COMBINED FREQUENCY AND SPATIAL DOMAINS POWER DISTRIBUTION FOR MIMO-OFDM TRANSMISSION

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1 The 8th nnual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) COMINED FREQUENCY ND SPTIL DOMINS POWER DISTRIUTION FOR MIMO-OFDM TRNSMISSION Wladiir ocquet, Kazunori Hayashi 2 and Hideaki Sakai 2 France Teleco R&D, Tokyo Laboratory, 3--3 Shinjuku, Tokyo, Japan 2 Kyoto University, Yoshida-Honachi, Kyoto, Japan STRCT In this paper, we propose to adapt siultaneously the transit both in the spatial and frequency doains using a heuristic expression of the bit error rate (ER) for each subcarrier and transit antenna. The ethod consists of grouping a certain nuber of subcarriers and perforing local adaptation in each subcarrier group and transit antenna. The subcarrier grouping is perfored in such a way that equalizes the average channel condition of each subcarrier group. The grouping and the local adaptation allow us to take advantage of the channel variations and to reduce the coputational coplexity of the schee. With the siplicity of the heuristic ER expression, we can obtain a closed for expression of the transit to be allocated. Siulation results show significant perforance gain in ter of ER copared to the. Keywords- OFDM, MIMO, Lagrangian ethod, global ER optiization. I INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) [] is a popular ethod for high data rate wireless transission. Cobining OFDM with antenna arrays at the transitter and receiver enhances the capacity gain. Robustness of MIMO- OFDM transission to ulti-path delay is obtained when appropriate guard interval is inserted in the transitted frae. In frequency selective fading environent, fading conditions strongly affect the channel gains of each subcarrier and transit antenna [2]. In this publication, we propose to cobine the adaptation of the transit in the frequency and spatial doains in ters of ER perforance. The ethod consists of grouping a certain nuber of subcarriers and then perfor local adaptation siultaneously in each subcarrier group and all the transit antennas. The rest of the paper is organized as follows. In Section II we review the conventional MIMO-OFDM schee. In Section III, we introduce the estiate value of ER for channel encoded sequence and we describe in detail the adaptation schee siultaneously in the frequency and spatial doains. Grouping ethod and calculation of the allocation in the frequency and the spatial doains will be explained. Section IV gives the experiental results over QPSK and QM odulations, different antenna configurations, and several coding gains. Finally, conclusions are drawn in Section V. II SYSTEM DESCRIPTION MIMO OFDM transit signal The principle of OFDM transission schee [] is to reduce bit rate of each sub-carrier and also to provide high bit rate transission by using a nuber of those low bit rate subcarriers. OFDM syste can provide iunity against frequency selective fading because each carrier goes through nonfrequency selective fading. Given the syste description of the OFDM syste, we can develop a MIMO-OFDM signal odel. In this paper, we will need both tie-doain and frequencydoain odels. Suppose that a counication syste consists of N t transit (T X) and N r receive (RX) antennas, denoted as N t N r syste, where the transitter at a discrete tie interval t sends an N t -diensional coplex vector and the receiver receives an N r -diensional coplex vector. n OFDM syste [] transits N odulated data sybols in the i-th MIMO-OFDM sybol period through N sub-channels. The transitted baseband MIMO-OFDM signal for the i-th block sybol, is expressed as s (l) i,k = N. N { pl, x (l) i, exp j 2π k } N where x (l) i, and p l, are respectively the odulated data sybol and the transit on the l-th transit antenna for the i-th OFDM sybol on the -th subcarrier. To cobat inter sybol interference (ISI) and inter carrier interference (ICI), guard interval (GI) [3] such as cyclic prefix (CP) or zero padding (ZP) is added to the OFDM sybols. In the case