JI Hui, Gheorghe Zaharia and Jean-François Hélard 13th International Symposium on Signals, Circuits and Systems(ISSCS 2015) 9- 10, July, 2015, Iasi, Romania Performance of DSTM MIMO Systems in Continuously Changing Rayleigh Channel JI Hui, Gheorghe Zaharia and Jean-François Hélard Institute of Electronics and Telecommunications of Rennes (IETR) Institut National des Sciences Appliquées (INSA), France My presentation is entitled …. This work has been carried out by from the Institute of … which is one of the research laboratories of INSA Rennes

Outline Models of SISO Rayleigh Radio Channel Clarke’s channel model Step channel model The new improved Rayleigh channel model Performance of the DSTM schemes over the new channel Conclusions The outline of this presentation will be the following: First of all, we will see two kinds of widely used Rayleigh channel models for wireless communications. Then, we will introduce a new improved channel model for Differential MIMO systems, which is more realistic and easy to simulate. A new scheme for Differential MIMO systems, which can be used with up to 8 transmit antennas will be presented and the performance of our new DSTM MIMO systems in this new channel model is evaluated. Finally, we will conclude. 1

Outline Models of SISO Rayleigh Radio Channel Clarke’s channel model Step channel model The new improved Rayleigh channel model Performance of the DSTM schemes over the new channel Conclusions 1

Rayleigh Channel Model Clarke’s mathematical channel model Proposed by Clarke in 1968 The correlation of h(t) is The PSD of h(t) is band-limited: |f – fc| ≤ fd Often used as sum-of-sinusoid based simulators such as Jakes’ model Jakes’ simulator is non-stationary and other improvements have some degrees of loss of the statistical properties such as deficiencies in higher-order statistics As we know, the mathematical multi-path Rayleigh channel model widely used today is established by Clarke in his fundamental work in 1968. The impulse response of this channel model can be expressed as this equation… The autocorrelation of h(t) is… In experiments, Jakes’ simulator is often used. It is derived from Clarke’s model by setting equal strength multipath components and choosing the N angles θn to be uniformly distributed. However Jakes’ simulator is non-stationary and other improvements have some degrees of loss of the statistical properties such as deficiencies in higher-order statistics 2

Rayleigh Channel Model Step channel model The channel is assumed to be constant during one frame and changes randomly for the next frame. One frame has L symbols: Too simple compared to real situation In the research of DSTM MIMO systems, another widely used channel model is step channel. This channel is assumed to be constant during one frame and changes randomly for the next frame which is too simple. 2

Outline Models of SISO Rayleigh Radio Channel Clarke’s channel model Step channel model The new improved Rayleigh channel model Performance of the DSTM schemes over the new channel Conclusions Thus we propose a new improved channel model for DSTM MIMO systems. 3

The new improved Rayleigh channel model Basic idea: Nyquist’s sampling theorem The channel h(t) can be reconstructed by random generated complex variables rk: Why it is possible? h(t) is band-limited, with sampling points derived from sampling frequency f0 ＞2fd, h(t) can be recovered. J0(x) has its first zero-point at x ≈ 2.4048. The channel coefficients separated by τ0 = 2.4048/(2πfd) ＜ 1/(2fd) are uncorrelated. If we assume that rk are separated by τ0, then h(t) can be recovered by rk . This new model is derived based on Nyquist’s sampling theorem by the sinc interpolation funciton. The channel h(t) can be reconstructed by random generated complex variables rk. Why it is possible? There are two reasons, the first and the most important, h(t) is band-limited and thus it can be recovered by sampling points generated by sampling frequency f0>2fd. Second, the channel coefficients separated by τ0 is uncorrelated and independent. Thus, h(t) can be reconstructed by K randomly generated independent complex Gaussian variables which are supposed to be separated by τ0. 3

The new improved Rayleigh channel model For continuously changing channel, the channel coefficient used for the ith symbol is: We propose a block-constant channel model in [8], the channel coefficients used for the ith matrix is: The channel coefficients used for the ith symbols is... We also proposed a channel model in a paper published last year in EW2014. The channel coefficients are supposed to be constant during the transmission of one matrix. However, in the block-constant channel model, the randomly generated complex variables rk are assumed to be separated by T0 = Tc ≈ 0.5/fd. [8] H. Ji, G. Zaharia, and J.-F. Hélard, “Performance of DSTM MIMO systems based on Weyl group in time selective channel,” in European Wireless, Barcelona, 2014. 3

The new improved Rayleigh channel model The differences between this new channel model, the step channel and the block-constant channel are presented in this figure. In step channel (red line), the channel coefficients are supposed to be constant for a fixed time duration (usually Tc) and changes randomly to another value for the next time duration. In block-constant channel (black line), the channel coefficients are supposed to be constant for a transmission matrix. And in the new channel model (blue line), the channel coefficient changes cotinuously for every two successive transmitted symbols. 3

Outline Models of SISO Rayleigh Radio Channel Clarke’s channel model Step channel model The new improved Rayleigh channel model Performance of the DSTM schemes over the new channel Conclusions We evaluate our new DSTM schemes over this new improved channel model. 4

