國 立 交 通 大 學
電信工程學系碩士班
碩士論文
用於多輸入多輸出正交分頻多工系統之
波束形成輔助多用戶適應性無線電資源管理技術
Beamforming Aided Multiuser Adaptive Radio
Resource Management for MIMO-OFDM Systems
研 究 生：蕭清文 Student: Ching-Wen Hsiao
指導教授：李大嵩 博士 Advisor:
Dr. Ta-Sung Lee
用於多輸入多輸出正交分頻多工系統之
波束形成輔助多用戶適應性無線電資源管理技術
Beamforming Aided Multiuser Adaptive Radio
Resource Management for MIMO-OFDM Systems
研 究 生：蕭清文 Student: Ching-Wen Hsiao
指導教授：李大嵩 博士 Advisor:
Dr. Ta-Sung Lee
國立交通大學
電信工程學系碩士班
碩士論文
A Thesis
Submitted to Institute of Communication Engineering
College of Electrical Engineering and Computer Science
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
in
Communication Engineering
June 2006
Hsinchu, Taiwan, Republic of China
用於多輸入多輸出正交分頻多工系統之
波束形成輔助多用戶適應性無線電資源管理技術
學生：蕭清文
指導教授：李大嵩 博士
國立交通大學電信工程學系碩士班
摘要
多輸入多輸出(Multiple-Input Multiple-Output, MIMO)為使用多天線於傳送和 接收端的可靠通訊技術，並被認為是符合第四代高速通訊需求的最佳方案之一， 透過空間多工的方式，多輸入多輸出技術可在空間中的獨立平行通道傳送不同資 料串流，藉以提昇系統的整體傳輸速率。另一方面，正交分頻多工（Orthogonal Frequency Division Multiplexing, OFDM）為一種具高頻譜效益，並能有效克服多 路徑衰落效應的調變技術。在本論文中，吾人將探討結合多輸入多輸出技術與多 用戶正交分頻多工系統的通訊系統架構。基於相同副載波對於不同用戶會展現不 同通道條件的現象，吾人將針對多用戶正交分頻多工系統提出一種動態副載波配 置演算法，此演算法考慮個別用戶對於服務品質與傳輸速率不同的需求，配置一 組最適當的副載波給每一用戶，藉以提昇系統的整體傳輸速率，並藉由波束形成 技術，使用多用戶能在不干擾彼此的情況下，同時在相同的副載波傳送。此外， 吾人更進一步針對多用戶多輸入多輸出正交分頻多工系統提出一種適應性傳收架 構及位元負載演算法，使系統能夠隨時間動態地在頻率與空間通道上調整傳輸參 數—例如，調變階數與傳輸能量—以便充分地利用空間、時間以及頻率通道上的 特性以維持系統的目標錯誤率和各用戶的傳輸速率要求，同時更進一步有效降低 系統所需要傳送能量。最後，吾人藉由電腦模擬驗證上述架構在多用戶無線通訊 環境中具有優異的傳輸表現。
Beamforming Aided Multiuser Adaptive Radio
Resource Management for MIMO-OFDM Systems
Student:
Ching-Wen
Hsiao Advisor:
Dr.
Ta-Sung
Lee
Institute of Communication Engineering
National Chiao Tung University
Abstract
Multiple-input multiple-output (MIMO) is a promising technique suited to the increasing demand for high-speed 4G broadband wireless communications. Through spatial multiplexing, the MIMO technology can transmit multiple data streams in independent parallel spatial channels, hence increase the total transmission rate of the system. On the other hand, orthogonal frequency division multiplexing (OFDM) is a high spectral efficiency modulation technique that can efficiently deal with multipath fading effects especially suited to multiuser systems. In this thesis, a new wireless communication system called the multiuser MIMO-OFDM system is considered. Based on the fact that the same subcarrier experiencing different channel conditions for different users, a dynamic subcarrier allocation algorithm is proposed. This algorithm enhances the overall transmission rate of the system by allocating the most appropriate subcarriers to each user under the constraints of user-specific quality of service (QoS). In addition, by using beamforming techniques, we also allow users to occupy the same subcarrier at the same time without interfering each other. It results in system performance enhancement. Then, an adaptive multiuser MIMO-OFDM transceiver architecture along with a bit loading algorithm is proposed, which dynamically adjusts the transmission parameters such as modulation order and transmit power over spatial and frequency channels, to fully exploit the properties of the space-time-frequency channels to meet the target bit error rate (BER) and each user’s rate requirement and further reduce the overall transmission power of the system. Finally, the performance of the proposed systems is evaluated by computer simulations, confirming that they work well in multiuser wireless communication environments.
