於多用戶多輸入多輸出系統中利用差動量化降低通道回饋量

國 立 交 通 大 學

電信工程研究所

碩士論文

於多用戶多輸入多輸出系統中

利用差動量化降低通道回饋量

CSI Feedback Reduction Based on Differential

Quantization in Multi-User MIMO Systems

研 究 生：鄭佳旻 Student: Chia-Min Cheng

指導教授：李大嵩 博士 Advisor: Dr. Ta-Sung Lee

吳卓諭 博士 Dr. Jwo-Yuh Wu

於多用戶多輸入多輸出系統中利用差動量化

降低通道回饋量

CSI Feedback Reduction Based on

Differential Quantization in Multi-User MIMO Systems

研 究 生：鄭佳旻 Student: Chia-Min Cheng

指導教授：李大嵩 博士 Advisor: Dr. Ta-Sung Lee

吳卓諭 博士

Dr. Jwo-Yuh Wu

國立交通大學

電信工程研究所

碩士論文

A Thesis

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

July 2010

Hsinchu, Taiwan, Republic of China

於多用戶多輸入多輸出系統中

利用差動量化

降低通道回饋量

學生：鄭佳旻

指導教授：李大嵩 博士

吳卓諭 博士

Chinese Abstract

國立交通大學電信工程研究所

摘要

CSI Feedback Reduction Based on Differential

Quantization in Multi-User MIMO Systems

Student: Chia-Min Cheng

Advisor: Dr. Ta-Sung Lee

Dr. Jwo-Yuh Wu

Institute of Communication Engineering

National Chiao Tung University

English Abstract

Abstract

Limited feedback techniques have drawn attention for many years and emerged as one of the key techniques in modern wireless communication systems. Under the finite-rate feedback link environment, the issue of how to quantize channel information efficiently becomes important for system design. In this thesis, we focus on reducing the required feedback rate while maintaining or even enhancing the performance of the overall system. We propose two methods: one is differential vector quantization (VQ) for channel state information (CSI) feedback, and the other is diagonal-wise full VQ for spatial correlation matrix feedback. The proposed differential VQ introduces Predictive Vector Quantization (PVQ) in the receiver end to compress the CSI by exploiting the temporal correlation of the channel. The diagonal-wise full VQ mainly exploits the spatial correlation of the channel to reduce the feedback quantization bits. Simulation results indicate that both methods can achieve a higher SINR and a smaller MSE compared to the existing schemes.

Acknowledgement

I am heartily thankful to my advisors, Dr. Ta-Sung Lee and Dr. Jwo-Yuh Wu, whose encouragement, guidance and support from the initial to the final level enabled me to develop an understanding of the subject. I learned a lot from their positive attitude in many areas. I also offer my regards and blessings to all members in the Communication System Design and Signal Processing (CSDSP) Lab. Last but not least, I would like to show my sincere thanks to my family for their invaluable love and support.

Table of Contents

Chinese Abstract ...I

English Abstract ... II

Table of Contents ... IV

List of Figures ... VI

List of Tables... VIII

Acronym Glossary ... X

Notations ... XI

Chapter 1 Introduction... 1

Chapter 2 System Model ... 4

2.1 Limited Feedback in Multiuser MIMO Systems ...5

2.2 Precoding Scheme...7

2.3 Correlated Channel Model...8

2.4 Quantization Criterion and Codebook Construction in LTE Release 8 ...10

2.5 Summary ...13

Chapter 3 Reduced CSI Feedback Based on Differential

Qunatizaion... 14

3.1 Motivation...15

3.2 Proposed Differential Quantization Method ...16

3.2.1 Incorporation of Predictor...19

3.2.2 Initial Full CSI and Error Accumulation...23

3.3 Computer Simulations ...25

3.3.1 Simulations in Time Domain CSI Feedback...26

3.3.2 Extension to MIMO OFDM system ...31

3.4 Summary ...37

Chapter 4 Discussion of Different Feedback Overheads and

Spatial Correlation Matrix Feedback... 38

4.1 Different Feedback Overheads in Multi-user Systems ...39

4.1.1 Simulation Demonstration ...39

4.1.2 Geometric Interpretations ...41

4.1.3 Discussions of Interferences and Precoding Gain ...43

4.1.4 Conclusions...45

4.2 Proposed Diagonal-wise Vector Quantization for Spatial Correlation Matrix Feedback ...46

4.2.1 Correlated Channel Model [28] ...47

4.2.2 Existing Solution: Entry-wise Differential Scalar Quantization [27] .49 4.2.3 Proposed Diagonal-wise Full Vector Quantization...50

4.2.4 Comparisons Between Entry-wise Differential SQ and Proposed Diagonal-wise Full VQ ...52

4.2.5 Computer Simulations ...53

4.3 Summary ...64

Chapter 5 Conclusion ... 65

List of Figures

Fig. 2-1 Limited feedback multi-user MIMO system...6

Fig. 2-2 Jakes’ fading simulator...9

Fig. 2-3 Autocorrelation of Jakes’ simulator underM =16...9

Fig. 2-4 Full vector quantization ...10

Fig. 3-1 Comparison of DPCM and Full Scalar Quantization...15

Fig. 3-2 Limited feedback system with proposed differential VQ scheme ...17

Fig. 3-3 Block diagram of PVQ system at receiver...17

Fig. 3-4 Mathematical expression of PVQ system at receiver ...18

Fig. 3-5 Block diagram of PVQ system at transmitter ...18

Fig. 3-6 Mathematical expression of PVQ system at transmitter ...18

Fig. 3-7 GLA training process [9, 10]...21

Fig. 3-8 Two-step codebook training...22

Fig. 3-9 SINR under different number of initial full scalar quantization bits ...24

Fig. 3-10 Error Accumulation...24

Fig. 3-11 Definition of Parameters ...25

Fig. 3-12 Differential vector quantization vs. Full vector quantization...29

Fig. 3-13 Differential vector quantization under different N T( )f ...29

Fig. 3-14 Differential vector quantization under different velocities ...30

Fig. 3-15 LTE resource grid...31

Fig. 3-16 Differential vector quantization vs. Full vector quantization...35

Fig. 3-17 Differential vector quantization under different N T( )_f ...35

Fig. 3-18 Differential vector quantization under different velocities ...36 Fig. 4-1 Simulation demonstration of different feedback overheads for different users 40