General System Model - M x N MIMO system The MIMO system model in the matrix form: M : number of transmit antennas N : number of received antennas T : symbol duration of each matrix : M × T transmission matrix : N × T received matrix : N × M channel coefficients matrix : N × T white Gaussian noise matrix The MIMO system model in the matrix form can be written : Xt is the M x T transmission matrix, where T denotes the normalized symbol duration of each matrix, which means that for each transmit antenna, T symbols are transmitted. Yt is the N x T received matrix. Ht is the N x M channel coefficients matrix. For step channel, it is constant for saveral transmission matrices. For block-constant channel, it is different for every two successive transmission matrices and for continuously changing channel, H is different for every two successive transmission column vectors. Wt is the N x M white Gaussian noise matrix. It is proved that for non-coherent MIMO systems, the capacities obtained with M > T and M = T are equal. Therefore, we choose M = T in our study. For convenience, at each time slot the total power over M transmit antennas is set to be 1 Assumptions: M = T. The total power over M transmit antennas is 1 at each time slot. 4

General System Model – transmitter and receiver Fundamental differential transmission relation: where is a unitary matrix which carries information The relation between two successively received signals: The maximum likelihood demodulation : The fundamental differential transmission relation is … And the received matrix at time (\tau +1) is… Applying the maximum likelihood detection criteria for the demodulation leads to determine the unitary matrix V which minimizes the quantity: Once the unitary matrix Vt is got, the inverse mapping is applied and the information bits can be recovered. 4

The constellation of the new scheme In the proposed scheme, the transmitted information matrices are selected from the Weyl group Gw . The Weyl group is a set that contains 12 cosets . Each coset contains 16 invertible matrix. (192 unitary matrices, 2×2). The first coset is a subgroup of Gw : with The 12 cosets of Gw are derived from C0 as follows: In our scheme, the transmitted matrices are based on the Weyl group. where the matrices are: 5

The scheme with 2 transmit antennas MIMO system with M = 2 transmit antennas – Each transmit matrix is sent during M = T = 2 symbol durations. Fundamental differential transmission relation: … For MIMO systems with 2 transmit antennas and R = 1 bps/Hz, 4 matrices are needed, the first 4 matrices of C0 are selected. So, we consider MIMO system with 2 transmit antennas and each transmit matrix is sent during M = T = 2 symbol durations. At time t = 0, we transmit a reference matrix X0 = M0 For R = 2 bps/Hz, we select C0 which has 16 matrices as the group to map the 4 bits information block. The maximum spectral efficiency is: 6

The scheme with 4 transmit antennas To design a MIMO system with 4 transmit antennas, the 2 x 2 Weyl group is not applicable. Main idea: to use the Kronecker product of two matrix to expand the Weyl group Example, for R = 1 bit/s/Hz system, we can make the Kronecker product between the first matrix M0 of C0 and all the matrix of C0: where Thus, 4 bits are mapped on to one of the 16 matrix of C00 and then sent during M = T = 4 symbol durations leading to a spectral efficiency R = 1 bit/s/Hz 7

The scheme with 4 transmit antennas To design a MIMO system with 4 transmit antennas, the 2 x 2 Weyl group is not applicable. Main idea: to use the Kronecker product of two matrix to expand the Weyl group. More generally, there are 192 2x2 matrix in the Weyl group, and we need to construct a set of 4x4 matrix. There are 192 x 192 = 36284 possible Kronecker products of all the matrix of the Weyl group. Actually, there are only 4608 distinct 4x4 matrix in the new group. Then, the maximum spectral efficiency is: 7

The scheme with 4 transmit antennas The performance can be improved considering the distance spectrum. For R = 2 bit/s/Hz system, we select the first 16 matrices from every successive 192 matrices of the group Gw4, i.e., the set is 8

The scheme with 8 transmit antennas To design a MIMO system with 8 transmit antennas, Kronecker product between Gw4 and Gw is used to generate a 8×8 set. 110592 matrices are distinct: Gw8. The maximum spectral efficiency is Considering the distance spectrum, for R = 1 bit/s/Hz system, new set C88 is selected: C44 = {M0, M1, M2, M3, } {M0, M1, M2, M3, }. 16 matrices. C88 = C0  C44. 256 matrices 6 For R = 2 bit/s/Hz system, we select the first 65536 matrices from the group Gw8 as the transmission mapping group. 9

Performance of the new DSTM schemes over the new channel This figure shows that for R = 1 bps/Hz, and L=Tc/Ts=200, DSTM schemes over continuous channel perform similar to those over step channel. However, DSTM schemes perform better than those over block-constant channel, which is resulted from the less value of discontinuity of the channel coefficients for two successively transmitted matrices compared to step channel. Performances of differential space-time schemes with R = 1 bps/Hz over different channel models. 10

Performance of the new DSTM schemes over the new channel Similar relative results for R = 2 bps/Hz M8N8, M4N4 and M2N2 schemes are obtained in this figure. As expected, the M8N8 scheme is more sensitive than the M4N4 and M2N2 schemes to the time selectivity of the channel Performances of differential space-time schemes with R = 2 bps/Hz over different channel models. 11

Performance of the new DSTM schemes over the new channel This figue presents the performance of M4N4 DSTM scheme with R = 1 bps/Hz over the step channel and over the new continuous channel with different normalized coherence time L. The faster the channel changes, the smaller the value of L. Consistent with our supposition, there is a trend that as L grows the BER performance becomes better. Performances of differential space-time schemes with R = 2 bps/Hz over different channel models. 12

Outline Models of SISO Rayleigh Radio Channel Clarke’s channel model Step channel model The new improved Rayleigh channel model Performance of the DSTM schemes over the new channel Conclusions 13

Conclusions Propose a new continuous time selective propagation channel based on Nyquist’s sampling theorem for differential space-time modulation schemes. The performances of DSTM schemes with 2, 4 and 8 transmit antennas in these channels are evaluated. 13

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