Non coherent coding for MIMO-OFDM systems

Transcription

1 Non coherent coding for MIMO-OFDM systems Raouia Ayadi Ecole supérieure des communications de Tunis Cité Technologie des Communications, 2083 El Ghazala, Ariana, Tunisia ayadi@telecom-paristech.fr Georges Rodriguez-Guisantes TELECOM ParisTech 46, rue Barrault, Paris Cedex 13, France rodriguez@telecom-paristech.fr Inés Kammoun LETI Department, ENIS BPW 3038 Sfax - Tunisia ines.kammoun@ieee.org Mohamed Siala Ecole suprieure des communications de Tunis Cité Technologie des Communications, 2083 El Ghazala, Ariana, Tunisia mohamed.siala@supcom.rnu.tn Abstract This paper deals with the problem of code design for non-coherent frequency-selective Multiple Input Multiple Output (MIMO)-Orthogonal Frequency Division Multiplexing (OFDM) fading links, where neither the transmitter nor the receiver knows the channel. A simple and classical approach for non coherent (NC) coding for MIMO-OFDM system consists of coding across antennas and time slots on each OFDM subcarrier. Only, this approach does not permit to exploit the diversity of frequencyselective channels. In our aim to exploit this diversity, we propose in this paper a strategy that consists of coding across antennas and OFDM subcarriers (space-frequency coding) within a single OFDM symbol. Our space-frequency (SF) codes are based on NC Grassmann space-time (ST) codes obtained via an exponential map from coherent ST codes. We determine the asymptotic Pairwise Error Probability (PEP) for each scheme in order to evaluate the diversity and the coding gains. Simulations show that compared with the classical approach, our NC-SF codes can get more diversity gain and better symbol error performance. I. INTRODUCTION The Multiple Input Multiple Output (MIMO) systems are considered as a good candidate for the fourth generation of wireless communications. These systems acheive a full spatial diversity. Several space-time (ST) codes have been developed over the flat fading channels. These codes exploit spatial and temporal diversity, leaving frequency domain unexploited. However, the MIMO channels present a selectivity for high wideband and cause the Inter-Symbol Interference (ISI). The Orthogonal Frequency Division Multiplexing (OFDM) is a powerful technique used in various wireless communication systems that endure the frequency selectivity of channels [1] [2]. Combined with MIMO systems, the OFDM proves a robustness while crossing frequency selective channels [3]. Most of the previous works introduced the space-frequency (SF) code in coherent MIMO-OFDM systems, i.e assume a perfect knowledge of the channel state information (CSI) at the receiver [4] [5]. In [4], some full-diversity SF codes are obtained from ST codes for arbitrary power delay profiles by a simple mapping. A code design criterion for SF-coded MIMO- OFDM systems was derived in [5]. Authors show in [5] that space-time codes introduced to achieve full spatial diversity in narrowand case will in general not achieve full spacefrequency diversity. In practice, the knowledge of the CSI, getting via the training sequences, decreases the spectral efficiency of the system. Otherwise, it is sometimes impossible to estimate the channel coefficients and especially in the frequency-selective case due to the presence of multiple paths. To combat this drawback, non coherent (NC) communication, where neither the transmitter nor the receiver has CSI, was considered recently [6]. In [6], Borgmann et al adress code design for NC frequencyselective MIMO-OFDM fading channels. They propose design criteria for SF unitary codebook and show that unlike in the coherent case, NC-ST codes designed to achieve full spatial diversity in the frequency-flat fading case can fail completely to exploit not only frequency diversity but also spatial diversity when used in frequency-selective fading environments. In this paper, we propose a non-unitary SF codebook suited for NC MIMO-OFDM links. These codes are based on unitary ST codes proposed in [7] for frequency flat fading channels. The ST codes introduced in [7] are obtained via an exponential