應用於相關性通道具位元配置有限回授系統之傳送器設計

國

立

交

通

大

學

電控工程研究所

碩

士

論

文

應用於相關性通道具位元配置有限回授系統之傳送器設計

Design of statistical precoder for correlated MIMO channel with limited feedback of bit allocation

研 究 生：歐士傑

指導教授：林源倍 教授

應用於相關性通道具位元配置有限回授系統之傳送器設計

Design of statistical precoder for correlated MIMO channel with limited feedback of bit allocation

研 究 生：歐士傑 Student：Shih-Jet Ou

指導教授：林源倍 Advisor：Yuan-Pei Lin

國 立 交 通 大 學

電控工程研究所

碩 士 論 文

Submitted to Department of Control Engineering College of Electrical Engineering

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Department of Electrical Engineering July 2011

Hsinchu, Taiwan, Republic of China

應用於相關性通道具位元配置有限回授系統之傳送器設計

研 究 生：歐士傑 指導教授： 林源倍

國立交通大學電控工程研究所碩士班

摘要

在本篇論文中，我們提出一個多輸入多輸出有限回授具位元配置系統

在萊斯通道。首先我們推算出能使系統達到最小錯誤率的最佳位元與

功率配置。接著我們根據相關性通道的統計特性設計出最佳的統計傳

送器達到最小化錯誤率上限。我們考慮線性接收器與判定回授接受器。

模擬結果我們顯示我們所提出的系統可以用較少的回授位元達到低

錯誤率。

致謝

感謝指導教授 林源倍老師在兩年研究過程中，有耐心給予我專業

領域上的教導，在研究上遇到困難時也會適時給我我幫助，讓我能順

利的完成碩士論文，遇到老師真的是我學生生涯中最幸運的一件事。

另外也要感謝林清安教授和蔡尚澕教授在百忙之中抽空參加我的口

試，並且在論文上所提出的建議，使我的論文更完善。

感謝實驗室的建樟大學長與虹君學姐在課業上、報告上、研究上的

幫助。本篇論文的概念是延續自人予學長的研究，為此要向人予學長

說聲謝謝。感謝實驗室的君維在課業上的幫助與討論。也謝謝謝實驗

室的超任、子軒在實驗室陪我度過歡樂的時光。謝謝徐瑜聰、姜智方

在交大的宿舍陪我度過快樂的時光。

最後要感謝我的父母和家人兩年來的支持和鼓勵還有為我加油打

氣的朋友們。

Design of statistical precoder for correlated

MIMO channel with limited feedback of bit

allocation

Shih-Jet Ou

Advisor: Dr. Yuan-Pei Lin

Department of Electrical and Control Engineering

National Chiao Tung University

Abstract

In this thesis, we design statistical precoder for precoded MIMO sys-tems over correlated Ricean channels with limited feedback of bit alloca-tion. We assume a reverse link of very low rate is available so that the receiver can send back the index of BA vector chosen from a codebook known to both transmitter and receiver. Furthermore we assume the cor-related channel is slow fading and the statistics of the channel are known to the transmitter. Based on statistical of the channel, we derive the op-timal statistical precoder so that bounds of the BER averaged over the random correlated channel is minimized. We will consider both linear and decision feedback receivers in the design of bit allocation codebook. The distribution of the bit allocation is taken into consideration. As a result, a nice tradeoff between performance and feedback rate can be achieved for correlated channels. Simulations show very good performance can be achieved when optimal precoder is used.

Contents

1 Introduction 7

1.1 Outline . . . 10

1.2 Notations . . . 10

2 General System Model 12 3 Previous Works 16 3.1 GTD Based System . . . 16

3.1.1 Formulating the Power Minimization Problem and Solution 16 3.1.2 QR Transceiver ZF-VBLAST . . . 17

3.2 Precoder System . . . 18

3.2.1 System Model . . . 18

3.2.2 Optimal Precoder for infinite-feedback rate . . . 18

3.2.3 Codebook construction . . . 20

4 The proposed BA system 22 4.1 Optimal Bit and Power Allocation . . . 22

4.2 Design of statistical precoders for minimum BER . . . 25

4.2.1 Optimal precoders design with Ricean channel . . . 27

4.3 Feedback of bit allocation . . . 33

4.4 Diversity Gain of BA system [45] . . . 38

6 Conclusion 60 Appendix . . . 61

List of Figures

2.1 MIMO system with limited feedback . . . 12 2.2 Propagation scenario for fading correlation. . . 15 4.1 Block diagram of the decision feedback receiver. . . 29 4.2 The transmitter of the BA system with (a) precoder F , and (b)

augmented precoder F′_{. . . . 35} 4.3 Block diagram of the decision feedback receiver based on cholesky

decomposition. . . 37 5.1 (a) Probability mass function of the bit allocation vectors for

Channel I, Mr = 5, Mt= 4, M = 4, and Rb = 12; (b) correspond-ing cumulative distribution function. . . 42 5.2 (a) Probability mass function of bit allocation vectors for Channel

II, Mr = 4, Mt= 5, M = 4 andRb = 8; (b) Corresponding CDF. . 43

5.3 (a) BER bound for Mr = 4, Mt = 5, M = 3 and Rb = 10 for

Channel II (b) BER bound for Mr = 5,Mt= 4,M = 4 and Rb = 12 for Channel IV . . . 45 5.4 (a) Different feebback bits with Mr = 4, Mt = 5, M = 4, Rb = 8

for Channel II (b) Different feebback bits with Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel V . . . 47 5.5 (a) BER for different precoder Mr = 4, Mt = 5, M = 4, Rb = 8

for Channel III. (b) BER for different precoder Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel IV. . . 49 5.6 BER with different Cb, Mr = 5, Mt = 4, M = 4, Rb = 12 for

5.7 (a) BER for linear receiver, Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel I (b) BER for decision feedback receiver for Channel I. . 52 5.8 BER of BA system for Mr = 3, Mt = 4, M = 3, Rb = 10, Channel

II is considered using the optimal BER criterion and suboptimal maximin criterion. . . 53 5.9 Mr = 3, Mt= 4, M = 3, Rb = 8 with linear receiver for Channel IV. 55 5.10 Mr = 3, Mt = 4, M = 3, Rb = 8 with decision feedback receiver

for Channel IV. . . 55 5.11 (a) Comparison of BER for Mr = 5, Mt = 4, M = 4, Rb = 12 for

channel II (b) Mr = 5, Mt = 4, M = 4, Rb = 12 for channel V. . . 57 5.12 (a) Comparison of BER for Mr = 4, Mt = 4, M = 3, Rb = 8 for

List of Tables

3.1 Table of Γth . . . 20 3.2 Table of Γth,l and Γth,h . . . 20

Chapter 1

Introduction

MIMO systems with limited feedback have received great interest recently [1]-[10]. The system performance in terms of transmission rate or error rate can be improved significantly with limited amount of feedback from the receiver through a reverse channel [1]. It is generally assumed that the transmitter has no knowl-edge of the forward link channel and only the receiver has knowlknowl-edge of the channel state information. The feedback of the complete channel information to the transmitter will require infinite number of bits. In practice the reverse channel can support only a limited transmission rate and it is desirable to have feedback rate as low as possible.

