A New Approach to Layered Space-Time Code Design

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1 A New Approach to Layered Space-Time Code Design Monika Agrawal Assistant Professor CARE, IIT Delhi Tarun Pangti Software Engineer Samsung, Bangalore Abstract The block layered space-time codes, with the intent of simultaneously achieving high coding gain and high code rate, have been suggested. These codes are based on the design principles of existing layered space-time codes and space-time block codes. The previously known concept of a layer has been modified to a block layer wherein a block layer has more than one antenna transmitting at the same instant and hence the name for this scheme. Each block layer incorporates Alamouti s transmit diversity scheme which decreases the decoding complexity. Alamouti s scheme is preferred as it is the simplest unique linear processing technique for achieving full diversity with two transmit antennas. The block layers are made transparent to each other by Diophantine approximation. The use of space-time codes improves the coding gain for the proposed code design scheme. This scheme has been implemented for four transmit antennas and four receive antennas but can easily be extended to any number of antennas depending on the code rate desired. The proposed codes achieve a normalized rate of -symbol/s. Simulation results show that the block layered space-time codes outperform threaded algebraic space-time (TAST) codes, which is the best known layering scheme that achieves full-rate and full-diversity with arbitrary number of transmit and receive antennas, in certain scenarios. Index terms Block Layered Space-Time Codes (BLST), Maximum Likelihood (ML), Diagonal Algebraic Space-Time Codes (DAST), Threaded Algebraic Space-Time Codes (TAST). Introduction The commonly known communication technologies like cell phones, Bluetooth, Wi-Fi operate either in Ultra High Frequency (UHF) band or lower microwave band. Path losses are an inverse function of wavelength and hence they become significant at such high frequencies. The losses in these Non-Line of sight environments occur mainly due to the presence of obstacles between transmitter and receiver and also due to interference from similar copies of the signal varying in phase transmitted over multiple paths. The solution to these issues lies in a recent but rapidly developing technology. Multiple Input and Multiple Output (MIMO) []-[] systems offer tremendous promise to resolve bottlenecks in traffic capacity and even make Gigabits/sec wireless links a reality and that too at no extra frequency spectrum. They offer significant improvements over traditionally used Single Input and Single Output (SISO) systems both in the data rate and quality by decreasing the bit error rate (BER). The performance improvement in MIMO[] systems is because of array gain, diversity gain, spatial multiplexing gain and due to interference reduction[3]- [5]. Array gain [] is due to the usage of more number of antennas at both transmitter and receiver with an intend to coherently process the signals to increase Signal-to-Noise ratio (SNR). Diversity is achieved by transmitting replicas of the signal independently over multiple paths either in frequency, time or space. It mitigates random fading by processing different copies of the same signal coherently. Spatial diversity [], a means of transmitting independent signals from different antennas separated spatially, is the most important contributor to the MIMO systems good performance. Spatial multiplexing is an attempt to increase the data rate with the intent of allowing many users to transmit simultaneously [6]. The bit streams are coded, interleaved, de-multiplexed over all transmit antennas and then transmitted. Beamforming is used to cancel co-channel interference resulting in improved performance. The objective of space-time coding is to design the code words, which use the total available spatial diversity in the best possible manner. A simple decoder, high data rate, low probability of error etc. are always the desirable features of a communicating system. MIMO codes have been broadly classified into two categories, space-time diversity codes [4] and spatial multiplexing codes [5]. The pioneering work of Alamouti [3] started the whole new era of space time block codes. He proposed codes with a normalized rate of symbol/s to be transmitted over two transmit antennas for two symbol periods. Subsequent work by various researchers resulted in its generalization to M transmit antennas and N receive antennas. Space-time trellis codes proposed by Tarokh et.al [5] provides a low BER but at the cost of higher decoding complexity. Space-time codes attempt to maximize the coding gain [7] whereas spatial multiplexing is done to increase the code rate [5].