Acknowledgement
I would like to express my deepest gratitude to my advisor, Dr. Ta-Sung Lee, for his enthusiastic guidance and great patience. I learned a lot from his positive attitude in many areas. Heartfelt thanks are also offered to all members in the Communication System Design and Signal Processing (CSDSP) Lab for their constant encouragement and help. Finally, I would like to show my sincere thanks to my parents for their inspiration and invaluable love.
Contents
Chinese Abstract I
English Abstract II
Acknowledgement III
Contents IV
List of Figures VII
List of Tables IX
Acronym Glossary X
Notations XII
Chapter 1 Introduction 1
Chapter 2 MIMO-OFDM Technique Overview 5
2.1 Introduction to MIMO-OFDM System... 6
2.2 Channel Capacity ... 7
2.2.2 SIMO and MISO Channel Capacity ... 10
2.2.3 MIMO Channel Capacity... 10
2.3 Diversity Based on MIMO Techniques ... 15
2.3.1 Receive Diversity... 16
2.3.2 Transmit Diversity: Space-Time Codes ... 17
2.3.2.1 Alamouti Scheme ... 18
2.4 Spatial Multiplexing Based on MIMO Techniques ... 20
2.4.1 Diagonal Bell Lab’s Layered Space-Time (D-BLAST) ... 21
2.4.2 Vertical Bell Lab’s Layered Space-Time (V-BLAST) ... 25
2.5 Beamforming Based on MIMO Techniques ... 29
2.5.1 Generic Beamforming... 29
2.5.2 Eigenbeamforming Technique ... 30
2.6 Review of OFDM ... 31
2.7 Summary ... 38
Chapter 3 Joint Beamforming and Subcarrier Allocation for
Multiuser MIMO-OFDM Systems 39
3.1 System Model and Problem Formulation ... 40
3.1.1 Singular Value Decomposition (SVD)... 41
3.1.2 System Model ... 42
3.2 Proposed Beamforming Scheme with Interference Nulling ... 43
3.3 Computer Simulations of ZF Algorithm... 45
3.4 Algorithm of Subcarrier Selection with Beamforming... 48
3.5 Computer Simulations ... 50
Chapter 4 Multiuser Adaptive Power and Bit Allocation for
OFDM Systems 52
4.1 System Model and Problem Formulation ... 53
4.2 Power and Bit Allocation Algorithm ... 54
4.2.1 Conventional Power and Bit Allocation Algorithm... 54
4.2.2 Proposed Power and Bit Allocation Algorithm with Low Complexity ... 56
4.3 Complexity Analysis... 67
4.4 Computer Simulations ... 68
4.5 Summary ... 71
Chapter 5 Beamforming Aided Multiuser Adaptive Radio
Resource Management 72
5.1 Proposed Beamforming Aided Multiuser Radio Resource Management Algorithm...73
5.2 Signaling Model in Adaptive Scenario ... 76
5.3 Computer Simulations ... 77
5.4 Summary ... 80
Chapter 6 Conclusion 81
List of Figures
Figure 2.1: (I) Conventional multicarrier technique (II) Orthogonal multicarrier
modulation technique... 7
Figure 2.2: (I) Receive diversity (II) Transmit diversity... 16
Figure 2.3: An illustration of a spatial multiplexing system ... 20
Figure 2.4: Diagonal Bell Labs’ Layered Space-Time encoding procedure ... 22
Figure 2.5: Diagonal Bell Labs’ Layered Space-Time decoding procedure. ... 23
Figure 2.6: Vertical Bell Labs’ Layered Space-Time encoding procedure. ... 26
Figure 2.7: Vertical Bell Labs’ Layered Space-Time decoding procedure. ... 26
Figure 2.8: The effect of guard period in multipath case... 34
Figure 2.9: Structure of complete OFDM signal with guard period... 34
Figure 2.10: Transceiver for OFDM systems... 36
Figure 3.1: Scenario of transmit and receive beamformer to null interference ... 40
Figure 3.2: Converting H into a parallel channel through the SVD ... 42
Figure 3.3: Block diagram of proposed MIMO-OFDM system with adaptive beamforming ... 42
Figure 3.4: BER performances of 2x2 ZF algorithm ... 47
Figure 3.5: BER performances of 4x4 ZF algorithm ... 47
Figure 3.6: Block diagram of beamforming aided subcarrier selection in single carrier case... 48
Figure 3.7: BER performances of beamforming aided subcarrier selection with 2x2
antennas and 16QAM modulation ... 50
Figure 4.1: Bit distribution after initial allocation ... 57
Figure 4.2: Comparison between optimal and the initial allocation ... 58
Figure 4.3: Bit distribution after second stage allocation ... 58
Figure 4.4: Procedure of bit-filling algorithm in two subcarrier case... 60
Figure 4.5: Flow chart of initial stage of proposed bit loading algorithm ... 62