Fig. 4-2 Geometric representation for two users case ...41

Fig. 4-3 Geometric representation of interference...43

Fig. 4-4 Geometric representation of precoding gain...44

Fig. 4-5 Angle parameters in MIMO channel model [14]...48

Fig. 4-6 Column wise vector quantization...50

Fig. 4-7 Diagonal-wise vector quantization...50

Fig. 4-8 Reducing the feedback frequency ...51

Fig. 4-9 Different number of quantization bits for Diagonal-wise full VQ...62

Fig. 4-10 Channel variation for high and low velocities ...62

List of Tables

Table 2-1 4-bit Codebook Construction [25] ...12

Table 3-1 Simulation Parameters ...26

Table 3-2 Codebook for v = 3 km/hr...27

Table 3-3 Codebook for v = 5 km/hr...27

Table 3-4 Codebook for v = 8 km/hr...27

Table 3-5 Codebook for v = 10 km/hr...28

Table 3-6 Codebook for v = 15 km/hr...28

Table 3-7 Simulation Parameters ...32

Table 3-8 Codebook for v = 3 km/hr...33

Table 3-9 Codebook for v = 5 km/hr...33

Table 3-10 Codebook for v = 8 km/hr...33

Table 3-11 Codebook for v = 10 km/hr ...34

Table 3-12 Codebook for v = 15 km/hr ...34

Table 4-1 Two cases in simulation ...39

Table 4-2 Simulation Parameters ...40

Table 4-3 Definitions of parameters ...41

Table 4-4 Parameter definitions ...47

Table 4-5 Values of Δ [30]...48 _k Table 4-6 Comparisons of diagonal-wise full VQ and enty-wise differential SQ ...52

Table 4-7 Simulation parameters...54

Table 4-8 Codebooks for third diagonal (1 1× vector) under time-invariant channel environment ...55

Table 4-10 Codebooks for main diagonal ( 3× vector) under time-invariant channel 1

environment ...57

Table 4-11 Codebooks for third diagonal (1 1× vector) under time-varying channel

environment ...58

Table 4-12 Codebooks for secondary diagonal ( 2× vector) under time-varying 1

channel environment ...58

Table 4-13 Codebooks for main diagonal ( 3× vector) under time-varying channel 1

Acronym Glossary

CSI channel state information PVQ predictive vector quantization SINR signal to interference and noise ratio MIMO multiple-input multiple-output MISO multiple-input single-output

VQ vector quantization

SQ scalar quantization

MSE mean squared error ZF zero-forcing

Notations

P total power constraint for single user

M number of transmit antennas

K number of mobile users

k

n _{complex Gaussian noise for the user k} 2

σ variance of complex Gaussian noise

n

h channel vector at the n th time instant

n

h quantized channel vector at the n th time instant

l

c _{codeword with index l}

n

e _{differential CSI at the n th time instant}

n

e _{quantized differential CSI at the n th time index}

i

v _{beamforming vector for the user i}

d

R number of quantization bits per channel vector for full CSI

f

R _{number of quantization bits per channel vector for differential CSI}

avg

R _{average number of quantization bits per cannel vector} α quantization angular error

Chapter 1

Introduction

Multiple-input multiple-output (MIMO) technologies have demonstrated the potential to significantly enhance the performance of wireless communications. In the case of multi-user MIMO systems, simultaneous transmissions of multiple user signals can be supported by space-division multiple access to provide a substantial gain in system throughput. Nevertheless multiple access introduces interferences in the system. Designing transmit vectors while considering the interference of other users is quite challenging. In this case, CSI at the transmitter enables the communication system to exploit the channel information and avoid interference. The precoders which can increase various performance gains for wireless communication [1] can be designed based on different forms of the channel information available to the transmitter [2-4]. When perfect CSI is available at the transmitter and receiver, the well-known dirty-paper coding is used to pre-cancel multiuser interference at the transmitter and hence achieves full channel capacity [5].

However, conveying perfect full CSI would impose a heavy burden on user feedback channel and the amount of feedback information increases with the number of users in service. For this reason, research on partial CSI feedback under limited feedback link has drawn much attention recently since it was proposed in [6, 7]. For

limited feedback systems, issues like transmission delays, channel estimation error at the receiver, quantization error, and non-ideal feedback link can cause problems to the overall system. Many research works have investigated the utilization of limited feedback systems and tried to solve the above problems [8-14]. Some of them focus on reducing CSI feedback overhead by designing an optimum codebook or quantizing the CSI efficiently [10, 12-14] since resource is valuable under the limited feedback environments.

In this thesis, our goal is to reduce the required feedback overhead by effectively quantizing the channel information. Considering multiuser MIMO systems (each users with a single antenna), we try to quantize the CSI via vector quantization. The proposed differential vector quantization adopts the Predictive Vector Quantization (PVQ) model [15] to perform differential quantization by exploiting the temporal correlation of CSI. The proposed scheme periodically feeds back the full CSI using a large number of quantization bits while using fewer bits to quantize the remaining differential CSI. We make the average number of quantization bits smaller than that of the conventional full vector quantization. We further extend the MIMO system to a MIMO OFDM system. Instead of quantizing the time domain CSI, frequency domain CSI, or subcarrier, requires much more quantization bits since at every time instant, the number of subcarrier may be up to 2048 in LTE systems. It is impossible to feed back all the subcarriers information. Hence, only some of the subcarriers are chosen to be fed back. Although the temporal correlation of subcarriers is not as obvious as the CSI in time domain, simulations show that the proposed differential vector quantization is still effective in MIMO OFDM systems.

In addition to CSI at the transmitter end, spatial correlation matrix can also be used to modify the precoding matrix to make the precoding matrix more suitable in

spatial correlation statistic information at the transmitter end, we propose a diagonal-wise full vector quantization scheme to quantize the matrix efficiently and thereby reducing the feedback overhead.

In this thesis, we also investigate the scenario where some users are granted a higher feedback overhead. From a geometrical point of view, we show a simple example to illustrate one user’ performance changes in the case that some other users are permitted to have a higher feedback overhead.

This thesis is organized as follows. Chapter 2 describes the system model and the conventional full VQ scheme. In Chapter 3, the proposed differential vector quantization is introduced along with our codebook design method and the incorporation of predictor. And in Chapter 4, we explain users’ performance under different feedback overheads and present the proposed diagonal-wise full vector quantization for spatial correlation matrix feedback. Numerical results illustrate the advantages of the proposed methods. Finally, we conclude this thesis in Chapter 5.

Chapter 2

System Model

In wireless communication environments, multiple-input multiple-output (MIMO) system provides significant increases in data throughput without additional bandwidth and transmission power [16]. There are two modes in a MIMO system, one is the open loop mode and the other is the closed loop mode. The basic idea of the closed loop mode is to use channel information to adapt the transmitted signal. If channel state information (CSI) is available at the transmitter, a precoder can be pre-designed to match the channel, and thereby offers diversity and yields a better performance. However, in limited feedback systems, it is impossible for the transmitter to know accurate and instantaneous CSI. The estimated CSI at the receiver end is quantized by a given codebook and the receiver will feed back the index of selected codeword to the transmitter.

Since limited feedback systems can only have a finite codebook size, codebook design becomes an essential issue and a difficult problem. A good codebook may reduce the feedback overhead while maintaining performance. The statistical distribution of the channel and quantization criterion must be taken into account to design a codebook. In [17], the random vector quantization (RVQ) method provides a simple way for codebook construction. Also, the problem of designing codebook is

shown to be equivalent to Grassmannian line packing in [18].

In this chapter, the multiuser MIMO system with limited feedback will be introduced first. And then the linear precoding method of the system is discussed. Finally, the codebook in LTE Release V8.7.0 (2009-05) will be introduced and its construction be presented.

