多輸入多輸出系統之位元配置有限回饋

(1)

國

立

交

通

大

學

電控工程研究所

碩士論文

多輸入多輸出系統之位元配置有限回饋

MIMO Systems with Limited Feedback of Bit Allocation

研究生：鄭人予

指導教授：林源倍教授

(2)

多輸入多輸出系統之位元有限回饋

MIMO Systems with Limited Feedback of Bit Allocation

研究生：鄭人予 Student：Panna Jeng 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

A Thesis

Submitted to Institute of Electrical 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

June 2010

(3)

i

多輸入多輸出系統之位元配置有限回饋

學生：鄭人予

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

本論文包含兩個部分，在第一個部分我們提出一個有限回饋

位元配置的多輸入多輸出系統。我們會證明此系統可以達到

全多樣性。在傳輸速率固定的假設下，我們推算出能達到最

小錯誤率的最佳位元配置，並證明達到最小錯誤率的位元配

置也是使用最低傳輸能量的位元配置。模擬結果顯示我們所

提出的系統可以使用較少的回饋位元達到低錯誤率。在第二

個部分我們針對傳統單階預先編碼器的有限回饋系統設計

低複雜度的編碼器選擇準則。我們也提出一個可以降低複雜

度的二階預先編碼器系統。模擬結果會展示出我們所設計的

系統的可用性。

(4)

MIMO systems with limited feedback of

bit allocation

Panna Jeng

Advisor: Dr. Yuan-Pei Lin

Department of Electrical and Control Engineering

National Chiao Tung University

January 4, 2011

Abstract

In this thesis we first proposed a limited feedback system which sends back only the bit allocation (BA) information. The system will be termed a BA system. we show that the proposed BA system can achieve full di-versity order. we will also derive the optimal bit allocation for minimum bit error rate when the transmission rate is given. Secondly, we develop low-complexity selection criteria for conventional one-step precoder system which feedbacks only the precoder information. A two-step system is pro-posed to reduce the number of searches. In simulations, the usefulness of the proposed systems will be demonstrated.

(5)

iii

誌謝

在電控所就讀的這段期間，不僅在學術上可以深入學習，

在人生經驗上也有所成長。特別感謝林源倍老師的悉心

教導，使我在這段重要的人生歷程中學到很多東西。感

謝林清安老師和蔡尚澕老師能撥冗來參加我的碩士論

文口試，並給予寶貴的建議，使我的論文能更加完善。

感謝我家人的支持與鼓勵。也感謝實驗室的學長姐與夥

伴：建樟、孟良、鈞麟、芳儀、素卿、懿德、士軒、虹

君、君維、士傑，感謝你們在學業上的討論、建議和實

驗室溫馨氣氛的經營。另外感謝在研究所期間給予我鼓

勵和幫助的同學朋友們。最後感謝芳霖多年的陪伴與支

持。沒有你們的協助，我是無法學習到如此多寶貴的知

識和留下如此多珍貴的回憶，由衷的感謝你們。

(6)

List of Figures

2.1 MIMO system with limited feedback . . . 6

4.1 The transmitter of the BA system with (a) precoder F, and (b)

augmented precoder F′ _{. . . .} ₂₂

4.2 MIMO wireless system with Mt transmit antennas and Mr receive

antennas . . . 35 5.1 Two step transmitter . . . 44 6.1 Probability mass function of the bit allocation vectors, where the

indexes of the vectors are ordered so that the probabilities are in nonincreasing order for Mr = 4, Mt = 4, M = 4 and Rb = 8 . . . . 48

6.2 Corresponding CDF . . . 48 6.3 Probability mass function of the bit allocation vectors, where the

indexes of the vectors are ordered so that the probabilities are in nonincreasing order for Mr = 3, Mt = 4, M = 3 and Rb = 8 . . . . 49

6.4 Corresponding CDF . . . 50 6.5 Bit error rate of BA system for Mr = 4, Mt= 4, M = 4 and Rb = 8 51

6.6 Bit error rate of BA system for Mr = 2, Mt= 4, M = 2 and Rb = 4 52

6.9 Comparison of BER for Mr = 4, Mt = 4, M = 4 and Rb = 8 . . . 55

(9)

6.13 BER of the BA system with different precoders for Mr = 3, Mt=

3, M = 3 and Rb = 8 . . . 59

6.14 BER of the BA system with different precoders for Mr = 3, Mt=

4, M = 3 and Rb = 8 . . . 60

6.15 BER of the efficient method proposed in Sec. 4.5 for Mr = 4,

Mt= 4, M = 4 and Rb = 8 . . . 61

6.16 BER of different selection criterion for Mr = 4, Mt = 4, M = 4

and Rb = 8 . . . 62

6.17 BER of different selection criterion for Mr = 4, Mt = 5, M = 4

and Rb = 8 . . . 63

6.18 BER of different selection criterion of FQ for Mr = 4, Mt = 5,

M = 4 and Rb = 8 . . . 64

6.19 BER of different selection criterion of FV for Mr4 =, Mt = 5,

M = 4 and Rb = 8 . . . 65

6.20 BER comparison of one-step and two-steps systems for Mr = 4,

(10)

List of Tables

3.1 Table of Γth . . . 12

(11)

Chapter 1 Introduction

Multiple input multiple output (MIMO) systems with limited feedback have at-tracted great interest recently [1–4]. These practical systems can improve perfor-mance metric such as transmission rate or error rate by sending limited amount of information bits through a reverse channel to transmitter [1]. It is gener-ally assumed that there is no channel state information at the transmitter and only the receiver has the perfect channel knowledge. To obtain complete channel knowledge at the transmitter may be unrealistic since it requires infinite number of bits. In practice the reverse channel can transmit only finite amount of bits and it is desirable to have feedback rate as low as possible.