Recently precoded spatial multiplexing systems with finite-rate feedback have been investigated extensively [2]- [10]. The receiver chooses the optimal precoder from a codebook and sends the index back to the transmitter. Optimal codebook designs of unitary precoders using Grassmannian subspace packing for different criteria are developed in [2]. In [3], randomly generated codebooks known to the transmitter and receiver a priori is proposed. The optimal unitary precoder for minimizing BER (bit error rate) using infinite feedback rate, i.e., full channel state information available to the transmitter, is given in [4] and generalized Lloyd algorithm is used for constructing codebooks. Capacity loss due to quantized feedback is thoroughly analyzed in [5]. A special form of precoding system is the antenna selection system [6] that chooses the best subset of transmit antennas to minimize BER. In this case the transmitter enjoys low complexity as the precoder

is a submatrix of the identity matrix. In multimode antenna selection [7], the number of substreams M or mode is allowed to vary with the channel and the bits are uniformly allocated to the M substreams. It is shown in [7] that with Mt bits of feedback, multimode antenna selection can achieve full diversity order MrMt, where Mt and Mr are respectively the number of transmit and receive antenna. In multimode precoding [8], the number of substreams M can also vary with the channel. In addition, a precoder codebook is designed for each possible M. The design of codebooks for multimode transmission over spatially correlated channels is developed in [9]. Generalized Lloyd algorithm is used in [10] to design capacity maximizing codebooks for multimode transmission.

Wireless communication over correlated fading be considered in [11]- [13]. The transmitter optimization be propose and determine a necessary and sufficient condition for maximize capacity in [11] and the special case that is used single antenna at receiver in [12]. In [13], a approximate minimum average symbol error rate precoder is designed for spatial multiplexing system with power allocation in Ricean channel.

A particular useful class of spatial multiplexing transceiver is the V-BLAST system that employs successive interference cancellation at the receiver [14]. The conventional V-BLAST system uses uniform bit/power allocation and thus no feedback is needed. It has been extended by incorporating power allocation or bit allocation when there is feedback [15]- [21]. In [15], approximate minimum BER power allocation was derived and the feedback is the power allocation information. An efficient algorithm for per antenna power and rate control of VBLAST system is developed in [16]. Joint optimization of bit allocation and detection ordering for minimizing outage probability is given in [17]. Successive quantization of power and bit allocation is proposed in [18]. Through the feedback of power and bit allocation, considerable gain can be achieved. Rate and power are optimized for uncoded error probability in [19]. As the receiver feedbacks only the ordering of detection to the transmitter, only a low feedback rate is needed. Average error probability is analyzed in [20] when power and bit allocation are taken into consideration. The optimal bit allocation is obtained by exhausting all possible

constellations subject to a sum rate constraint. Several optimal designs of MIMO transceivers with decision feedback and bit loading are proposed in [21]. These optimal designs have similar performance when the channel state information is available to the transmitter. For the case of limited feedback, the use of identity precoder combined with feedback of only bit allocation is suggested therein as it intuitively requires less feedback. In earlier works of V-BLAST systems with bit allocation and a sum rate constraint [16] [18] [21], an exhaustive listing of all possible constellation combinations is used and thus a moderate feedback rate may be needed. Using capacity as a criterion statistical bit loading is considered in [22]. When the channel statistics are available to the transmitter but not the current state of the channel, the precoder can be designed according to the channel statistics. For example, optimal beamforming for maximizing average capacity of correlated channels has been designed in [23] [24]. There have also been a lot of research on designing statistical precoders of various design criteria for spatial multiplexing. Precoder for minimizing error probability are derived in [25] [26] [27]. The optimal precoder that minimizes the sum of mean squared error is given in [28]. A unified framework for solving a number of transceiver design problems for correlated channel is presented in [31]. The method can be applied when the cost belongs to a useful class of functions of subchannel mean squared error. In these works, a uniform bit allocation is assumed. Optimization of precoders with a fixed bit allocation vector have been considered in [29] [30].

In [45] the so called the BA system is proposed for the transmission over uncorrelated MIMO channels with feedback of bit allocation. For a given channel, a bit allocation vector is chosen from a codebook whose codewords (bit allocation vectors) satisfy the target transmission rate. The index of the selected codeword is feedback to the transmitter. The transmitter allocates bits to the modulation symbols according to the bit allocation vector and perform spatial multiplexing (precoding) using a precoder known to the transmitter and receiver a priori. In [45] it is shown that a uncorrelated channel the optimal precoder can be an arbitrarily unitary matrix for a uncorrelated channel and the BA system can achieve full diversity order.

In this thesis, we consider the transmission for Ricean channel (mean and covariance information) with feedback of bit allocation. Linear and decision feedback receiver be considered. We assume transmitter knows statistics of the correlated channel via a feedback link. We derive the optimal statistical precoder to minimize the bounds of BER averaged over the random channel. Simulations will show the BER performance is improved with optimal statistical precoder and detection order.

1.1

Outline

• Chapter 2: General system model is presented.

• Chapter 3: Previous works are reviewed in this chapter. In section 3.1 we re-view a spacial case of GTD based that is QR based system by P.P Vaidyanathan and C.C. Wang. Section 3.2 introduces a BER criterion and optimal unitary precoder for precoded spatial multiplexing system with infinite feedback rate proposed by S. Zhou and B. Li.

• Chapter 4: The proposed BA system over correlated channel for both co-variance feedback and mean feedback are presented in this chapter. The optimal bits and power allocation are derived in 4.1. optimal statistical precoders are designed in 4.2. Feedback of bits allocation using a codebook in 4.3. In 4.4, we show that BA system can achieve full diversity.

• Chapter 5: Simulation examples are presented in this chapter. • Chapter 6: A conclusion is given in this chapter.