2.1 Limited Feedback in Multiuser MIMO

Systems

The communication system with a limited feedback link requires cooperation between the transmitter and receiver. The limited feedback multiuser MIMO system is illustrated in Figure 2.1. The MIMO system has M transmit antennas and each receiver has a single antenna. The broadcast channel can be described as

, 1,..., H i i i y =h s+n i= K (2.1) where h h1, 2,...,h (K 1 M i C × ∈

h ) are the channel vectors of users 1 ~ K , the vector 1

M

C × ∈

s is the transmitted signal, and n₁,...,n are independent complex _K Gaussian noise with variance σ . 2 H=

[

]

K×Mmatrix with the ith row equal to the channel vector of the ith receiver. The transmitted signal has a total power constraint of P , that is E[s 2]≤P.

Fig. 2-1 Limited feedback multi-user MIMO system

Each receiver is assumed to have perfect and instantaneous knowledge of its own channel vector, i.e., h , and the feedback link is zero-delay. The receiver quantizes its _i channel by a vector quantization algorithm which is designed to minimize some distortion function like mean squared error (MSE) between the channel vector and quantized vector (i.e., codeword). The codebook with 2B codewords

{

}

C  c c c is known at both transmitter and receiver. Quantized CSI h i

is fed back from each mobile receiver to the transmitter for further processing.

Feedback Link

n

n

Transmitter

User 1

User K

H

h

h

Vector

Quantization

Scheme

Vector

Quantization

Scheme

2.2 Precoding Scheme

The performance of linear precoding depends on the choice of beamforming vectors. One simple choice of beamforming vectors is the zero-forcing vectors, which are chosen such that no multiuser interference is experienced at any of the receivers. Zero-forcing precoding is a low complexity precoding scheme whose performance is asymptotically optimal among the linear precoders at high SNR[19]. If the perfect CSI is known at the transmitter, zero-forcing can be used to completely eliminate multiuser interference. This creates a interfering-free channel to each of the K receivers and thus leads to a multiplexing gain of K . In the finite rate feedback system, the transmitter only knows the partial information about CSI. For example, the beamforming vectors of the zero-forcing precoder are selected based on the quantized channel vectors.

Let h denote the quantized channel vector of mobile useri . These quantized _i vectors are compiled into a matrix 1 2 ...

H K

⎡ ⎤

= ⎢_⎣ ⎥_⎦

H h h h . The unnormalized

precoding matrix is V=H HHH( H −) 1. The normalized precoding matrix is 1 2 ... K

⎡ ⎤

= ⎢_⎣ ⎥_⎦

V v v v , where v is the corresponding unit norm beamforming i

vector for user i . Also, the transmitted signal is

1 K j j j x = =

∑

intended for the i user. Thus the received signal at useri is th

1 , K H H i i i i j j i j y n x n = =h s+ =h

∑

and the SINR at useri is

2 i ₂ 2 SINR . H i i H i j i j P K P K σ ≠ = +

∑

2.3 Correlated Channel Model

The Rayleigh fading process of the mobile radio channel follows WSS uncorrelated scattering model. The mobile user is moving at a speed of v while the transmitter is fixed. The theoretical power spectral density of the received fading signal has the well-known U-shaped form[20]

2 1 , 1 ( ) , 0 , elsewhere d d d f f f f S f f π ⎧⎪⎪ ≤ ⎪⎪ _{⎛ ⎞} ⎪ _{⎜ ⎟} ⎪ _{− ⎜ ⎟} = ⎨_{⎪ ⎜ ⎟⎜⎝ ⎠}⎟ ⎪⎪ ⎪⎪ ⎪⎩ (2.4)

where f is the maximum Doppler frequency, given by d d

v f

λ

= , v is the speed of the mobile user and λ is the wave length of the carrier wave. The normalized continuous time autocorrelation of the received fading signal is given by the zeroth-order Bessel function of the first kind[20] r( )τ =J₀

(

)

In this thesis, Jakes’ channel simulator (see Figure 2.2) is used as the temporal correlated channel. One of the reasons of using Jakes’ simulator is that the autocorrelation and, hence, the power spectral density of received signal can be very close to those of the theoretical Rayleigh fading WSS US channel (see Figure 2.3). The model generates fading channel as a sum of M sinusoids defined by the following equation:

1

1

( ) 2{[2 cos cos 2 2 cos cos 2 ] [2 sin cos 2 2 cos cos 2 ]} ( ) ( ) , M n n D n M n n D n I Q h t f t f t j f t f t h t jh t β π α π β π α π = = = + + + = +

∑

∑

Fig. 2-2 Jakes’ fading simulator -300 -200 -100 0 100 200 300 -0.5 0 0.5 1

Normalized time delay

Norm al iz ed A u to c o rrel a ti on Autocorrelation Jakes' theoretical

2.4 Quantization Criterion and Codebook

Construction in LTE Release 8

Different from conventional vector quantization criterion which often minimizes the mean squared error h−c_l 2, the quantization criterion in limited feedback systems is aimed to quantize the direction of the channel vector [22-24]. The user chooses the codeword which maximizes the correlation or, equivalently, minimizes the angle between the channel vector and selected codeword, and then feeds back the codeword index to the transmitter (Fig. 2-4). The index of selected codeword is

0,...,2 1 arg max_B H . i l l q = − = h c (2.6)

Fig. 2-4 Full vector quantization

Clearly, the codebook generation is always a crucial issue because codebook is a dominant factor for the quality of CSI provided to the transmitter. In this thesis, we introduce LTE 4-bit codebook [25] for our limited feedback system as shown in Fig.

2-4. The user selects a codeword from LTE codebook based on the quantization

criterion in (2.6). We call this quantization scheme in our limited feedback system “Full Vector Quantization” hereinafter.

h

h

q

=

Generally, each mobile user uses different codebooks to prevent multiple users from quantizing their channels to the same codeword. The codebook construction in LTE Release 8 is given in Table 2-1. There are sixteen generating vectors

0 1 15

{ , ,...,u u u } and these vectors result in sixteen 4× matrices 4 W which is _n defined by 2 , H n n n H n n u u W I u u = − (2.7)

where I is the 4× identity matrix. In Table 2-1, 4 W_n{ }p denotes the matrix defined by the columns given by the set { }p . For example, Wn{13} is a 4× matrix 2

formed from the first and the third columns of W . _n

For single user case ( 1 layer), Wn{1} , i.e., the first columns of all

( 0,1,...,15)

n

W n = , total sixteen 4× vectors, are used as the codebook. For two 1 users ( 2 layers), W_n{(1)}are used as the codebook for the user 1 and W_n{(2)} are used for user 2 where

{ }

2.5 Summary

In this chapter, the limited feedback MIMO system is presented. We introduce the 4-bit codebook in LTE Release 8 for full vector quantization in our system. The user feeds back optimum codeword (i.e., quantized CSI) to the transmitter and transmitter performs Zero-Forcing precoding based on these received quantized CSI from different users. Moreover, Jakes’ channel simulator is given as our temporal correlated channel model and we will further exploit this temporal correlation of CSI to reduce the number of feedback bits by proposed differential quantization scheme in the next chapter.

