1. Bold face upper case letters represents matrices. Bold face lower case letters represents matrices. The notation A† denotes transpose-conjugate of A.
The notation AT denotes transpose of A.
2. The function E [y] denotes the expect value of a random variable y.
3. The notation Im is used to represent the m × m identity matrix.
4. The notation Wmis used to represent the m×m unitary DFT matrix given by,
[Wm]kn= √1me−j2πmkn for 0 ≤ k, n ≤ m − 1. (1.1) 5. The notation C(n, k) is used to denote the chosen function of n and k.
Chapter 2
General System Model
The finite-rate feedback Mr×MtMIMO system is shown in Fig. 2.1. The channel
bits to
Figure 2.1: MIMO system with limited feedback
is modeled by a Mr × Mt memoryless matrix with an channel noise vector q of size Mr × 1. It is supposed that the channel is block fading, which means the channel remains constant over sufficiently long period before independently taking a new realization. The noise vector q is assumed to be additive white Gaussian with zero mean and variance N0. The system can process M substreams, where M ≤ min(Mr, Mt). The input vector s is an M × 1 vector which consists of M modulation symbols. The symbols sk are assumed to be zero mean and uncorrelated, hence the autocorrelation matrix R = E[ss†] is a diagonal matrix.
Rb is the number of bits transmitted during each symbol period. Assume the total transmission power is P0 and the precoder F is an unitary Mt× M matrix.
The total transmission power P0 = E[x†x] can be written as P0 = E[x†x] = trace(FRsF†) = trace(Rs) =
M −1X
k=0
σ2sk, (2.1)
where we have used the trace property trace(AB) = trace(BA) for two matrices A and B and the fact that F†F = IM. The channel output vector r is therefore
r = HFs + q (2.2)
The M × Mr receiving matrix G can be zero forcing receiver or minimum mean square error (MMSE) receiver [22]
G =
(F†H†HF)−1F†H†, zero-forcing receiver,
RsF†H†(HFRsF†H†+ N0IMr)−1, MMSE receiver. (2.3) The error vector e at the output of receive matrix G is
e = ˆs − s = Gr − s (2.4)
The autocorrelation matrix of error vector Re = E[ee†] given by [22] is Re=
N0(F†H†HF)−1, zero-forcing receiver Rs− RsF†H†(HFRsF†H†+ N0IMr)−1HFRs, MMSE receiver
(2.5) Generally, it is assumed the transmitter has no channel state information and the channel is perfectly estimated at the receiver. The reverse link can feedback B bits. At the receiver, transmission information for enhancing desired perfor-mance is derived from the full channel knowledge. Based on this information, an index is selected from the codebooks which are known to both the transmitter and the receiver. Then the index is sent back through the reverse channel to the transmitter. According to the feedback information, the transmitter adapts the transmission settings and sends the signals into channel. Transmission infor-mation extracted from the full CSI at receiver such as precoding matrix, power loading, and bit allocation are used by different system designs. Using these in-formation, various performance like BER, capacity and transmission bit rate can
be improved. In this thesis, the efficiency between BER performance and the amount of feedback bits is the main topic of our work.
In the following we discuss the system model for the precoder system (no bit allocation) and the system model for the BA system (with bit allocation) separately.
Precoder system. In the precoder system there is no bit allocation, the trans-mitted bits are assumed to be equally allocated on M symbols. Each modulation symbol carries RMb bits and RMb is assumed to be integer. Assuming QAM modu-lation, the symbol error rate for k-th subchannel is well approximated by [23]:
SERk= 4(1 − 1
and βk is the unbiased SNR of the k-th subchannel. For zeroforcing and MMSE linear receiver, βk can be expressed respectively as,
βk =
σ2sk
σ2ek, zero-forcing receiver,
σ2sk
σ2ek − 1, MMSE receiver. (2.7) When Gray code is used, the BER for k-th subchannel can be approximated by
BERk≈ SERk
(Rb/M).
So, when precoder matrix F is used the average BER for a given channel H can be approximately expressed as
Since the bit allocation is set to be uniformly loaded, the error performance is independent of bit allocation and is decided by the unbiased SNR βk. In the precoder system, the receiver sends back the information of the precoder back to
BA system. In the BA system, the symbols can carry different number of bits. Suppose bk bits are carried by the k-th modulation symbols. Thus, the transmitted bits per channel use is
Rb =
be the bit allocation vector. When the input symbols sk are bk-bits QAM symbols, the k-th symbol error rate is approximated by [23]:
where βk is the unbiased SNR of k-th subchannel (2.7). Using Gray code, the BER can be approximated by BERk ≈ SERk/bk. Given a channel H and the precoding matrix F, the average BER can be approximately computed using
BER(b, F, H) ≈ 1
In addition, the system without bit allocation can be considered as having a uniform bit allocation vector b whose entries
b0 = b1 = · · · = bM −1= Rb
M. (2.12)
Chapter 3
Previous Works
In this chapter, previous works for minimizing error performance are reviewed.
Section 3.1 presented a limited feedback precoder ststem with BER selection criterion and codebook design proposed in [7]. Optimal unitary precoder for infinite feedback rate is also derived. In section 3.2 multomode antenna selection [17] is introduced. Section 3.3 recaps multimode precoding [18].