1.2

Notations

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

H

F

s

x

q

r

s

ê

M

M

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 = I} M. 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}

matrix is G = (F†_H†_HF)−1_F†_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 = _N0kwikPt/M2. 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]. SERk = 4(1− 1 2bk/2)Q s 3σs2 k (2bk− 1)σ e2 k ! , (2.2) where Q(y) = √1 2π R_∞ y e−t 2_/2

dt, y_{≥ 0. Note that for the decision feedback}

receiver σsk

σ_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 Rb M −1_X k=0 bkBERk = 1 Rb M −1_X k=0 SERk (2.3)

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 + 1HwR 1/2 t , (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, where ar =  1 ej2πdrsin θr _{· · · e}j2πdr(Mr−1) sin θr T at =  1 ej2πdrsin θt _{· · · e}j2πdt(Mt−1) sin θt T

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)/2_X 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_{− 1}θt× 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

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 σ}1, 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: min F,G,B,bk PT = M X k=1 ck2bk[F†F]kk[GG†]kk. (3.3) constrainted byPM_k=1bk = Rband GHF−B = I. Where ck= N0₃ (Q−1(Pe(k)/4))2. The following are solution for GTD-based method to construct the transceiver matrices F,G,B [21]. F = [P]_Mt×M (3.4) G = (diag([R]_{M ×M}))−1[Q†]_{M ×Mr} (3.5) B = (diag([R]_{M ×M}))−1[R]_{M ×M − I} (3.6) bk = log2( ck M2 Rb/M₍ 1 QM k=1 )1/M)− log2ck+ log2([R]2kk) (3.7) 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= P0_MIM. 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 M M X l=1 log₂[GG†]ll− log2[GG†]kk+ Rb M (3.8)

(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.

3.2

Precoder System

3.2.1

System Model

Based on the general system model at chapter 2, the system in [4] 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 R_Mb bits. The power is also equally allocated for each symbols, Rs= P0_MIM. 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 (2.3). The

BER-base selection criterion is b

F = arg min

F_∈CFBER(F, H). (3.9)

To choose a precoder matrix by BER selection criterion, we need to compute the BER formula (2.3) for each precoder matrix in _CF. Therefore, 2B computations of (2.3) are required to complete BER selection criterion.

3.2.2

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

performance for finite-rate precoder feedback system. Assuming the singular

value decomposition of H = UΛV†_{, where U and V are respectively M}

r × 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 3.2.2. 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., |[Q]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 F = VM  QK1 0 0 I_{M −k1}  (3.10)

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 3.2.2.

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   QK1 0 0 0 I_{M −K1−K2} 0 0 0 QK2   (3.11) M 2 4 8 16 32 64 128 256 Γth 1.5 3 9.01 14.93 38.46 62.50 166.7 250.0 Table 3.1: Table of Γth M 2 4 8 16 32 64 128 256 Γth,l 0 0 0.579 0.247 0.326 0.264 0.330 0.271 Γth,h 0 0 7.621 13.72 37.46 61.50 165.7 249.0

Table 3.2: Table of Γth,l and Γth,h

3.2.3

Codebook construction

From [2] it is shown that the precoder codebook design problem can be related to Grassmanian subspace packing. Thus, in [4], 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, (3.12)

where _{k · k}F 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}Ntrn=1. 2. Iterate following steps until it converges.

(a) Assign Vn to one of the regions{Ri}2 B₋₁

i=0 using the rule

(b) For each region Ri, find the centroid as Fcentroid_i = arg min

F 1 Ntr X Vn∈Ri d2_c(Vn, F) (3.14) = arg min F 1 Ntr X Vn_∈Ri trace(IM − F†VnV†nF) (3.15) = arg max F trace(F †_{RF) (3.16)} where R is defined as R = 1 Ntr X Vn_∈Ri VnV†n. (3.17)

Let the eigendecomposition of R as

R = URΛRU†R. (3.18)

ΛR is a diagonal matrix whose diagonal elements are in nonincreasing order. It is easy to show that Fcentroid

i is a Mt× M matrix obtained

by keeping the first M columns of UR. (c) Set_CF ={Fcentroidi }2

B −1

i=1 . During each iteration, The codebook will be record if the minimum chordal distance of _CF

min

0≤i<j≤2B₋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.

Chapter 4

The proposed BA system

In this chapter we propose the design of statistical precoder for correlated MIMO channel with limited feedback of bit allocation . The proposed system will be termed a BA system. We will derive optimal unconstrained bit allocation and statistical precoders for both linear and decision feedback receiver for minimizing BER. We also show that proposed BA system can achieve full diversity.

4.1

Optimal Bit and Power Allocation

In this section, we will consider the BA system when there is no integer constraint on bit allocation. For a given precoder, we will derive the optimal bit allocation that minimizes the BER. We will see that the solution has the same form as that given in [21] in which the bit allocation is optimized for minimum transmission power. The BER obtained with optimal bit allocation will be used in the next section to design statistical precoders for minimum BER. The results derived in this chapter are valid at linear and decision feedback receivers for correlated channel in chapter 2. The optimal bit allocation will also be useful in chapter 5 for efficient codeword selection in practical applications where feedback rate is limited.

Assume the transmission rate is high and the number of bits loaded on the kth subchannel bk is large enough so that 1− 2−bk/2 ≈ 1 and 1 − 2−bk ≈ 1, then

the symbol error rate expression in (2.2) can be approximated by SERk≈ 4Q s 3 2bk σ2 sk σ2 ek ! (4.1) For the convenience of derivation, we define the function

f (y) = Q(1/√y), y≥ 0, (4.2)

The function f (y) is monotone increasing and it can be verified that f (y) is

convex for y ≤ 1/3 and concave for y ≥ 1/3. Using f(·), we have SERk ≈

4f (2bk_σ2

ek/(3σsk2 )). Therefore the BER in (2.3) can be written as

BER_≈ 4

Rb M −1_X

k=0

f 2bkσ_ek2 /(3σ_sk2 )
(4.3)

Let us consider the high SNR case in which the convexity of f (·) holds and the low SNR case in which the concavity of f (·) holds.

