6.1 Assumptions for Numerical Simulation Runs
6.2.2 Optimal/Orthogonal Training: NMSE Performance
e h for the time-domain linear MMSE counterpart.
6.2.1 Arbitrary Training: NMSE Performance
As shown in Fig. 3, the time-domain linear MMSE estimate delivers substantially lower NMSE than that of frequency-domain LS estimate as the former exploits more CSI. The examples can be divided into two major groups depending on value of K. The group with K=4 consists of (3, 2, 4) and (4, 2, 4) configurations while another one consists of (5, 2, 8), (6, 2, 8) and (7, 2, 8). For both of the groups, simulation results validate the summaries that the higher the value of Mt is, the worst the NMSE of channel estimate becomes as we expected. We used N=10 and L=5 for simulations in Fig. 3.
6.2.2 Optimal/Orthogonal Training: NMSE Performance
As depicted in Fig. 4, the performances of channel estimates based on optimal/
orthogonal training outperform those based on arbitrary training in terms of NMSE generally. The minor inconsistency for linear MMSE at larger antenna numbers is due to the fact that the FDE-based trainings already enjoy high diversity plus partial channel information and that (5.31) is a rough approximation. In general, the results verify the summaries in chapter 5. Here we simply make N=K for simulations in Fig. 4 due to the fact that (5.16) can be easily satisfied in this situation, as we explained in the previous chapters. Note that we set L=2 for both (3, 2, 4) and (6, 2, 8) system configurations.
Chapter 7
Conclusion
In this thesis, we propose an instructive derivation for the generalized block-level orthogonal space-time block encoder, capable of achieving full spatial diversity via frequency-selective fading environment, provided that channel order is known. Instead of dealing with special case and then extending the results intuitively, we provide an alternative by starting with the general signal model with multiple transmit and multiple receive antennas, from which a general form of block-level orthogonality is established. In particular, transmit diversity with more than two transmit antennas can be achieved without compromise by means of frequency-domain equalization, in contrast to the QO-STBC-based approach. However, the cost is that the proposed scheme only has nearly 1/2 symbol rate when discarding CP overheads. Pairwise error probability analysis are derived, under certain assumption which is numerically supported by simulation results, for analytical verifications of our claim on full diversity, inclusive of transmit-receive diversity and the multipath one. Hence, we are able to counterbalance the deduction[26] that the “CP-only” scheme based on the GOSTBC extension cannot exploit full multipath diversity. It is seen from the simulation results that the proposed scheme does stand a big chance of delivering full multipath diversity.
Moreover, the encoder structure enables us to generalize a training-based channel estimation technique, originally proposed for flat-fading scenario, to the frequency- selective fading scenario. Surprisingly we even obtain similar optimality criteria for optimal training block design which in our case, the signal block are fixed as OSTBC-based and the design derivation reduces to derive optimal power constraint over the training blocks. The optimality criteria for the training blocks are easy to satisfy when randomness of signal constellation is not a concern. Simulation results validate our discussion of the behaviors of the least-squares and linear MMSE channel estimates. In contrast to the involved derivations provided in [16], which considers general training blocks for MIMO frequency-selective fading, we provide an alternative by generalizing the work in [3] to achieve the same purpose in a straightforward manner
Appendix
A1. Proof of Block-Level Orthogonality part I
1 1 1 1
Note that the constituent matrices have the following properties:
(:, ) (:, ) (:, ) (:, ) 0, for 1 .
A2. Proof of block-level orthogonality part II
By the same orthogonalities used in Appendix A1.
(:,( 1) : 1) (:,( 1) : 1)
A3. OSTBC construction
To be self-contained, we hereby review some basics about orthogonal design. Recall the OSTBC proposed by V. Tarokh [1].
t
is real-valued symbol block.
are the constituent matrices, 1 .
, 1 GOSTBC. Therefore the subsequent block-level discussion applies whenever the corresponding symbol-level GOSTBC design exists. The construction of GOSTBC is as follows:
is complex-valued symbol block.
, are the constituent matrices, 1 . , 1
Again it is easy to verify the orthogonality, 2
simply adopt the complex orthogonal design introduced in [1], where GMt( )x is closely related to GMt( )x by GMt( )x = GMTt( )x GMTt( )x* T. Here GMt( )x is an OSTBC design having identical constituent matrices as the aforementioned ones but taking complex objectives in lieu of real-valued ones. It can be shown that the construction of ΧΧ is ΧΧAi
for the constituent matrices and transmitted symbols with kronecker product and using index K in lieu of T, we immediately arrive at the BGOSTBC structure. For illustrational purposes in the following chapters, we give two examples:
1
1 2 3 4 5
1 2
Fig. 1 Transmission scheme based on BGOSTBC encoder and the associated FDE based on training-based channel estimation.
(4)
Fig. 2 A BGOSTBC encoder employing K=4, M=3 configuration.
0 1 2 3 4 5 6 7 8 9 10 x 105 0
5 10 15 20 25 30 35 40
Condition number K(Dps) over 106 runs
iterations K(Dps)
Number of singular occurrence = 0
Fig. 3-1 Condition numbers generated for 106 independent simulation runs under (N,Mr,Mt,L,K)=(10,1,3,1,4).
0 1 2 3 4 5 6 7 8 9 10
x 105 0
5 10 15 20 25 30
Condition number K(Dps) over 106 runs
iterations K(Dps)
Number of singular occurrence = 0
Fig. 3-2 Condition numbers generated for 106 independent simulation runs under (N,M,M,L,K)=(16,1,3,1,4).
0 1 2 3 4 5 6 7 8 9 10 x 105 0
5 10 15 20 25
Condition number K(Dps) over 106 runs
iterations K(Dps)
Number of singular occurrence = 0
Fig. 3-3 Condition numbers generated for 106 independent simulation runs under (N,Mr,Mt,L,K)=(16,1,3,2,4).
0 1 2 3 4 5 6 7 8 9 10
x 105 1
1.5 2 2.5 3 3.5
Condition number K(Dps) over 106 runs
iterations K(Dps)
Number of singular occurrence = 0
Fig. 3-4 Condition numbers generated for 106 independent simulation runs under
0 1 2 3 4 5 6 7 8 9 10
Condition number K(Dps) over 106 runs
iterations K(Dps)
Number of singular occurrence = 0
Fig. 3-5 Condition numbers generated for 106 independent simulation runs under (N,Mr,Mt,L,K)=(10,2,6,3,8).
NMSE of channel estimations (dB)
Mt=3, Mr=2, K=4-LS-based(arbitrary training)
Fig. 3 Compare NMSE of LS and linear MMSE estimates, with arbitrary training.
0 2 4 6 8 10 12 14
NMSE of channel estimations (dB)
Mt=3, Mr=2, K=4-LS-based(arbitrary training)
Fig. 4 Compare NMSE of arbitrary training and optimal/orthogonal training.
-6 -4 -2 0 2 4 6
Mt=3, Mr=1, K=4, L=1-Known CSI(arbitrary training) Mt=3, Mr=1, K=4, L=2-Known CSI(arbitrary training)
Fig. 5 Demonstration of full multipath diversity at high SNR region, averaged over 100
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