In the proposed BA system, only bit allocation will be sent back to the trans-mitter. The information of the precoder is not fed back to the transtrans-mitter. We discuss the feedback of bit allocation for two cases (i) precoder is square with M = Mt (implicitly Mt ≤ Mr), and (ii) precoder is rectangular with M ≤ Mt, separately in Section 4.2.1 and Section 4.2.2. Although the first case is a special case of the second, it is more convenient to discuss the simpler case M = Mtfirst.
4.2.1 M = M
tCase
In this case the precoding matrix F in the transmitter of the BA system shown in Fig. 4.1(a) is a fixed Mt×Mtmatrix. When we consider bit allocation in practical
Bits to
Figure 4.1: The transmitter of the BA system with (a) precoder F, and (b) augmented precoder F′
applications, the bits assigned to the symbols are typically integer-valued. When the number of bits transmitted per channel use Rb is given, the components of the bit allocation vector b satisfies
b0+ b1+ · · · + bM −1= Rb, where bi ∈ Z+, (4.4) where Z+ denotes the set of nonnegative integers. The number of such nonneg-ative integer bit allocation vector is (pp. 337, [25])
C(Rb+ Mt− 1, Rb), (4.5)
where C(·, ·) denotes the choose function. Feedback of all these possible bit allocation vectors requires
B0 = ⌈log2(C(Rb+ Mt− 1, Rb))⌉ , (4.6) 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 B0 = 8. To reduce the number of feedback bits, we can quantize the bit allocation vector.
Quantization of bit allocation. Suppose we are given B feedback bits and a codebook Cbof 2Bbit allocation vectors. The vectors in Cbsatisfy the transmission rate constraint in (4.4) so that the number of bits transmitted for each channel use is Rb. We can choose the best bit allocation vector ˆb ∈ Cb that minimizes the BER. The BER expression in (4.3) is a function of bit allocation vector and we can choose
ˆ
The actual number of transmitted symbols can be smaller than M as some of the symbols may be assigned with 0 bits. The selection criterion in (4.7) requires the computation of BER for all possible bit allocation vectors in the codebook, so BER(b, H) is evaluated 2B times. When the codebook size is small (i.e. low feedback rate), for example, B = 2, 3, the number of searches is small as well. As we will see in the simulation examples, we can get good BER performance using a small codebook size.
4.2.2 M ≤ M
tCase
For M ≤ Mt, we can start off with an augmented initial precoder F′ of size Mt× Mt. The corresponding augmented input vector s′ and bit allocation vector b′ are of size Mt× 1. For a given M, we can choose M columns out of F′ to form the actual Mt× M precoder F, i.e., (M − Mt) columns of F′ are removed. As we choose M columns from F′, there are C(Mt, M) possible choices. The entries of s′ and b′ corresponding to the removed columns of F′ are equal to zero. s and b are M × 1 vectors which is formed by removing the zero entries of s′ and b′ so that F′s′ = Fs. The transmitter with the augmented precoder and augmented input vector s′ is shown in Fig. 4.1(b). The augmented bit allocation vector b′ satisfies
b′0+ b′1+ · · · + b′M −1= Rb, where b′i ∈ Z+, (4.8) 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. In this case the number of symbols transmitted is at most M, carrying a total of Rb bits. To count the number of integer bit allocation vectors satisfy (4.8), let us first consider the case that b′ has exactly k zeros, where k ≥ Mt−M.
Then Rb will be distributed among Mt− k symbols, each with at least one bit.
There are C(Mt, k)C(Rb− 1, Mt− 1 − k) such combinations [25]. Thus the total number of possible integer bit allocation vectors satisfying (4.8) is
MXt−1 k=Mt−M
C(Mt, k)C(Rb− 1, Mt− 1 − k). (4.9)
For example, when Mt = 4, M = 3 and Rb = 8, the number is 130. To feedback all these vectors requires 8 bits. To have a smaller feedback rate, we can use a codebook Cb′ of augmented bit allocation vectors. Each b′ ∈ Cb′ satisfies (4.8).
The BER can be obtained by a slight change of the summation in (4.3), BER(b′, H) = 1
Rb
MXt−1 k=0,b′k6=0
SERk. (4.10)
We can choose the best bit allocation vector from Cb′ to minimize BER, bb′ = arg min
b′∈C′b
BER(b′, H). (4.11)
Note that 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 i-th column from F′ if b′i = 0. The transmitter can then use the resulting Mt× M0 submatrix as the precoder, where M0 is the number of nonzero entries in b′.
The optimal augmented precoder. In the BA system, the augmented precoder F′ is a fixed square unitary matrix. It does not vary with the channel; only the bit allocation does. A question that arises naturally here is this: What is the optimal channel-independent augmented precoder? It turns out that any Mt×Mtunitary matrix will yield the same performance if the entries of the channel matrix H are independent, identically distributed circularly symmetric Gaussian random variables with zero mean. For example, choosing F′ as the normalized DFT matrix in (1.1) or the identity matrix will give us the same result. To see this let us view the BA system as having precoder F′ and input s′. (In the case M = Mt, F′ = F and s′ = s). Let the auto correlation matrix of s′ be Rs′. It can be verified that the corresponding Mt× Mt error autocorrelation matrix Re′ can be obtained from (2.5) by replacing F with F′ and Rs with Rs′,
H. That is, the entries of HF′ are independent, identically distributed circularly symmetric Gaussian random variables with zero mean. Therefore, for any fixed unitary F′, HF′ is statistically equivalent to H and hence the same performance is achieved.
Fixed Mt × M precoder. In the above discussion, we have used augmented initial precoder when M < Mt. The actual precoder F is not a fixed Mt× M matrix. The reason for not using a fixed precoder F is as follows: If 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 allowing F to be an arbitrary Mt × M submatrix of F′. There is no such problem for the case M = Mt because the column space of any Mt× Mt unitary F is CMt, where CMt is the set of all Mt× 1 vectors of complex numbers. Note that with B feedback bits, for a given channel, using augmented precoder F′ is not guaranteed to be better than using a fixed F. This is because for a given number of feedback bits B, the codebook Cb′ for BA system with augmented F′ is different from Cb
for a fixed Mt× M precoder. 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 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.
The case F′ = IMt. When the initial precoder is the identity matrix, the BA system implicitly employs a form of antenna selection at the transmitter [12], in which the best M antenna are chosen to minimize the BER. But unlike conventional antenna selection, the symbols transmitted on the chosen antennas do not carry the same amount of bits. For the BA system, the feedback of antenna selection at the transmitter is embedded in the feedback of bit allocation. There is no need to tell the transmitter which antennas to use other than the index of bit allocation vector. When F′ = IMt, we can also view the BA system as a
extension of the multimode antenna selection [17], which also chooses a subset of transmit antennas, but the number of antenna used is allowed to vary with the channel. As the bits are uniformly loaded [17], the number of antenna used should divided Rb. There is no such condition for the BA system.