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1 Tite Anaysis and Compensation of Transm OFDMA and SC-FDMA Systems Author(s Yoshida, Yuki; Hayashi, Kazunori; S Wadimir Citation IEEE Transactions on Signa 3129 Process Issue Date URL IEEE. Persona use of this m Permission from IEEE must be obtain any current or future media, incud this materia for advertising or pr Rightnew coective works, for resae or ists, or reuse of any copyrighted other works.; This is not the pubi the pubished version. この論文は出版社版であり引用の際には出版社版をご確認ご利用ください Type Journa Artice Textversion author Kyoto University

2 1 Anaysis and Compensation of Transmitter IQ Imbaances in OFDMA and SC-FDMA Systems Yuki Yoshida, Student Member, IEEE, Kazunori Hayashi, Member, IEEE, Hideaki Sakai, Feow, IEEE, and Wadimir Bocquet, Member, IEEE 1 Abstract One imiting issue in impementing high-speed wireess systems is the impairment associated with anaog processing due to component imperfections. In upink transmission of mutiuser systems, a major source of such impairment is IQI introduced at mutipe transmitters. In this paper, we dea with OFDMA and SC-FDMA which have received attention in recent years as physica ayer protoco in WiMAX and 3GPP LTE, and anayze the effect of the transmitter (Tx IQ imbaances on OFDMA and SC-FDMA receivers. To cope with the inter-user interference probem due to Tx IQ imbaances, we propose a widey inear receiver for OFDMA and SC-FDMA systems and aso propose a nove subcarrier aocation scheme, which has high toerance to such Tx IQ distortion. Index Terms IQI, OFDMA, SC-FDMA, IFDMA A. Introduction I. INTRODUCTION Recenty, orthogona frequency division mutipexing (OFDM [1 based mutipe access schemes are attracting much attention for upink physica ayer protocos in highspeed wireess networks. The most prominent exampe of this is orthogona frequency division mutipe access (OFDMA empoyed in IEEE802.16e (WiMAX [2. Other significant exampe is singe carrier frequency division mutipe access (SC- FDMA [3, incuding intereaved frequency division mutipe access (IFDMA [4, which resuts from the appication of discrete Fourier transform (DFT preceding to the moduation process in OFDMA. Due to the ow peak-to-average-powerratio (PAPR of the moduated signas, SC-FDMA can eases the requirement on the power ampifiers of the transmitters. Therefore SC-FDMA has been empoyed for upink physica ayer protoco in 3GPP Long Term Evoution (LTE [5, where ower PAPR greaty benefits mobie handsets in terms of power consumption. A ow-cost impementation of such physica ayers is desirabe in view of mass depoyment, but chaenging due to impairments associated with the anaog components. A major source of anaog impairments in high-speed wireess communications systems is the In-phase/Quadrature-phase imbaance (IQI [6, [7. The IQI is the mismatch between I and Q baances due to the anaog imperfection and introduced both in the up- and down- conversion at the transceivers. In genera, it is difficut to efficienty and entirey eiminate such imbaances in the anaog domain due to power consumption, size and cost 1 Copyright (c 2008 IEEE. Persona use of this materia is permitted. However, permission to use this materia for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. of the devices. Therefore, efficient compensation techniques in the digita baseband domain are needed for the transceivers [8. In this paper, we start with investigating the impact of IQIs introduced at the mutipe transmitters in OFDMA and SC- FDMA upinks. Firsty, we describe the base band received signa modes for the systems with genera, intereaved and ocaized subcarrier aocations in the presence of transmitter (Tx IQIs and aso derive user-by-user received signato-interference ratio (SIR after user separation, where the interference is due to the Tx IQIs. Through our formuations, it wi be reveaed that the Tx IQIs sometimes cause severe interuser interference (IUI at the receiver and the appearance of the IUI uniquey determined by the empoyed subcarrier aocation, e.g., in the OFDMA system with intereaved subcarrier aocation, the IUIs occur between certain pairs of users, whie specific two users can avoid such IUI. Heretofore the anaysis on the impact of Tx and/or receiver (Rx IQIs and some usefu compensation methods have been reported for the OFDM receivers [8 [14. However, to the best of our knowedge, few works has been done focusing on the such unfair IUI probem due to the Tx IQIs in OFDMA and SC-FDMA systems. Based on the anaysis, we consider signa processing techniques to compensate for such distortions in OFDMA and SC-FDMA receivers. For the OFDM systems, the widey inear (WL fitering approach [16, [17, which