Various methods have been proposed to exploit the use of feedback bits. For precoded spatial multiplexing systems with finite-rate feedback, the receiver se-lects a transmitting matrix (or precoder) from a set of matrices (precoder code-book) known to both transmitter and receiver. Then the corresponding index is sent back to the transmitter using finite number of bits. Different criteria of pre-coder selection and unitary prepre-coder codebook designs are developed in [5]. For the criteria considered in [5], it has been show that with some approximations the design of optimal codebook can be converted to a problem of Grassmannian sub-space packing. Randomly generated codebooks known to both transmitter and receiver is proposed in [6], and the method is called random vector quantization (RVQ). In [7], Using bit error rate (BER) as a criterion of selecting precoder ma-trix from the codebook is proposed and the optimal unitary precoder for infinite

(12)

feedback rate, i.e., full channel knowledge at transmitter, is given. Generalized Lloyd algorithm is employed to construct precoder codebooks. An iterative ap-proach of searching a codebook for maximum mutual information is proposed in [8]. Capacity loss due to quantized feedback is thoroughly analyzed in [9]. Spatial multiplexing for two substreams using simple rotation is designed in [10]. A special form of precoding systems is the antenna selection system [11, 12] that chooses the best subset of transmitt antennas to minimize BER. In this case the transmitter possess the advantage of low complexity since the precoder is a submatrix of the identity matrix.

In addition to precoder information, quantized power allocation information can be also fed back for improving system performance. In this case there are two codebooks, one for quantized precoder and one for quantized power allocation. The index of precoder and the index of power allocation are both sent back to the transmitter. Usually a higher feedback rate is required. In [13], power loading codebook is designed separately and the performance is significantly improved. In [14], based on parameterizations, two efficient methods for precoder quantization are proposed. Combined with feedback of power loading, the proposed system’s capacity is very close to the case when full channel state information is available at the transmitter. In some recent work, bit loading information is also sent back to the transmitter. In [15], the optimal unquantized precoder is factorized via Given’s rotations and the parameters in the rotation matrices are quantized. Thus the complexity of precoder quantization is low. Feedback of bit loading, power loading and precoder is considered in [16] to improve the system throughput. In these works, bit loading is not quantized.

In most of the previously mentioned works, the number of subchannels (or substreams) M is fixed and does not change with the channel. Multimode antenna selection [17] allows the number of sucstreams M or ”mode” to vary with the channel. The transmission bits are uniformly allocated on the M substreams. It is shown in [17] that with Mt feedback bits, the system can achieve diversity

(13)

substreams M to alter in accordance with the channel. Transmission bits are equally allocated too. In addition, precoder codebooks are constructed for each modes. With judicious design, multimode precoding can achive diversity order MrMt with log2Mt bits. The design of codebooks for multimode precoding over

spatially correlated channel is developed in [19]. Generalized Lloyd algorithm is applied to design capacity maximizing codebooks for multimode transmission in [20]. In [21] a quantized principal component selection precoding scheme for capacity maximizing is proposed. The achieved performance by [21] can be close to the capacity obtained with full channel state information.

In this thesis, we consider two feedback scenarios. In the first scenario, the receiver feedbacks only bit allocation and in the second scenario the receiver feedbacks only the precoder information. The system that sends back only infor-mation of bit allocation (BA) is called BA system. Given a channel realization, receiver selects a bit allocation vector that minimizes the BER from a bit al-location codebook whose codewords satisfy the target transmission rate. The index correspond to this BER-minimizing codeword is sent back to transmitter through a reverse channel. According to the feedback information, the transmit-ter allocates bits to the modulation symbols and perform spatial multiplexing (precoding) using a unitary precoder known to the transmitter and receiver a priori. We will show that BA system can achieve full diversity order MrMtusing

log2Mt bits. Moreover, we will derive the optimal bit allocation that minimizes

the BER when the bit allocation vector is not constrained to be from a codebook and it can be real nembers. In this case, the BER performance of the BA system always outperforms the optimal BER-minimizing unitary precoder system which employs uniform bit loading and has complete channel knowledge at the trans-mitter. Furthermore, we will show that the unconstrained optimal bit allocation for BER minimization also minimizes the transmission power for a given error rate. To reduce the complexity of bit allocation vector selection, we develop an efficient quantization method. Simulation will be presented to show the useful-ness of the proposed BA system, especially for MIMO systems with low feedback rate.

(14)

The system that feedbacks only the precoder information is called a precoder system in this thesis. For a given precoder codebook, we propose a simple selec-tion criterion whose BER performance is very close to the method in [7] which requires exact BER computation. In addition, we propose a two-step design. The design is motivated by crucial properties of the optimal unquantized precoder. Namely, the total mean squared error (MSE) is minimized and the subchannel error variances are equalized.

In the proposed two-step design, the precoder F is a product of the form FVFQ. When there is unlimited feedback, FV and FQ can be chosen so that F

is the optimal precoder. When the feedback rate is finite, FV and FQ are chosen

from their respective codebooks; FV is chosen to minimize total MSE while FQ

is chosen to equalize subchannel error variances. The indexes of codewords for FV and FQ are sent back to the transmitter. If the codebooks for FV and FQ

contains respectively 2BV _{and 2}BQ _{codewords, the required number feedback bits} is B = BV + BQ, while the number of searches for selecting the precoder is

2BV + 2BQ.

Simulation results show that the performance of the proposed two-step design is comparable to the conventional design for the same feedback rate but the complexity of selecting precoder is much lower.

1.1 Outline

• Chapter 2: General system model is presented.

• Chapter 3: Previous works are reviewed in this chapter. Section 3.1 intro-duces a BER criterion and optimal unitary precoder for precoded spatial multiplexing system with infinite feedback rate proposed by S. Zhou and B. Li. In section 3.2, we review multiple antenna selection proposed by R. W. Heath, Jr. and D. J. Love. Section 3.3 introduces multimode precoding which is also proposed by R. W. Heath, Jr. and D. J. Love.