map from coherent codes and they were called Grassmann codes since they are set of points in the Grassmannian G T,Mt,the set of the M t -dimensional vector subspaces of C Mt, where T is the coherence time of the channel and M t it the number of transmit antennas. At the receiver the decision is made according to the Generalized Likelihood Ratio Test (GLRT), which does not require the knowledge of the channel statistics. These codes have been designed for flat fading channels and don t exploit the frequency diversity of a selective channel. Our objective in this paper is to design SF codes for NC communications using the Grassmann NC-ST codes. Comparison will be done with a simple approach which consists of tansmitting a Grassmannian NC-ST codeword over each flat fading subcarrier. The rest of this paper is organized as follows. In Section 2, we introduce the system model. We describe the NC-ST coded MIMO-OFDM in section 3 and give its diversity and coding gains. In section 4, we present our approch to design the NC- SF code. In section 5, we provide some simulation results. Finally, section 6 contains our conclusions /09/ $ IEEE 3243

2 II. SYSTEM MODEL In this section, we describe the MIMO-OFDM system in wireless communications. We consider a system model with M t transmit antennas, M r receive antennas and N subcarriers. Suppose that the Rayleigh fading channels between each pair of transmit and receive antennas have L independent delay paths and the same power delay profile. The M t M r frequency response matrix of the length-l MIMO channel from the i-th transmit antenna and the j-th receive antenna for the n-th subcarrier is defined as L 1 [H n ] i,j = h i,j (l)e j 2π N ln, (1) l=0 where h i,j (l) is the path gain coefficient of the l-th path between the transmit antenna i and the receive antenna j. We note that the elements of h i,j (l),l = 0,..., L 1 are i.i.d. circularly symmetric zero-mean complex Gaussian. The OFDM modulator applies a N-point inverse fast Fourier transform (IFFT) and added a cyclic prefix (CP) to the parallelto-serial-converted OFDM symbol. In this paper, we consider that the channel order is lower than or equal to the length of the CP, L L CP. The receiver discards the cyclic prefix, and then applies an N-point FFT to each of the M r received signal. The 1 M r received signal vector on the n-th subcarrier is given by y n = x n H n + w n. (2) where x n 1 M t is the transmitted vector on the n-th subcarrier. We note that w n is an 1 M r additive complex Gaussian noise vector satisfying with N 0 the power density of w n E{w n w n } = N 0δ n,n, (3) III. NC-ST-OFDM SCHEME A. Description of the scheme In this section, we assume that the frequency-band used for the transmission over each subcarrier is narrow enough so that the channel can be assumed flat fading. Let T denotes the interval coherence. That is, we assume that the channel coefficients for each block are generated independently from the previous blocks. We will consider here a NC-ST coded MIMO-OFDM system which uses the NC-ST coding proposed in [7] for each separately flat fading subcarrier. Denoting the T M t codeword transmitted on the n-th subcarrier as X n C, where C is a codebook of unitary T M t matrices, the T M r received signal on the n-th subcarrier can be written as Y n = X n H n + W n, (4) where W n is an T M r additive complex Gaussian noise matrix satisfying E{W n W n } = N 0I T δ n,n, (5) and I T denoting the identity matrix of size T. The matrix X n will be taken as a codeword from the Grassmann codebook (C) obtained as in [7] from a coherent code ST block code (V ) as follows 0 αv n X n = exp I T,Mt, (6) αv n 0 where V n is a M t (T M t ) matrix obtained from M t (T M t ) information QAM symbols by using a non linear transformation of the Grassmann manifold G T,Mt (C). Coherent code in this work means a code designed to be used with a coherent receiver. The common positive scalar α, which will be called homothetic factor, is chosen to satisfy the invertibility of the exponential map and to optimize the performance of the code, at the same time [7]. At the receiver