the approximated smmation, the same argments describing the convergence presented for SCACOV apply also in this case. Hence, it is garanteed that the objective vale of the SCA with 4) converges monotonically towards the local optimm of the original problem. B. PAPR constraint for OFDMA Similarly to SC-FDMA, the PAPR constraint for OFDMA can be written as b l P,l + l δ P,l + l n,n n >n d n,n n >n d + mn n P,n P,n mn n ) P,n P,n, 5) + where d mn n, d mn n R,, m, n, n. A detailed derivation can be fond in Appendix D. The major difference between the OFDMA s PAPR constraint and the SC-FDMA s PAPR constraint presented in ) is in the factors ˆη n + n m, ˆη n n m, d+ mn n and d mn n. Similarly to Sections V-A and V-A, constraint 5) can be sccessively approximated as SCACOV or SCAGP. SCACOV and SCAGP derivation can be fond in Appendices E and F, respectively. VI. POWER VARIANCE CONSTRAINT Another way to redce the PAPR is to redce the variance of the power of the transmitted time domain signal [7], [8]. The variance is taken over all possible symbol seqences and therefore, nlike in instantaneos PAPR constraint, the variance constraint does not depend on the transmitted symbol seqence. Note that redcing the power variance leads to statistically decreased PAPR. In this paper, we investigate this relationship of PAPR and power variance by deriving a constraint that restricts the power variance below a desired threshold. A. Power variance constraint for SC-FDMA Assming E{ b n },, n and E{b n b n } 0, n n, the power variance constraint can be written as n,n,n 3,n 4 S ) P,l ) l n,n S P,n P,n + P,n P,n P,n3 P,n4 + P,l ) σ N F 3, l 6) where σ is the preset pper bond of the variance of transmitted power for the th ser. The details, inclding the definition of the smmation sets S and S, can be fond in Appendix G. Both sides of 6) are posynomials. Ths, SCA is needed for the RHS. Both, SCACOV and SCAGP are applied for approximating 6) in Appendices H and I, respectively.

8 7 B. Power variance constraint for OFDMA In the case of OFDMA, the variance constraint is written as N F N F n,n n >n P,nP,n σ m P,m ). 7) The details are shown in Appendix J. Again, both sides of 7) are posynomials. Ths, SCA is needed for the RHS of 7). Both, SCACOV and SCAGP are applied for approximating 7) in Appendices K and L, respectively. VII. NUMERICAL RESULTS In this section, the reslts obtained in the simlations are presented. A. Simlation setp The reslts are obtained with the following parameters: 8, qadratre phase-shift keying QPSK) N Q ) and 6-ary qadratre amplitde modlation 6QAM) N Q 4) with Gray mapping, and systematic repeat accmlate RA) code [6] with a code rate of /3 and eight internal iterations. Uniform diagonal sampling [6] is sed for EXIT sampling in the QoS constraint, and the nmber of samples is K 5. The SNR per ser and per RX antenna averaged over freqency bins is defined by SNR tr{p}/un R σv). We consider two different channel conditions, namely, a static 5-path channel, where path gains were generated randomly, and a qasistatic Rayleigh fading 5-path average eqal gain channel. The stopping criterion of the optimization algorithms is that the change in the objective fnction becomes less than or eqal to a small specific vale between two sccessive iterations. In simlations, the stopping threshold vale was set at 0.05 for TX-RX alternations and 0.0 for SCAs. Let ÎE and I E denote the MI at the otpt of the eqalizer of the th ser and at the otpt of the decoder of the th ser, respectively. The QoS target sed in the simlations is the MI I E,target target after the trbo iterations in the RX denoted as and ÎE,target. MI point can be converted to bit error probability BEP) by sing [7, Eq. 3)]. The PAPR cannot be considered as the only performance metric since there is often a tradeoff between the PAPR and the average power, i.e., a decrease in PAPR may lead to an increase of the average power and vice versa. The peak power of the