Figure 4.6: Flow chart of second stage of proposed bit loading algorithm ... 63
Figure 4.7: Example of proposed bit loading algorithm ... 67
Figure 4.8: BER performance of adaptive bit and power allocation compared with fixed modulation ... 69
Figure 4.9: BER performances of proposed bit loading algorithm with different bits requirements... 70
Figure 4.10: BER performances of proposed bit loading algorithm with different BER constraints ... 70
Figure 5.1: Block diagram of downlink multiuser adaptive MIMO-OFDM systems .. 75
Figure 5.2: Transmitter architecture of beamforming aided radio resource management algorithm... 75
Figure 5.3: Signaling model in adaptive scenario... 77
Figure 5.4: BER performances of beamforming aided multiuser resource allocation compared to single user case ... 79
Figure 5.5: BER performances of beamforming aided multiuser resource allocation with channel estimation error ... 79
List of Tables
Table 3.1: Simulation parameters of ZF algorithm ... 46 Table 4.1: Complexity of proposed bit and power loading algorithm compared with
other conventional algorithms ... 68
Table 4.2: Example of complexity analysis ... 68 Table 5.1: Simulation parameters of beamforming aided multiuser adaptive radio
Acronym Glossary
ADC analog-to-digital conversion
AWGN additive white Gaussian noise
BER bit error rate
BLAST Bell Lab Layered space time BPSK binary phase shift keying
BS base station
CCI co-channel interference
CNR channel gain-to-noise ratio
CP cyclic prefix
CRC cyclic redundancy check
CSI channel state information
D-BLAST diagonal Bell labs’ layered space-time
DFT discrete Fourier transform
FDMA frequency division multiple access
FFT fast Fourier transform
FSK frequency shift keying
ICI intercarrier interference
IEEE institute of electrical and electronics engineers IFFT inverse fast Fourier transform
ISI intersymbol interference
LOS line of sight
MAI multiple access interference
MIMO multiple-input multiple-output
MISO multiple-input single-output
MMSE minimum mean square error
MS mobile station
OFDM orthogonal frequency division multiplexing QAM quadrature amplitude modulation
QoS quality of service
QPSK quaternary phase shift keying
RF radio frequency
SD spatial diversity
SIMO single-input multiple-output
SISO single-input single-output
SM spatial multiplexing
SNR signal-to-noise ratio
STC space-time coding
SVD singular value decomposition
TDD time division duplex
TDMA time division multiple access
V-BLAST vertical Bell laboratory layered space-time
WF water filling
Notations
b number of bits can be assigned in the system
B total number of assigned bits
req
B rate requirement per OFDM system
C channel capacity b E bit energy s E symbol energy , i j
h channel gain between the jth transmit and ith receive antenna
H channel matrix
K number of users in one cell
M modulation order
N number of subcarriers cp
N number of guard interval samples t
N number of transmit antenna r
N number of receive antenna
P allocated power s T symbol duration sample T sampling period R
w receive beamforming vector T
w transmit beamforming vector 2
n
σ noise power
δ subcarrier indicator
Chapter 1
Introduction
The increasing popularity of enhanced communication services, such as wireless multimedia, telecommuting, fast Internet access, and video conferencing, has promoted the growing demand of high data rate, high mobility, and high quality of service (QoS) requirements for users. However, the limited available bandwidth drives the wireless communication technology towards the emerging issue of high spectral efficiency. Besides, in wireless channels, the time-selective and frequency-selective fading caused by multipath propagation, carrier frequency/phase shift, and Doppler shift limit the developments of high data rate and reliable communications. As a remedy, some efficient modulation and coding schemes such as coded multicarrier modulation, multiple-input multiple-output (MIMO) technology [1]-[11], and adaptive resource allocation [12]-[14] are proposed to enhance the spectral efficiency and quality of wireless communication links.