jointy eaborate the received signa and its conjugate, have been expicity or impicity adopted to the equaization probem in the presence of IQI. We aso resort to the WL approach and propose WL equaization and muti-user detection method for OFDMA and SC-FDMA based on the zero-forcing (ZF and minimummean-square error (MMSE criteria. Especiay for the system with intereaved subcarrier aocation, we propose an efficient pairwise equaizer which resuting from our anaysis on the user dependent IUI. Finay, we propose a nove subcarrier aocation method, which has high toerance to the Tx IQ distortions. Recaing, the subcarrier aocation dependent IUI component is the main source of performance degradation caused by the Tx IQIs. Therefore, by empoying some specific subcarrier aocations, we can avoid the IUI and efficienty suppress the i effect of the Tx IQIs at the OFDMA receiver. This paper is organized as foows. Firsty, we introduce some matrices and their characteristic which wi be used in the rest of the paper. Sec. II formuates the effect of Tx IQIs on the received OFDMA and SC-FDMA symbos. In Sec. III, the joint channe equaization and muti-user detection methods in the presence of Tx IQIs are proposed. Then we propose the nove subcarrier aocation method to avoid the

3 2 IUI probem due to the Tx IQIs for OFDMA and SC-FDMA transmitters in Sec. IV. In Sec. V, the vaidity of our anaysis and the effectiveness of the proposed methods are evauated via computer simuations. Concusions are given in Sec. VI. B. Notations and Groundwork The notations used in this paper are as foows: the usua bod capita etters are used to denote coumn vectors or matrices and (, ( T, and ( H, are compex conjugate, transpose, and Hermitian transpose of ( respectivey. An M M identity matrix is denoted by I M and 0 M N is an a-zero matrix of size M N. We aso use diagx} as a diagona matrix whose diagona eements consist of the vector x, denotes the Eucidean norm of vector, and E[ is the expectation operation. Further on, et us introduce some matrices and their properties which wi be used in the paper. A KM KM matrix W represents KM-point DFT matrix and its (m, n eement is given by W(m, n = 1 (m 1(n 1 j 2π e KM, (1 KM where m, n = 1, 2,, KM. On the other hand, W denotes M M DFT matrix. We aso consider the square of the DFT matrix: W 2 1 (m = n = 1 or m = KM n + 2 (m, n =, 0 (otherwise (2 where m, n = 1,, KM. Thus W 2 is a permutation matrix. U is defined as KM M expander matrix, whose Km-th row is equa to the m-th of I M and the other rows are zero vectors. Left mutipication of U to a matrix represents an expand operation,i.e., inserting K 1 a zero rows between two neighboring rows of the matrix and increase the number of rows K times. On the other hand, eft mutipication of U H corresponds to decimation, which is the operation of picking up every K-th row. It can be verified that U can be expressed by using DFT matrix as [20: U = 1 W H K I Ṃ.. I M W = 1 W K I Ṃ.. I M W H (3 We aso introduce a KM KM down-shifting matrix Π whose (m, n eements Π(m, n, with m, n = 1,, KM, is given by 1 (m = n + 1 mod KM Π(m, n =. (4 0 (otherwise Left mutipication of Π denotes the coumn wise circuar down-shift operation. Since the shifting matrix is a circuant matrix, we can diagonaize it using DFT matrix as Π = KMW H diagw(2}w, (5 = KMWdiagW H (2}W H, (6 where W(2 denotes the second coumn of W. Since Π T is coumn wise up-shifting matrix, we have Π T = Π H = Π 1. What is more, it is cear that Π n and Π n represent n-times down shifting matrix and up shifting matrix respectaby (Here we assume n < KM for convenience. From (5, we have Π n = KMW H diagw(n + 1}W (7 Π n = KMW H diagw H (n + 1}W. (8 Thus, from (2, (3, and (7, it is cear that W 2 U = U W 2, (9 W 2 Π n = Π n W 2. (10 II. SIGNAL MODEL WITH IQI In this section, signa modes for OFDMA and SC-FDMA in the presence of Tx IQI wi be derived. First, we describe the IQ distorted received signa mode for OFDMA with genera subcarrier aocation. Then the impact of the interference due to the Tx IQIs at the conventiona receiver is investigated. Furthermore, we specify the modes to the systems with two major subcarrier aocations, namey intereaved and ocaized, and show the unfair IUIs depend on empoyed subcarrier aocations. After that, a system mode for SC-FDMA incuding IFDMA wi be shown. In the foowings, we assume a system with K users and the avaiabe KM subcarriers are divided into K subcarrier groups equay. A. OFDMA with Genera Subcarrier Aocation Fig. 1 depicts the bock diagram of the OFDMA system of our interest. Let s k = [s k (0, s k (1,, s k (M 1 T designate a bock of M data symbos transmitted by a user with index k (k = 