transmission is defined as P max db) P avg db) + P AP RdB), where P avg SNR N R σv denotes the average power of ser. If the peak power can be redced, the average power can be increased and ths, we can se the metric SNRdB) + P AP RdB) to compare the algorithms in terms of the range of the transmission. B. Initialization To employ the SCAs presented in this paper, it is necessary to find a feasible starting point for the iterative algorithm. In the case of SC-FDMA, it can be fond by setting the power to be eqal for all sbcarriers. The power level has : Calclate the ZF matrix for the freqency domain channel. : Find the power allocation satisfying QoS constraint. 3: repeat 4: Set P,n P,n + ϵ, for all, for some n {,,..., }, and ϵ > 0. 5: Calclate PAPR for all the sers PAPR. 6: ntil PAPR δ Fig. 4. Initialization of the optimization algorithm in OFDMA. to be high enogh to satisfy the QoS constraints and it can be fond by sing a bisection algorithm [5]. In eqal allocation, the PAPR is 0 db and.55 db for QPSK and 6QAM, respectively, which are the modlation schemes considered in the simlations. As long as the target PAPR is above this vale, the reslt obtained by eqal allocation satisfies the PAPR constraint for SC-FDMA. In the case of OFDMA, a feasible starting point for the iterative algorithm can be fond, for example, with the help of spatial zero forcing ZF) [4] RX. It is straightforward to max show that, for OFDMA, m s m avg[ s m ] when P,n for some n. Increasing P,n does not violate the SINR constraint becase ZF RX removes all the interference. The feasible initialization method is smmarized in Fig. 4. Step can be performed by allocating the same power for all the sbcarriers. The power level can be fond by sing a bisection algorithm [5]. This initialization method presented above applies also with appropriate modifications for a power variance-constrained problem. Nmerical reslts revealed that in this particlar case the optimization is not highly sensitive to initializations. It is worthwhile to notice that in all reslts presented in this paper, SCACOV and SCAGP converge towards the same objective vale. C. PAPR constraint To demonstrate the operational principle of the PAPR constraint, EXIT simlations were carried ot in a static channel for a fixed symbol seqence. The EXIT crve of the decoder is obtained by sing 00 blocks for each a priori vale, with the size of a block being 6000 bits. The EXIT chart of the trbo eqalizer when precoding with instantaneos PAPR constraint is presented in Fig. 5. SC-FDMA and OFDMA denote the schemes withot the PAPR constraint, i.e., with the QoS constraint only. The SC-FDMA reslt is obtained via SCAGP approximation. Clipping denotes the case where the signal is clipped when the power exceeds the peak vale calclated from the PAPR threshold. The minimm gap between the EXIT crves of the eqalizer and the decoder of ser can be controlled by changing the parameter ϵ. It can be also seen from Fig. 5 that, with SC-FDMA, the minimm gap between the EXIT crves can be sppressed down to ϵ according to the convergence constraint. For OFDMA, the gap is larger than ϵ, which reslts in significantly larger SNR reqirements compared to SC-FDMA. This can be seen in Table II, where the corresponding SNR and The reader is gided to [6] for more detailed information on CCPA.

9 OFDMA withot PAPR constr OFDMA SCAGP 3 db OFDMA SCACOV 3 db SC-FDMA withot PAPR constr. SC-FDMA SCAGP 3 db SC-FDMA SCACOV 3 db Î E,k / I A,k 0.5 BEP Decoder OFDMA withot PAPR constr. OFDMA with clipping 3 db 0. OFDMA SCAGP 3 db OFDMA SCACOV 3 db SC-FDMA withot PAPR constr. 