MIMO systems, which use multiple antennas at both transmitter and receiver, provide spatial diversity that can be used to mitigate signal-level fluctuations in fading channels [1]-[3]. In narrowband channels, MIMO systems can provide a diversity advantage in proportion to the product of the number of transmit and receive antennas. When the channel is unknown to the transmitter, diversity can be obtained by using
space-time codes [1]-[3]. When channel state information (CSI) is available at the transmitter, however, diversity can be obtained using a simple approach known as transmit beamforming and receive combining [4]-[11]. Compared with space-time codes, beamforming and combining achieves the same diversity order as well as additional array gain; thus, it can significantly improve system performance. This approach, however, requires knowledge of the transmit beamforming vector at the transmitter. When the uplink and downlink channels are not reciprocal (as in a frequency division duplexing system), the receiver informs the transmitter about the desired transmit beamforming vector through a feedback channel.
The high data rate wireless transmission over the multipath fading channels is mainly limited by intersymbol interference (ISI). Orthogonal frequency division multiplexing (OFDM) [17]-[18] has been considered as a reliable technology to deal with the ISI problem. The principle of the OFDM technology is to split a high data rate stream into a number of low data rate streams which are simultaneously transmitted on a number of orthogonal subcarriers. By adding a cyclic prefix (CP) to each OFDM symbol, both intersymbol and intercarrier interference can be removed and the channel also appears to be circular if the CP length is longer than the channel length. The multicarrier property of OFDM systems can not only improve the immunity to fast fading channels, but also make multiple access possible because the subcarriers are independent of each other.
OFDM combining antenna arrays at both the transmitter and receiver, which leads to a MIMO-OFDM configuration, can significantly increase the diversity gain or enhance the system capacity over time-variant and frequency-selective channels. Typical MIMO-OFDM systems can be categorized into two types: those based on spatial multiplexing (SM), [16] and those based on spatial diversity (SD) schemes. The former system is a layered spatial transmission scheme, in which different data streams
are transmitted from different transmit antennas simultaneously and received through nulling or canceling to mitigate the co-channel interference (CCI). The latter one uses beamforming techniques to improve the transmission reliability. Beamforming can provide power gain and increase the effective transmission rate without sacrificing the bandwidth.
In the multiuser MIMO-OFDM system, each of the multiple users’ signals may undergo independent fading due to different locations of users. Therefore, the subcarriers in deep fade for one user may not be in the same fade for other users. In fact, it is quite unlikely that a subcarrier will be in deep fade for all users. By beamforming technique, the system can allow multiple users to occupy the same subcarrier without interfering each other. Hence, for a specific subcarrier, the groups of users with the best channel quality can use the subcarrier to transmit data yielding multiuser diversity effects [19]. Recently, methods for dynamically assigning subcarriers to each user have been widely investigated [20]-[21]. These dynamic subcarrier allocation algorithms can be geared to decrease the power consumption for a given achievable data rate or to increase the data rate when the available power is limited. In this thesis, a beamforming aided dynamic subcarrier allocation algorithm suited to the multiuser MIMO-OFDM system is developed to take both user-specific data rate and BER requirements into account and allocate to each user the most appropriate subcarriers with ZF beamforming.
As the instantaneous channel state information (CSI) is determined beforehand, the multiuser MIMO-OFDM system incorporating the adaptive modulation technique can provide a significant performance improvement. Adaptive modulation can dynamically adjust transmission parameters to alleviate the effects of channel impairments. Subcarriers with good channel qualities can employ higher modulation
employ lower modulation order or even no transmission. In addition, the adaptive modulation technique must take into account the additional signaling dimensions explored in future broadband wireless networks [15]. More specifically, the growing popularity of both MIMO and multiuser OFDM systems creates the demand for link adaptation solutions to integrate temporal, spectral, and spatial components together. In this thesis, an adaptive wireless transceiver called beamforming aided multiuser adaptive radio resource management is developed to effectively exploit the available degrees of freedom in wireless communication systems.