(15)

4.1, we give the MIMO system model for BA system. Feedback of bit allocation is presented in Section 4.2. The diversity order of the proposed system is given in Section 4.3. Optimal bit allocation for minimum BER without constraining the bit allocation vector to be from a codebook is derived in Section 4.4. In Section 4.5, an efficient method of bit allocation vector selection is discussed.

• Chapter 5: We consider the precoder system in this chapter. Section 5.1 introduces the system model for precoder system and the BER optimal precoder. Section 5.2 presents two simple selection criterion for precoder system. Two-step system is given in Section 5.3.

• Chapter 6: Simulation examples are presented in this chapter. • Chapter 7: 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 Wmis used to represent the m×m unitary DFT matrix given

by,

[Wm]kn= √1_me−j

2π

mkn for 0 ≤ k, n ≤ m − 1. (1.1)

(16)

Chapter 2 General System Model

The finite-rate feedback Mr×MtMIMO system is shown in Fig. 2.1. The channel

bits to symbols mapping

F

G

symbols to bits demap ping

-H

M

_t

M

r

M M bit stream bit stream full CSI index selection B bits feedback link

Transmission setting Transmitter Receiver

s

b

s

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

(17)

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 −1_X

k=0

σ2

sk, (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 = I}

M. 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)−1_F†_H†_, _{zero-forcing receiver,}

RsF†H†(HFRsF†H†+ N₀I_M_r)−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†+ N₀I_M_r)−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

(18)

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 Rb

M bits and Rb

M 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 2Rb/2M)Q s 3 (2Rb/M − 1)βk ! , (2.6) where Q(y) = √1 2π R_∞ y e−t 2_/2 dt, y ≥ 0,

and βk is the unbiased SNR of the k-th subchannel. For zeroforcing and MMSE

linear receiver, βk can be expressed respectively as,

βk =    σ2 sk σ2 ek, zero-forcing receiver, σ2 sk σ2 ek − 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 BER(F, H) ≈ _R1 b M −1_X k=0 Rb MBERk= 1 M M −1_X k=0 SERk. (2.8)

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

(19)

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 = M −1_X k=0 bk. (2.9) Let b = b0 b1 · · · bM −1 T

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]: SERk = 4(1 − 1 2bk/2)Q s 3 (2bk − 1)βk ! . (2.10)

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) ≈ _R1 b M −1_X k=0,bk6=0 bkBERk = 1 Rb M −1_X k=0,bk6=0 SERk. (2.11)

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

(20)

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 R_Mb bits. The power is also equally

allocated for each symbols, Rs = P_M0I_M. 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

(21)

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 b

F = arg min

F_∈C_F 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 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 3.1.2.

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/

√

(22)

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 −k}1 (3.2)

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   QK1 0 0 0 I_{M −K}₁_−K₂ 0 0 0 QK2   (3.3) 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

(23)

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}2

B

−1

i=0 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

Fcentroid_i = arg min

F 1 Ntr X V_n_∈R_i d2_c(Vn, F) (3.6) = arg min F 1 Ntr X V_n_∈R_i trace(IM − F†VnV†nF) (3.7) = arg max F trace(F †_RF) _(3.8) where R is defined as R = 1 Ntr X V_n_∈R_i VnV†n. (3.9)

Let the eigendecomposition of R as

(24)

ΛRis 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.

3.2 Multimode Antenna Selection

This section is organized as follows. Section 3.2.1 introduces the system model and the diversity of multimode antenna selection system. Section 3.2.2 presents the selection criteria.

3.2.1 System Model

Based on the general model in chapter 2, multimode antenna selection design a system whose number of subchannels M varies according to the channel H and M ≤ min(Mr, Mt). Assuming target transmission rate Rb is unchanged and

in-dependent of channel H, the bit loading for each subchannel is bk = R_Mb and the

power is uniformly divided among M symbols, Rs= P_M0IM. For sending M

sym-bols, antenna selection system selects M antennas from Mt transmit antennas to

perform transmission. Therefore, there are C(Mt, M) possible antenna

combina-tions. This is equivalent to select a precoder matrix from a set WM, where the

(25)

assume Mt = 3, W1 =      1 0 0   ,   0 1 0   ,   0 0 1     , W2 =      1 0 0 1 0 0   ,   1 0 0 0 0 1   ,   0 0 1 0 0 1     , and W3 =      1 0 0 0 1 0 0 0 1     .

And WM = {WM,1, WM,2, · · · , WM,C(Mt,M )}. For each M, WM’s size is C(Mt, M).

Suppose it is allowed to select from the complete precoder codebook CF =

{WM}MM =1t , the total number of precoder matrices is Mt)

X

m=1

C(Mt, M) = 2Mt − 1 (3.11)

which requires Mt bits to feedback.

Given a channel, the receiver decide what the number of subchannels M is and which precoder should be chosen from WM. Then the corresponding index

is sent back to transmitter. The transmission is adapt based on this information.

Diversity. Selection diversity provides full diversity MrMt [11]. Since selection

diversity is equivalent to selecting a precoder matrix from W1 which is included in

the complete precoder codebook CF = {WM}MM =1t , the diversity gain of multimode

antenna selection can only be better than selection diversity system. Thus, the diversity order of multimode antenna selection is MrMt.

3.2.2 Selection Criteria

Various selection criteria is designed in this paper [17]. Simulations in [17] shows that these selection criteria all yield approximately identical performance. Here we introduce a suboptimal, low-complexity selection criterion that is proposed in this paper. This selection criterion decides M∗_{, number of using subchannels,}

first, then selects precoder matrix F∗ _{from W} M.

Eigenmode Based Selection. Choose M∗ such that

M∗ = arg max

1≤M≤Mt

(26)

where λk(H) is the k-th largest singular value of H, and d2min(M, Rb) is the

normalized minimun distance in QAM constellation defined as d2_min(M, Rb) =

6

(2Rb/M − 1)/M. After the M∗ _{is determined, F}∗ _{is chosen as}

F∗ = arg max

F_∈W

M∗

λ2(HF). (3.13)

3.3 Multimode Precoding

This section is organized as follows. In Section , we show the system model and diversity of multimode precoding system. The selection criteria are given in Section 3.3.2. And Section 3.3.3 reviews the criteria of codebook size allocation and construction.