0. SC-FDMA with clipping 3 db SC-FDMA SCAGP 3 db SC-FDMA SCACOV 3 db Î,k A / I,k E SNR db) Fig. 5. EXIT chart for δ 3 db. U, N R, N Q, ÎE,target E,target 0.789,,, I ,,, ϵ 0.0,,, b / [ i, i, i, + i, i, + i, i, + i] T, b / [ + i, + i, i, + i, + i, i, + i, + i] T. Fig. 6. BEP comparison with δ 3 db. U, N R, N Q, ϵ 0.,,. PAPR are listed together with the smmation of SNR and PAPR for each algorithm sed. The larger SNR reqirement of OFDMA compared to SC-FDMA is de to the difference in convergence constraints. In the case of OFDMA, the SINR reqirement is the same for all sbcarriers, nlike in SC- FDMA where the average of SINRs over the sbcarriers is sed. On the other hand, there is no intra-ser interference in OFDMA, nlike in SC-FDMA for which the starting point in the EXIT chart is interference-limited. Hence, the target point in the case of OFDMA can be achieved even with linear RX by simply increasing the power in all the sbcarriers. Clipping redces the SNR bt convergence to the desired MI point is not garanteed. In the case of SC-FDMA with clipping, the EXIT crves intersect at MI point I, E ÎE ) 0.936, 0.54), which corresponds to BEP vale It can be seen from Table II that the PAPR threshold sed in Fig. 5 is not exceeded with the PAPR constraint. The sm of SNR and PAPR describes the actal power gain achieved by the proposed algorithms, which helps to improving the QoS for cell edge sers. It can be seen that in the case of OFDMA the improvement when sing the PAPR constraint is 9.3 db 8.44 db 0.87 db. In the case of SC-FDMA, the improvement is 3.6 db and 3.7 db for SCAGP and SCACOV, respectively. In Fig. 6, the reqired SNR verss BEP is presented, where the reslts are obtained by averaging over 00 channel realizations. For different BEP target vales are considered for,, namely 0 3, 0 4, 0 5 and 0 6. It can be seen that for SC-FDMA, the reqired SNR is roghly the same with and withot the PAPR constraint, i.e., the PAPR can be sppressed to 3 db withot a significant increase in transmit power. For OFDMA, the reqired SNR is increased by db, depending on the BEP target and algorithm sed. Complementary cmlative distribtion fnctions CCDF) ProbP AP R > δ) for SC-FDMA and OFDMA withot PAPR constraints and with a BEP target of 0 5 are plotted in ProbPAPRs )>δ) 0 0 OFDMA SC-FDMA δ db) Fig. 7. CCDFs for SC-FDMA and OFDMA with BEP target 0 5. U, N R, N Q, Î E,target E,target 0.789,,, I ,,, ϵ 0.,,. Fig. 7. CCDFs are calclated sch that 0 5 randomly generated symbol seqences of length for each ser are sent over 00 channel realizations. Obviosly, for the algorithms with PAPR constraint, the CCDF is 0 when the PAPR is larger than the PAPR threshold. For a CCDF vale of 0 5, the corresponding PAPRs are 7.66 db and 8.5 db for SC- FDMA and OFDMA, respectively. From Figs. 6 and 7, we can calclate the SNR+PAPR gains for SC-FDMA to be 4.63 db and 4.9 db for SCAGP and SCACOV, respectively. Similarly for OFDMA, the gains are 4.9 db and 4.00 db, respectively. D. Power variance constraint CCDFs for the OFDMA scheme when precoding with a variance constraint is shown in Fig. 8. It can be seen that the PAPR can be significantly redced by decreasing the variance. In fact, the PAPR approaches the theoretical limit i.e.,.55 db for 6 QAM when the variance target approaches zero. However, becase the per-sbcarrier SINR constraint is

10 9 TABLE II SNR AND PAPR COMPARISON IN A STATIC CHANNEL Algorithm SNR db) PAPR db) SNR + PAPR db) OFDMA OFDMA with clipping OFDMA SCAGP OFDMA SCACOV SC-FDMA SC-FDMA with clipping SC-FDMA SCAGP SC-FDMA SCACOV ProbPAPRs ) > δ) 0 0 OFDMA, SNR6.33 db SCAGP, σ 0.0, SNR9.07 db SCACOV, σ 0.0, SNR9.8 db SCAGP, σ 0.00, SNR. db SCAGP, σ 0.000, SNR3.43 db SCAGP, σ , SNR5.5 db ProbPAPRs ) > δ) 0 0 SC-FDMA, SNR4.05 db SCAGP, σ 0.00, SNR4.43 db SCACOV, σ 0.00, SNR4.44 db SCAGP, σ 0.000, SNR4.79 db SCACOV, σ 0.000, SNR4.99 db SCAGP, σ , SNR5.00 db SCAGP, σ , SNR5.0 db δ db) δ db) Fig. 8. CCDFs for OFDMA with variance constraint. BEP target0 5, U 4, N R 4, N Q 4, ÎE,target E,target 0.789,, I ,, ϵ 0.,. Fig. 9. CCDFs for SC-FDMA with variance constraint. BEP target0 5, U 4, N R 4, N Q 4, Î E,target E,target 0.789,, I ,, ϵ 0.,. sed as a QoS constraint, the SNR increase is high compared to the PAPR redction. CCDFs for the SC-FDMA scheme when precoding with a variance constraint are shown in