1, 2,, K. The data symbos may resut from appication of a moduation scheme ike PSK or QAM to either forward error contro coded or to uncoded data bits. In this paper, we discard the time index for convenience, since we wi consider bock by bock transmissions and equaizations, and never refer to previous or subsequent bocks. The assignment of the data symbos s k to the user specific set of M subcarriers can be represented by KM M mapping matrix M k and a KM-point IDFT matrix W H. The mapping matrix M k has a unique eement of one in each coumn and has properties of M T 0 M M ( k M k = I M ( = k, (11 and M M T k = 0 KM KM ( k diagm k } (k =, (12 where the KM 1 vector m k denotes the pacement of the occupied subcarriers by the k th user, whose eements consist of M 1s and (K 1M 0s. Using M k, the transmitted signa of the k th user is given by x k = W H M k s k. (13 Before transmission, the baseband signa x k is up-converted to the radio frequency signa by using a oca osciator

4 3 K transmitters s Subcarrier IFFT: x Tx IQ ~ 1 Mapping: distortion: x 1 W H α 1 x 1 +β 1 x * 1 s k M 1 Subcarrier Mapping: M k IFFT: W H x k Tx IQ distortion: ~ α κ x k +β κ x * k x k Channe: h 1 Channe: h k r FFT: W Subcarrier Demapping: M 1 H Subcarrier Demapping: M 1 H r 1 r k Fig. 1. The OFDMA system with Tx IQIs and the notation used in the derivations. Note that the insertion and remova of the cycic prefix are not shown in this figure for ease of notations. (LO of the carrier frequency. Ideay, the LO outputs for the I and Q branches (representing the rea part Rex k } and imaginary parts Imx k } shoud have equa ampitudes and phase difference of π/2. However, in practice, the matching of I and Q signas is habituay imperfect and this eads to the ampitude and phase imbaance between the I and Q signas. Such impairments are known as Tx IQ imbaance or mismatch and this severey imits the performance of the receiver, especiay if cheap components or architectures, e.g., direct conversion architecture [18, are empoyed. The IQ distorted signa of the k th user can be modeed using the origina signa x k and its conjugate [6, [7 as x k = α k x k + β k x k, (14 where two compex scaers α k and β k are given by α k = cos θ k + j ϵ k sin θ k, (15 β k = ϵ k cos θ k j sin θ k, (16 where θ k and ϵ k denote the phase and ampitude imbaances between the I and Q branches of the transmitted signa of the k th user, respectivey. When the matching of I and Q baances is idea, i. e., θ k = 0 and ϵ k = 0, then α k = 1 and β k = 0. The degree of the imbaance is evauated by using the image rejection ratio (IRR, for the k th user s transmitted signa, which is defined by IRR k = E[ α kx k 2 E[ β k x k 2 = α k 2 β k 2. (17 Let h k (t (t = 0, 1,, L 1 denote a channe impuse response of the k th user where L 1 is the order of the channe, and the KM 1 vector h k is defined as h k := [h k (0, h k (1,, h k (L 1, 0 1 KM L T. Before the transmission over the channe h k, cycic prefix (CP is appended to x k, which is removed at the receiver before the demoduation. When the time duration of CP is onger than that of the channe, it is we known that insertion of the CP, transmission over the channe, and remova of the CP at the receiver can be described by an equivaent KM KM circuant channe matrix H k, whose first coumn is given by h k. Since the circuant matrix can be diagonaized by DFT matrix, the received signa bock after CP remova is described as, r = K H k x k + n = W H α k Λ k M k s k + β k Λ k W 2 M k s k + n, (18 where Λ k := diagwh k }, and the KM 1 vector n denotes additive noise. In addition, even if the ength of CP is insufficient, we can reduce the i effect of insufficient CP by using some advanced signa processing at the receiver, e.g., [15. It shoud be mentioned that the IQ distortion is aso introduced at frequency down-conversion at the receiver. In this paper, since we consider the upink transmission, we assume that the IQI at the receiver, e.g., a base station, is negigiby sma compared with that of the transmitters, e.g., mobie terminas, or has a priori compensated with some advanced digita signa processing techniques such as [8, [11, and we ony dea with the Tx IQIs. In addition, though we ony resort to the frequency-fat IQI as in (14, the frequency-seective IQI [22 together with carrier frequency offset (CFO is aso notabe anaog impairments in wideband systems. We can find some joint compensation methods for IQI and CFO in [23 and [24. In this paper, we mainy study the IUI probem due to Tx IQIs in OFDM based mutipe access schemes and ony resort to the frequency-fat case for simpicity. Next, we describe the per-user received signa after mutiuser detection. Here, we assume that the user separation is performed in the conventiona manner, i.e., the received signa component on the set of subcarriers assigned to a user is extracted out by using the corresponding aocation matrix in discrete frequency domain. The received signa of the th user ( = 1, 2,, K after user separation is given by r = M T Wr = α Λ s + β k Λ k M T W 2 M k s k + n, (19 where Λ k = diagm T Wh k}, and n = M T Wn. The second term in the right-hand side of (19 represents the interference due to the Tx IQIs. Recaing (2 and M k is the mapping matrix, M W 2 M k consist of 0 or 1 eements and hence the appearance of the interference argey depend on the