of CP, the last N g saples of every OFDM sybol are copied and added to the heading part. The transit signal can be described as follows: s (l) i,k = { s (l) i,n N g +k () for 0 k < N g s (l) (2) i,k N g for N g k < N + N g MIMO OFDM received signal We assue that the syste is operating in a frequency selective Rayleigh fading environent [4] and the counication channel reains constant during a packet transission. One data frae duration is assued to transit within one coherent tie of the wireless syste. In this case, channel characteristics reain constant during one frae transissions and ay change between consecutive frae transissions. We suppose that the fading channel can be odeled by a discrete-tie baseband equivalent (L )-th order finite ipulse response (FIR) filter where L represents tie saples corresponding to the axiu delay spread. In addition, an additive white Gaussian /07/$25.00 c 2007 IEEE

2 The 8th nnual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) noise (WGN) with N r independent and identically distributed (iid) zero ean, coplex Gaussian eleents is assued. When the axiu delay spread does not exceed GI, since ISI does not occur on MIMO OFDM sybol basis, the frequency doain MIMO OFDM signal after reoval of GI is described by: Nt y (q) i, = h (q,l) p l, x (l) i, + n(q) i, (3) where y (q) j, is the received signal at the q-th received antenna for the i-th OFDM sybol and the -th sub-carrier and h (q,l) is the channel paraeter fro the l-th transitting antenna to the q-th receiving antenna which coposes the MIMO channel atrix. In addition, n (q) j, denotes the WGN for the q-th received antenna. In this paper, we liit the study to the linear detection schee. Thus, output of the equalizer can be described by: z i, = G y i, (4) where z i, = [z (0) i,,, z(n t ) i, ] T and y i, = [y (0) i,,, y(n r ) i, ] T respectively denote the output of the equalizer and the received signal. In the case of zero forcing (ZF) detection, the equalizer eleents are equal to with and G = H = III G = (H H H ) H H (5) g (0,0) g (0,Nr ) g (N t,0) g (N t,n r ) h (0,0) h (0,Nt ) h (N r,0) h (N r,n t ) POWER LLOCTION SCHEME, (6). (7) The schee is based on a siple procedure which consists of perforing frequency doain optiization of the transit in function in ter of the CSI and the expression of the approxiated ER [5]. approxiation Recently, it has been in [6] and [7] that the heuristic expression to approxiate the ER is obtained as: f(β l,, p l, ) a exp { b β l, p l, } (8) where β l, is equal to: β l, = (9) (2 N ).σ 2 n. Nr g (l,n) 2 n=0 Table : Transission odes for QPSK odulation Modulation QPSK QPSK QPSK Coding Rate /2 3/4 Rate (bits/syb.).5 2 a b Table 2: Transission odes for 6-QM odulation Modulation 6-QM 6-QM 6-QM Coding Rate /2 3/4 Rate (bits/syb.) a b 6.73 where N is the nuber of bits per sybol(n = 2 for QPSK, N = 4 for 6-QM and N = 6 for 64-QM) and β l, denotes an equivalent received signal-to-noise ratio (SNR), which depends on the odulation schee and the equalizer weights on the -th subcarrier and l-th transit antenna. The paraeters a and b are to be deterined in a heuristic way, naely, via coputer siulations. In this paper, we consider the following two MIMO-OFDM systes [7]: -Uncoded OFDM: Without forward error correction (FEC) with QPSK and QM odulations. -Convolutionally coded OFDM: With the convolutional code in [8]. The generator polynoial of the other code is g = [33, 7]. The coding rates are obtained fro the puncturing pattern described in [8]. Tables, 2 and 3 respectively suarize the paraeters of a and b for QPSK, 6-QM and 64-QM obtained via coputer siulations. In the tables, the coding rate of eans the uncoded MIMO-OFDM syste. Proposed principle The basic principle of the frequency doain allocation for MIMO-OFDM signal is to cobine spatial and frequency doains optiization of the transit in function of the channel state inforation (CSI) and the expression of the heuristic ER expression [5] defined in (8). The Lagrangian optiization ethod will be to obtain analytical value of the allocation for each load subcarrier. Furtherore, constraint is added in order to keep constant the global transit at the transitting part. The optial case is to consider the allocation schee through one MIMO OFDM sybol which is represented by (N N t ) eleents. However, due to the coputation coplexity to perfor allocation schee, we propose to perfor it through a liited nuber of subcarrier denoted N s and then we repeat