the decision of the codeword on the n-th subcarrier is made according to the GLRT, which does not require the knowledge of channel statistics since it is defined as ˆX n = arg max sup p(y n X n, H n ). (7) X n C H n In our case, that is i.i.d. fadings and unitary codebook, the GLRT is equivalent to the Maximum Likelihood (ML) criterion, and it takes the form ˆX n = arg max X Trace(Y n C nx n X ny n ). (8) B. Diversity and coding gains Let P (X i X j X i ) be the PEP of the transmitting X i and deciding in favor of another codeword X j at the decoder. By considering the GLRT metric, the PEP at high SNR is given by [7] ( ) 2Mt M γ MtMr r 1 M t M r P (X i X j X i ), (9) M t sin 2 θ m,i,j m=1 where γ is the average signal to noise ratio and (θ 1,i,j,θ 2,i,j,..., θ Mt,i,j) the principal angles between the two subspaces Ω i and Ω j generated by the columns of, respectively, X i and X j. Then, the diversity gain and the coding gain that can be achieved with NC ST coded MIMO-OFDM are 1) Diversity gain : In order to maximize the diversity gain offered by the flat fading channel, the minimum number of nonzero principal angles taken on each subcarrier over all distinct codewords pairs X i and X j should be maximized. The diversity gain of the NC-ST OFDM coded in transmission is given by [7] d NC ST = M t max dim(ω i Ω j ). (10) i,j 2) Coding gain : If a maximum diversity gain is achieved, [ Mt ] 1/Mt. the coding gain is m=1 sin2 θ m,i,j 3244

3 Fig. 1. MIMO-OFDM model For small homothetic factors, it was proved in [7] that the full diversity of the coherent code (V ) for the ML receiver gurantees the full diversity of the corresponding NC code. That s why we consider in our simulations the full diversity coherent codes proposed in [10]. IV. NC-SF-CODED MIMO-OFDM SYSTEMS The NC-ST coded can be applied to MIMO-OFDM systems as an attractive solution to high transmission rate over flat fading channels. But, this approach does not permit to exploit the frequency diversity of the channel. We propose in this section to code across antennas and OFDM tones (spacefrequency coding) within a single OFDM symbol in order to exploit the frequency diversity. Our idea is to obtain NC-SF codes from NC-ST Grassmann codes of [7] by replacing the temporal domain by the frequency domain. A. NC-SF coded system Using equation (2) and by arranging the transmitted vectors x n in a N M t matrix X =[x 0, x 1,..., x N 1 ], the transmission of an entire OFDM symbol can now be written as L 1 Y = D l XH l + W = F XH + W, (11) l=0 The N N diagonal matrix D is given by } D = diag {e j 2π N n,n=0,..., N 1, and H l (l =0...L 1)istheM t M r channel impulse response taps matrix defined as H l =(h i,j (l)),i=1,..., M t and j =1,..., M r. The N M r matrix corresponds to the additive noise. We define the N LM t SF codeword as F X =[X, DX,..., D L 1 X], and the LM t M r stacked channel impulse response taps matrix as H =[H 0, H 1,..., H L 1 ]. We propose to construct X as a concatenation of N/F matrices as follows X =[X 1,..., X N/F ], X k, k =1,..., N/F is a F M t matrix constructed as in (6) while replacing the temporal domain and the codeword length T by the frequency domain and F, respectively. At the receiver the decision of the codeword is made according to the GLRT Y ˆX = arg min X FX Ĥ 2 (12) = arg max X Trace(Y F X (F X F X) 1 F X Y). We note here that the SF codewords F X are not unitary. B. Diversity and coding gains Let F X and FˆX two different NC-SF codewords. In order to derive the asymptotic PEP P (X ˆX X) of transmitting X and deciding in favor of another codeword ˆX, we follow the approach of Brehler et al. in [8] by interpreting the OFDM symbol length as the coherence interval T of a quasi-static Rayleigh fading channel. We can easily show that for the nonunitary codebook (F) and with i.i.d Gaussian additive noise, the asymptotic PEP for the GLRT decision (12) is given by ( )( ) 2LMt M ρ LMtMr r 1 1+ MXX Mr LM t M r MˆXˆX Mr P (X ˆX X) ( ) Mr, det M XX M XˆX M 1 ˆXˆX MˆXX (13) where ρ is the average signal to noise ratio and [ ] [ ] MXX M XˆX F X [ ] = MˆXX MˆXˆX F ˆX FX FˆX. (14) For ( any two