5 4 empoyed subcarrier mappings. In order to carify the effect of the interference on the performance of OFDMA receivers, we derive the per-subcarrier SIRs for each user. By assuming, E[s k s H σ 2 = si M (k = 0 M M (otherwise, (20 we have R (s R (i = E[r (s (r (s H = α 2 Λ 2 Λ (21 = E[r (i (r (i H = K β k 2 Λ 2 k Λ kdiagm T W 2 m k }. (22 where r (s = α Λ s and r (i = K β kλ k M T W2 M k s k. Therefore, the SIR of the m th subcarrier (m = 1,, M in the th user s received signa is given by SIR (m = R(s (m, m R (i (m, m = α 2 β 2 m (m + K (k β k 2 Λ k(m 2 Λ (m 2 m k (m, (23 where we set m k = diagmt W2 m k } and Λ k (m denotes the m th diagona eement of Λ k. In the denominator of (23, the first term corresponds to the interference due to the th user s own Tx IQI. Meanwhie, the second term comes from the interference caused by other users IQIs, namey the IUI. For comparison, we aso consider the OFDM system in the presence of the Tx IQI, or equivaenty K = 1 and the mapping matrix M 1 = I M in our formuations. In this case, the received signa is given by r 1 = α 1 Λ 11 s 1 + β 1 Λ 11 W 2 s 1 + n 1. (24 Thus, the per-subcarrier received SIR resuts in SIR 1 (m = α 1 2 β 1 2. (25 Obviousy, there is no IUI probem in OFDM system and the SIR 1 (m is the same as the IRR. Therefore high IRR eve at the transmitter s anaog front-end directy means ow interference power due to the Tx IQI at the receiver. Meanwhie, in the OFDMA system, the per-user received signa suffer from the IUI depending on the empoyed subcarrier aocation and the resuting SIR (m is far from the IRR. Furthermore, since the impact of such IUI is proportiona to the corresponding channe gains Λ k(m 2 Λ (m as in (23, SIR 2 (m possiby comes down to the serious eve due to the IUI in seective fading environments. This suggests essentia need of Tx IQI compensation in OFDMA systems. Heretofore, the Tx IQI compensation is considered at the transmitter to simpy improve the quaity of IQ moduation, or at the receiver to avoid the additiona compexity at the transmitter [9. Here, in OFDMA systems, we can newy say that the Tx IQI compensation at the receiver is much more effective than at the transmitters, because achieving the high IRR eve at the transmitter is not necessary meaning the ow SIR at the Fig. 2. Index of subcarrier k M+k 2M+k (K-1M+k Subcarrier aocation of k th user in the intereaved OFDMA receiver. From this perspective, we wi propose the simpe Tx IQI compensation methods for OFDMA receivers in Sec. III. Before that, for further anaysis of the impact of the IUI, we specify the signa modes for OFDMA systems with two major subcarrier aocations, i.e., intereaved and ocaized aocations and aso derive the received SIR for each subcarrier aocation. B. Intereaved OFDMA Intereaved OFDMA, aso known as distributed OFDMA has the user dependent comb shape subcarrier aocation depicted in Fig. 2. Such mapping matrix M k for the k th user is given by M k = Π k 1 U. (26 Thus from (19, we have the th user received signa for the intereaved OFDMA as r = α Λ s + β k Λ k U T Π +1 W 2 Π k 1 Us k + n = α Λ s + β k Λ k U T Π k+2 U W 2 s k + n (27 From (3 and (7, M U T Π n U = K W H diagp} W, (n = 0,, 2K 2 where the m th eement of the vector p is given by Therefore p(m = K 1 KM K WH (m, 1 (n = 0 (m 1+kMn j 2π e KM = K WH (m, 2 (n = K. (28 0 M 1 (otherwise I M (n = 0 U T Π n U = Π 1 (n = K, (29 0 M M (otherwise where Π i := M W H diag W H (i + 1} W denotes the i- times up shifting matrix of M M. As a resut, we can further simpify the per-user received signa as r =α Λ s + β f( Λ f( Π g( W2 s f( + n, (30

