This section is organized as follows. In Section , we show the system model and diversity of multimode precoding system. The selection criteria are given in Section 3.3.2. And Section 3.3.3 reviews the criteria of codebook size allocation and construction.
3.3.1 System Model
Founded on the general system model in chapter 2, multimode precoding assumes Rb is the fix target transmission rate, the bit loading is uniformly allocated bk =
Rb
M, for k = 1 · · · M, and transmission power is equally divided for M symbols, Rs = PM0IM. Similar to multimode antenna selection in section 3.2, the multimode precoding system allows the number of subchannels M to vary according to the channel H and M ≤ min(Mr, Mt). In addition, a codebook FM is prepared for each mode M. Since multimode precoding requires RMb to be integer, thus only some modes can support transmission. The set of these supported modes is denoted as M. For example, if Rb = 8 bits and Mr = Mt= 4, then M = {1, 2, 4}.
Based on the channel H, the receiver determines the number of subchannels M and selects the precoder matrix from the complete precoder codebook CF = {FM}MM =1t . Subsequently, the index represented this selection is fed back to the transmitter. The transmitter adjusts the transmission setting according to the feedback information.
Diversity. let NM denoted the number of precoder matrices in FM. It is proved
codebook size of F1, is greater than or equal to Mt and the vectors in F1 span CMt. Selecting vector from F1 = {f1, f2, · · · , fN 1} is equal to a beamforming system with finite beamforming feasible set [24]. From [24], we know that such a beamforming system has full diversity order equal to MrMt if the span of F1 is equal to CMt. Therefore, the multimode precoding has full diversity order if above mentioned condition is satisfied.
3.3.2 Selection Criteria
Two selection criteria are proposed in this paper. One is for minimizing proba-bility of error. The other is for maximizing capacity.
Probability of Error Selection Criterion. The selection is divided in two step. For every M ∈ M, first step selects the FM∗ from each precoder codebooks FM using the following selection criterion,
FM∗(H) = arg max
F∈FMλ2M(HF), (3.14) where λk(H) is the k-th largest singular value of H. The second step determines the number of subchannels M∗ by
M∗(H) = arg max
Capacity Selection Criterion. Assuming uncorrelated Gaussian signaling on each substream, the mutual information is known to be
CU T(FM) = log2det
Similar to above selection criterion, for every M ∈ M, first step select the FM∗
from each precoder codebooks FM using the following selection criterion, F∗M = arg max
F∈FMCU T(F). (3.17)
Then, M∗ is decided by
M∗ = arg max
M ∈MCU T (F∗M) . (3.18)
3.3.3 Allocation Criterion and Codebook Construction
Given B feedback bits, there are total 2B codewords for complete codebook CF. Some criterions are designed in [18] to distribute 2B codewords among the modes in M. Under the assumption that the probabilities of selecting each mode in M are equal, the codeword allocation criteria for maximizing capacity and minimiz-ing probability of error are given as follows.
Probability of Error Allocation Criterion. Define the cost function as A(N1, · · · , NMt) = X
M ∈M
d2min(M, Rb)
M N
−2 Mt(Mt+1)
M (3.19)
• For B ≤ log2(Mt+ 1), set NM t = 1 and N1 = 2B− 1.
• For B > log2(Mt+1), find the (N1, · · · , NMt) that minimizes A(N1, · · · , NMt) such that N1 ≥ Mt, NMt = 1, and P
M ∈MNM = 2B This minimization can be done using a numerical search or by using convex optimization tech-niques.
Capacity Allocation Criterion.
• For B ≤ log2(Mt+ 1), set NM t = 1 and N1 = 2B− 1.
• For B > log2(Mt+ 1), if B ≤ log2(Mt(|M| −1) + 1), set NMt = 1, N1 = Mt, and Nk = (2B|M|−2−Mt−1), for k ∈ M, k 6= 1, Mt. If B > log2(Mt(|M| − 1) + 1), set NMt = 1 and Nk= |M−1|2B−1 for k ∈ M, k 6= Mt.
After the sizes for each modes’ codebooks are allocated. The codebook for each mode is construct using the method in [5]. The work in [5] can approximately convert the problem of precoder codebook construction into Grassmannian
sub-Probability of Error Design Criterion. From [5], the projection two-norm distance is defined as
dproj(Fi, Fj) = kFiF†i − FjF†jk2,
where k·k denotes 2-norm of a matrix. For minimizing probability of error, design FM such that
δproj = min
Fi,Fj∈FM:Fi6=Fjdproj(Fi, Fj) is maximized.
Capacity Design Criterion. The Fubini-Study distance is defined in [5] as dF S(Fi, Fj) = arccos | det(F†iFj)|.
For maximizing capacity, design FM such that δF S = min
Fi,Fj∈FM:Fi6=FjdF S(Fi, Fj) is maximized.
Chapter 4
The Proposed BA system
In this chapter we propose the feedback of only bit allocation (BA) for MIMO systems with limited feedback. The proposed system will be termed a BA sys-tem. We show that the proposed BA system can achieve full diversity order. We also derive the optimal bit allocation for minimum BER when the transmission rate is given and the bit allocation vector is not constrained to be from a code-book. It turns out that the optimal bit allocation that minimizes the BER is also the optimal solution for minimizing the transmission power. Using the optimal unconstrained bit allocation, an efficient method for selection BA is developed.
4.1 System Model
Based on the general system model in chapter 2, we assume the total transmission power P0 is equally divided among all symbols carrying nonzero bits. So sk has variance given by
σ2s =
P0/M0, bk > 0,
0. bk = 0, (4.1)
where M0 is the number of symbols carrying nonzero number of bits. As the power is equally divided among symbols with nonzero bits, the autocorrelation matrix of the error vector for the MMSE case (2.5) can be simplified. Removing the symbols with zero bits from s, we obtain a reduced vetor s0 of size M0 × 1.
If we remove the corresponding columns of F, the result is an Mt× M0 matrix, say F0. Then using precoder F0 with input s0 gives the same transmitter output
(x = F0s0 = Fs). The vector s0 has the autocorrelation matrix Rs0 = MP0
0IM0. The autocorrelation matrix of the corresponding error vector e0 is
Re0 =
( N0(F†0H†HF0)−1, zero-forcing receiver, (N1
0F†0H†HF0+ P 1
0/M0IM0)−1, MMSE receiver. (4.2) In our proposed system, the precoder matrix F in the transmitter is determined beforehand. Therefore, when the channel H is given, the average BER formula in (2.11) depends only on the bit allocation vector b, which can be optimized to minimize BER.
The receiver feedbacks only the bit allocation vector b to the transmitter. When the bit allocation vector b has integer entries, in principle the whole vector can be sent back to the transmitter using finite-rate feedback. However, in a system with low feedback rate it may not be possible to feedback the complete informa-tion of b without quantizainforma-tion. In this case the bit allocainforma-tion vector is chosen from a codebook Cb and the index of the bit allocation vector is fed back to the transmitter as we will see in the next section.