Fig. 9. It can be seen that the PAPR can be significantly redced with a minor increase in SNR by decreasing the power variance. Similarly to the OFDMA case, the PAPR approaches the theoretical limit when the variance target approaches zero. VIII. CONCLUSIONS In this paper, we have formlated PAPR constrained power allocation problem for mlticarrier transmission with iterative MMSE mltiser mltiantenna RX. We derived an analytical expression of PAPR as a fnction of transmit power allocation for SC-FDMA and OFDMA. The derived PAPR constraints are applicable for any normalized data modlation format. In addition, a statistical approach considering the transmission power variance constrained power allocation was derived. Two different sccessive convex approximations were derived for all the proposed constraints. Nmerical reslts indicate that instead of amplitde clipping, the PAPR constraint is of crcial importance to garantee the convergence of an iterative eqalizer. It was also observed that the proposed techniqes can significantly improve the efficiency of the transmission of power limited sers. Hence, the constraints derived in this paper are especially beneficial for the sers on the cell edge becase the coverage area of the cell can be increased for a given QoS target. The PAPR and the variance constraints depend only on the local information, i.e., the power allocation and the transmitted symbol seqence in the case of instantaneos PAPR constraint, and the power allocation only in the case of the power variance constraint. If the QoS constraint reqires centralized processing, the intition is that the power variance constraint is a better alternative becase it does not reqire information abot the transmitted symbol seqence. However, if the QoS constraint can be handled by sing the local information, instantaneos PAPR constraint can be sed becase the information abot the symbol seqence is available.

11 0 APPENDIX A THE INSTANTANEOUS POWER OF THE TRANSMITTED SIGNAL IN SC-FDMA The power of the transmitted waveform of ser at time instant m is calclated as s m gm,nb n N F + n n n,n n n N F b nb n ) ) P,l a lnm l l ) b n b n P,l a lnm) )) P,l a lnm b n P,l + n b n n + + n,n n n n,n n n n,n n n l l P,n P,n a nnn a mn n bn b n P,l a ln q bn b n l n,n n n P,n P,n a n n ma n nm ), 8) where gm,n q P,q a qmn and a qmn e jπq )m n). The second term of 8) can be rewritten as where b n n n,n n n P,n P,n a nnn a mn n n,n n n βmn n P,n P,n, 9) βmn n b n R[a nn n ]R[a mn n ]+ n ) I[a mn n ]I[a nn n ]. 30) Operators R and I in 30) take the real and imaginary part of a complex argment, respectively. The third term of 8) can rewritten as N F n,n n n bn b n l P,l a ln n N F P,l d l, 3) l where d l Denoting n,n n >n R[a ln n ]R[b n ]R[b n ] + I[b n ]I[b n ])+ ) I[a ln n ]R[b n ]I[b n ] I[b n ]R[b n ]). 3) η n n m n 4,n 3 n 3 >n 4 R[b n 3 ]R[b n 4 ] + I[b n 3 ]I[b n 4 ]) R[a nn 3ma n n 4 m] + R[a nn 4ma n n 3 m]) I[b n 3 ]R[b n 4 ] R[b n 3 ]I[b n 4 ])I[a n n 3 ma n ) n 4m] I[a n n 4 ma n n 3 m]), 33) the last term of 8) can be expressed as N F n,n n n bn b n n,n n n P,n P,n a n n ma n n m n,n n >n ηn n m P,n P,n. 34) Sbstitting 9), 3) and 34) to 8), the signal power is expressed as N F n,n n >n s m N F l ) κ + d l P,l + β mn n + η n n m) P,n P,n, 35) where κ n b n. The term κ + d l can be rewritten as κ + d l l ) ) b na ln b na ln 0. 36) n n However, the factor βmn n + ηn n m can be negative, depending on the symbol seqence and the power allocation. Let ˆη n + n m max{0, βmn n + ηn n m} and ˆη n n m min{βmn n + ηn n m, 0}. In sch a case, the instantaneos PAPR constraint can be written as ) κ + d l P,l + + ˆη n N n m P,n P,n F δ P,l + n,n n >n ˆη n n m) P,n P,n, l n,n n >n m,,...,,,,..., U, 37) where all the terms in each smmation are non-negative. It shold be noted that the nmber of smmation terms in 35) increases in the order of 4 + n n) + + n n), where the negative term is de to the ineqality sign in smmation limits in 35).