2 Another family of codes, known as the layered space-time codes [5], use the optimized efficient algebraic codes designed for Single Input Single Output (SISO) systems, to construct codes for MIMO systems. Layers [8] are made independent so that each user can use them independently. Efficient algebraic codes provide good coding gain to these layered codes. We propose Block Layered Space-Time (BLST) codes to provide high coding gain and high code rate. The idea is to also provide diversity gain to layered codes. The concept of a layer has been modified to a block layer, so as to allow more than one transmit antenna to belong to a single block layer at a given time. The incorporation of Alamouti s scheme in a block layer, as space time block code ensures a simple decoder. This approach incurs no penalty but improves the performance. It also provides the flexibility to trade off between the coding gain and the code rate. The rest of the paper is organized as follows. Section outlines the Block Layered Space-Time code construction and Section 3 shows some simulation results. Finally conclusions and future work proposed are presented in Section 4.. Block layered space time codes A layer [8] is defined as a set l, l = (w, t) ε {,,,M} x {t,t,, t T }, where the former set represents the transmit antenna and the latter represents the symbol interval. Also if (w, t) ε l and (w, t ) ε l, then either t t or w=w (i.e. w is a function of t). The concept will become clear from Fig. which shows an example of 4 layers for 4 transmit antennas. the latter represents the symbol interval. Here the necessary condition of layering concept that the more than one antenna cannot transmit symbols of a given layer at a given time instant has been relaxed. A group of transmit antennas may now belong to the same block layer for given symbol period. As an example two block layers (L=) constructed for four transmit antennas (M=4) are shown in Fig.. Like in Threaded Algebraic Space-Time (TAST) and Diagonal Algebraic Space-Time (DAST) scheme, the idea is to map each block layer to a different subspace so that they are as far away from each other as possible. With the concept of block layers, the total number of layers becomes less and consequently a less number of diophantine numbers [6] are required which increases the coding gain. Also, the real or complex rotated [6]-[7], [9]-[0] symbols are used to further increase the coding gain. In a block we use Alamouti s transmit diversity scheme [3] that ensures simple decoding at the receiver. Symbol interval Antenna Index Figure : Two block layers (L=) for four transmit (M=4) antennas. The numerals in the figure refer to the indexes of the block layers. Symbol interval Antenna index Figure : [6] Four layers (L=4) for four transmit (M=4) antennas. The numerals in the figure refer to the indexes of the layers In this paper, the layering concept of [8] has been extended to a block layer. A block layer is indexed by a set b, b = (w, t) ε {,,,M} x {t,t,,t T }, where the again former set again represents the transmit antenna and The transmitted symbol α j corresponding to source information symbol u j over j th block layer is () α j ( u j ) = Φ j s j = Φ j M j u j, j=,,l where L represents the total number of block layers and s j =M j u j are the rotated information symbol vectors. Here M j is an M x M real or complex rotation and the numbers Ф j, j=,,l are the Diophantine numbers. In matrix form the BLST code, for M transmit antennas, L layers and R number of bits per channel usage, is represented by B M,L,R. A BLST code which uses p PSK or QAM signal constellation has R = p for N=M=4 and L=. As an example B 4,,4 is shown in equation () where the BLST code uses a 4-PSK signal constellation

3 Ф s -Ф s where Φ and Φ represent the two diophantine numbers used and s,,s 8 represent the rotated information symbol vectors to be transmitted. By using a rotated signal constellation recovering the symbol vectors becomes easier. This is achieved because rotation maximizes the minimum product distance [6] among constellation points. In general, one may rotate the different block layers by different rotation matrices. A general matlab relation to generate rotation matrix M [7] for dimension d on the number field Ω(cos π/8d) is given by M=sqrt(/d)cos(π/4d)(4[:d] -)(([:d]-) (3) To construct a rotation matrix M d [7] of higher dimensions in d the following recursive approach can be used. M d = Ф s 7 -Ф s 8 Ф s Ф s Ф s 8 Ф s 7 Ф s 5 -Ф s 6 Ф s 3 -Ф s 4 Ф s 6 Ф s 5 Ф s 4 Ф s 3 - where is the rotation for dimension d/ and is its orthogonal transformation. The Diophantine approximation Φ =, Φ = Φ /M,, Φ L = Φ L-/M intends to achieve full diversity and maximize the coding gain [6] for the block layered space-time code where Φ is an algebraic integer if the set {, Φ,, Φ L- } is algebraically independent over the algebraic number field Ω(θ)... Decoding The received signal can be written as X = HB M,L,R + W (5) where H is the NxM complex gaussian random channel matrix with elements h ij, i=,,,n, N being the no. of the receive antennas and j=,,,m and W is a complex Gaussian random noise vector added by the channel. Let x = vec[x T ] (6) = [ x x x 3... x NM ] Let us define a vector y s.t. y = [ x x... x NM ] (7) On simplifying equation (5) and (8) we obtain y = H new Φ new M new u + w (8) () (4) where M new, Ф new and H new are the equivalent rotation matrix, Diophantine number matrix and the channel matrix given in (9), (0) and () for the BLST code B 4,,4. w is obtained by converting W T into a column vector after stacking its columns one after another and u is a vector containing the source information symbols. M new = where A is a 4x4 zero matrix. 3. Results Simulations on computer generated data is performed to study the performance of the suggested codes with that of TAST [6] codes. Both the coding schemes are implemented for four Ф new = H new = M A A M Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф Ф (0) h 0 h h 3 0 h h 0 -h h 4 0 -h h 3 0 h h 0 h h 4 0 -h h 0 -h h 0 h h 3 0 h h 0 -h h 4 0 -h h 3 0 h h 0 h h 4 0 -h h 0 -h h 3 0 h h 33 0 h h 3 0 -h h h h 33 0 h h 3 0 h h h h 3 0 -h 3 h 4 0 h h 43 0 h h 4 0 -h h h h 43 0 h h 4 0 h h h h 4 0 -h 4 () (9)