3.3.1 System Model

Founded on the general system model in chapter 2, multimode precoding assumes Rb is the fix target transmission rate, the bit loading is uniformly allocated bk =

Rb

M, for k = 1 · · · M, and transmission power is equally divided for M symbols,

Rs = P0

MIM. Similar to multimode antenna selection in section 3.2, the multimode

precoding system allows the number of subchannels M to vary according to the

channel H and M ≤ min(Mr, Mt). In addition, a codebook FM is prepared

for each mode M. Since multimode precoding requires Rb

M to be integer, thus

only some modes can support transmission. The set of these supported modes is denoted as M. For example, if Rb = 8 bits and Mr = Mt= 4, then M = {1, 2, 4}.

Based on the channel H, the receiver determines the number of subchannels M and selects the precoder matrix from the complete precoder codebook CF =

{FM}MM =1t . Subsequently, the index represented this selection is fed back to the

transmitter. The transmitter adjusts the transmission setting according to the feedback information.

(27)

codebook size of F1, is greater than or equal to Mt and the vectors in F1 span

CMt_{. Selecting vector from F}

1 = {f1, f2, · · · , fN 1} is equal to a beamforming

system with finite beamforming feasible set [24]. From [24], we know that such a beamforming system has full diversity order equal to MrMt if the span of F1

is equal to CMt. Therefore, the multimode precoding has full diversity order if above mentioned condition is satisfied.

3.3.2 Selection Criteria

Two selection criteria are proposed in this paper. One is for minimizing proba-bility of error. The other is for maximizing capacity.

Probability of Error Selection Criterion. The selection is divided in two step. For

every M ∈ M, first step selects the F∗

M from each precoder codebooks FM using

the following selection criterion,

F_M∗(H) = arg max

F_∈F_Mλ 2

M(HF), (3.14)

where λk(H) is the k-th largest singular value of H. The second step determines

the number of subchannels M∗ _by

M∗_{(H) = arg max} M ∈M λ2 M{HF∗M(H)} M d 2 min(M, Rb), (3.15) where d2 min(M, Rb) is defined as d2_min(M, Rb) = 6 M(2Rb/M − 1).

Capacity Selection Criterion. Assuming uncorrelated Gaussian signaling on each

substream, the mutual information is known to be CU T(FM) = log2det IM + P0 MN0 F†_MH†_HF M . (3.16)

Similar to above selection criterion, for every M ∈ M, first step select the F∗ M

from each precoder codebooks FM using the following selection criterion,

F∗_M = arg max

(28)

Then, M∗ _{is decided by}

M∗ = arg max

M ∈MCU T (F ∗

M) . (3.18)

3.3.3 Allocation Criterion and Codebook Construction

Given B feedback bits, there are total 2B _{codewords for complete codebook C} F.

Some criterions are designed in [18] to distribute 2B _{codewords among the modes}

in M. Under the assumption that the probabilities of selecting each mode in M are equal, the codeword allocation criteria for maximizing capacity and minimiz-ing probability of error are given as follows.

Probability of Error Allocation Criterion. Define the cost function as

A(N1, · · · , NMt) = X M ∈M d2 min(M, Rb) M N −2 Mt(Mt+1) M (3.19)

• For B ≤ log2(Mt+ 1), set NM t = 1 and N1 = 2B− 1.

• For B > log2(Mt+1), find the (N1, · · · , NMt) that minimizes A(N1, · · · , NMt) such that N1 ≥ Mt, NMt = 1, and

P

M ∈MNM = 2B This minimization can

be done using a numerical search or by using convex optimization tech-niques.

Capacity Allocation Criterion.

• For B ≤ log2(Mt+ 1), set NM t = 1 and N1 = 2B− 1.

• For B > log2(Mt+ 1), if B ≤ log2(Mt(|M| −1) + 1), set NMt = 1, N1 = Mt, and Nk = (2 B −Mt−1) |M|−2 , for k ∈ M, k 6= 1, Mt. If B > log2(Mt(|M| − 1) + 1), set NMt = 1 and Nk= 2B₋₁ |M−1| for k ∈ M, k 6= Mt.

After the sizes for each modes’ codebooks are allocated. The codebook for each mode is construct using the method in [5]. The work in [5] can approximately convert the problem of precoder codebook construction into Grassmannian

(29)

sub-Probability of Error Design Criterion. From [5], the projection two-norm distance

is defined as

dproj(Fi, Fj) = kFiF†i − FjF†jk2,

where k·k denotes 2-norm of a matrix. For minimizing probability of error, design FM such that

δproj = min

F_i_,F_j_∈F_M_:F_i_6=F_jdproj(Fi, Fj)

is maximized.

Capacity Design Criterion. The Fubini-Study distance is defined in [5] as

dF S(Fi, Fj) = arccos | det(F†iFj)|.

For maximizing capacity, design FM such that

δF S = min

F_i_,F_j_∈F_M_:F_i_6=F_jdF S(Fi, Fj)

(30)

Chapter 4 The Proposed BA system

In this chapter we propose the feedback of only bit allocation (BA) for MIMO systems with limited feedback. The proposed system will be termed a BA sys-tem. We show that the proposed BA system can achieve full diversity order. We also derive the optimal bit allocation for minimum BER when the transmission rate is given and the bit allocation vector is not constrained to be from a code-book. It turns out that the optimal bit allocation that minimizes the BER is also the optimal solution for minimizing the transmission power. Using the optimal unconstrained bit allocation, an efficient method for selection BA is developed.