High SNR case

Assume SNR is large enough so that the arguments of f (_{·) are smaller than} 1/3 and the convexity of f (_{·) holds. Using the convexity of f(·), we have}

BER≈ 4 (Rb/M) 1 M M −1_X k=0 f 2bkσ_ek2 /(3σ_sk2 )
(4.4) ≥ _(R4 b/M) f ( 1 3M M −1_X k=0 2bk_σ2 ek/σ2sk) (4.5) ≥ 4 (Rb/M) f 2 Rb/M 3 ( M −1_Y k=0 σ_e2_k)1/M( M −1_Y k=0 1 σ2 sk )1/M ! (4.6) ≥ 4 (Rb/M) f 2 Rb/M 3Pt/M ( M −1_Y k=0 σ_ek2 )1/M ! (4.7) , BER0 (4.8)

The second inequality is obtained by using the fact that PM −1_i=0 bi = Rb and the AM-GM (arithmetic mean-geometric mean) inequality and also using the mono-tone increasing property of f (·). We can obtain the third inequality using the

AM-GM inequality (QM −1_k=0 σ2

sk)1/M ≤ 1/M

P_{M −1}

k=0 σsk2 = Pt/M and the monotone increasing property of f (·). Notice that the lower bound BER0 in (4.8) is in-dependent of bit allocation and power allocation. The optimal bit allocation and power allocation are such that the three inequalities in (4.8) become equal-ities. Due to the convexity of f (_{·), the first equality in (4.5) holds if and only if} 2bkσ2

ek/σ 2

sk are of the same value for all k. The same set of conditions is also neces-sary and sufficient for equality to hold in the second inequality as f (_{·) is monotone} increasing. The third equality is achieved if σ2

s0 = σs12 = . . . = σ2sM −1 = Pt/M. The optimal bit allocation for minimizing the BER is thus

bk = 1 M M −1_X l=0 log₂(σ_e2_l)_{− log2}(σ2_ek) + Rb M (4.9)

With the above optimal bit allocation and uniform power allocation, the lower bound BER0 is achieved. We can see that the symbols with smaller error vari-ances σ2

ek are allocated with more bits. When bits are allocated as in (4.9), 2bk_σ2

ek/σsk2 are the same for all k. This means the symbol error rates are equal-ized for all transmitted symbols. The bit allocation formula in (4.9), derived using the criterion of minimum BER, has the same form as that designed for minimum transmission power in [21].

Low SNR case

Assume SNR is low enough so that the arguments of f (_{·) are larger than 1/3} and the concavity of f (_{·) holds.}

BER≈ 4 (Rb/M) 1 M M −1_X k=0 f 2bkσ2_ek/(3σ_sk2 )
≤ 4 (Rb/M) f 1 3M M −1_X k=0 2bkσ2_ek/σ_sk2 ! (4.10) The inequality follows from the concavity of f (·). Similar to the high SNR case, the quantity on the right hand side is minimized if uniform power allocation is used and bit allocation is chosen according to (4.9). In this case the upper bound on the right hand side is equal to BER0 and at the same time the inequality in (4.10) becomes an equality, ie., BER _{≈ BER}0

Summarizing, for both high and low SNR regions the BER with optimal bit allocation and uniform power loading is approximately BER0. The results can be used for both linear and decision feedback receivers. The quantity BER0 is different for different types of receivers. We will use BER0 in the next section to derive optimal statistical precoder of BA system over correlated channel.

4.2

Design of statistical precoders for minimum

BER

In this section we consider the design of optimal statistical precoders over cor-related channels model described in chapter 2. Assume Mr > M. To consider

the average BER performance, we average BER0 computed in (4.8) over the

random channel H, BER0 = E[BER0] = E " 4 Rb/M f c M −1_Y k=0 σ_e2/M_k !# (4.11) where we have used c = _{3(Pt/M )}2Rb/M . To simplify the expression further, we define the geometric mean function

h(y) = M −1_Y i=0 y1/M_i (4.12) y = y0 y1 · · · yM −1 T

and yi > 0. Let yi = cσei2, then BER0 = _(Rb4_{/M )}f (h(y)). To analyze BER0, we first derive the Hessian matrix of f (h(y)), which is an M _{× M matrix with the (i, j)th entry given by [H}ess]i,j = ∂2f (h(y))/∂yi∂yj, for0_{≤ i, j < M. We can verify that [H}ess]ij is given by

[Hess]i,j =  0.5/M2_f′_(h(y))y−1 i yj−1(1− h(y))) ,i6= j 0.5/M2_f′_(h(y))y−2 i (1− (1 + 2M)h(y)), .i = j (4.13) It is derive in appendix. It is known that [37] a function is convex (concave) if and only if the Hessian matrix is positive (negative) semi definite. In the following we discuss the behavior of BER0 for the high and low SNR cases.

Consider Pt/N0 ≫ 1 such that the arguments of f(·) are much smaller than unity, ie., h(y) = _{3(Pt/M )}2Rb/M QM −1_k=0 σe2/Mk ≪ 1. We can approximate the ith diagonal element of the Hessian matrix in (4.13) as 1/(2M2_)f′_(h(y))y−2

i . Defining the M×1 vector u with ith element ui = 1/yi, we have Hess ≈ 1/(2M2)f′(h(y))uuT, which is a positive semidefinite matrix. Applying Jensens inequality, we get E[f (h(y))] & f (h(E[y])). Therefore we have

E[BER0] = E " 4 Rb/M f (c M −1_Y k=0 σ_ek2/M) # ≥ 4 Rb/M f c M −1_Y k=0 ¯ σ_ek2/M ! , BERbd (4.14) where ¯σ2 ek = E[σ 2/M

ek ] is the kth error variance averaged over the channel H. The right hand side BERbd is a lower bound of BER0.

Low SNR case

A property of f (h(y)) that is useful for studying E[BER0] in low SNR region is presented in the following lemma.

Lemma 1. Let f (x) and h(y) be as defined in (4.2) and (4.12), respectively. Then the composite function f (h(y)) for yi > 0 is concave when h(y)≥ 1/3. Proof. The Hessian matrix in (4.13) can be rewritten as

Hess = 1/M2f′(h(y))h(y)[0.5(1/h(y)− 1)uuT − MD]

, where u is M _{× 1 with ith element u}i = 1/yi, and D is a diagonal matrix with [D]ii = 1/yi2. We examine the quadratic form vTHv for an arbitrary M × 1 vector v. It can be rearranged as

vT_{Hv =} 1

M2f′(h(y))h(y)[(v

T_uuT_{v− Mv}T_{Dv) + 0.5(1/h(y)− 3)v}T_uuT_v].

The first term in the bracket vT_uuT_{v− Mv}T_{Dv is equal to (}PM −1

k=0 vkuk)2 − MPM −1_k=0 v2

ku2k, which is always non positive due to Cauchy-Schwartz inequality. The second term in the bracket, equal to 0.5(1/h(y)_{− 3)(}PM −1_k=0 vkuk)2, it is non positive if (1/h(y) _{− 3) ≤ 0 ie., h(y) ≥ 1/3. Therefore we can conclude that}

when h(y) ≥ 1/3, the Hessian matrix of f(h(y)) is negative semidefinite and

thus f (h(y)) is concave. 

