In this proposed BA system, only bit allocation is adapted according to the vary-ing random channel. The precoder is chosen as F = Ut,M. based on the the results in the previous section. Such a precoder depends only on the channel statistics and the information of the precoder need not be fed back to the trans-mitter frequently. The transmission power is uniformly distributed among the subchannels loaded with nonzero bits. When we consider bit allocation in prac-tical applications, the bits assigned to the symbols are typically integer-valued.
The components of the bit allocation vector b satisfy the sum rate constraint
b0 + b1 + . . . + bM −1 = Rb where bi ∈ Z+ and Z+ denotes the set of nonneg-ative integers. The number of such nonnegnonneg-ative integer bit allocation vectors is C(Rb + M − 1, Rb), where C(·, ·) denotes the choose function. This requires B0 =⌈log2C(Rb+ M− 1, Rb)⌉ bits, where ⌈x⌉ denotes the smallest integer larger than or equal to x. For example Rb = 8, M = Mt = 4, the required number of feedback bits is 8. The approach of using all possible constellation combinations is adopted in earlier works that employs bit allocation subject to a sum rate constraint [20] [21]. To reduce the feedback rate, the codebook is trimmed by imposing some constraints on the vectors [21].
Codeword selection. Suppose we are given B feedback bits and a code-book Cb of 2B bit allocation vectors. The vectors in Cb satisfy the sum rate constraint so that the number of bits transmitted for each channel use is Rb. The BER expression in (2.3) is a function of bit allocation vector. For a given channel H, we can choose the best bit allocation vector bb ∈ Cb that minimizes the BER, bb = arg minb∈CbBER(b, H), where BER(b, H) denotes the BER when the channel is H and the bit allocation vector is b. To make codeword selection more efficient, we can choose (suboptimal) codewords based on the optimal bit allocation given in (4.9). The criterion of minimizing the largest subchannel error rate will be considered. Suppose the optimal bit allocation vector computed from (4.9) is b∗. Given a bit allocation vector b ∈ Cb, the kth subchannel symbol error rate associated with b is
SERk ≈ 4Q(
Let us call this subchannel independent quantity A. Then we have SERk ≈ 4Q(√
A2b∗k−bk) . Therefore the largest subchannel error rate can be minimized by choosing the bit allocation vector b∈ Cb that has the largest mink(b∗k− bk).The optimal bit allocation is derived under the assumption that all M subchannels are loaded with nonzero bits. To remove the assumption, we can compute BER0 in (4.8) for each M0with 0 ≤ M0 ≤ M where M0 is the number of subchannels used, and choose the M0 that has the smallest BER0. We can then apply quantization
Bits to
Figure 4.2: The transmitter of the BA system with (a) precoder F , and (b) augmented precoder F′.
on the corresponding optimal bit allocation using the above maximin criterion maxb∈Cbmink(b∗k− bk) Such a suboptimal selection criterion does not require the computation of BER for each bit allocation in the codebook. Simulations in chap-ter5 will demonstrate that the use of the suboptimal maximin criterion leads to only a minor degradation compared to the optimal BER criterion
Augmented precoding [45]. We have used a fixed Mt× M matrix F as the precoder. When M < Mt and the channel matrix is such that the column space of F is contained in the null space of H, then there is zero signal power at the receiver. This can be avoided by starting off with an augmented initial precoder F′ of size Mt× Mt. For a given M, we can choose M columns out of F′ to form the actual Mt×M precoder F , i.e., (Mt−M) columns of F′ are removed.
The corresponding augmented input vector s′ and bit allocation vector b′ are of size Mt× 1.The entries of s′ and b′ corresponding to the removed columns of F′ are all equal to zero so that the transmitter output F′s′ is equal to Fs. As we choose M columns from F′, there are C(Mt, M) possible choices for precoder F.
The transmitter with the augmented precoding scheme is shown in Fig. 4.2(b).