4.1 System Model

Based on the general system model in chapter 2, we assume the total transmission power P0 is equally divided among all symbols carrying nonzero bits. So sk has

variance given by

σ2s =

P0/M0, bk > 0,

0. bk = 0, (4.1)

where M0 is the number of symbols carrying nonzero number of bits. As the

power is equally divided among symbols with nonzero bits, the autocorrelation matrix of the error vector for the MMSE case (2.5) can be simplified. Removing the symbols with zero bits from s, we obtain a reduced vetor s0 of size M0 × 1.

If we remove the corresponding columns of F, the result is an Mt× M0 matrix,

(31)

(x = F0s0 = Fs). The vector s0 has the autocorrelation matrix Rs₀ = P0 M0IM0. The autocorrelation matrix of the corresponding error vector e0 is

Re₀ = ( N0(F†0H†HF0)−1, zero-forcing receiver, ( 1 N0F † 0H†HF0+ _P₀_/M1 ₀IM0)−1, MMSE receiver. (4.2) In our proposed system, the precoder matrix F in the transmitter is determined beforehand. Therefore, when the channel H is given, the average BER formula in (2.11) depends only on the bit allocation vector b, which can be optimized to minimize BER. BER(b, H) ≈ _R1 b M −1_X k=0,bk6=0 bkBERk= 1 Rb M −1_X k=0,bk6=0 SERk, (4.3)

The receiver feedbacks only the bit allocation vector b to the transmitter. When the bit allocation vector b has integer entries, in principle the whole vector can be sent back to the transmitter using finite-rate feedback. However, in a system with low feedback rate it may not be possible to feedback the complete informa-tion of b without quantizainforma-tion. In this case the bit allocainforma-tion vector is chosen from a codebook Cb and the index of the bit allocation vector is fed back to the

transmitter as we will see in the next section.

4.2 Feedback of Bit Allocation

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

t

Case

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

(32)

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 4.1: 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

(33)

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}

t

Case

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 M}

t× 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(M}

t, 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 ≥ M} t−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

M_Xt−1

k=Mt−M

(34)

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 C′

b 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 M_Xt−1 k=0,b′ k6=0 SERk. (4.10)

We can choose the best bit allocation vector from C′

b to minimize BER, b b′ _{= 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, · · · , M}

t− 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 = M}

t,

F′ _{= F and s}′ _{= s). Let the auto correlation matrix of s}′ _{be R}_s′. 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 R}_s _{with R}_s′,

Re′ = N0(F′†H†HF′)−1, zero-forcing receiver, Rs′− R_s′F′†H†(HF′R_s′F′†H†+ N₀I_M r)−1HF′Rs′, MMSE receiver. (4.12)

(35)

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 M_t × 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 M}

t× 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 C′

b 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 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 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′ = I_M_t. 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

(36)

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.

4.3 Diversity Gain of BA System

In this section, we show that the BA system can achieve diversity order MrMt

for a system with Mr receive antennas and Mttransmit antennas if the codebook

is properly 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(Mt, Mr)). The augmented bit allocation vector b′ is of size Mt× 1.

It has at most M nonzero entries andPMt−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), (4.13)

where BER(b′_{, H) is the BER in (4.3). Assume the bit allocation codebook C}′ b

contains the set of codewords

Cb∗ = {Rbe0, Rbe1, · · · , RbeMt−1}, (4.14)

where ei are standard vectors of size Mt× 1, i.e., [ei]i = 1 and [ei]j = 0 for j 6= i.

The following theorem 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.

Theorem 1. 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′

(37)

Proof. _{As C}_b∗ _{is a subset of C}_b′, we have

BERmin(H) = min

b′ ∈C′ b BER(b′_{, H) ≤ min} b′ ∈C∗ b BER(b′, H). (4.15)

The BER averaged over the channel H is denoted as BER = E[BERmin(H)].

Using (4.15), it 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 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

choos-ing the best beamformchoos-ing vector from the columns of F′ _{to maximize the}

re-ceived SNR. In other words, the equivalent codebook of beamforming vectors is Cf = {f0′, f1′, · · · , fM′ t−1}, where f

′

i is the i-th column of F′. The zero-forcing

receiver in (2.3) performs maximal ratio combining. From [24], we know such a beamforming system has diversity order equal to MrMt if the span of Cf is equal

to CMt_{. Therefore the BA system has diversity order M}

rMt as well. The result

holds as long as the codebook C′

b contains the simple vectors in (4.14).

We have shown that the BA system achieves full diversity order if the code-book has the codewords in (4.14). However, when C′

b has only Mt codewords,

the codewords in C∗

b are not necessarily the best choices as we will see in the

simulations.

Alternative proof of Theorem 1. Suppose the initial precoder F′ _{= I}

Mt and bit allocation vector is chosen from C∗

b. As F′ has only one nonzero entry in each

column and b′ _{∈ C}∗

b has only one nonzero entry, only one transmit antenna is

used and this becomes an antenna selection system that chooses only one antenna. The right hand side of (4.15) corresponds to the BER of the system in which the transmitter chooses the best transmit antenna and receiver uses maximal ratio combining. Such a system has been shown to achieve a diversity order equal to

(38)

MrMt [27]. So the BA system with identity F′ achieves diversity order MrMt.

From the discussion of optimal augmented precoder in the previous section we know any Mt× Mt unitary F′ lead to the same average performance. Therefore

we can arrive at the result given in Theorem 1.

4.4 BA system with Unconstrained Bit

Alloca-tion

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 when the bit allocation is not constrained. Although the unconstrained optimal bit allocation requires infinite feedback rate, the corre-sponding BER performance provides insightful observations as we shall see. The optimal bit allocation is given in section 4.4.1. The connection of zero-forcing

BA system with M = Mt to precoder system and power-minimizing BA system

are given in Section 4.4.2.

4.4.1 Optimal Bit Allocation

We first consider the case when the precoder F is a fixed Mt× M unitary matrix.

The number of bits transmitted per channel use is Rb and b0+b1+· · ·+bM −1= Rb.