Chapter 3

Reduced CSI Feedback Based on

Differential Quantization

In limited feedback systems, perfect CSI feedback is impossible due to finite-rate feedback link. How to reduce CSI feedback overhead in limited feedback systems while maintaining performance has became a crucial issue. Differential pulse code modulation (DPCM) gives us the motivation to apply the concept of differential quantization in limited feedback systems to reduce or, equivalently, to compress the number of feedback bits. We further extend DPCM to “Predictive Vector Quantization” (PVQ) [15] to implement differential quantization and thereby reduce the number of feedback bits required. The architecture of PVQ and its components, including a vector quantizer and a predictor, are presented in this chapter.

First, we use differential quantization to feedback CSI by exploiting its temporal correlation. Then, extend to feedback the subcarrier information of MIMO OFDM systems by exploiting the temporal correlation of subcarriers. Simulation results have shown that limited feedback systems with differential quantization scheme can achieve higher SINR under highly correlated channel environment.

3.1 Motivation

Differential quantization is frequently used in data compression or source coding to reduce the number of quantization bits. The most commonly used technique for differential quantization is “differential pulse code modulation＂ (DPCM) [26]. Intuitively, we can apply DPCM in limited feedback systems to reduce the number of feedback overhead. In Fig. 3-1, 1 bit per element, real and imaginary part separately, is assumed (i.e., 8 bits for each channel vector h∈C4 1× ,M = ) for both DPCM 4 and full scalar quantization. Fig. 3-1 shows higher SINR can be achieved by DPCM compared to the conventional full scalar quantization. However, DPCM is an “element-wise” quantization scheme which requires a large number of bits to quantize a complex channel vector. Therefore, we propose the idea of using differential vector quantization by “Predictive Vector Qautization” (PVQ) in limited feedback systems to further reduce the feedback overhead.

2 3 4 -5 0 5 10 15 20 25 SI NR Number of users DPCM

Full Scalar Quantization

3.2 Proposed Differential Quantization Method

The system model of limited feedback systems with proposed differential VQ scheme is presented in Fig. 3-2. We introduce PVQ to implement differential VQ. The block diagrams of PVQ at the transmitter and receiver are shown in Fig. 3-5 and Fig.

3-3 respectively. A vector quantizer and a predictor are incorporated at the receiver end

while the transmitter end only includes a predictor. The corresponding mathematical expressions of Fig. 3-3 and Fig. 3-5 are illustrated in Fig. 3-4 and Fig. 3-6 respectively. Instead of quantizing h (full CSI), the difference of channel vector and n

predicted channel vector, e_n =h_n −h (differential CSI), is quantized. Note that _n

n is denoted as the time index and user index i is ignored here for simplicity. The

quantization criterion is 2 ˆ_n arg min _n j = − _j e e c (3.1)

wherec , _j j =1,..., 2Rd _{is the codeword from codebook C (}

d

R is the number of

quantization bits for differential CSI). The received quantized e is added by the _n predicted channel vector h to recover the quantized channel vector _n h _n

Fig. 3-2 Limited feedback system with proposed differential VQ scheme

Fig. 3-3 Block diagram of PVQ system at receiver

Feedback Link

n

n

Transmitter

User 1

User K

H

h

h

VQ

⊕

⊕

p

h

e

e

h

,...,

h

h

Fig. 3-4 Mathematical expression of PVQ system at receiver

Fig. 3-5 Block diagram of PVQ system at transmitter

Fig. 3-6 Mathematical expression of PVQ system at transmitter 1 p n k n k k − =

= −

∑

h

A h

=

+

h

e

h

⊕

p

order

Predictor

e

h

,...,

h

h

h

= −

∑

h

A h

=

−

e

h

h

h

=

e

+

h

3.2.1 Incorporation of Predictor

The LMMSE Predictor [27] is incorporated in the proposed differential VQ system. The optimal predictor of LMMSE can be obtained by the “orthogonality principle.” Suppose the order of the predictor is p , we can express the orthogonality principle as

0 for 1,..., ,

H n n i

E ⎡⎣e h − ⎤ =⎦ i = p (3.2)

that is E _⎣⎡_⎢

(

)

∑

where A_k ∈CM M× . We can rewrite (3.2) as

1 0 p H n k n k nn ii k E _{− −−} = ⎡⎛_{⎜⎢ ⎞⎟} ⎤_⎥ + ⎟ = ⎜ _⎟ ⎢_⎜⎝ ⎟_⎠ ⎥ ⎣ h

∑

∑

R A R Thus A can be found by the following _k equation 11 12 1 1 01 21 22 2 02 1 0 H p H H p pp p p ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ _{= −}⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ R R R A R R R A R R R A R … (3.4)

where R_ij =E[h_{n i−}hH_{n j−} ]. Since the autocorrelation function of Rayleigh fading WSS US is given by zeroth-order Bessel function of the first kind

(

)

0

( ) 2 d s , 0,1,2,...,

r τ =J π τf T τ = .

3.2.2 Design of Differential VQ Codebook

For the design of the vector quantization codebook, Generalized Lloyd Algorithm (GLA), which is a widely used codebook generation technique [15, 26], is adopted in our system. The distortion function of GLA is the overall Mean Squared Quantization Error (MSQE) between vectors and quantized vectors.

2 1 1 D ( ) M m m m Q M = =

∑

where

{

}

Fig. 3-7. At the beginning, the initialization of codebook is randomly chosen from the sample vectors. A threshold ε is selected to stop iterations when D( )k −D(k−1) < ε (D represents the distortion (MSQE) at k th iteration). GLA training process is ( )k mainly based on the two rules:

1. Nearest Neighbor Condition: The encoding partition S should consist of all _n vectors that are closer to codeword c than any other codewords. n

{

}

'

: '

n n n

S = x x−c ≤ x−c ∀n ≠n (3.6)

2. Centroid Condition: The codeword c should be average of all the vectors that _n are in encoding partition S . n

1,2,..., 1 m n m n m S n S n N ∈ ∈ =

∑

∑

From Fig. 3-7, we learn that GLA requires a large number of sample vectors for training the codebook. If we take a closer look at Fig. 3-3, there is one problem: quantized CSI h is required to generate _n e . (i.e., codebook is needed before _n e _n generation). The solution for this problem can be solved by two-step codebook training as shown in Fig. 3-8.

Fig. 3-7 GLA training process [15, 26]

Step3: Update the codebook based on

Centroid Condition.

Step1: Start with

random codebook

( represents

codeword) and select a threshold ε

Step2: Classify the sample

vectors according to Nearest

Neighbor Condition.

Training samples

Final codebook

Step4: Go back to

step2 to classify the vectors again. Iterations

(step2 and step3)

Step 1 is to train the initial codebook by feeding real channel vectors 1,..., n p

− −

n

h h into the predictor instead of quantized channel vectors hn−1,...,hn−p.

Then the initial codebook can be used to generate another bunch of e samples for _n

Step 2 training. Because the initial codebook is generated by feeding the real channel vectors, the magnitudes of training samples e are smaller than the practical _n implementation and hence the magnitudes of codewords of initial codebook are also smaller. By Step 2 training, the codebook would be closer to the practical optimum codebook.