12 APPENDIX B SCACOV FOR PAPR CONSTRAINT IN SC-FDMA Denoting P,l e α,l,,,..., U, l,,...,, constraint ) becomes l κ + d l )e α,l + δ l e α,l + n,n n >n ˆη n,n n >n + ˆη n n me α,n +α,n ) n n m)e α,n +α,n ). 38) The smmation of exponentials is convex, and hence both sides of 38) are convex fnctions. Let T m α ) δ e α,l + l ˆη n,n n >n n n m)e α,n +α,n ), where α [α,, α,,..., α, ] T. The best concave approximation of T m α ) at a point ˆα is given by T m ˆT m α, ˆα ) T m ˆα )+ ˆα )α,k ˆα,k ). 39) α,k The partial derivative Tm α,k k is derived as T m δ e α,k + α nk+ ˆη knm )e α,k+α,n),k + k n ˆη nkm )e α,n+α,k ). 40) The best convex approximation of 38) at a point ˆα is written as l κ + d l )e α,l + n,n n >n + ˆη n n me α,n +α,n ) ˆT m α, ˆα ),,,..., U, m,,...,. 4) Applying [6, Eq. 36)] 3 to A m P ) yields a lower bond 43), where θ ) P,l l l P,,l θ ) n n m ˆη n n m P,n P,n n,n n >n ˆη n n m P,n P,n, 44) and τ ) m and τ ) m are given in 45). Hence, constraint ) can be approximated, m sing 46). APPENDIX D PAPR CONSTRAINT FOR OFDMA Similarly to SC-FDMA, the average power in the case of OFDMA is avg[ s m ] P,l, 47) i.e., the same as in the case of SC-FDMA. The power of the m th transmitted waveform can be calclated as s m l b l P,l + l n,n n >n d mn n P,n P,n, where ) d mn n R[a mn n ] R[b n ]R[b n ] + I[b n ]I[b n ] I[a mn n ] R[b n ]I[b n ] I[b n ]R[b n ]) ). 48) 49) The nmber of smmation terms in 48) increases in the order or n n. The PAPR constraint for OFDMA can be written as b l P,l + d + mn n P,n P,n l δ P,l + l n,n n >n d n,n n >n mn n ) P,n P,n, 50) where d+ mn n max{0, d mn n } and d mn n min{ d mn n, 0}. Let APPENDIX C SCAGP FOR PAPR CONSTRAINT IN SC-FDMA A m P ) δ P,l + l ˆη n,n n >n n n m) P,n P,n. 4) APPENDIX E SCACOV FOR PAPR CONSTRAINT IN OFDMA Changing the variables as P,m e α,m,, m, the approximation of 5) is written as b l e α,l + l n,n n >n d + mn n e α,n +α,n ) ˆT m α, ˆα ), 5) 3 This bond is derived sing the ineqality of weighted arithmetic mean and weighted geometric mean.