Assume the transmission rate is high and bkis large enough so that 1−2

bk

2 ≈ 1 and

1 − 2bk ≈ 1, then the symbol error rate expression in (2.10) can be approximated by SERk ≈ 4Q r 3 2bkβk ! . (4.16)

With the high bit rate assumption, bk > 0, for all k and thus Rs = P₀/MI_M. For

the convenience of derivation, we define the function f (y) = Q( 1

(39)

The function f (y) is monotone increasing and it can be verified that f (y) is convex for y ≤ 1/3. Using f(·), we can express SERk as

SERk≈ 4f(

2bk 3βk

). (4.18)

Therefore the average BER in (4.3) can be written as

BER(b) ≈ _R4 b M −1_X k=0 f (2 bk 3βk ). (4.19)

where we have dropped the dependence of BER function on the channel H for convenience. Assume the arguments of f (·) are smaller than 1/3 so that the convexity of f (·) holds (we will see later why this assumption is reasonable). Using the convexity of f (·), we have

1 M M −1_X k=0 f (2 bk 3βk) ≥ f( 1 M M −1_X k=0 2bk 3βk ). (4.20) It follows that BER(b) ≈ _(R 4 b/M) 1 M M −1_X k=0 f 2bk 3βk (4.21) ≥ 4 (Rb/M) f 1 3M M −1_X k=0 2bk βk ! (4.22) ≥ _(R 4 b/M) f  2Rb/M 3 M −1_Y k=0 1 βk !1/M  (4.23) , BERBA. (4.24)

The second inequality is obtained by using the fact that Rb = b0+ b1+ · · ·+ bM −1

and the AM-GM (arithmetic mean-geometric mean) inequality 1 M M −1_X k=0 2bk βk ≥ M −1_Y k=0 2bk βk !1/M = 2Rb/M M −1_Y k=0 1 βk !1/M . (4.25)

and also using the monotone increasing property of f (·).

Notice that the lower bound in (4.23) is independent of bit allocation. The optimal bit allocation is such that the two inequalities in (4.22) and (4.23) become

(40)

equalities. Due to convexity of f (·), the first inequality (4.22) holds if and only if 2bk/(3β

k) are of the same value for all k. The same set of conditions is also

necessary and sufficient for equality to hold in the second inequality as f (·) is monotone increasing. When both inequalities hold, the lower bound in (4.23) is achieved. Therefore the optimal bit allocation for minimizing the BER is such that 2bk/β k= 2Rb/M(QM −1_l=1 1/βl)1/M, i.e., bk= log2(βk) + Rb M − 1 M M −1_X l=0 log₂βl. (4.26)

We can see that the symbols with larger SNR βk are allocated with more bits.

We have denoted the BER lower bound in (4.23) as BERBA, where the subscript

is a reminder which notifies that it is the BER of the BA system. Note that BERBA is obtained when the bits are allocated as in (4.26) and there is no

in-teger constraint on bit allocation in the above derivation. The bit allocation bk

computed in (4.26) are not integers in general. Nonetheless BERBA gives useful

insight on the performance of the BA system and connections with other system as we will see in Section 4.4.2

Remarks

1. In the above derivation, we have assumed that the argument of f (·) in (4.19) is larger than 1/3 so that the convexity of f (·) can be used in (4.20). We now examine the validity of such an assumption. When the argument 2bk/(3βk)_{= 1/3, the corresponding SER}

kis SERk≈ 4Q(

√

3) ≈ 0.17, a large symbol error rate that may not be useful. In practical applications, it is more reasonable to have smaller error rate, which requires 2bk/(3β

k) < 1/3.

2. When bits allocated optimally as in (4.26), 2bk/βk are the same for all k. This means the symbol error rates are equalized for all transmitted symbols. 3. The actual number of symbols transmitted may be smaller than M if some symbols are allocates with 0 bits. However the number of bits transmitted

(41)

Now let us consider the case F is not fixed, but an Mt× Mt augmented

pre-coder F′ _{(implicitly M < M}

t in this case). The imput s′ is an augmented Mt× 1

vector and bit allocation vector b′ _{is M}

t× 1 as in Section 4.2.2. For a given M,

we can choose M columns out of F′ _{to form the actual M}

t× M precoder. As

we choose M columns of F′_{, there are C(M}

t, M) possible choices. For each of

these choices, we can compute the optimal bit allocation and the corresponding BER using (4.23), and choose the best precoder. In this case the BA system with augmented precoder F′ _{is always better than the BA system with a fixed}

precoder F if F us a submatrix of F′_.

Bit allocation for optimal number of substreams

In the above discussion of optimal bit allocation, we assumed all symbols carry nonzero bits and transmission power is loaded on all M symbols. In the end some of the symbols may be assigned zero bits while take up 1/M of the total power. To make efficient use of power, we can allocate power to only the symbols that carry nonzero bits. To do this, we can compute the optimal bit allocation for all possible number of symbols with nonzero bits and choose the best one. To be more specific, let us illustrate this in another viewpoint. We start out with an Mt× Mt initial precoder F′ as before. The precoder F can be any Mt× M0

submatrix of F′_{, where M}

0 = 1, 2, · · · , M. There are

PM

M0=1C(Mt, M0) possible precoders. We collect all these possible precoder in a set SF. For each F ∈ SF,

we can use (4.29) to compute the BER under optimal bit allocation. The error rate BERBA given in (4.23) depends on the precoder used. For convenience let

us use the notation BERBA(F) to indicate the dependence on F. The best F is

Fopt = arg min

F_∈S_FBERBA(F). (4.27)

The resulting minimum BER is given by BERBA,opt = min

F_∈S_FBERBA(F). (4.28)

When the optimal precoder is obtained this way, all the symbols will carry nonzero bits. The reason is as follows: Let the optimal precoder Fopt be Mt× l, and the

(42)

optimal l × 1 bit allocation be bopt. Suppose one of the symbols is assigned with

zero bits. The actual number of symbols transmitted is l − 1. Let us remove from Fopt the column corresponding to the symbol with zero bit and call the remaining

Mt× (l − 1) submatrix F0. Also remove from bopt the element equal to zero and

call the reduced vector b0. Then using precoder F0 with bit allocation b0 gives

a smaller BER for the same transmission power as the power is now distributed among (l − 1) symbols instead of l symbols. So Fopt can not be optimal if one

symbol is assigned 0 bits. We can therefore conclude that all symbols carry nonzero bits in the optimal system that uses Fopt as precoder.