The above lemma means that BER0 is concave in σ2ei when

h(y) = 2 Rb/M 3(Pt/M) M −1_Y k=0 σ2 ek !1/M ≥ 1/3, (4.15)

which holds in low SNR case, ie., small Pt/N0. When f (h(y)) is concave, we can apply Jensens inequality E[f (h(y))] _{≤ f(h(E[y])) to obtain}

E[BER0]≤ 4 Rb/M f c M −1_Y k=0 ¯ σ2/M ek ! , BERbd. (4.16)

Now BERbdbecomes an upper bound of BER0. In both high SNR and low SNR

regions, we would like to have the bound BERbd minimized, which is discussed for linear receivers and decision feedback receivers for Ricean channel.

4.2.1

Optimal precoders design with Ricean channel

Suppose A is a Mr× Mt matrix each row of which is independently drawn from a Mt-variate normal distribution with zero mean each row of A is independently and let the ith column of A†_{be g}

i , then the autocorrelation matrix of gi is equal to Rt. It is known that A†A =

P_Mr−1

i=0 gigi† has a complex Wishart distribution with Mrdegrees of freedom, denoted asWMt(Rt, Mr) [38]. When B has a Wishart distribution, we say B−1 _{has inverse Wishart distribution. For Ricean channel} model, the channel be considered as

H = r K K + 1Hsp+ r 1 K + 1HwR 1/2 t . (4.17)

It is known H†_{H has a complex non-central Wishart distributionN W}

Mt(Rt, M, Mr)

[39], where M =q K

K+1Hsp, is called non-centrality parameter matrix means the expectation of H. Mr is degree of freedom and Rt is the autocorrelation matrix of HwR1/2t . This non-central Wishart distribution can be approximated by a

Wishart distribution [39]

N WMt(Rt, M, Mr)∼ WMt( bRt, Mr), (4.18)

where bRt= Rt+ M†M/Mr.

Linear receiver We can obtain σ2ek by averaging the error covariance matrix Re= N0(F†H†HF)−1 over the channel. If F is nonsingular matrix then the ma-trix F†_H†_{HF is W}

M(F†RbtF, Mr) and so the matrix R−1e = 1/N0F†H†HF has a Wishart distribution _WM(N0−1F†RbtF, Mr). Then Re has an inverse Wishart distribution. It has been shown in [43] that when a matrix B is Wishart distri-bution_Wp(A, r) with r > p , then E[B−1] = 1/(r− p)A−1. Using this result and assuming Mr> M, Re= E[Re] is given by

Re= N0

Mr− M

(F†RbtF)−1. (4.19)

Let the eigen decomposition of bRt be bRt = bUtΛbtUb†t, where bΛt is a diagonal matrix and the diagonal elementsλt,i are the eigen value of bRt. Let the diagonal elements of bΛt be ordered such that λt,0 ≥ λt,1 ≥ . . . ≥ λt,Mt−1 and assume λ_t,Mt−1 > 0

Theorem 1. For the linear receiver, the BER bound BERbd in (4.14) satisfies

BERbd ≥ BERbd,lin, whereBERbd,lin = 4M Rb Q   v u u t 3Pt/M 2Rb/MN₀(Mr− M) M −1_Y i=0 λ1/M_t,i   (4.20) The inequality becomes an equality when F = bUt,M, where bUt,M is the submatrix of bUt that consists of the first M column vectors of bUt.

Proof. Majorization theorem [36] will be used to prove the theorem. For com-pleteness, some related definitions are given below.

(1)Given a sequence a[0], a[1], . . . , a[M −1],the notation a[k] refers to the permuted sequence such that a[0] ≥ a[1] ≥ . . . ≥ a[M −1]. (2) Given two real vectors a =  a0 a1 · · · aM −1

T

and b =  b0 b1 · · · bM −1 T

, we say that a majorizes b if the following two conditions are satisfied: PM −1_k=0 ak =

P_{M −1} k=0 bk

and Pn_k=0a[k] ≥Pn_k=0b[k] , 0≤ n ≤ M − 2 . Let g(y) be a real-valued function of a real vector y. We say that g(y) is Schur-concave if g(a)≤ g(b) whenever a majorizes b.

The function g(x) = QM −1_i=0 xi, for xi > 0 is known to be Schur concave [41]. As σ2

ei are the diagonal elements of Re, the sequence {σ2ei}M −1i=0 is majorized by _{λi(Re)}M −1i=0 , where we have used λi(A) to denote the i-th largest eigen value of A. So QM −1_i=0 σ2

ei ≥

Q_{M −1}

i=0 λi(Re) and the equality holds when Re is a diagonal matrix. The matrix R−1_e is the inverse of Re, their eigen val-ues are related by λi(Re) = 1/λM −1−i(R−1e ). As R

−1

e = Mr−MN0 F†RbtF and F is unitary, we can apply the Poincare separation theorem to bound the eigen values of R−1_e using the eigen values of bRt. Poincare separation theorem says λi(B) ≥ λi(C†BC),i = 0, 1, . . . , r− 1, for any n × n Hermitian matrix B and

any n × r unitary matrix C with C†_{C = I}

r. Using this theorem, we have

QM −1 i=0 Mr−MN0 λi( bRt)≥ QM −1 i=0 Mr−MN0 λi(R −1 e ). Thus M −1_Y i=0 σ2_{ei ≥} M −1_Y i=0 λi(Re) = M −1_Y i=0 1 λi(R−1e ) ≥ M −1_Y i=0 N0 Mr− M 1 λi( bRt) (4.21) In (4.20),The lower bound QM −1_{i=0 Mr−M}N0 _λi(Rt)1 can be achieved by choosing F = bUt,M. Using the above inequality and the monotone increasing property of f (·), we can establish the inequality in (4.20). Therefore, to minimize the BER

bound BERbd the optimal precoder is F = bUt,M. 

detector Mr r S

G

B

Figure 4.1: Block diagram of the decision feedback receiver.