3.1 Precoder System
This section is organized as follows: Section 3.1.1 introduces the system model and presents the BER-based selection criterion. Optimal precoder for infinite feedback rate is given in Section 3.1.2. And Codebook construction is showed in Section 3.1.3.
3.1.1 System Model
Based on the general system model at chapter 2, the system in [7] assumes the number of subchannels M is fixed and all M subchannels are used. The system is without bit allocation design. Thus, the bit loading is uniform and the target bit rate Rb is divisible for M. Each symbol carries RMb bits. The power is also equally allocated for each symbols, Rs = PM0IM. For the reverse channel, it is constrained to send B bits. In this paper, the feedback information is the precoder matrix.
Therefore, a precoder codebook CF of size 2B is prepared. After the estimation
of forward channel, a precoder matrix is selected using a BER-based selection criterion from CF and the corresponding index is fed back to the transmitter.
The BER-based selection criterion will be reviewed as follows.
BER selection criterion. Under the assumption of uniform bit allocation, the average BER for each precoder matrix in CF can be computed by (2.8). The BER-base selection criterion is
F = arg minb
F∈CF
BER(F, H). (3.1)
To choose a precoder matrix by BER selection criterion, we need to compute the BER formula (2.8) for each precoder matrix in CF. Therefore, 2B computations of (2.8) are required to complete BER selection criterion.
3.1.2 Optimal Precoder for infinite-feedback rate
With infinite feedback bits, it can be assumed that the transmitter has full chan-nel knowledge. The optimal precoder Fopt with BER-based criterion can be derived directly from H. The optimal precoder Fopt can provide a benchmark performance for finite-rate precoder feedback system. Assuming the singular value decomposition of H = UΛV†, where U and V are respectively Mr× Mr and Mt×Mtunitary matrices. The Mr×Mtmatrix Λ is a diagonal matrix whose diagonal elements are the singular values of H in a nonincreasing order. And let βk be the k-th largest subchannel SNR. The optimal precoders for zero forcing and MMSE receiver are given respectively as follows.
Zero-forcing case. Consider a rectangular/square QAM constellation with size M is applied for ¯b. Constellation-specific threshold Γth is shown in table 3.1.2.
1. When β1 ≤ Γth, Fopt = VM, where VM is the Mt× M matrix obtained by keeping the first M columns of V.
2. When βM ≥ Γth, Fopt = VMQM, where QM is an M × M unitary that has equal magnitude property, i.e., |[QM]m,n| = 1/√
M , for 0 ≤ m, n ≤ M − 1.
3. When conditions in 1 or 2 do not hold, the optimal precoder Fopt can’t be found analytically. Suppose that K1 subchannels’ SNR are larger than Γth. Then one suboptimal precoder that is better than VM can be constructed as
MMSE case. Consider a rectangular/square QAM constellation with size M is applied for ¯b. Two constellation-specific thresholds Γth,l, Γth,h are shown in table 3.1.2.
1. When Γth,l ≤ βM and β1 ≤ Γth,h, Fopt = VM. 2. When β1 ≤ Γth,l or βM ≥ Γth,h, Fopt = VMQM.
3. When conditions in 1 or 2 do not hold, the optimal precoder Fopt can’t be found analytically. Suppose that K1 subchannels’ SNR are larger than Γth,h
and K2 subchannel SNRs are smaller than Γ(th, l). Then one suboptimal precoder that is better than VM can be constructed as
F = VM
3.1.3 Codebook construction
From [5] it is shown that the precoder codebook design problem can be related to Grassmanian subspace packing. Thus, in [7], generalized Lloyd algorithm is used to construct a precoder codebook by minimizing a chordal distance cost function.
The chordal distance between two unitary Mt by M matrices, Fi and Fj is dc(Fi, Fj) = 1
√2
FiF†i − FjF†j
F , (3.4)
where k · kF denotes Frobenius norm. Suppose that V is an isotropically dis-tributed Mt× M matrix. The following algorithm quantizes V to 2B matrices.
Starting with an initial codebook CF = {F0, F1, · · · , F2B−1} (obtained from ran-dom computer search or using the currently best codebook if available), the codebook design steps are as follows.
1. Generate a training set with Ntr samples {Vn}Nn=1tr. 2. Iterate following steps until it converges.
(a) Assign Vn to one of the regions {Ri}2i=0B−1 using the rule
Vn∈ Ri, if dc(Vn, Fi) < dc(Vn, Fj), ∀j 6= i. (3.5)
(b) For each region Ri, find the centroid as Fcentroidi = arg min
Let the eigendecomposition of R as
R = URΛRU†R. (3.10)
ΛRis a diagonal matrix whose diagonal elements are in nonincreasing order. It is easy to show that Fcentroidi is a Mt× M matrix obtained by keeping the first M columns of UR.
(c) Set CF = {Fcentroidi }2i=1B−1. During each iteration, The codebook will be record if the minimum chordal distance of CF
0≤i<j≤2minB−1dc(Fi, Fj)
is larger than the currently best.
3. Go back to 1, generate another training set, then execute the next steps.
The algorithm will stop if there is no further improvement on the minimum chordal distance.