The augmented bit allocation vector b′satisfies b′0+b′1+. . .+b′M −1 = Rb, b′i ∈ Z+, with the additional constraint that at most M of the components can be nonzero as it is assumed that the transmitter and receiver can process at most M sub-streams.It can be verified that the total number of possible integer bit allocation
vectors satisfying the sum rate constraint is
MXt−1 k=Mt−M
C(Mt, k)C(Rb− 1, Mt− 1 − k) (4.40)
As in the non augmented case we can design a smaller codebook Cb′ b to have a smaller feedback rate. There is no need to feedback the information of the actual precoder F used. The information is embedded in the augmented bit allocation vector b′. For i = 0, 1, . . . , Mt − 1, the transmitter removes the ith column from F′ if bi = 0. The transmitter can then use the resulting Mt× M0
submatrix as the precoder, where M0 is the number of nonzero entries in b′. Note that for a given channel, using augmented precoder F′ is not guaranteed to be better than using a fixed F because the codebooks are different.Suppose F is a submatrix of F′. Let us consider the special case that the codewords of Cb′ is obtained by inserting appropriate zeros in the codewords of Cb.Then the system with augmented precoder has the same performance as the one with a fixed precoder, but not better. Nonetheless the simulations in chapter5 will demonstrate that when M < Mt the system of augmented precoder outperforms the one with a fixed precoder for the same number of feedback bits.
Optimal detection ordering for decision feedback receiver. When all the subchannels use the same constellation, the optimal detection ordering for the decision feedback receiver is to maximize the post detection SNR ρi in each recursion [14]. Such an approach minimizes the worst subchannel error rate. It is not same for the case with bit allocation and bit allocation needs to be taken into consideration. Suppose the bit allocation is given. In the second step of the recursive procedure we need to choose the nonzero row vector of Gi to maximize
µki = 1
(2bki − 1)kwik2, f or ki ∈ S, (4.41) where S = {j : bj > 0} is the collection of subchannels that are used for trans-mission. This can be proved by following a procedure similar to that in [14]. The maximization of µi (also called rate-normalized SNR) in each recursion has been shown to minimize the outage probability in [17]. Note that there is no need
for the receiver to feedback the detection order; the transmitter only needs to know the bit allocation but not the detection ordering. For each bit allocation in the codebook, we can perform the recursive procedure to maximize the rate-normalized SNR. Then the best bit allocation and corresponding detection order can be selected.
y
G
+P
TB
+à I detector
x
Figure 4.3: Block diagram of the decision feedback receiver based on cholesky decomposition.
Reduce complexity for optimal ordering. Above detection ordering, we need to take Moore-penrose inverse after each detected. It will raise complex-ity. In [44] V-BLAST is proposed to reduce the complexity by applying cholesky decomposition with symmetric permutation. It derive new algorithm based on a specific receiver structure in Figure4.3, where G is feedforward matrix, B is feedback matrix and P is permutation matrix that recover original ordering. Let the cholesky decomposition of Re be LDL†, where L is a Mt× Mt unit lower triangular matrix and D is a Mt× Mt diagonal matrix with diagonal element [D]ii= diiand diiis the error variance of the ith detected of subchannel input xi. the feedforward matrix G and feedback matrix B are given, respectively, by [44]
B†= L−1 (4.42)
G†= DL†PH†R−1e (4.43)
where Re = N0(F†H†HF)−1.
The algorithm with maximizing rate-normalized SNR is shown as follow
• step 1: Re = N0(F†H†HF)−1 , P = IMt , D = 0Mt
• step 2: for i = Mt, . . . , 1
q = arg minq′Re(q′, q′)(2bq′ − 1)
Pi = IMt, whose ith and qth rows are exchanged P = PiP , Re = PiRePTi , b = Pib
D(i, i) = Re(i, i) , Re(i : Nt, i) = Re(i : Nt, i)/D(i, i) for j = i + 1, . . . , Mt
Re(j : Mt, j) = Re(j : Mt, j)− Re(j : Mt, i)R∗e(j, i)D(i, i) Re(j, j : Mt) = Re(j : Mt, j)†
L = tril(Re)
• step 3: B† = L−1 , G†= DL†PH†R−1e
By using this algorithm, we don’t need to take matrix inverse after each detection so we can successfully reduce the complexity.