Fig. 3-8 Two-step codebook training

Train initial codebook

⊕

order

p

Predictor

h

e

h

,...,

h

h

Step 1

VQ

initial codebook

⊕

⊕

order

p

Predictor

h

e

e

h

,...,

h

h

Step 2

3.2.2 Initial Full CSI and Error Accumulation

Initial Full CSI Quantization

Before differential CSI (e ) feedback, the first p channel vectors ( p is the n

order of predictor) have to be fed back by full CSI (h ) quantization. For differential _n quantization, the accuracy of initial full CSI is a dominant factor of the system performance. Therefore, we quantize initial full CSI “element-wise” by uniform scalar quantization. Separate the real and imaginary parts of each entry in h ∈CM×1. Then each channel vector h has 2M real elements, and these 2M elements are quantized by uniform scalar quantization. Intuitively, the more quantization bits are, the better performance is. Fig. 3-9 (Two users, the rest of parameter settings are in

Table 3-1) shows SINR goes up with the number of quantization bits of initial full CSI, R , and tends to saturate when _f R is larger than 56 bits (i.e., 7 bits per _f element).

Error Accumulation Due to Differential Quantization

One major problem for differential quantization is error accumulation. Because the recovered quantized channel vector h is based on the past quantized differential _n CSI vectors (e ), the quantization errors are accumulated over time as shown in _n

Fig. 3-10 (Two users, the parameter settings are in Table 3-1 except

64 bits, 4 bits

f d

R = R = ). Our solution is to feedback full CSI periodically to maintain CSI quality and correct the past accumulated error. To sum up, we use large number of bits to quantize periodically fed-back full CSIs and few bits to quantize remaining differential CSIs. The average number of feedback overhead is roughly the same as that in Full VQ.

Fig. 3-9 SINR under different number of initial full scalar quantization bits

Fig. 3-10 Error Accumulation

0 200 400 600 800 1000 9 10 11 12 13 14 15 16 17 18

Time Index

SI

NR

Diff. VQ

Full VQ

R

f

SI

NR

3.3 Computer Simulations

In this section, we will give some simulations to demonstrate the advantage of the proposed differential vector quantization scheme. This section is separated into two main simulation environments. The environment in first subsection is the time domain CSI feedback and the environment in second subsection is the frequency domain CSI, i.e., subcarrier information, feedback. Before the demonstration of simulation results, some parameters are defined below.

Definition of Parameters

y Differential CSI feeds back every T sec. d

y Full CSI feedback every T_f =N T× _d sec.

y R : Number of quantization bits per channel vector for full CSI f

y R : Number of quantization bits per channel vector for differential CSI _d

y R : Average number of quantization bits per channel vector avg

y p : Predictor order

Fig. 3-11 Definition of Parameters

… … … … … Full CSI Quantization Differential CSI Quantization Differential CSI Quantization Full CSI Quantization time d R f R Rf Rd , _{f d} N T =N T× Td

(

)

(

)

3.3.1 Simulations in Time Domain CSI Feedback

The time domain CSI feedback is considered in this section. Table 3-1 lists all parameters used in our simulation. The simulation is based on a 4× wireless 1 system. Perfect channel knowledge known at the receiver and the error-free feedback channel are also assumed in the simulation. We compare the proposed differential vector quantization to the full vector quantization scheme. The codebook in LTE Release 8 is applied in full VQ. As for the proposed differential VQ, the system uses different self-trained codebooks for different velocities since the users with same velocity should have similar magnitude of codewords. The corresponding codebooks are listed in Table 3-1.

Table 3-1 Simulation Parameters

Parameter Value

Channel Rayleigh fading channel

Number of Transmit antennas (M) 4

Predictor Order p 2 N 100 d T 5ms d R 3 bits f R _{56 bits}

(

)

Codebooks (Full VQ) LTE Release 8 (Table 2-1)

Codebooks (Differential VQ) Self-Trained Codebooks are listed in

Table 3-2 codebook for v = 3 km/hr

index codewords 1 0.01861- 0.00444i -0.00301 - 0.00463i 0.00527+ 0.03416i -0.03592 - 0.01361i

2 -0.00973 - 0.01458i -0.02546 + 0.05598i 0.00356 - 0.00238i 0.00522 - 0.01194i 3 -0.01701 + 0.05485i 0.01718 + 0.01455i -0.02505- 0.00429i -0.00938 - 0.00136i 4 -0.00934- 0.00726i -0.03289 - 0.02495i -0.02330- 0.009009i 0.01166 - 0.02058i 5 0.00879 + 0.00057i -0.00280 - 0.00547i -0.00072 + 0.00785i -0.00044 + 0.05364i 6 0.02906 - 0.01368i 0.01717 + 0.00356i -0.00995 - 0.00238i 0.01279 + 0.00022i 7 -0.00414 + 0.00953i -0.00462 - 0.01268i 0.04349 - 0.03858i -0.01125 - 0.00078i 8 -0.03182 - 0.01367i 0.02247 - 0.01104i 0.01172 + 0.01060i 0.01901 - 0.00563i

Table 3-3 codebook for v = 5 km/hr

index codewords 1 -0.00254 + 0.01459i -0.00247 - 0.02514i 0.00051 - 0.01239i 0.034216 - 0.06769i 2 0.01244 + 0.00347i -0.01007 - 0.02681i -0.01971 - 0.08106i -0.003504 + 0.01975i 3 0.00665 + 0.00853i 0.00021 - 0.01303i 0.01479 + 0.00724i -0.08061 - 0.00688i 4 0.02305 - 0.009834i -0.00868 - 0.03118i 0.03563 + 0.03017i 0.019476 + 0.01758i 5 -0.00351- 0.00070i 0.00546 - 0.00642i -0.08181 + 0.04070i 0.00876 + 0.00964i 6 0.01351 + 0.00103i 0.07769 + 0.04163i 0.02180 - 0.00622i -0.00251 + 0.00760i 7 -0.06896 - 0.03703i 0.00003 + 0.00010i 0.01173 + 0.00265i 0.01042 + 0.01587i 8 0.01214 + 0.01653i -0.05697 + 0.06416i 0.00512 + 0.00583i 0.00654 + 0.00204i

Table 3-4 codebook for v = 8 km/hr

index codewords 1 0.00035 - 0.01122i 0.01596 - 0.001174i 0.00201 + 0.00473i 0.11816 + 0.12526i 2 0.15035 + 0.04310i -0.00192 - 0.00409i -0.01544 + 0.01145i -0.01265 + 0.00132i 3 -0.11170 + 0.11171i -0.01319 - 0.00398i 0.00558 + 0.01114i -0.00569 + 0.00489i 4 -0.00363 - 0.02638i 0.02106 - 0.03701i -0.08108 + 0.11219i -0.00409 - 0.02105i 5 -0.00838 - 0.02139i -0.14004 + 0.03852i -0.01182 - 0.01149i -0.01468 - 0.01528i 6 -0.01055 - 0.01615i 0.07633 + 0.1015i -0.04741 - 0.05341i 0.00459 - 0.02524i 7 0.00171 - 0.01931i 0.01484 - 0.03722i 0.08274 - 0.01451i 0.06673 - 0.08474i 8 -0.01458 - 0.04106i 0.020802 - 0.04022i 0.04235 - 0.04929i -0.10540 + 0.04854i