13 A mp ) δ l τ ) m P,l θ ) l ) l ) ) ˆη τ m n,n n >n n n m ) P,n P,n θ ) n n m τ ) m ) n n m ) τ ) m 43) τ ) m τ ) m δ l δ l P,l θ ) l ) l n,n n >n P,l θ ) l ) l δ l + n,n n >n P,l θ ) l ) l ˆη ˆη n n m ) P,n P,n θ ) n n m + n,n n >n ˆη n n m) P,n P,n θ ) n n m ) n n m n n m) P,n P,n θ ) n n m ) n n m ) n n m 45) δ l τ ) m P,l θ ) l ) l N F l κ + d l )P,l + ) ) ˆη τ m n,n n >n N F n,n n >n n n m ) P,n P,n θ ) n n m τ ) m + ˆη n n m P,n P,n ) n n m ) τ ) m 46) where ˆT m α, ˆα ) is given in 39), T m α ) is the RHS of 5) after change of variables COV), and the partial derivatives are given as T m α,k δ e α,k + k n nk+ d mkn )e α,k+α,n) + d mnk )e α,n+α,k). 5) APPENDIX F SCAGP FOR PAPR CONSTRAINT IN OFDMA Applying [6, Eq. 36)] to RHS of 5) yields a constraint 53). where θ ) P,l l l P,,l θ ) mn n d n,n n >n mn n P,n P,n d mn n P,n P,n, 54) and τ ) m and τ ) m are given in 55). Hence, constraint 5) can be approximated, m sing SCA with 53). The LHS is a posynomial and RHS is a monomial and hence, 53) is a valid GP constraint. APPENDIX G POWER VARIANCE CONSTRAINT FOR SC-FDMA Let the average power of the transmitted signal of the th ser be denoted as µ l P,l. Assming E{ b n },, n and E{b n b n } 0, n n, the variance of the otpt power is given by Σ P ) E[ s k 4 ] µ ) k [ gk,m ) k The first term redces to m k m m g k,m 4 ] µ. 56) gk,m ) µ. 57) The second term can be expressed as a fnction of power allocation as k m g k,m 4 µ + N 3 F 3 n,n,n 3,n 4 S n,n S P,n P,n + P,n P,n P,n3 P,n4, 58) { where S n, n {,,..., } : n n, n n } { ± / and S n, n, n 3, n 4 {,,..., } : n n, n 3 n 4, n, n ) n } 3, n 4 ), n 4 n 3 {n n, + n n, +n n }. Sbstitting 57) and 58) in 56)

14 3 N F b l P,l + l N F n,n n >n d + mn n P,n P,n δ l τ ) m P,l θ ) l ) l ) τ ) m n,n n >n ) ) θ mn d mn n ) P,n P n,n θ mn ) n τ ) m ) τ ) m 53) τ ) m τ ) m δ l δ l P,l θ ) l ) l n,n n >n P,l θ ) l ) l δ l + n,n n >n P,l θ ) l ) l d d mn n ) P,n P,n θ mn ) n + n,n n >n d mn n ) P,n P,n θ ) mn n ) mn n mn n ) P,n P,n θ ) mn n ) mn n ) mn n 55) we get Σ P ) N 3 F l 3 n,n,n 3,n 4 S P,l ) N 3 F n,n S P,n P,n P,n P,n P,n3 P,n4. 59) The nmber of smmation terms in n,n,n 3,n 4 S is 3. Hence, the nmber of smmation terms in 59) increases in the order of 3 +. The objective is to control the variance of the normalized power, and hence P,l in 59) is divided by n P,n, l. Hence, the constraint for power variance is written as Σ P ) σ P,l ), 60) where σ R + is the preset pper bond of the variance of transmitted power for the th ser. Plgging 59) into 60) the constraint can be written as n,n,n 3,n 4 S l l ) P,l ) n,n S P,n P,n + P,n P,n P,n3 P,n4 + P,l ) σ N F 3. l APPENDIX H SCACOV FOR POWER VARIANCE CONSTRAINT IN SC-FDMA 6) Changing the variables as P,m e α,m,, m, both LHS and RHS of 6) are convex fnctions. The linear pper bond of the convex RHS is approximated a convex constraint ) e α,l ) ˆT α, ˆα ), 6) l where ˆT α, ˆα ) is given in 39), T ˆα ) is the RHS of 6) after change of variables, and the partial derivatives are given as T α,k n n k n k± / n,n,n 3 n n,n 3 k n,n ) n 3,k) k n 3 S l e α,n+α,k + e α,n +α,n +α,n3 +α,k ) + σ N