6 5 Fig. 3. km+1 Index of subcarrier (k+1m Subcarrier aocation of k th user in the ocaized OFDMA where f( and g( are given by 1 ( = 1 f( = K + 2 (otherwise, (31 0 ( = 1 g( = 1 (otherwise. (32 For exampe, when K = 4, the reation ship between and f( is given as foows, f( (33 Under the same assumption in (20, the per-subcarrier received SIR of the th user is given by SIR (m = E[ α Λ (ms (m 2 E[ β f( Λ f( (m( Π g( W 2 s f( (m 2 = α Λ (m 2 β f( Λ f( (m 2, (34 where m = 1,, M. Interestingy, in the intereaved OFDMA system, the th user s received signa interfered ony by the f( th user s image component. In particuar, when f(, the interference can be considered as an IUI and, with respect to user fairness, such IUI is quite undesirabe since the impairment in IQ moduation of a user termina make troube not on its own performance but on the other s. What is more, from (34, the power of such IUI is proportiona to the ratio of the channe gains and the resuting SIR (m (m = 1, 2,, M is far from IRR. On the other hand, the 1 st and the (K/2 + 1 th users successfuy avoid the IUI and the resuting SIR is equivaent to the IRR. We evauate this unique phenomenon ater in our computer simuations. C. Locaized OFDMA The ocaized OFDMA has bock-wise subcarrier aocation as in Fig. 3, and the mapping matrix for the k th user is given by M k = Π (k 1M V, (35 where KM M matrix V is defined as [ I V = M. (36 0 (K 1M M Therefore, the per-user received signa is represented by r = α Λ s + β k Λ k V T Π ( 1M W 2 Π (k 1M Vs k + n = α Λ s + β k Λ k V T Π (k+ 2M W 2 Vs k + n. Here [I V T Π nm M 0 M (K 1M = [ 0 M n M I M 0 M (K n 1M (37 (n = 0 (otherwise, (38 where n := n mod K, and W 2 V denotes KM M matrix consist of the first M coumns of W 2, i.e., J W 2 V = 0 (K 2M M, (39 W 2 J where the (m, n eement of M M matrix J is defined as 1 (m = n = 1 J(m, n =. (40 0 (otherwise Therefore, we can simpify (37 as r = α Λ s + β f( Λ f( Js f( + β e(λ e( ( W 2 Js e( + n,, (41 where For exampe, when K = 4, e( = K + 1. ( e( (43 The resuting per-subcarrier SIR of the th user is given by α Λ (m 2 β SIR (m = f( Λ f( (m (m = 1 2 α Λ (m 2 β e( Λ e( (m (m > 1, (44 2 where m = 1,, M. In the ocaized OFDMA, the interference on the th user received signa consist of the IUIs from f( and e( th user s image components. Actuay, from (44, the IUI from the e( th user is the main source of the performance degradation when M is sufficienty arge. D. SC-FDMA A system mode for SC-FDMA is easiy derived in the context of DFT-precoded OFDMA [3, i.e., to repace s k by Ws k. Thus from (19, the k th user s transmitted signa is given by x k = W H M k Wsk. (45

7 6 From (18, the received signa is represented by r =W H α k Λ k M k Wsk + β k Λ k W 2 M WH k s k + n. (46 Correspondingy, the received signa component of the th user is represented as r = W H M T Wr = α WH Λ Ws + β WH k Λ k M T W 2 M k Ws k + n, (47 where the noise vector n = W H M T Wn. Two major choices of M k are ocaized and intereaved aocation. It shoud be mentioned that, when the intereaved subcarrier aocation is empoyed, the subcarrier assignment for each user can be further simpified, i.e., x k = W H Π k 1 U Ws k M = K diagwh (k} I Ṃ.. I M s k. (48 This shows that the transmitted signa is easiy designed based on repetition and subsequent user dependent frequency shift of a moduated signa. In such case, the intereaved SC-FDMA is known as IFDMA [4. IFDMA is one promising aternative to OFDMA due to the ow compexity of user separation as we as the ow PAPR of the transmitted signa, especiay for the mobie upink. In regard to the per-subcarrier received SIR, since the DFT matrix W is a unitary, thus we have E[(Ws k (m(ws k H (m = E[s k (ms H k (m under the same assumption in (20 and hence the per-subcarrier SIRs for the SC-FDMA resuts in the same as those of OFDMA. III. WIDELY LINEAR EQUALIZATION AND USER DETECTION METHOD IN THE PRESENCE OF TX IQI Here, we consider the equaization and the mutipe user detection probem at the OFDMA receiver in the presence of Tx IQIs. From (18, the received signa r is not the inear function of s k (k = 1,, K due to the Tx IQI, however, the regression is inear both in s k and s k. Such system is known as widey inear (WL or conjugate inear system. The WL fitering approach [17, which jointy eaborate the received signa and its conjugate, can be efficienty appied to the equaization and the user detection probems in such WL systems. In this section, first, the WL equaization and user detection method for the OFDMA receiver with genera subcarrier aocation wi be proposed. One attractive feature of the conventiona OFDMA receiver is the efficient user separation and channe equaization in discrete frequency domain [2, [19. The proposed receiver is aso impemented in discrete frequency domain, and can jointy compensate the Tx IQI and the channe. Moreover, we propose a simpe pairwise equaizer for the OFDMA receiver with intereaved subcarrier aocation. In the previous section, we have shown the IUI due to the Tx IQIs appears ony between certain pairs of users in the OFDMA. Making use of the fact, we consider a pairwise equaization scheme for