Table 3-5 codebook for v = 10 km/hr

index codewords 1 0.02692- 0.02778i 0.05072 - 0.11008i 0.03964+ 0.09216i 0.07518- 0.10204i

2 -0.20369- 0.08143i -0.00424- 0.00608i 0.00546+ 0.01879i -0.01307+0.00941i 3 0.00951+ 0.00001i -0.01817- 0.02675i -0.20294- 0.03991i -0.023612- 0.07295i 4 0.01735- 0.023523i 0.01399- 0.01443i 0.10748- 0.18637i 0.04081- 0.01840i 5 0.03052- 0.02662i 0.14073+ 0.08023i 0.03114+ 0.01050i -0.10101+ 0.00621i 6 0.02303+ 0.21302i 0.00811+ 0.02800i 0.01418+ 0.00739i 0.01157- 0.01202i 7 0.01571- 0.01906i 0.00910+ 0.01425i -0.02651+ 0.02517i 0.11569+ 0.18245i 8 0.05089- 0.02973i -0.16741+ 0.02444i 0.01857+ 0.05147i -0.05736+ 0.01484i

Table 3-6 codebook for v = 15 km/hr

index codewords 1 -0.03156+ 0.01836i 0.23321- 0.27631i 0.00668+ 0.07667i 0.06117+ 0.07148i

2 -0.19797+ 0.29578i -0.09071+ 0.04809i -0.05678+ 0.07583i 0.06777- 0.02833i 3 0.05109+ 0.02838i -0.06168- 0.00099i 0.00341+ 0.03938i -0.37495+ 0.10158i 4 0.05547 - 0.01414i -0.10691- 0.03118i -0.21091- 0.28494i 0.08396+ 0.05085i 5 -0.18560- 0.32995i -0.08078 - 0.03426i -0.01932+ 0.04005i -0.00943- 0.1251i 6 0.27523- 0.02563i -0.04658+ 0.01468i -0.04051+ 0.18578i 0.08386- 0.11775i 7 -0.01383- 0.03967i 0.23025+ 0.34098i -0.03386- 0.05087i 0.04909+ 0.01794i 8 0.02344+ 0.03002i -0.04260+ 0.004195i 0.36572- 0.11770i 0.03537+ 0.02099i

From the above tables, we can observe that the magnitude of the codewords is larger in the higher speed codebook since for high speed users, the magnitude difference between two consecutive CSI is larger.

Following are three simulation results. The first simulation result in

Fig. 3-12 shows the proposed differential VQ successfully exploits the temporal correlation of channel vectors and can achieve higher SINR compared to full VQ under 5 km/hr environment. If we extend the time of full CSI feedback, average number of bits decrease but SINR degrades. Fig. 3-14 shows the SINR for different velocities. Because higher velocity results in smaller temporal correlation, smaller correlation of CSI degrades the performance of differential quantization. Unlike

differential VQ which suffers from error accumulation, full VQ quantizes channel vector at every time instant independently, so the performance of full VQ is unvarying under different velocities.

2 3 4 -2 0 2 4 6 8 10 12 14 Number of Users SI N R

diff.-R_avg=4.06 bit LTE codebook, R=4bit

Fig. 3-12 Differential vector quantization vs. Full vector quantization

2 3 4 -2 0 2 4 6 8 10 12 14 Number of Users SI N R

Different Length (N) of Full CSI Feedback

Diff. VQ: N=100, R_avg=4.06 bits Diff. VQ: N=200, R_avg=3.53 bits Diff. VQ: N=300, R_avg=3.35 bits Full VQ: R=4 bits

2 3 4 -2 0 2 4 6 8 10 12 14 16 18 Number of Users SI N R

Different Velocities Diff.VQ: v = 3 km/hr Diff.VQ: v =5 km/hr Diff.VQ: v=8 km/hr Diff.VQ: v=10 km/hr Diff.VQ: v=15 km/hr Full VQ: v=3 km/hr Full VQ: v=5 km/hr Full VQ: v=8 km/hr Full VQ: v=10 km/hr Full VQ: v=15 km/hr

3.3.2 Extension to MIMO OFDM system

In LTE frame structure, one radio frame is 10 ms long and consists of 20 slots of length 5 ms. A subframe is defined as two consecutive slots, that is one radio frame is composed of 10 subframes. A physical resource block is defined as N_symb ( 7)= consecutive OFDM/SC-FDMA symbols in the time domain and NscRB ( 12)=

consecutive subcarriers in the frequency domain. The smallest resource unit is denoted a resource element. Since there are many subcarriers (may up to 1024) in a time instant, it is impossible to feedback all the subcarriers information to the transmitter. Usually, k consecutive resource blocks share one codeword and k depends on different modes in LTE [25]. So only the subcarrier in the mid of k consecutive resource blocks is fed back.

Fig. 3-15 LTE resource grid

Subcarrier (frequency)

OFDM symbol (time)

RB RB sc N ×N RB sc N Resource element smyb N

One radio frame (10ms)

Resource block resource elements 7 12 resource elements = × = × RB smyb sc N N

Table 3-7 Simulation Parameters

Parameter Value

Channel Rayleigh fading channel

Number of transmit antennas (M) 4

Number of users 2-4 Predictor Order p 2 N 100 d T 5ms f d T =N T× _500ms d R 3 bits f R _{56 bits}

(

)

Codebooks (Full VQ) LTE Release 8 (Table 2-1) Codebooks (Differential VQ) Self-Trained Codebooks are listed in

3 km/hr: Table 3-8

5 km/hr: Table 3-9

8 km/hr: Table 3-10

10 km/hr: Table 3-11

Table 3-8 codebook for v = 3 km/hr

index codewords 1 0.01850+ 0.01663i 0.15751+ 0.10612i -0.075845+ 0.09921i 0.12522- 0.18442i

2 0.40755 - 0.26827i 0.18229+ 0.02534i 0.07001+ 0.21347i 0.08667+ 0.07725i 3 0.10392+ 0.24114i 0.09846+ 0.10685i -0.11746+ 0.06245i -0.10948+ 0.12052i 4 0.03421+ 0.0204i -0.15147+ 0.14220i -0.06990- 0.09554i 0.15874+ 0.22629i 5 -2.00152- 1.08612i 2.39253+ 0.61204i -1.92461- 1.81288i 0.58650- 0.87042i 6 0.00092- 0.11588i 0.03269 - 0.18963i 0.15093+ 0.01060i -0.07942- 0.11052i 7 -2.81571- 0.071022i 0.79602- 0.96609i 1.46382- 0.36982i 0.92291+ 1.14374i 8 -0.19272+ 0.13480i -0.04540+ 0.07487i -0.08165- 0.09375i -0.11690- 0.13523i