Decision feedback receiver To consider the precoder design for a decision

QR decomposition of HF [21] [28]. This corresponds to the case of a reverse detection ordering. Let the QR decomposition of HF be QR, where Q is an Mr×M unitary matrix and R is an M ×M upper triangular matrix with diagonal element [R]ii= rii. The feedforward matrix G and feedback matrix B are given, respectively, by [32]

G = (r−1₀₀ r−1₁₁ · · · r−1

M −1,M−1)Q† (4.22)

B = (r00−1 r−111 · · · r−1_{M −1,M−1})R− IM (4.23) Assuming there is no error propagation, the kth subchannel error ek = bsk − sk has variance σ2

ek = N0r −2

kk, k = 0, 1, . . . , M − 1. The average error variance is σ2ek = N0E[r

−2

kk]. The value E[rkk−2] as been shown to be related to the Cholesky decomposition of F†_Rb

tF in [28]. The result is summarized in the following lemma. Lemma 2. [28] When K = 0, H = HwR1/2t , the following result was derived. Let the Cholesky decomposition of F†_R

tF be LDL† where L is a lower triangular matrix with unity diagonal elements and D is diagonal. For Mr > M, E[r−2_kk] = d−1_kk/(Mr− k − 1), for k = 0, 1, . . . , M − 1 where dkk is the kth diagonal element of D.

Using lemma 2 and the approximation in (4.18), the results in Lemma 2 allows us to establish the following bound for BERbd

Theorem 2. For the decision feedback receiver with Mr > M, the BER bound BERbd in (4.14) satisfies BERbd ≥ BERbd,df, BERbd,df = 4M Rb Q( v u u t 3Pt/M 2Rb/MN₀ M −1_Y k=0 (Mr− k − 1)1/M M −1_Y k=0 λ1/M_t,k ), (4.24) where λt,k = λk(F†RbtF). The inequality becomes an equality when F = bUt,M, where bUt,M is the submatrix of bUt that consists of the first M column vectors of

b Ut.

Proof. Using Lemma 2, we can obtain σ2 ek = N0d−1kk/(Mr − k − 1) . Thus Q_{M −1} k=0 σ2ek = Q_{M −1} k=0 N0d−1kk/(Mr−k−1). Note thatQM −1k=0 dkk=QM −1k=0 λk(F†RbtF). The BERbd in(4.14) can be expressed as

BERbd = 4M Rb f cN0 M −1_Y k=0 (Mr− k − 1)−1/M M −1_Y k=0 λ−1/M_t,k (F†RbtF) ! (4.25) Applying the Poincare separation theorem (also stated in the proof of Theorem 1), we have the inequality

BERbd = 4M Rb f cN0 M −1_Y k=0 (Mr− k − 1)−1/M M −1_Y k=0 λ−1/M_k (F†RbtF) ! (4.26) ≥ 4M Rb f cN0 M −1_Y k=0 (Mr− k − 1)−1/M M −1_Y k=0 λ−1/M_t,k ! (4.27) = BERbd,df (4.28)

The lower bound BERbd,df can be achieved when F = bUt,M 

Rt= I case

In this special case, we assume no correlation at transmitter and receiver. Channel

is considered as H = q K K+1Hsp+ q 1 K+1Hw, where Hsp , ara T t, ar and at are LOS array response at transmitter and receiver described in chapter 2. Let H† spHsp = VΛV†, we have b Rt= INt+ cH†spHsp = V(INt + cΛ)V†, (4.29) where c = K (K+1)Mr. Note that H†_spHsp = a∗ta†raraTt =kark2katk2aet∗aetT = λ0vv†,

where λ0 = kark2katk2, eat = _kaat_tk, v = eat∗, we can see that λ0 is the only nonzero eigenvalue, the other eigenvalues are 0 and the eigenspaces of λ0 and 0 are othogonal. Because Hsp = araTt = [ar0at ar1at. . . arNrat]

T_{, we can see} that when we take the hermitian of the first row and normalize, it is equal to ea∗

t. When Rt= I, the first column of optimal precoder F is the hermitian of the first

row of Hsp with normalization, the other columns of F can be arbitrarily chosen, except for the restriction that the columns of F are orthonormal. That is

F = h_katkat f2 · · · fM i

, (4.30)

where f2,· · · fM are arbitrarily vectors such that F is unitary. Such a precoder has been shown in [11] to maximize capacity of a beamforming system.

No line of sight (K = 0) case

In this case, we consider the special case that K = 0 such that H in (4.17) becomes to

H = HwR1/2t

it is known H†_{H has a complex Wishart distribution with M}

r degree of freedom, denoted as _WMt(Rt, Mr) instead of complex non-central Wishart distribution so we don’t need take approximate. Let the eigen decomposition of Rt be Rt = UtΛtUt, where Λt is a diagonal matrix and the diagonal elementsλt,i are the eigen value of Rt. Let the diagonal elements of Λt be ordered such that λt,0 ≥ λt,1 ≥ . . . ≥ λt,Mt−1 and assume λt,Mt−1 > 0.

Linear receiver Using the property of Wishart distribution , the matrix

F†_H†_{HF is W} M(F†RtF, Mr) and we have Re = N0 Mr− M (F†_R tF)−1 (4.31)

Using the proof of theorem 1 , we have M −1_Y i=0 σ2ei ≥ M −1_Y i=0 λi(Re) = M −1_Y i=0 1 λi(R−1e ) ≥ M −1_Y i=0 N0 Mr− M 1 λi(Rt) ,

thus BERbd = 4M Rb f (c M −1_Y k=0 σ2/M_ek ) (4.32) ≥ 4M_R b f c M −1_Y k=0 λ1/M_i (Re) ! (4.33) ≥ 4M Rb f c N0 Mr− M M −1_Y k=0 1 λ1/M_i (Rt) ! (4.34) = BERbd,lin. (4.35)

When F = Ut,M , the lower bound BERbd,lin can be achieved .

Decision feedback receiver We can use lemma 2 and theorem 2. Let the

Cholesky decomposition of F†_R tF be LDL†, thus we have BERbd = 4M Rb f cN0 M −1_Y k=0 (Mr− k − 1)−1/M M −1_Y k=0 λ−1/M_k (F†RtF) ! (4.36) ≥ 4M Rb f cN0 M −1_Y k=0 (Mr− k − 1)−1/M M −1_Y k=0 λ−1/M_t,k ! (4.37) = BERbd,df (4.38)

With the same result as mean feedback, then F = Ut,M the bound BERbd can

be minimized.

4.3

Feedback of bit allocation

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 min_b∈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 ∈ C}

b, the kth subchannel symbol error rate associated with b is

SERk ≈ 4Q( s 3 2bk σ2 sk σ2 ek ) = 4Q( s 3 2b∗ k σ2 sk σ2 ek 2b∗_k−bk ) (4.39)

As shown in Sec. 4.1 the optimal bit allocation b∗_{equalizes the quantity 3σ}2 sk/(2b

∗ kσ2

ek).