4.4.2 BER performance of Zero-forcing BA system When

M = M

t

In this subsection, we will examine the BER lower bound BERBA derived in

(4.24) when the receiver is zero forcing and M = Mt. Connection between the

BA system with two other systems, the precoder system [7] and power minimizing BA system, will be studied.

For M = Mt, the precoder F is an Mt×Mt unitary matrix. When the receiver

is zero-forcing, the k-th SNR βk is equal to P0/(Mσe2k,BA), where we have added a subscript to the error variances to indicate that these are the error variances in

the BA system. The BER lower bound BERBA in (4.24) can be written as

BERBA = 4 Rb/M Q s 3P0/M 2Rb/M 1 (QM −1_l=0 σ2 el,BA) 1/M ! . (4.29)

We can see from the above expression that BERBA depends on the geometric

mean of {σ2 el,BA}

M −1

l=0 , which is in turn determined by the given precoder.

Al-though the geometric mean of {σ2 el,BA}

M −1

l=0 depend on the choice of precoder,

the arithmetic mean does not. This is due to unitary property of the precoder. To see this, we can use the expression Re = N0(F†H†HF)−1 given in (4.12) for

zero-forcing receiver. It follows that the average error Err is

E = 1

M −1_X

(43)

Using FF† _{= I}

M and trace(AB) = trace(BA), we have

Err =

1

MN0trace(H

†_H)−1_. _(4.31)

As N0trace(H†H)−1 does not depend on the precoder F, we come to the

con-clusion that the average error Err does not depend on the precoder. It is the

same quantity for any square unitary precoder regardless of bit allocation. This property allows us to show that the BER of the BA system is always smaller than the BER-minimizing precoder system, which is briefly reviewed below.

BER-minimizing precoder system[7,22]. In the precoder system [7], the precoder

is optimized to minimize BER. Referring to Chapter 3.1, the power and bits are uniformly loaded on all M symbols (M ≤ min(Mt, Mr)). Suppose Rb bits

are transmitted using total power P0 for each channel use; each sk, for k =

0, 1, · · · , M −1, is a QAM symbol with variance P0/M that carries Rb/M bits. Let

the singular value decomposition of H be UΛV†_{, where U and V are respectively}

Mr× Mr and Mt× Mt unitary matrices. The Mr× Mt matrix Λ is diagonal,

whose diagonal elements are singular values of H in a nonincreasing order. From Section 3.1.2, the optimal Mt× M unitary precoder that minimizes the BER for

large SNR is given by [7, 22]

Feq = VMQ, (4.32)

where VM is the Mt× M matrix obtained by keeping the first M columns of V

and Q is an M × M unitary matrix that has the equal magnitude property. For the optimal precoder given in (4.32), the subchannel error ek = bsk− sk

has the property that variance σ2

ek are equalized [7, 22], σ2 e0 = σ 2 e1 = · · · = σ 2 eM −1. (4.33)

Now consider the case M = Mt and the receiver is zero forcing. We know from

(4.31) that all square unitary precoders lead to the same average error variance. That is, the BER-minimizing precoder yields the same average error variance as the BA system. Therefore when the optimal precoder in (4.32) is used, all error variances are equal to Err given in (4.31) and hence identical BER for all

(44)

symbols transmitted. Using the approximation in (4.16), the minimized BER of the precoder system can be expressed as

BER ≈ 4 Rb/M Q   s 3P0/M 2Rb/M 1 Err   , BERprecoder. (4.34)

Using the fact that Err is also equal to _M1 PM −1_l=0 σe2l,BA and applying AM-GM inequality to {σ2 el,BA} M −1 l=0 , we get Err= 1 M M −1_X l=0 σ2 el,BA ≥ M −1_Y l=0 σ2 el,BA !1/M . (4.35)

As Q-function is monotone decreasing we arrive at BERprecoder = 4 Rb/M Q s 3P0/M 2Rb/M 1 1 M P_{M −1} l=0 σe2l,BA ! (4.36) ≥ _R4 b/M Q s 3P0/M 2Rb/M 1 (QM −1_l=0 σ2 el,BA) 1/M ! = BERBA (4.37)

We recognized that the right hand side of the above inequality is the BER of

the BA system given in (4.29). Therefore when M = Mt the BA system with

optimal bit allocation and an arbitrary fixed precoder has a smaller BER than the precoder system with an optimal precoder.

Unlike the M = Mt case, the BER of the BA system for M < Mt is not

guaranteed to be smaller than the precoder system. We can see this using the case M = 1 as an example, i.e., beamforming transmission. When M = 1, the precoder system corresponds to the beamforming system with maximal ra-tio transmission [28] at the transmitter and maximal rara-tio combining at the receiver, which achieves the smallest error rate among all beamforming systems. As M = 1, all Rb bits are loaded on one symbol. For the BA system, all the bits

are allocated to only one symbol as well but the choices of the beamforming vec-tors are limited to the Mt columns of F′. If the number of symbols transmitted

in each channel use can not exceed one, the precoder system is better than BA system.

(45)

H

G

F

s

x

q

r

s

ê

M

_r

M

_t

Figure 4.2: MIMO wireless system with Mt transmit antennas and Mr receive

antennas

Connection with power-minimizing BA system

In Section 4.4 bit allocation is optimized to minimize BER. Suppose, instead of BER criterion, we optimize the bit allocation to minimize the transmission power for a given symbol error rate constraint ǫ and transmission rate Rb. When

the receiver is zero-forcing, we now show that the optimal bit allocation derived in Section 4.4 for minimum BER is also optimal for minimizing transmission power. Consider the MIMO system in Fig. 4.2. Let the total transmission power be PT.