Table 3-9 codebook for v = 5 km/hr

index codewords 1 0.30445+ 0.22635i 0.15751+ 0.10612i -0.07584+ 0.09921i 0.12522 - 0.18442i

2 0.40755- 0.26827i 0.18229+ 0.02534i 0.07001+ 0.21347i 0.08667 + 0.07725i 3 0.10392+ 0.24114i 0.09846+ 0.10685i -0.11746+ 0.06245i -0.10948+ 0.12052i 4 0.03421+ 0.02045i -0.15147+ 0.14220i -0.06990- 0.09554i 0.15874+ 0.22629i 5 -2.00152 - 1.08612i 2.39253+ 0.61204i -1.92416 - 1.81288i 0.58650- 0.87042i 6 0.00092- 0.11588i 0.03269- 0.18963i 0.15093 + 0.01060i -0.07942 - 0.110529i 7 -2.81571 - 0.07102i 0.79602 - 0.96609i 1.46382 - 0.36988i 0.92291+ 1.14374i 8 -0.19272+ 0.13480i -0.04540+ 0.07487i -0.08165- 0.09375i -0.11690- 0.13523i

Table 3-10 codebook for v = 8 km/hr

index codewords 1 -0.30445+ 0.22635i -0.08192 + 0.30640i -0.13432+ 0.199751i 0.41735 - 0.14419i

2 -0.15031+ 0.01269i 0.35441+ 0.11945i 0.04724+ 0.20804i 0.08446+ 0.14228i 3 -0.28148- 0.06025i -0.17323 - 0.00949i 0.17891- 0.19109i -0.08100- 0.23096i 4 0.10639+ 0.01218i 0.04763+ 0.06343i -0.0409 - 0.43678i 0.24865+ 0.29275i 5 0.017640- 0.07233i -0.22372+ 0.28215i 0.063256+ 0.00006i -0.22293+ 0.24969i 6 0.04519+ 0.19604i -0.00388- 0.10754i -0.06925+ 0.27971i -0.35732- 0.13852i 7 0.29571- 0.15511i -0.26035- 0.35558i 0.06717- 0.01712i -0.17588+ 0.02205i 8 0.03268+ 0.06397i 0.31217- 0.31186i -0.23381- 0.12263i 0.17470- 0.45801i

Table 3-11 codebook for v = 10 km/hr

index codewords 1 0.38819- 0.05640i 0.279081- 0.084417i -0.085771- 0.228474i 0.294287+ 0.50050i 2 -0.26705+ 0.09830i -0.10912+ 0.27842i -0.18124- 0.39648i -0.15851 + 0.34360i 3 0.27077+ 0.36627i 0.00919+ 0.09340i 0.27123+ 0.19372i -0.37589+ 0.29274i 4 0.033498 - 0.28856i -0.23501- 0.04900i 0.56626- 0.32576i -0.06516- 0.04615i 5 0.12185+ 0.30981i 0.02615+ 0.04254i -0.02207+ 0.25378i 0.43702- 0.31196i 6 -0.35221+ 0.03415i 0.22407- 0.03079i -0.00789+ 0.168390i -0.35287- 0.31445i 7 -0.06664 - 0.39172i -0.21855- 0.15347i -0.02613+ 0.39965i 0.00177- 0.04736i 8 -0.18204 - 0.47617i 0.13907- 0.05957i -0.29671- 0.264254i 0.34720- 0.24472i

Table 3-12 codebook for v = 15 km/hr

index codewords 1 0.31149- 0.51042i -0.84816- 0.35939i 0.00875- 0.03616i -0.55162- 0.09785i

2 0.14076+ 0.32218i -0.01956+ 0.94223i 0.29629+ 0.24979i 0.116174- 0.74335i 3 -0.11393+ 0.72280i -0.11386- 0.20279i -0.41209+ 0.70975i -0.06094+ 0.49142i 4 -0.62541- 0.14539i 0.50913- 0.50412i -0.37464 - 0.15517i -0.03885- 1.05382i 5 0.81650+ 0.38587i 0.38412+ 0.43180i -0.25944 - 0.55869i -0.15011+ 0.17496i 6 0.15015- 0.75979i 0.83052+ 0.15415i 0.12825+ 0.12149i 0.43432+ 0.31504i 7 -0.27572+ 0.48021i 0.31749- 0.67221i 0.68262 - 0.49347i 0.36509+ 0.07361i 8 -1.07468- 0.17525i -0.35833+ 0.58647i 0.20414- 0.10759i -0.38812+ 0.22483i

Similarly, the codebook for higher velocity users has the larger magnitude of codeword since the temporal correlation of channel is weaker. In MIMO OFDM systems, the temporal correlation is not obvious so the performance tends to degrade compare to that in MIMO systems. But our proposed differential quantization scheme can still attain higher SINR under highly correlated channel environments as shown in

Fig. 3-16, Fig. 3-17 and Fig. 3-18. Intuitively, the reason for differential VQ is

superior to full VQ is that the magnitude of codewords in differential VQ is smaller than those in full VQ. The same size of codebook can represent the differential CSI more precisely compared to full CSI. Therefore, the quantization error will be smaller in our proposed differential VQ scheme.

2 3 4 -4 -2 0 2 4 6 8 10 12 Number of Users SI NR Diff. VQ: R avg=4.06 bit Full VQ: R=4 bits

Fig. 3-16 Differential vector quantization vs. Full vector quantization

2 2.5 3 3.5 4 -4 -2 0 2 4 6 8 10 12 14 Number of Users SI N R

Different Length(N) of Full Quantization Feedback

Diff. VQ: N=50, R_avg=5.12 bits Diff. VQ: N=100, R_avg=4.06 bits Diff.VQ: N=150, R

avg=3.7 bits Diff. VQ: N=200, R_avg=3.53 bits Full VQ: R=4 bits

2 3 4 -4 -2 0 2 4 6 8 10 12 14 Number of Users SI N R Different Velocities Diff.VQ: v=3 km/hr Diff.VQ: v=5km/hr Diff.VQ: v=8 km/hr Diff.VQ: v=10 km/hr Diff.VQ: v=15 km/hr Full VQ: v=3 km/hr Full VQ: v=5 km/hr Full VQ: v=8 km/hr Full VQ: v=10 km/hr Full VQ: v=15 km/hr

3.4 Summary

In this chapter, we apply the concept of differential quantization in data compression to limited feedback systems. The proposed differential vector quantization scheme is presented. We first incorporate the model of PVQ in the limited feedback system. Also, LMMSE predictor is used in our PVQ model. The codebook is trained by GLA with some modifications. The full CSI are periodically fed back in between differential CSI to correct the accumulated error. For full CSI, a large number of quantization bits are used while fewer quantization bits are used for differential CSI. The average feedback overhead is roughly the same as in full VQ. To sum up, the proposed differential vector quantization scheme can attain higher SINR at the expense of a slight increase in feedback overhead compared to full VQ.