4 transmit antennas (M=4) and four receive antennas (N=4). The BER is studied as a function of signal to noise ratio (SNR). Channel is assumed to be Rayleigh fading and complete channel state information is assumed to be present at the receiver. Here maximum likelihood (ML) decoding is performed to study the performance. The BER was averaged over over 0 5 symbols. The diophantine number of Φ = (exp(iπ/6)) /4 is used. The real rotation of [7] used is given in eq. () SNR s upto 4 db. T 4,,8 codes perform better than our codes for higher SNR s. Next we compared the performance of B 4,,6 scheme with T 4,4,6 and T 4,,6. The T 4,4,6 employs a 6-PSK signal constellation while T 4,,6 and B 4,,6 use a 56-PSK signal constellation. The BER curves indicate that BLST code perform the best again, but at high SNR s of over db four layered T 4,4,6 performs better () 0 0 M=N=4,R=4 bits per channel use 0 - TAST(L=) BLST(L=) and the complex rotation of [0] used is given in eq. (3) θ θ θ 3 -θ θ -θ 3 (3) iθ -θ -iθ 3 -iθ -θ iθ where θ = exp (i Π/8). Here i = (-) /. All the simulations are done on Matlab 7.0. It was ensured while comparing the two coding schemes that the no. of bits transmitted per channel usage i.e. their spectral efficiencies were the same. A TAST code for M transmit antennas, L layers and R bits transmitted per channel usage is represented as T M,L,R. The T 4,4,R code has a normalized rate of 4 symbols per use and T 4,,R has a normalized rate of symbols per use while the proposed B 4,,R codes have a normalized rate of symbols per use. As a result constellations of different dimensions were used for each coding scheme keeping bits per channel use i.e. R same. Fig 3 compares the performance curves for BLST codes and TAST codes for R=4 bits per channel use. T 4,4,4 code employs a single bit modulation scheme while T 4,,4 and B 4,,4 employ 4-PSK. B 4,,4 codes outperform both T 4,4,4 and T 4,,4 for SNR s upto 8 db but T 4,4,4, performs the best at SNR s higher than 0 db. Figure 4 shows the performance curves for BLST codes and TAST codes for R=8 bits per channel use. The T 4,4,8 code uses a 4-PSK signal constellation while T 4,,8 and B 4,,8 employ a 6-PSK signal constellation. As in the previous figure, the proposed B 4,,8 code performs the best for Figure 3: B 4,,4 vs. T 4,4,4 and T 4,,4 0 0 M=N=4,R=8 bits per channel use TAST(L=) BLST(L=) Figure 4: B 4,,8 vs. T 4,4,8 and T 4,,8

5 4. Conclusion and future work In this paper we have suggested a new family of block layered space-time codes which outperform codes from TAST scheme under certain scenarios. The proposed codes use the design framework of layered space-time codes and space-time block codes. Modifications are made to the originally known layered theory to design a block layer which differentiates our coding scheme from previously known DAST and TAST schemes. Alamouti s scheme is incorporated in a block layer which makes decoding simpler at the receiver and consequently makes it advantageous over TAST. 0 0 M=N=4,R=6 bits per channel use TAST(L=) BLST(L=) Figure 5: B 4,,6 vs. T 4,4,6 and T 4,,6 We implemented the proposed Block Layered Space- Time Coding scheme for four transmit antennas and four receive antennas and we intend to generalize it for any number of transmit and receive antennas. We hope to improve the performance of the suggested BLST codes further by improving its design. This may be achieved by suitably choosing the Diophantine approximation and effectively implementing the rotated constellations. 5. References [] A. J. Paulraj, A. Gore R.U. Nabar and H. Bolcskei, An overview of MIMO communications- A key to Gigabit wireless, Proceedings of IEEE, vol 9, Feb [] D. Gesbert, Mansoor Shafi, Da-shan Shiu, P.J. Smith, A. Naguib, From Theory to Practice-An Overview of MIMO Space-Time Coded Wireless Systems, IEEE J. Select. Areas Commun.,vol., no. 3, April 003. [3] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Select. Areas Commun., vol. 6, pp , Oct [4] V. Tarokh, H. Jafarkhani, and A. Calderbank, Space time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol. 45, pp , July 999. [5] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inform. Theory, vol. 44, pp , Mar [6] H. El. Gamal, M. O. Damen, Universal space-time coding, IEEE Trans. Inform. Theory, vol. 49, no. 5, May 003. [7] M. O. Damen, K. Abed-Meraim and J.-C. Belfiore, Diagonal algebraic space-time block codes, IEEE Trans. Inform. Theory, vol. 48, pp , Mar 00. [8] G. J. Foschini, Layered space time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs Tech. J., vol., no., pp. 4 59, 996. [9] J. Boutros, E. Viterbo, Signal space diversity-a Power and bandwidth-efficient diversity technique for the Rayleigh fading channel, IEEE Trans. Inform. Theory, vol. 44, no. 4, July 998. [0] X.Giraud, E. Boutillon, J.C. Belfiore, Algebraic Tools to build Modulation Schemes for Fading Channels, IEEE Trans. Inform. Theory, vol. 43, no. 3, May 997.