F 3 e α,l+α,k ), 63) where S {n n, + n n, + n n }. APPENDIX I SCAGP FOR POWER VARIANCE CONSTRAINT IN SC-FDMA Similarly to Appendix C, applying [6, Eq. 36)] to the RHS of 6) yields a constraint 64), where the weights are given in 65) and θ ) n n θ ) n n n 3n 4 θ 3) l P,n P,n n,n S P,n P,,n P,n P,n P,n3 P,n4 n,n,n 3,n 4 S Pn P,n P,n P, 3,n 4 P,l l P,l, θ 4) n n P,n P,n n,n n >n P,n P,n. 66) APPENDIX J POWER VARIANCE CONSTRAINT FOR OFDMA In the case of OFDMA, the first term of 56) redces to gk,m ) µ, 67) k m

15 4 N F ) P,l ) l n,n S P,n P,n θ ) n n τ ) n,n,n 3,n 4 S P,n P,n P,n3 P,n4 θ ) n n n 3 n 4 τ ) ) n n ) n n n 3 n 4 ) ) τ σ N F 3 n,n n >n ) τ ) σ N 3 F l τ 3) P,n P,n θ n 4) n τ 4) P,l θ 3) l 3) l ) τ 3) ) 4 n n ) τ 4) 64) τ ) τ ) τ 3) τ 4) n,n S P,n P,n n,n S P,n P,n + n,n,n 3,n 4 S P,n P,n P,n3 P,n4 + l P,l) σ N F 3 n,n,n 3,n 4 S P,n P,n P,n3 P,n4 n,n S P,n P,n + n,n,n 3,n 4 S P,n P,n P,n3 P,n4 + l P,l) σ N 3 F σ N 3 F l P,l n,n S P,n P,n + n,n,n 3,n 4 S P,n P,n P,n3 P,n4 + l P,l) σ N 3 F σ N 3 F n,n n >n P,n P,n n,n S P,n P,n + n,n,n 3,n 4 S P,n P,n P,n3 P,n4 + l P,l) σ N 3 F 65) while the second term is simplified to k m g k,m 4 N F Sbstitting 67) and 68) in 56) we get Σ P ) N F m P,m. 68) P,nP,n. 69) n,n n >n The smmation terms in 69) increases in the order of ) n n. After normalization, the variance constraint is written as N F n,n n >n P,nP,n σ m P,m ). 70) APPENDIX K SCACOV FOR POWER VARIANCE CONSTRAINT IN OFDMA Changing the variables to P,m e α,m,, m, constraint 7) can be approximated as N F n,n n >n α e,n +α,n ˆT α, ˆα ), 7) where ˆT α, ˆα ) is given in 39), T ˆα ) is the RHS of 6) T after COV, and the partial derivatives are given as α,k σ m eα,m+α,k. APPENDIX L SCAGP FOR POWER VARIANCE CONSTRAINT IN OFDMA Applying [6, Eq. 36)] to the RHS of 7) yields a constraint 7), where the weights are given in τ ) τ ) and θ ) m m m ) ) P θ m,m + θ m ) P,m θ ) m n,n N n >n F ) n P,n, θ ) n P,m θ ) m ) m n,n N n >n F P,n P,n θ n ) n m + n,n N n >n F n n REFERENCES P,n P,n θ n ) n ) n n P,n P,n n,n n >n P,n P,n θ n ) n ) n n ), n n 73) P,n P,n. 74) [] F. Pancaldi, G. Vitetta, R. Kalbasi, N. Al-Dhahir, M. Uysal, and H. Mheidat, Single-carrier freqency domain eqalization, IEEE Signal Processing Mag., vol. 5, no. 5, pp , 008. [] 3rd Generation Partnership Project 3GPP); Technical Specification Grop Radio Access Network Evolved Universal Terrestrial Radio Access E-UTRA, Physical channels and modlation 3GPP TS 36. Version..0 Release ), Tech. Rep., 04. [3] R. V. Nee and R. Prasad, OFDM for Wireless Mltimedia Commnications. Norwood, MA: Artech Hose, 000. [4] D. Tse and P. Viswanath, Fndamentals of Wireless Commnication. Cambridge, U.K.: Cambridge Univ. Press, 005. [5] J. Karjalainen, M. Codrean, A. Tölli, M. Jntti, and T. Matsmoto, EXIT chart-based power allocation for iterative freqency domain MIMO detector, IEEE Trans. Signal Processing, vol. 59, no. 4, pp , Apr. 0. [6] V. Tervo, A. Tölli, J. Karjalainen, and T. Matsmoto, Convergence constrained mltiser transmitter-receiver optimization in single carrier FDMA, IEEE Trans. Signal Processing, vol. 63, no. 6, pp , 05.