Chapter 4

Discussion of Different Feedback

Overheads and Spatial Correlation

Matrix Feedback

In this chapter, two main topics are addressed; one is discussion of users’ performances under different feedback overheads and the other is feedback reduction for spatial correlation matrix via vector quantization. In the previous chapter, all users are permitted to feedback CSI under the same overhead but in the practical communication system, some users might be allowed to have higher feedback overhead. Therefore, the first part of this chapter is mainly to discuss the system performance in different feedback overhead scenarios.

Also, in Chapter 3, the proposed differential vector quantization has been shown to be an effective way to quantize CSI. So, the second part of this chapter aims to further find an effective method to quantize the spatial correlation matrix to reduce the number of quantization bits. Here, we propose diagonal-wise full vector quantization by exploiting the channel spatial correlation. The proposed scheme can achieve a smaller MSE compared to existing methods such as entry-wise differential scalar quantization.

4.1 Different Feedback Overheads in Multi-user

Systems

In LTE systems, there are different reporting modes for CSI feedback. Some reporting modes are triggered by scheduling grants and allow mobile users to feedback CSI with higher overhead,. In this section, we are going to discuss about the influence on all mobile users under the scenario in which only some specific users are permitted to have higher feedback overhead. First, a simulation result demonstrates the performance for users with different feedback overheads. Then, discussion of the result will be given under a geometrical point of view.

4.1.1 Simulation Demonstration

In the simulation, we have two simple cases as shown in Table 4-1. Only the first

user in Case 1 has higher feedback overhead. Other users in Case 1 have the same feedback overhead as the users in Case 2. The simulation parameters are defined in

Table 4-2. As we can see from Fig. 4-1, only first user in Case 1 benefits from the

higher cost of feedback overhead. Other users in Case 1 have the same SINR as those users in Case 2. This implies that user performance does not change if some other users in the system are allowed to feedback CSI with higher overheads.

Table 4-1 Two cases in simulation

Case 1 First user: 4 bits/ 12 subcarriers Other users: 4 bits/ 48 subcarriers Case 2 All the users: 4 bits/ 48 subcarriers

Table 4-2 Simulation Parameters

Parameters Value

Quantization Scheme Full vector quantization

Channel Multipath Rayleigh fading channel

Tap 1 2 3 4 5 6

Relative delays (ms) 0 10 20 30 40 50

Number of transmit antennas 4

Carrier frequency 2.5G Hz Velocity 5 km/hr Sampling time 5 ms Number of subcarriers 512 2 3 4 -4 -2 0 2 4 6 8 10 12 14 Number of users SI NR ( d B )

First user in Case 1 Other users in Case 1 All users in Case 2

4.1.2 Geometric Interpretations

In the following sections, we will analyze the simulation result in Fig. 4-1 under

a geometrical point of view. The geometric representation for a two-user case is shown in Fig. 4-2, and the corresponding definitions of parameters are listed in Table 4-3.

Fig. 4-2 Geometric representation for two users case

Table 4-3 Definitions of parameters

Parameters Definitions 

i

h Normalized channel vector for user i

i

h Quantized channel vector for user i

i

v _{Beamforming vector for user i}

αi Quantization angular error between h and _i h _i

βi Angle between h and i v i

Since the transmitter performs zero-forcing precoding, the beamforming vector

j

v is orthogonal to the quantized channel vector h for i i≠ j (i.e.,

,

i ⊥ j i ≠ j

h v ). Also, we have two assumptions here,

α

₁

β

₁

β

₂

α

₂

h

h

v

v

h

h

1. 2 i π α < . If 2 i π _{<α < , the magnitude of π} 2 ₂ cos i i = αi h h is equal to cos2

(

)

π−α < . We can always find the equivalent angle which is smaller

than 2

π

. 2. βi is small.

If h and _i h_j ( i≠ ) are closely aligned, βj _i would be large (close to 2

π

) because hi ⊥vj which implies hj and v are nearly orthogonal. And if j h i

and hj ( i ≠ ) are closely aligned, the problem of singularity, or rank deficiency j

would happen when the system performs pseudo-inverse in zero-forcing precoding. If we exclude the singular case and h and _i h_j (i ≠ ) are nearly j orthogonal, we can reasonably assume that βi is small.

In the following sections, the interferences h vH_{i j} 2 and precoding gain h vH_{i i} 2 in SINR of user i 2 2 2 2 2 2 2 2 SINR H H i i i i i i H H i j i i j i j i j P P K K P P K K σ σ ≠ ≠ = = +

∑

∑

4.1.3 Discussions of Interferences and Precoding Gain

a. Interferences

Fig. 4-3 shows the geometric representation of interferences where Ω is the

angle between v and j h . The interferences for user i result from i

2 i j h v for all ≠j i . Define 2 ₂ cos , for . H i j = Ω i ≠ j h v (4.1)

Fig. 4-3 Geometric representation of interference

From the assumption that

2 i π α < , Ω is bounded by

π

π

α

α

− ≤Ω≤ + .

2

2

Socos2Ω ≤sin2αi, and the interferences have the upper bound as

2 ₂

sin .

H

i j ≤ αi

h v (4.3)

From (4.3), we can conclude that the interferences for user i is only bounded by the sine of its own quantization angular error αi and independent of other users’

quantization errors.

α

_i

v

h

h

Ω

b. Precoding Gain

As for the precoding gain, its geometrical representation is illustrated in Fig. 4-4.

Now Ω is the angle between v and i h . The precoding gain for user i is i

2 ₂

cos

H

i i = Ω

h v .

Fig. 4-4 Geometric representation of precoding gain

From the assumption that βi is small and

π α +β <

2

i i , we can infer that

α

β

Ω≤ + .

So cos2Ω ≥cos2

(

)

bound as 2 ₂ cos . H i i ≥ αi h v (4.5)

Similar to the upper bound of interferences, we can conclude that the precoding gain of user i is only bounded by the cosine of its own quantization angular error α_i and independent of other users’ quantization errors.

α

_i

v

h

h

Ω

β

_i

4.1.4 Conclusions

From above analysis of interferences and precoding gain, we can derive the lower bound of SINR as

( )

(

)

∑

∑

4.2 Proposed Diagonal-wise Vector Quantization

for Spatial Correlation Matrix Feedback

Generally, the codebook design is mainly based on typical communication environments. However, it is impossible to consider all cases. So, channel spatial correlation at the transmitter end can provide the current statistic information of the channel and therefore can be used to refine the codewords of the codebook, such that the refined codewords are more suitable for the current communication scenario. Methods of refining the codebook by using the channel spatial correlation can be found in [28]. In [28], the average spatial correlation matrix as defined in (4.7) is fed back to the transmitter. The average spatial correlation matrix is

, , 1 1 1 , T L H n n T t l n T t l t l T L = = × + × + = ×

∑∑

where t is time index, l is subcarrier index and n means to take average over

n th time slot. Since the spatial correlation matrix R is a Hermitian matrix, we need

only quantizing the upper triangular part of R .

In the following sections, the spatially and temporally correlated channel model is given first, followed by the introduction of one of the existing methods for spatial correlation matrix feedback: entry-wise differential scalar quantization [29]. Finally, the proposed a diagonal-wise vector quantization scheme which feeds back spatial correlation matrix by exploiting the spatial correlation of channels is presented.