user-separated signas in the intereaved OFDMA receiver, which can further reduce the compexity. Both for the genera and intereaved OFDMA systems, the WL receiver based on ZF and MMSE criteria wi be derived where we assume a the channe state information and the Tx IQI parameters are a priori known to the receiver by using some sophisticated signa processing techniques such as [9 or [11. It shoud be mentioned that, in this section, we mainy dea with OFDMA. However, the proposed methods can be easiy appied to the SC-FDMA in the context of the DFT-precoded OFDMA as in Sec 2. D. A. The WL receiver for OFDMA with genera subcarrier aocation Here, we propose the WL receiver for the OFDMA system with genera subcarrier aocation. From (18, r = W H diagγ}s + W H diagδ}w 2 s + n, (49 where K s = M k s k, K γ = α k Λ k m k, K δ = β k Λ k W 2 m k Furthermore, by stacking r and r, we have [ [ r W r = H diagγ} W H diagδ}w 2 Wdiagδ }W 2 Wdiagγ } [ s s + (50 [ n n. (51 Ceary, we can separate each user s transmitted signa from s by using (11, e.g., s k = M T k s. Therefore, in the foowings, we discuss the estimation of s based on the ZF and MMSE criterion in the system (51. 1 ZF equaizer: In the absence of the noise, the WL ZF condition is given by [ r s = G ZF r, (52 where KM 2KM matrix G ZF represents the WL equaizer. From (51, we can derive G ZF as G ZF = [ IKM 0 KM KM [ W H diagγ} W H diagδ}w 2 where Wdiagδ }W 2 Wdiagγ } 1 = [ diagξ}w diagξ δ (W 2 γ 1 }W, (53 ξ = W 2 γ (γ W 2 γ δ W 2 δ 1. (54 In the above equations, denotes the eement wise mutipication of vectors, i.e., x y = diagx}y, and each eement of vector x 1 is given by the reciproca of the corresponding eement of x, i.e., x 1 (m = 1/x(m. Eq. (53 shows that, when KM is power of 2, the proposed WL ZF equaizer can be efficienty impemented by using FFT. In fact, it needs

8 7 twice the computationa effort as the one-tap FDE used in conventiona OFDMA systems [1, [19. It shoud be mentioned that when we empoy noncircuar moduation schemes incuding rea moduation format such as BPSK, m-ask or OQAM, we can find a certain reationship between s and s and this enabes us to gain the additiona degrees of freedom in choosing WL equaizer [21. The expoitation of such noncircuar property of the transmitted signa is one notabe advantage of WL approach over conventiona inear approach. However, in practica OFDMA and SC- FDMA systems, circuar moduation schemes such as QPSK and QAM scheme where s and s have the orthogonaity, i.e., s cannot be obtained from s by a simpe inear reation, are mainy empoyed. Therefore, in this paper, we ony dea with the circuar moduation schemes and don t resort to expoiting non-circuarity of the signa moduation. In addition, the proposed WL ZF receiver is cosey reated to the two per-tone equaizer in [23. In [23, the compensation method for the Tx IQ distortion together with the Rx IQI and the carrier frequency offset at the OFDM and OFDMA receivers has been proposed. The structure of the proposed ZF WL receiver resuts in the same as that of their two per-tone equaizer. In this paper, we expicity derive the tap weights of the receiver. Furthermore, we wi anayticay show the optima WL receiver in terms of the MMSE criterion is aso reaized by the same structure as the ZF receiver. 2 MMSE equaizer: As in [17, the best WL equaizer in terms of minimum error covariance, i. e., G MMSE = arg min G is obtained as [ [ E s s G [ r 2 r, (55 where a = W 2 γ (γ W 2 γ δ W 2 δ + σ2 n 2 γ } ν 1, (62 b = δ (γ W 2 γ δ W 2 δ + σ2 n 2 W 2 δ} ν 1, (63 ν = σ2 n 2 ( γ + δ + W 2 γ + W 2 δ + γ W 2 γ δ W 2 δ + σ4 n 4. (64 As the proposed WL ZF equaizer, the proposed MMSE equaizer is aso efficienty impemented in discrete frequency domain. B. The proposed pairwise equaizer for the intereaved OFDMA system Next, we consider the channe equaization in the intereaved OFDMA receiver with the conventiona user separation. As in Sec. 2.2, when we empoy the intereaved subcarrier aocation, the th user s received signa after the user separation ony interfered with the f( th user s image component. Taking the fact into consideration, we propose the pairwise equaizer for the intereaved OFDMA based on ZF and MMSE criteria. From (30, we have the pairwise received signa: [ [ [ [ r diagγ } diagδ }Φ s n = diagδf( }Φ diagγf( } + n f(, r f( where s f( (65 G MMSE = [ (Rsr C sr R 1 rr C rrd 1 rr (C sr R sr R 1 rr C rr D 1 rr, (56 diagγ } = α Λ, (66 diagδ } = β f( Λ f(, (67 Φ = Φ g( = Π g( W2, (68 where the correation matrix R xy = E[xy H and the compementary correation matrix C xy = E[xy T and D rr = R rr C rr R 1 rr C rr. Here we consider the system with circuar moduation schemes and eventuay assume R ss = I 