3 The 8th nnual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) Table 3: Transission odes for 64-QM odulation Modulation 64-QM 64-QM 64-QM Coding Rate /2 3/4 Rate (bits/syb.) a b the allocation schee N/N s ties. Let first define the transit atrix of the t-th subcarrier group as p (t) o = p 0,t N s,o p 0,(t+) N s,o p N t,t N s,o p N t,(t+) N s,o (0) Then, we consider for the t-th subcarrier group under the condition that the average transit are kept constant to be P. The optiization proble can be given by p (t) o s.t. N t = arg in N t s s f(β l,t Ns+,p l,t Ns+ ) N s N t p l,t N s + = N s N t P () One possibility to solve this optiization proble is to apply the Lagrangian procedure. Defining: J(p (t) o ) = t + λ ( N t s s f(β l,t Ns +, p l,t Ns +) N t N s p l,t Ns + N t N s P ), (2) The optial solutions are obtained by solving for 0 l < N t and 0 < N s : ( Nt p l, t N s s f(β l,t N s+,p l,t N s+) N t N s ) + λ = 0 p l,t Ns + N t N s P = 0 (3) fter calculation and rearrangeent for 0 l < N t and 0 < N s, we finally obtain: p l,t Ns +,o = + b t u=0 [ Nt u=0 s v=0 s v=0 ] [ β l,t Ns+ N t N s P log ( βl,t Ns+ ) ] (4) Due to the specificity of the Lagrangian calculation (only atheatical solutions are obtained), we need to add a constraint when the output of the Lagrangian optiization does not reflect any physical solution, typically when we obtain: p l, 0. In this case we propose to apply the conventional schee. We repeat the allocation process for each subset of subcarriers and transit antenna which copose the MIMO-OFDM sybol and each transit antenna. IV EXPERIMENTTION We now evaluate the perforance of the allocation ethod for MIMO-OFDM schee in a utli-path fading environent with ZF detection. We assue perfect knowledge of the channel variations both at the transitting and receiving parts. n exponentially decaying (-d decay) ulti-path odel is assued and carrier frequency is equal to 2.4GHz. The IFFT/FFT size is 64 points and the guard interval is set up at 6 saples [8]. Table 4: Siulation Paraeters Carrier Frequency 2.4 GHz andwidth 20 MHz (N t, N r ) (2,2), (4,4) Modulations QPSK, 6-QM, 64-QM Channel encoder No code and convolutional code Coding gain R=/2, 3/4 and Channel estiation Perfect CSI Nuber of data subcarrier 64 Guard Interval length 6 Channel odel 0-path, Rayleigh Fading Saple period 0.05µs Nuber of data packet 40 Subgroup size (N s ) 2, 4, 8, 6, 32, 64 Uncoded MIMO-OFDM Effect of schee for several subset sizes is highlighted for the antenna configuration N t = N r = 4, without channel encoder R= for QPSK and QM odulations. Figs., 2, and 3 show the ER versus the total received SNR (d) of the schee with various subcarrier group sizes N s. The ER perforance of the conventional schee () is also plotted in the sae figure. The siulation results in Fig. shows that, for QPSK odulation at average ER=0 4, 2.8, 4.5, 5.2, 6., and 8.7 d gains are respectively obtained for N s = 2, 4, 8, 6 and 64. Figs. 2 and 3 show the ER perforance with 6-QM and 64-QM odulations, respectively for several subcarrier group sizes. Fro these figures, we can see that the schee can achieve significant perforance gain also for QM odulations. It is shown that the subcarrier grouping size strongly affects the perforance of the schee for both QPSK and

