1. Hui Ji, 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
Hui Ji, 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

2. Outline General MIMO system The step channel model
The new improved Rayleigh channel model
Performance of the DSTM schemes
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

3. 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 : number of columns of each transmitted matrix
: M × T transmitted matrix
: N × T received matrix
: N × M channel 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
Narrowband or OFDM MIMO systems (flat channel) 2

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. 3

5. The constellation of the new scheme
In the proposed 2x2 MIMO scheme, the information matrices are selected from the Weyl group Gw.
The Weyl group is a set that contains 12 cosets Each coset contains 16 invertible matrices (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: 4

6. 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: 5

7. The scheme with 4 transmit antennas
To design a MIMO system with 4 transmit antennas, the 2 x 2 Weyl group can not be used.
Main idea: to use the Kronecker product of two matrices to expand the Weyl group
Example, for R = 1 bps/Hz system, we can make the Kronecker product between the first matrix M0 of C0 and all the matrices of C0:
where
Thus, 4 bits are mapped onto one of the 16 matrices of C00 and then sent during M = T = 4 symbol durations 6

8. The scheme with 4 transmit antennas
To design a MIMO system with 4 transmit antennas, the 2 x 2 Weyl group can not be used.
Main idea: to use the Kronecker product of two matrices to expand the Weyl group.
More generally, there are x2 matrices in the Weyl group, and we need to construct a set of 4x4 matrices.
There are 192 x 192 = possible Kronecker products of all the matrices of the Weyl group.
Actually, there are only 4608 distinct 4x4 matrices in the new group Gw4. Hence, the maximum spectral efficiency is: 7

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

10. 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 set of 8×8 matrices. Only matrices are distinct: Gw8.
The maximum spectral efficiency is
Considering the distance spectrum, for R = 1 bps/Hz system, a new set C88 is selected:
C44 = {M0, M1, M2, M3, } {M0, M1, M2, M3, }: 16 matrices
C88 = C0  C44 : 256 matrices
For R = 2 bps/Hz system, we select the first matrices from the group Gw8 as the transmission mapping group. 9

11. Rayleigh Channel Model
Step channel model
The channel is assumed to be constant during one frame and changes randomly from one frame to the next frame.
One frame has L symbols:
Tc = coherence time of the channel, Ts = duration of a transmitted symbol
Too simple compared to real situation
Two successive channel matrices are different, so the differential modulation can not be continued; for the simulation, a new identity matrix must be transmitted when the channel matrix changes randomly (this is not the real case)
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. 10

12. The new improved Rayleigh channel model
Basic idea: Nyquist’s sampling theorem
The channel h(t) can be reconstructed from 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.
For h values computed between rk and rk+1, K/2 - 1 random values are placed before rk and K/2 - 1 random values are placed after rk+1
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. 11

13. 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. 12

14. 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.
Performance of differential space-time schemes with R = 1 bps/Hz over different channel models. 13

15. 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
Performance of differential space-time schemes with R = 2 bps/Hz over different channel models. 14

16. 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.
Performance of DSTM M4N4R1 scheme over the new continuous channel with different L 15

17. Conclusions
A new continuous propagation channel based on Nyquist’s sampling theorem is proposed for differential space-time modulation schemes.
The performance of DSTM schemes with 2, 4 and 8 transmit antennas in these channels is computed.
The degradation of the performance of DSTM schemes for fast Rayleigh channel is evaluated. 16

18. Thank you!