2 KM and C ss = 0 KM KM. Aso by assuming n is a zero-mean white circuar compex Gaussian noise, n N c (0 KM 1, σni 2 KM, we have R sr = σ 2 sdiagγ }W, (57 C sr = σ 2 sdiagw 2 δ}w, (58 R rr = W 2 H diag γ + δ + σ2 n 2 1}W, (59 C rr = W 2 H diagγ W 2 δ + δ W 2 γ}w, (60 where x = x x and 1 denotes a KM 1 a one vector. Consequenty, the WL MMSE equaizer is given by G MMSE = [ diaga}w diagb}w, (61 1 ZF equaizer: In the absence of the noise, the ZF criterion is given by [ [ r s Z ZF r = f( s, (69 f( where the 2M 2M matrix Z ZF equaizer. Thus we have denotes the inear ZF [ 1 diagγ } diagδ }Φ Z ZF = diagδf( }Φ diagγf( [ } diagc } diagd }Φ = diagd f( }Φ diagc f( }, (70 where M size vector c and d denotes equaizer weights and they are given by c = Φ γ f( (γ Φ γ f( δ Φ δ f( 1, (71 d = δ (γ Φ γ f( δ Φ δ f( 1. (72

9 8 Index of subcarrier Fig. 4. Subcarrier aocation of the 2nd user in the proposed IUI free OFDMA where K = 4 and M = 4 2 MMSE equaizer: Under the same assumption in Sec. 3.1.B, we have R s r = σ 2 sdiagγ }, (73 C s r f( = σ 2 sdiagφ δ f( }Φ, (74 R r r = diag γ 2 + δ + σ2 n 2 1}, (75 C r r f( = diagγ 2 Φ δ f( + δ Φ γ f( }Φ. (76 The optimum inear equaizer Z MMSE based on MMSE criterion, i.e., [ [ [ Z MMSE = arg min E s r 2 R s R f( r, (77 f( can be derived as [ [ 1 Rs r Z MMSE = C s r Rr f( r C r r f( C s f( r R s f( r f( C r f( r R r f( r f( [ diage } diagf }Φ = diagff( }Φ diage f( }, (78 where and e = Φ γ f( (γ Φ γ f( δ Φ δ f( + σ2 n 2 γ } ν 1, (79 f = δ (γ Φ γ f( Φ δ f( δ + σ2 n 2 Φ δ f( } ν 1, (80 ν = σ2 n 2 ( γ + δ + Φ γ f( + Φ δ f( + γ Φ γf( δ Φ δf( + σ4 n 4. (81 From (70 and (78, it is cear that proposed ZF and MMSE pairwise equaizers are aso efficienty impemented by using FFT. IV. IUI FREE SUBCARRIER ALLOCATION Here, we propose a nove subcarrier mapping method which can significanty suppress the performance degradation due to the Tx IQI for OFDMA and SC-FDMA. In Sec. 2, we show the IUI component in the received signa is the significant source of the performance degradation due to the Tx IQI and aso the appearance of the IUI is uniquey determined by the empoyed subcarrier aocation M k. Therefore, it is possibe to choose a certain set of subcarrier assignments in which a the users can avoid the IUI due to their Tx IQIs. Ceary, from (19, if β k Λ k M T W 2 M k s k = 0 M 1 (k (82 hods, there is no IUI due to the Tx IQI. The sufficient condition for this is M T W 2 M k = 0 M M (k. (83 Actuay, there are many choices for a set of M k (k = 1,, K which hods (83 and (11. One simpe soution is to assign certain two intereaved aocations for one user based on (30: First we temporariy consider the system with K = 2K users where each user occupies M/2 subcarriers and design 2K intereaved mapping matrices M k (k = 1,, 2K (Here we assume K and M are power of 2 for simpicity. The resuting IUI ony appears between certain users k and K k + 2 as in (31. Therefore, in the system with K users, we can derive a set of IUI free KM M subcarrier mapping matrices as [ M 1 M K+1 (k = 1 M k = [ (k 1, (84 M k M 2K k+2 where k = 1,, K. An exampe of subcarrier aocation in the proposed IUI free OFDMA is iustrated in Fig. 4 where K = 4 and M = 4. By the use of such IUI free subcarrier aocations, we can efficienty avoid the performance deterioration due to Tx IQIs without knowing the imbaance parameters and channe coefficients a priori. This feature is quite attractive in practica systems because the IQ distortion parameter estimation is chaenging task when the transceiver suffer from severa anaog imperfections such as carrier frequency offset or phase noise simutaneousy [12, [23. V. SIMULATION RESULTS Here, we evauate the BER performances of OFDMA and SC-FDMA systems with the proposed receivers via computer simuations. In our simuations, we have empoyed 16QAM with coherent detection for the moduation/demoduation scheme, and set the number of users K = 4, where a the users are assigned the same transmit power. The number of subcarriers per-user M = 128, therefore the FFT size is KM = 512 and the ength of CP is set to be 32. The channes and Tx IQIs are randomy generated for each iteration. We considered channes of ength L = 10 with i.i.d circuar compex Gaussian coefficients, i.e., h k N c (0 L 1, 1 L I L (k = 1, K. On the other hand, Tx IQI parameters are drawn from uniform distributions, e.g., ϵ k U(0, 0.1. For the receiver, we test the proposed WL receiver as in Sec. III and the conventiona one where user separation is performed as in (19 and the one-tap FDE [1, [3 is empoyed. A the channe response and IQI parameters are assumed to be known to the receiver and the equaizer weights based on ZF criterion are used both the proposed