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_{∈ C}b 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 symbols mapping _M Mt s x F Bits to_symbols mapping M M_t x F0 s0

bit stream bit stream

(a) (b)

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 M}

t× 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(M}

t, 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 Mt−1_X

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 _C′

b 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, . . . , M}

t − 1, the transmitter removes the ith 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′. 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 _C′

b 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 (2b_{ki − 1)kw}

ik2

, f or ki ∈ S, (4.41)

where S = _{{j : b}j > 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

B

_{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−1_e (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′R_e(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−1 e

By using this algorithm, we don’t need to take matrix inverse after each detection so we can successfully reduce the complexity.

4.4

Diversity Gain of BA system [45]

we show that the BA system can achieve diversity order MrMt for a system with Mr receive antennas and Mt transmit antennas if the codebook is prop-erly designed and has at least Mt codewords. Let the initial precoder F′ be an

Mt × Mt unitary matrix (F′ = F and M = Mt). The number of bits to be

transmitted in each channel use is Rb, which is distributed among M symbols (M _{≤ min(M}t, Mr)). The augmented bit allocation vector b′ is of size Mt× 1. It has at most M nonzero entries and PMt−1_i=0 b′

i = Rb. Suppose the bit allocation codebook is _C′

b. The minimum achievable BER is

BERmin(H) = min

b′_∈C′ b

BER(b′, H), (4.44)

where BER(b′_{, H) is the BER in (2.3) Assume the bit allocation codebook C}′ b contains the set of codewords

C∗

b ={Rbe0, Rbe1,· · · , RbeMt−1}, (4.45) where ei are standard vectors of size Mt× 1, i.e., [ei]i = 1 and [ei]j = 0 for j 6= i. The following lemma shows that the BA system can achieve full diversity order

using the bit allocation vectors in C∗

b. Therefore to achieve a diversity order of MrMt we can use a codebook of size Mt, which requires only log2Mt feedback bits.

Lemma 3. For a finite-rate feedback MIMO channel with Mr receive antennas

and Mt transmit antennas, the BA system with an Mt× Mt augmented unitary precoder F′ _{achieves diversity order M}

rMt if the bit allocation codebook Cb′ con-tains the Mt vectors in (4.44).

Proof. As _C∗

b is a subset of Cb′, we have

BERmin(H) = min

b′_∈C′ b BER(b′, H)≤ min b′_∈C∗ b BER(b′, H). (4.46)

The average BER is bounded by

BER≤ E[ min

b′_∈C∗ b

BER(b′, H)]. When the bit allocation b′ _{is chosen from C}∗

b,all the Rb bits are allocated to the same symbol and this system becomes a beamforming system. For example,

when b′ _{= [ R}

b 0 · · · 0 ]T, the beamforming vector is the 0-th column of F′_{. When we choose b}′ _{∈ C}∗

b to minimize the BER, we are actually choosing the best beamforming vector from the columns of F′ _{to maximize the received} SNR. In other words, the equivalent codebook of beamforming vectors is _Cf = {f′

0, f1′,· · · , fMt−1′ }, where fi′ is the i-th column of F′. From [40], we know such a beamforming system has diversity order equal to MrMt if the span ofCf is equal to CMt _{. Because F}′ _{is an M}

t× Mt unitary matrix, the span of Cf is the same as CMt_{.Therefore the BA system has diversity order M}

rMt when codebook Cb′

Chapter 5

Simulations

In our simulations, the channel is of the form H = r K K + 1Hsp+ r 1 K + 1HwR 1/2

t for Ricean channel.

and

H = HwR1/2t for no line of sight. and

H = Hw for uncorrelated channel.

Consider different channel case as following Channel I Uncorrelated channel.

Channel II No line of sight with low correlation for dt = 2, θt= 40◦. Channel III No line of sight with high correlation for dt= 2, θt = 8◦.

Channel IV Ricean channel with low correlation for dt = 2, θt= 40◦, dr = 1, θr = 20◦, K = 5.

Channel V Ricean channel with high correlation for dt= 1, θt= 20◦, dr = 1, θr = 10◦, K = 3.

We have used 106 _{channel realizations in the Monte Carlo simulations. The} error rates are computed using (2.3) for both linear and decision feedback re-ceivers. For the decision feedback receiver, the detection order is determined

using the criterion of maximizing the rate-normalized SNRs mentioned in Sec 4.3. Antennas with spacing dt, dr and plane-wave span an angular spread of θt, θr at transmitter and receiver respectively.

Example 1. Distribution of bit allocation vectors.

In this example, the Channel I is considered. The number of receive antennas Mr is 5, and the number of transmit antennas Mtis 4. we compute the empirical distribution of bit allocation vectors. For a given channel realization, the best bit allocation vector in the codebook is chosen using the BER criterion. The number of bits transmitted per channel use is Rb = 12 and the number of substreams that the transmitter and receiver can process is M = 4. The corresponding opti-mal precoder F is the identity matrix and the receiver is linear. The number of possible integer bit allocation vectors is 455. We include in the codebook all 455 integer bit allocation vectors. Fig. 5.1(a) shows the distribution of the bit alloca-tion vectors, where the indexes of the vectors are ordered so that the probabilities are in decreasing order. The cdf (cumulative distribution function) is shown in Fig 5.1(b). We can see that some bit allocation vectors are far more probable than others. The probability of the 53 most probable bit allocation vectors is more than 99%. The distribution of the bit allocation vectors is highly skewed, rather than uniform. In all following examples with quantize bit allocation, we will choose the most probable 2B _{bit allocation vectors obtained in experiments} like this example and use them as codewords when the number of feedback bits is B.

0 100 200 300 400 0 0.02 0.04 0.06 0.08 0.1 0.12 k (a) 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 k (b)

Figure 5.1: (a) Probability mass function of the bit allocation vectors for Channel I, Mr = 5, Mt = 4, M = 4, and Rb = 12; (b) corresponding cumulative distribution function.

Example 2. Precoder and distribution of bit allocation.

The correlated Channel II with zero mean is considered for Mr = 4, Mt= 5, M = 4. The number of bits transmitted per channel use is Rb = 8. We condider two type of the precoder F = Ut,M and F =



I 0 used, the receiver is linear. The number of possible integer bit allocation vector is 460. The codebook contains all 460 integer bit allocation vectors. Fig. 5.2(a) shows the distribution of the bit allocation vectors. The cumulative distribution function (cdf) is shown in Fig.