pairs of distinct ) codewords, quantity Θ = X,ˆX det M XX M XˆX M 1 ˆXˆX MˆXX must never be equal to zero. In order to acheive maximal diversity order offered by the frequency-selective channel, the matrix M XX M XˆX M 1 ˆXˆX MˆXX should have the full-rank (LM t ) over all pairs of codewords X and ˆX. In this case, the maximum acheviable diversity order in a NC-SF-coded MIMO-OFDM system is given by dmax NC SF = LM t M r. The problem now is to find the relationship between the quantity Θ and the diversity of the matrices X X,ˆX k constructed as in (6). This relationship is rather complicated to express and will not be reported here. In the following, the diversity of our NC-SF code will be checked by simulation. We will prove that 3245

4 if the homothetic factor α in (6) is chosen with a great care, that it guaranteed the full diversity of the NC-ST Grassmann code, we guarantee the full diversity of the corresponding NC- SF code with GLRT receiver. V. SIMULATION RESULTS In this section, we compare the error performance of the two schemes NC-ST coded MIMO-OFDM and NC-SF coded MIMO-OFDM when M t = M r =2and the number of the OFDM subchannels is fixed at N = 128. For the first scheme, on each subcarrier we transmit a Grassmann codeword from G 4,2 (C) or from G 6,2 (C) constructed as in (6). For the Grassmann code on G 4,2 (C), we consider the two coherent codes V 1 and V 2 introduced in [9] and [10], respectively. The codewords of Code V 1 are given by V 1 = s 1 + θs 2 φ(s 3 + θs 4 ), (15) φ(s 3 θs 4 ) s 1 θs 2 where φ 2 = θ = e i π 4 and symbols s i are drawn from a 4- QAM constellation. We prove via numerical simulations that the optimal homothetic factor which is α = 0.3 for this constellation. The coherent code V 2 is constructed by the golden code and its codewords are given by V 2 = 1 φ r (s 1 + rs 2 ) φ r (s 3 + rs 4 ), (16) 5 φ r (s 3 rs 4 ) φr (s 1 + rs 2 ) where r =(1+ 5)/2, r =(1 5)/2, φ r =1+i(1 r) and φ r =1+i(1 r). We prove via numerical simulations that the optimal homothetic factor is α =0.39. For the Grassmann code on G 6,2 (C), we will consider the coherent codeword V built as follows V = s1 + θs2 φ(s3 + θs4) φ2 (s 5 + θs 6) φ 3 (s 7 + θs 8), φ 3 (s 7 θs 8) s 1 θs 2 φ(s 3 θs 4) φ 2 (s 5 θs 6) (17) where φ 2 = θ = e i π 4 and the symbols s i are drawn from a 4-QAM constellation. We chose for this case α =0.23. Figures 2 and 3 show the symbol error rate (SER) performance comparison results of the proposed NC-SF-coded MIMO- OFDM and the NC-ST-coded MIMO-OFDM systems. We consider in Fig.2 the case of NC-ST and NC-SF codewords obtained via an exponential map from the coherent code V 1 and transmitted over Rayleigh flat fading channel. The proposed NC-SF-coded MIMO-OFDM scheme can transmit four symbols through four sub-carriers, so the transmission rate is one per channel use (pcu). The NC-ST-coded MIMO- OFDM scheme transmit four symbols during four OFDM time slots through every subcarrier. For the same transmission rate, four neighbor subcarries are taken into account in the proposed NC-SF scheme. So this scheme exploit the frequency diversity over frequency-selective channels. Hence, it can get higher diversity. For the proposed NC-SF scheme, we consider low delay spread case (i.e. one path) and high delay spread case (i.e. two and three paths). Figure 2 shows that the performance achieved with NC-SF- OFDM code is roughly equal to the performance obtained with NC-ST-OFDM code over flat fading channel when L =1. We show in Fig. 3 the performance of the proposed NC- SF-coded MIMO-OFDM scheme obtained from the coherent codes given in (15) and (16) over Rayleigh frequency-selective channel when L =1, L =2and L =3. We note that the performance of NC-SF scheme for L =1is the same for NC- ST scheme. Figure 3 shows that the proposed scheme get much more diversity gain than NC-ST-coded OFDM scheme over Ryleigh frequency-selective channels for high delay spread case (L =2and L =3). The NC-ST-coded OFDM scheme does not exploit the diversity of frequency-selective channels when L =2and L =3. For a further characterization of the proposed NC-SF-coded OFDM scheme, we