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 C(n, k) is used to denote the chosen function of n and k.
Chapter 2
General System Model
Consider the wireless system with Mt transmit antenna and Mr receiver antenna in Figure 2.1. The channel is modeled by an Mr× Mt memoryless matrix with
F H
Figure 2.1: MIMO system with limited feedback
channel noise vector q of size Mr × 1. The noise vector q is assumed to be additive white Gaussian with zero mean and variance N0. Suppose 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 Rs= E[ss†] is a diagonal matrix. Assume the total transmission power is Pt and F is an unitary Mr× M matrix. The total transmission power can be written as E[x†x] = E[s†F†Fs] = PM −1
k=0 σs2k, where we have used the fact that F†F = IM. We will consider linear and decision feedback receiver in this paper. Define the error vector at the output of receiver as e = ˆs− s. When the receiver is linear and zero forcing, the receiver output ˆs = Gr, where the M × Mr receiver
matrix is G = (F†H†HF)−1F†H† [32]. The error vector at the output of G has autocorrelation matrix Re = E[ee†] given by [32]
Re= N0(F†H†HF)−1 (2.1)
When there is decision feedback at the receiver, the part from previously detected symbols are subtracted from the received signal and this is also called successive interference cancellation. The decision feedback receiver can be described as a recursive procedure [14]. First initializes r0 = r, A0 = HF and i = 0. The steps in the recursions are as follows. (1) Let Gi be the Moore-Penrose inverse of Ai. Find the row vector of Gi that has the smallest 2-norm. Call the index of the row vector wi. (2) Compute yi = wTi ri , apply symbol detection on yi, and call the output ˆsi. (3) Subtract from ri the contribution of the kith subchannel, ri+1 = ri − ˆskiaki, where aki is the kith column of A0 and zero the kith column of Ai to obtain Ai+1. When all the subchannels are of the same constellation, the post detection SNR of the kith subchannel is ρki = NPt/M
0kwik2. In this case, the above procedure is optimal in the sense that the worst subchannel error rate is minimized.
Assuming the inputs sk are bk-bit QAM symbols, the kth symbol error rate is well approximated by [42].
ek is the post detection SNR and (2.2) is the error rate assuming there is no error in detecting previous symbols. When Gray code is used, the BER can be approximated by BERk ≈ SERk/bk. Using this approximation, the BER for a given channel H can be computed using
BER≈ 1
For a given channel H, the BER depends on the bit allocation and power allo-cation, which will be optimized to minimize BER in chapter 4. The channel is
well known Ricean model [13] or mean information model [11]. In the Ricean model, the flat fading channel is composed of a line-of-sight(LOS) component and a Rayleigh component. We can express H as
H =
r K
K + 1Hsp+
r 1
K + 1HwR1/2t , (2.4) where K is Ricean factor defined as the power ratio of LOS signal to diffused scattered signal, Hw is an Mr × Mt matrix of i.i.d, zero mean, unit variance complex Ganussion random variable and Rt is the Mt× Mt correlation matrix.
Hsp can be expressed as
Hsp = ar× aTt,
are the line-of-sight(LOS) array responses at receiver and transmitter with angle of arrival θr and angle of departure θt respectively and a Uniform Linear Array is considered. If K is large then a pure LOS channel in environment. Such a model assumes correlation only exists at transmitter, this assume is useful for downlink transmission [33]. We also discuss two special case for (2.4) as follow.
1) No line of sight (K = 0)
In a Environment full of obstacles, the multipath components is enough then ricean factor K will approach 0, thus the channel model becomes to
H = HwR1/2t . (2.5)
It is well known covariance information model [11].
2) Rt = IMt
No correlation at transmitter assumption, the H becomes H =
r K
K + 1Hsp+
r 1
K + 1Hw. (2.6)
For transmitter correlation matrix Rt, we consider a uniform linear array of Mt antennas with spacing dt. The plane wave departure directions of these signals span an angular spread θt and uniformly distributed, we find [34] [35].
[Rt]m,k = 1 S
i=(S−1)/2X
i=−(S−1)/2
e−2jπ(k−m)dtcos(π2+θt,i) (2.7) where S is the number of scatterers with corresponding directions of arrival θt,i
θt,i = 1
S− 1θt× i, i =−(S − 1)/2 . . . (S − 1)/2. (2.8) when θtor dtis large, Rtwill converge to the identity matrix which is uncorrelated fading. When θtor dtis small,the correlation matrix becomes rank deficient which is full correlated fading.
d t
θt
Tx
Figure 2.2: Propagation scenario for fading correlation.
Chapter 3
Previous Works
In this chapter, we review two referred works in the literature. Section 3.1 presents a GTD based system for optimal transceiver design and a special case is the QR based system proposed in [21]. Section 3.2 presents a limited feedback precoder system with BER selection criterion and codebook design proposed in [4].
3.1 GTD Based System
3.1.1 Formulating the Power Minimization Problem and Solution
The generalized triangular decomposition (GTD), proposed in [21]. Let H be a M× N rank-K matrix with singular values σ1, σ2. . . σK in descending order. let h = [σ1, σ2. . . σK] and r = [r1, r2. . . rK] be a given vector which satisfies
ˆr≺× h (3.1)
Then there exist matrices R, Q, P such that
H = QRP† (3.2)
where ≺× is multiplicative majorization [36], R is a K × K upper triangular matrix with diagonal terms equal to rk, and Q ∈ CM ×K, P ∈ CN ×K both have orthonormal columns.
The problem of minimizing the transmitted power subject to the specified BER and total bit rate constraints, and the ZF constraint can be written as follows: The following are solution for GTD-based method to construct the transceiver matrices F,G,B [21]. With above choice, the minimum transmission power can be achieve.
3.1.2 QR Transceiver ZF-VBLAST
The QR Transceiver is a special case of GTD based system. Based on the general system model at chapter 2, the system in [21] has decision feedback receiver and precoder is identity. Assume the number of subchannels M are used. This system has bit allocation, the optimal power loading is equally that Rs= PM0IM. Because the precoder matrix is identity and only bit allocation vector need to be known.
The channel matrix be written as H = QR, where Q has orthonormal columns, and R is upper triangular. |R(k, k)|−2 is error variance corresponding to kth subchannel. The receiver can compute {bk} from [21]
bk = 1
(3.8) is called the optimal bit loading formula. we will quantize it to the bit allocation vector nearest to the vector in pre-determined codebook Cb, and feed back the index of that vector to the transmitter.