4 The 8th nnual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) Ns= Ns= e e Figure : it error rate perforance for QPSK odulation and coding rate R= Figure 3: it error rate perforance for 64-QM odulation and coding rate R= Ns= e e Figure 2: it error rate perforance for 6-QM odulation and coding rate R= QM odulations. However, due to the structure of the schee, there is a trade-off between the subcarrier grouping size and the coputational coplexity. Coded MIMO-OFDM In Figs. 4, 5, and 6, the benefit of perforing the schee, in function of the total received SNR, is highlighted for the specific case of R = /2, N t = N r = 4, and QPSK and 64-QM odulations. The siulation results show that at average ER=0 4, between 2.5d and 6d gains are obtained depending on the subcarrier grouping size (N s =2, 4, and 64) for QPSK odulation. In the case of 6-QM and 64-QM odulations, the allocation with ordering allows to obtain between 2.5 and 7d gain depending on the size of the subcarrier grouping, N s. The benefit in ter of gain for QM odulation is coparable to the QPSK odulation. In Figs. 7 and 8, the benefit of perforing the allocation schee on coded MIMO OFDM syste with N t = N r = 2, R = 3/4, QPSK and QM odulations is presented. Results are presented for a wide range of subcarrier grouping size, N s fro 2 to 64. The siulation results Figure 4: it error rate perforance for QPSK odulation and coding rate R=/2 show that at average ER=0 4, significant gain are obtained by siply perforing the schee. In the schee, the ipact of the subcarrier group size strongly affects the ER perforance. So the series of results, presented with coputer siulations, highlight the fact that trade off between group size order and perforance should be considered to define the ost appropriate selection of the paraeter N s. V CONCLUSION This paper proposes a ethod for MIMO-OFDM that adapts the transit in both the spatial and the frequency doains using a heuristic expression of the ER for each subcarrier and transit antenna. closed for expression of the optiu to be allocated for each subcarrier and transit antenna has been presented for uncoded MIMO-OFDM transission as well as coded schee. Proposed schee allows us to reduce the coputational coplexity, by siply including a subcarrier grouping ethod with local adaptation in each subcarrier group. The siulation results show signifi-

5 The 8th nnual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) 0. Ns=6 0. Ns=6 Ns= e e Figure 5: it error rate perforance for 6-QM odulation and coding rate R=/2 Figure 7: it error rate perforance for QPSK odulation and coding rate R=3/4 0. Ns=6 0. Ns= e e Figure 6: it error rate perforance for 64-QM odulation and coding rate R=/2 Figure 8: it error rate perforance for 6-QM odulation and coding rate R=3/4 cant iproveent of ER perforance both in the uncoded and coded cases for QPSK and QM odulations copared to the equal allocation schee. Furtherore, cobining the schee to any ful detection such as V- LST or axiu likelihood detection (MLD) can be also ipleented. In this paper, we have liited the channel coding schee to the convolutional code, however, other ful coding schees such as the Turbo Codes (TC) or Low Density Parity Check (LDPC) can also be included in the odulated transission. Finally, future orientation for this work would include the introduction of the error in the channel estiation. [5] Zh. Jiang et al., Max-Utility wireless resource anageent for best-effort traffic, IEEE Wirel. Coun. Mag., vol.4, no., pp.00, [6] X. Qiu et al., On the perforance of adaptive odulation in cellular syste, IEEE TRNS. On Counications, vol.47, no.6, pp , [7] Q. Liu, S. Zhou, and G. Giannakis, Cross-Layer Coibining of daptive Modulation and Coding With Truncated RQ Over Wireless Links, IEEE Trans. Wireless Coun., vol. 3, no. 5, pp , Septeber [8] Draft IEEE 802.g standard, Further Higher Speed Physical Layer Extension in the 2.4GHz and, 200. REFERENCES [] L.J. Ciini., nalysis and siulation of digital obile channel using orthogonal frequency division ultiple access, IEEE Tans. Coun., pp , 995. [2]. Van Zelt et al., Space Division Multiplexing (SDM) for OFDM systes, Proc. of the IEEE VTC-Spring, Tokyo, Japan, [3]. Muquet et al., Reduced coplexity equalizers for zero-padded OFDM transissions, Proc. of the ICSSP, Istanbul, Turkey, [4] J. Proakis, Digital Counications, 3rd ed. Singapore: McGraw-Hill, 995.