show in Figure 4 the performance (in SER versus the average SNR) when the SF codewords are built with matrices X k, that form the matrix X, obtained by applying the exponential map (6) on the matrix V given in (17). SER 10 3 G 4,2, code V 1 NC ST OFDM coded NC SF OFDM coded SNR(dB) Fig. 2. SER performance comparison result of the proposed NC-SF-coded OFDM and the NC-ST-coded OFDM systems with symbols s i drawn from 4-QAM constellation, M t = M r =2, T =4and N = 128. VI. CONCLUSION We have introduced in this paper a new NC-SF coded OFDM transmission scheme. This scheme consists of coding across antennas and OFDM subcarriers within a single OFDM symbol. It is a strategy for improving the performance of a NC-ST-OFDM scheme which transmits each NC-ST code on a single subcarrier. For these two schemes, we apply first a coherent coding to the modulated transmission symbols, then we apply an exponential map to get a codeword matrix which is a point on the Grassmann manifold. For the NC-ST-coded OFDM scheme, each matrix is transmitted over a subcarrier. For the NC-SF-coded OFDM scheme, each line of this matrix is transmitted over a subcarrier. At the receiver, the symbols have been recovered with a GLRT detection. We prove that under some conditions the new NC-SF OFDM scheme can get 3246

5 SER L=1, code V 1 L=1, code V 2 L=2, code V 1 L=2, code V 2 L=3, code V 1 L=3, code V SNR Fig. 3. Performance of NC-SF-coded MIMO-OFDM system for different number of paths with symbols s i drawn from 4-QAM constellation, M t = M r =2, T =4and N = G 6,2 L=1 L=2 L=3 [3] Y. Li, J. H. Winters and N. R. Sollenberger, MIMO-OFDM for Wireless Communications : Signal Detection With Enhanced Channel Estimation, IEEE Transactions on communications, vol.50, no.9, pp , September [4] W. Su, Z. Safar, M. Olfat and K. J. Ray Liu, Obtaining Full-Diversity Space-Frequency Codes from Space-Time Codes via Mapping, IEEE Transactions on signal processing, vol. 51, no. 11, pp , November [5] H. Bolcskei and A. J. Paulraj, Space-Frequency coded broadband OFDM systems, IEEE on wireless communications and networking conference, vol.1, no., pp.1-6, [6] M. Borgmann and H. Bolcskei, Noncoherent SpaceFrequency Coded MIMO-OFDM, IEEE Journal on selected areas in comunications, vol. 23, no. 9, pp , September [7] I. Kammoun, A. M. Cipriano and J.-C. Belfiore, Non coherent codes over the Grassmannian, IEEE Transactions on wireless communications,vol. 6, no. 9, September [8] M. Brehler and M. K. Varanasi, Asymptotic error probability analysis of quadratic receivers in Rayleigh fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers, IEEE Transactions on information theory, vol. 47, no. 6, pp , September [9] M. O. Damen, A. Tewfik and J.-C. Belfiore, A construction of a spacetime code based on the theory of numbers, IEEE Transactions on Information Theory, vol.48, no.3, pp , March [10] J.-C. Belfiore, G. Rekaya, and E. Viterbo, The golden code : a 2*2 fullrate space-time code with non-vanishing determinants, IEEE Transactions on Information Theory, vol.48, no.3, pp , March SER SNR Fig. 4. Performance of NC-SF-coded MIMO-OFDM system for different number of paths with symbols s i drawn from 4-QAM constellation, M t = M r =2, T =6and N = 132. the maximum available diversity gains in frequency-selective quasistatic fading channels. Compared with the NC-ST-coded OFDM scheme, the proposed scheme can achieve more diversity gains and better performance on SER. In order to get available spatial, frequency and temporal diversity, it will be interesting to propose space-time-frequency (STF) codes. For STF coded MIMO-OFDM, frequency diversity is exploited by coding across frequency tones of one OFDM symbol, time diversity is extracted by coding across multiple OFDM symbols and space diversity is provided thanks to coding across antennas. This will be the topic of a future work. REFERENCES [1] J. G. Proakis, Digital Communications, New York : McGraw-Hill, 3rd ed, [2] S. Hara and R. Prasad Multicarrier techniques for 4G mobile communications, Boston, NJ : Artech House, 2003 The Artech House universal personal communications series. 3247