10 9 and conventiona receiver in OFDMA, whie we empoy the MMSE weights in SC-FDMA. In a the figures, horizonta axes denotes the received SNR in db and vertica axes is the average of per-user BER over 3000 iterations. First, we test the vaidity of our anaysis. For this purpose, we consider the situation where ony the 1st and 2nd users transmitter suffer from IQI with ϵ 1, ϵ 2 } U(0, 0.04 and θ 1, θ 2 } U( 0.04π, 0.04π and the other s have idea anaog front-ends, i.e., ϵ 3 = ϵ 4 = θ 3 = θ 4 = 0. Resuting IRR 1 and IRR 2 are about 35 db. Fig. 5 shows the user-byuser BER performance of the intereaved OFDMA system with the conventiona receiver. From the figure, we can see the significant performance deterioration ony on the 4th user. As we have shown in (33, the 4th user s received signa suffer from the IUI due to the 2nd user s image component, meanwhie the image component of the 1st user appears among its own subcarriers. Therefore, the resuting SIR 4 (m (m = 1,, M possiby become fata according to the channe conditions, whie SIR 1 (m = IRR 1 and the interference of 35 db sighty degrade the performance of the 1st user. Fig. 6 represents the performance of the ocaized OFDMA with the conventiona receiver in the same scenario as Fig. 5. In this case, as in (43, the most of IUI due to the Tx IQIs of the 1st and 2nd transmitter appears on the 4th and 3rd user s received signa respectaby. Therefore we can see the significant performance degradation on the 3rd and 4th users. Next, we show the impact of Tx IQI in the OFDMA with genera subcarrier aocation where the IQ distortion occurs at a the transmitters. Here the set of K mapping matrices is randomy generated for each iteration and we set ϵ 1, ϵ 2, ϵ 3, ϵ 4 } U(0, 0.04 and θ 1, θ 2, θ 3, θ 4 } U( 0.04π, 0.04π. Fig. 7 show the BER performance of the OFDMA with the conventiona receiver. In the figure, we aso incude the performance of OFDM scheme with the conventiona ZF equaizer in the same simuation setting for comparison. Recaing that the IRR of 35 db is sufficienty sma and such Tx IQIs cause a sight performance degradation uness the IUI, the performance of the OFDM receiver deteriorates inconsideraby. On the other hand, in the OFDMA, the IUI possiby occurs between a the users and the BER performances are seriousy degraded. Fig. 8 represents the effectiveness the proposed WL ZF receiver in the OFDMA system. From the figure, we can see the significant performance improvement. On the other hand, the performance of the intereaved SC- FDMA or equivaenty IFDMA in the presence of the Tx IQI can be seen in Fig. 9 and 10. Fig. 9 shows the BER performance of IFDMA with the conventiona receiver with MMSE equaizer weights [19 where we set the Tx IQI parameters as in Fig. 7. From the figure, though we empoy the MMSE equaizer, the 2nd and 4th users BER performance sti suffer from error foor, whie the 1st and 3rd user can achieve the significant performance improvement due to the diversity gain. Fig. 10 is the performances of the IFDMA system using the proposed pairwise MMSE equaizer. As in Fig. 10, the proposed pairwise equaizer efficienty compensate the interference due to IQI and the channe distortion, and significanty improve the BER performances. Finay, we test the advantage of the proposed IUI free Fig. 5. The BER performance versus SNR of the intereaved OFDMA with the conventiona receiver Fig. 6. The BER performance versus SNR of the ocaized OFDMA with the conventiona receiver subcarrier aocation. In particuar, we empoy the aocation (84 and the Tx IQI parameters are set the same as in Fig. 7. Fig. 11 shows the performance of OFDMA with the proposed subcarrier aocation and the conventiona ZF equaizer. From Fig. 11, by empoying the proposed aocation, the received signa efficienty avoid the IUI due to Tx IQIs and achieve the comparabe performance to the OFDMA receiver with the proposed WL equaizer. In this case, there is no need to know the Tx IQIs parameters a priori. VI. CONCLUSION In this paper, the effect of the Tx IQIs at the OFDMA and SC-FDMA receiver is investigated. We revea the IUI due to the IQI is the major source of performance degradation and such IUI emerges depend on the subcarrier mapping used in the system. We proposed the ow compexity WL receivers for the OFDMA and SC-FDMA receivers and show the performance improvements with the proposed equaizers via computer simuations. In addition, a nove subcarrier aocation method is proposed to cope with the severe IUI due to the Tx IQ distortion. Under Tx IQIs, we can significanty improve the BER performance of the OFDMA receivers by just empoying the proposed mapping.

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