From (2.1), we have PT = trace(E[xx†]) =PM −1k=0 σ2sk. Suppose the k-th symbols sk is loaded with bk and b0 + b1+ · · · + bM −1 = Rb. In the power minimization

problem, we allow PT and symbol variance σs2k to vary so that the given symbol error rate constraint ǫ can be satisfied. For a zero forcing receiver, the error variance σ2

ek can be computed using Re = N0(F

†_H†_HF)−1 _{in (2.5). If σ}2 sk and σ2

ek are given, the number of bits that can be loaded is well approximated by [29]

bk = log2 1 + σ 2 sk σ2 ekΓ , (4.38)

where Γ, called SNR gap, depends on the given symbol error rate ǫ. In our problem σ2

sk is not given. Let us rearrange the above equation to get σ

2 sk = Γ(2bk− 1)σ2

ek, which gives the required symbol variance when the k-th symbol is loaded with bk bits. Using high bit rate assumption 2bk − 1 ≈ 2bk, we have

σs2 k ≈ Γ2

bk_σ2

(46)

Using the approximation in (4.39), we have PT ≈ Γ M −1_X k=0 2bk_σ2 ek. (4.40)

Applying the AM-GM inequality to the above summation, we get

PT ≈ Γ M −1_X k=0 2bk_σ2 ek ≥ MΓ M −1_Y k=0 2bk_σ2 ek !1/M = MΓ2Rb/M M −1_Y k=0 σ_e2_k !1/M (4.41) The right hand side is a lower bound that is independent of bit allocation. The minimum transmission power can be achieved by allocating the bits bk such that

AM-GM inequality becomes an equality, i.e., 2bkσ2

ek are equalized. This in turns means σ2

sk are identical and thus 2

bkσ2

ek/σ

2 sk = 2

bk/β

k are the same for all k. It

follows that bkare as given in (4.26). So the optimal bit allocation for minimizing

BER of zero forcing BA system is also optimal for minimizing transmission power.

4.5 Efficient Method of Selecting Bit Allocation

Vector

In this section we consider efficient search of bit allocation vector from the code-book C′

b. Suppose the feedback bits is B, so the codebook size is 2B. To

ob-tain bb′ _{= arg min}_b′

∈C′

bBER(b

′_{, H) given in (4.11), exhaustive search can be}

ap-plied by computing BER formula (4.3) for each bit allocation vector in C′ b, thus

BER(b′_{, H) is evaluated 2}B _{times. When B is large, such an exhaustive search}

requires lots of computations. Using the unconstrained optimal bit allocation in (4.26), an efficient method is developed to reduce the complexity of selecting bit allocation in Cb′. The development of our method can be easier to understand if

we explain the basic idea first.

Quantization of b′_opt. The basic idea of the proposed method is described as

follows. Rather than evaluating BER formula for 2B _{bit allocation vectors, we}

can compute the optimal unconstrained bit allocation b′

opt first. Let b′opt be the

多輸入多輸出系統之位元配置有限回饋

國

立

交

通

大

學

電控工程研究所

碩 士 論 文

多輸入多輸出系統之位元配置有限回饋

MIMO Systems with Limited Feedback of Bit Allocation

研 究 生：鄭人予

指導教授：林源倍 教授

多輸入多輸出系統之位元有限回饋

MIMO Systems with Limited Feedback of Bit Allocation

國 立 交 通 大 學

電控工程研究所

碩 士 論 文

多輸入多輸出系統之位元配置有限回饋

學生：鄭人予

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

本論文包含兩個部分，在第一個部分我們提出一個有限回饋

位元配置的多輸入多輸出系統。我們會證明此系統可以達到

全多樣性。在傳輸速率固定的假設下，我們推算出能達到最

小錯誤率的最佳位元配置，並證明達到最小錯誤率的位元配

置也是使用最低傳輸能量的位元配置。模擬結果顯示我們所

提出的系統可以使用較少的回饋位元達到低錯誤率。在第二

個部分我們針對傳統單階預先編碼器的有限回饋系統設計

低複雜度的編碼器選擇準則。我們也提出一個可以降低複雜

度的二階預先編碼器系統。模擬結果會展示出我們所設計的

系統的可用性。

MIMO systems with limited feedback of

bit allocation

Panna Jeng

Advisor: Dr. Yuan-Pei Lin

Department of Electrical and Control Engineering

National Chiao Tung University

January 4, 2011

誌謝

在電控所就讀的這段期間，不僅在學術上可以深入學習，

在人生經驗上也有所成長。特別感謝林源倍老師的悉心

教導，使我在這段重要的人生歷程中學到很多東西。感

謝林清安老師和蔡尚澕老師能撥冗來參加我的碩士論

文口試，並給予寶貴的建議，使我的論文能更加完善。

感謝我家人的支持與鼓勵。也感謝實驗室的學長姐與夥

伴：建樟、孟良、鈞麟、芳儀、素卿、懿德、士軒、虹

君、君維、士傑，感謝你們在學業上的討論、建議和實

驗室溫馨氣氛的經營。另外感謝在研究所期間給予我鼓

勵和幫助的同學朋友們。最後感謝芳霖多年的陪伴與支

持。沒有你們的協助，我是無法學習到如此多寶貴的知

識和留下如此多珍貴的回憶，由衷的感謝你們。

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Outline

1.2

Notations

Chapter 2

General System Model

F

G

-H

M

t

M

r

s

b

s

Chapter 3

Previous Works

3.1

Precoder System

3.1.1

System Model

碩士論文

研究生：鄭人予

指導教授：林源倍教授

國立交通大學

碩士論文

_t

_{M ≤ M}