0 50 100 150 200 250 300 350 400 450 0 0.02 0.04 0.06 0.08 0.1 0.12 k F=U t,m F=[I M 0] T (a) 0 50 100 150 200 250 300 350 400 450 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k F=U t,m F=[I M 0] T (b)

Figure 5.2: (a) Probability mass function of bit allocation vectors for Channel II, Mr = 4, Mt = 5, M = 4 andRb = 8; (b) Corresponding CDF.

5.2(b). From Fig. 5.2(a) we can see when F = Ut,M is used, the distribution of bit allocation vectors is more concentrated.

Example 3. BER bound.

In Fig. 5.3(a), Channel II is used. Mr = 4, Mt = 5, M = 3, Rb = 10, the

precoder is F′ _{= U}

t. We show the BER bounds BERbd,lin andBERbd,df. We

have also computed BER0 in (4.8) over 106 channels for a linear receiver and for a decision feedback receiver. The results are called, respectively, BER0,lin and BER0,df. The gap between BERbd,lin and BERbd,df is around 3.5dB. We can see that the curve BERbd,df is an upper bound for BER0,df in low SNR and a lower bound for BER0,df in high SNR, consistent with what we have shown in Sec. 4.2. The same can be observed for the case of linear receiver. In Fig. 5.3(b) Channel IV with both mean and covariance information is used. Mr = 5, Mt= 4, M = 4, Rb = 12. We use the approximation in (4.18) and choose F′ = bUt. In. 5.4 shows the same set of curves. We can have conclusions similar to those for correlated Channel II with zero mean.

5 10 15 20 25 30 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER BER 0,lin BER 0,df BER bd,lin BER bd,df (a) 8 5 10 15 20 25 30 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER 0 BER₀,_lin BER₀,_df BER_bd,_lin BER_bd,_df (b)

Figure 5.3: (a) BER bound for Mr = 4, Mt= 5, M = 3 and Rb = 10 for Channel II (b) BER bound for Mr = 5,Mt= 4,M = 4 and Rb = 12 for Channel IV

Example 4. BER for different feedback bits.

In Fig. 5.4(a), Mr = 4, Mt = 5, M = 4, Rb = 8, Channel II is considered, the precoder is F′ _{= U}

t. We shows the BER performance of the BA system

for different number of feedback bits. The codewords are selected to minimize BER. The performance is shown for both linear and decision feedback receivers for different number of feedback bits. The BER is improved when the number of feedback bits B increases. We can see that BER of B = 5 is close to that of B = 9, in which case all the integer bit allocation codewords are used. Observe that the curves correspond to B = 7 and B = 9, are indistinguishable in the figure. We can understand this by examining the distribution plot in Fig. 5.2

The cdf is very close to one for k ≥ 150. When we increase B from 7 to 8

to 9, the added codewords are almost never chosen so the performance has no improvement. Fig. 5.4(b) also shows BER of the BA system when Channel V is considered with Mr = 5, Mt= 4, M = 4 and Rb = 12. The precoder is chosen as F′ _{= bU}

t. For the case B = 9 which considers all integer bit allocation codewords, the gain of the decision feedback receiver over the linear receiver is around 3.5dB, similar to the gap between BERbd,df and BERbd,lin observed in Fig. 5.3(a).

0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 Po/No(dB) BER lin B=3 lin B=4 lin B=5 lin B=7 lin B=9 df B=3 df B=4 df B=5 df B=7 df B=9 (a) 8 10 12 14 16 18 20 22 24 26 28 10−6 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER lin B=3 lin B=4 lin B=5 lin B=9 df B=3 df B=4 df B=5 df B=9 (b)

Figure 5.4: (a) Different feebback bits with Mr = 4, Mt = 5, M = 4, Rb = 8 for Channel II (b) Different feebback bits with Mr = 5, Mt= 4, M = 4, Rb = 12 for Channel V

Example 5. BER for different Precoders.

In Fig. 5.5(a), Mr = 4, Mt = 5, M = 4, Rb = 8, B = 9 and channel III

be considered. The BER plots are given for four different types of Mt × Mt precoders and decision feedback at receiver. (1) the identity matrix, (2) the normalized DFT matrix , (3) the DCT matrix and (4) F = Ut . We can see that Uthas the best performance among. Fig. 5.5(b) shows the same set of curves for four precoders with linear receiver. Channel IV be considered. It has the same result as covariance feedback case that optimal precoder is F = bUt.

6 8 10 12 14 16 18 20 22 10−5 10−4 10−3 10−2 10−1 100 Po/No(dB) BER F=identity F=DFT F=DCT F=U_t (a) 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 Po/No(dB) BER F=identity F=DFT F=DCT F=U t (b)

Figure 5.5: (a) BER for different precoder Mr = 4, Mt = 5, M = 4, Rb = 8 for Channel III. (b) BER for different precoder Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel IV.

Example 6. BER for different Cb.

In Fig. 5.6, Mr = 5, Mt = 4, M = 4, Rb = 12 and F = Ut, Channel II and linear receiver are considered. In this case we show BER for two codebook, one trained using H and one trained using HF. Even though the precoder is chosen as F = Ut, the performance of the codebook trained using HF is better than the other for about 1dB for the same feedback rate. So we can conclude codebook training is important for system performance.

12 14 16 18 20 22 24 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER trained using H, B=3 trained using H, B=5 trained using HF, B=3 trained using HF, B=5 5.6 Figure 5.6: BER with different _Cb, Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel II

Example 7. BER for MMSE and ZF receivers.

In this case, Mr = 5, Mt = 4, M = 4, Rb = 12 and F = IMt, Channel I is considered. We show the BER performance of MMSE and ZF receivers with linear and decision feedback receivers. Fig. 5.7(a) is linear receiver. In each case, the codebook is trained based the channel at receiver. From Fig. 5.7(a) we can see the ZF receiver is close to that of MMSE receiver. Fig. 5.7(b) show the two curves again when the receiver has decision feedback. We can draw conclusions similar to that for the linear receiver case.

8 10 12 14 16 18 20 22 24 26 10−6 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER MMSE B=4 MMSE B=5 MMSE B=9 ZF B=4 ZF B=5 ZF B=9 (a) 8 10 12 14 16 18 20 22 24 26 10−6 10−5 10−4 10−3 10−2 10−1 100 P t/N0 BER MMSE B=4 MMSE B=5 MMSE B=9 ZF B=4 ZF B=5 ZF B=9 (b)

Figure 5.7: (a) BER for linear receiver, Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel I (b) BER for decision feedback receiver for Channel I.