This thesis is organized as follows. In Chapter 2, the channel capacity and basic technique of a MIMO communication link are described first. Secondly the multiple access concepts of OFDM system is given. In Chapter 3, a beamforming aided subcarrier allocation algorithm is proposed which allows users to choose the most appropriate subcarriers according to their requirements and the channel qualities. This algorithm fully utilized the advantage of spatial diversity and power gain by beamforming and frequency diversity by subcarrier allocation. In Chapter 4, the adaptive modulation concepts are introduced and an optimal bit loading algorithm with low complexity suited to the multiuser MIMO-OFDM system is proposed to further decrease the total transmit power and still meet the target bit error rate (BER) and rate requirement. In Chapter 5, we combine the algorithm described in Chapter 3 and 4 which adaptively adjust the beamforming vectors, subcarrier, modulation orders and power for each user. Finally, Chapter 6 gives concluding remarks of this thesis and leads the way to some potential future works.
Chapter 2
MIMO-OFDM Technique Overview
Digital communication using multiple-input multiple-output (MIMO) has recently emerged as one of the most significant technical breakthroughs in wireless communications. The technology figures prominently on the list of recent technical advances with a chance of resolving the bottleneck of traffic capacity in future Internet intensive wireless networks.
In recent years, there has been substantial research interest in applying orthogonal frequency division multiplexing (OFDM) to high speed wireless communications due to its advantage in mitigating the severe effects of frequency-selective fading [22]-[23]. In this chapter, the basic ideas and key feature of MIMO–OFDM system will be introduced.
This Chapter focuses on the MIMO-OFDM techniques. An overview of the MIMO-OFDM system will first be given. Then we introduce MIMO techniques followed by introduction of MIMO-OFDM system which include the capacity, spatial multiplexing and diversity view. Finally OFDM technique is given in this chapter.
2.1 Introduction to MIMO-OFDM System
OFDM has long been regarded as an efficient approach to combat the adverse effects of multipath spread, and is the main solution to many wireless systems. It converts a frequency-selective channel into a parallel collection of frequency flat subchannels, which makes the receiver simpler. The time domain waveforms of the subcarriers are orthogonal, yet the signal spectrum corresponding to the different subcarriers overlap in frequency domain. Therefore, the available bandwidth is used very efficiently, especially compared with those systems having intercarrier guard bands, as shown in Figure 2.1. In order to eliminate inter-symbol interference (ISI) almost completely, a guard time is introduced for each OFDM symbol. Moreover, to eliminate inter-carrier interference (ICI), the OFDM symbol is further cyclically extended in the guard time, resulting in the cyclic prefix (CP). Otherwise, multipath remains an advantage for an OFDM system since the frequency selectivity caused by multipaths can improve the rank distribution of the channel matrices across those subcarriers, thereby increasing system capacity. We summarize the advantages of OFDM as follows [24]:
¾ High spectral efficiency
¾ Simple implementation by FFT
¾ Robustness against narrowband interference
¾ High flexibility in terms of link adaptation for having many subcarriers ¾ Suitability for high-data-rate transmission over a multipath fading channel
MIMO systems where multiple antennas are used at both the transmitter and receiver have been also acknowledged as one of the most promising techniques to achieve dramatic improvement in physical-layer performance [25], [26]. Moreover, the use of multiple antennas enables space-division multiple access (SDMA), which
allows intracell bandwidth reuse by multiplexing spatially separable users [27], [28]. Channel variation in the spatial domain also provides an inherent degree of freedom for adaptive transmission. To sum up, after OFDM is combined with MIMO techniques, MIMO-OFDM can be a potential candidate for the next generation wireless communication systems.
Ch.1 Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10 (a) Frequency Ch.1 Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10 (a) Frequency Saving of bandwidth (b) Frequency Saving of bandwidth (b) Frequency
Figure 2.1: (I) Conventional multicarrier technique
(II) Orthogonal multicarrier modulation technique
2.2 Channel Capacity
A measure of how much information that can be transmitted and received with a negligible probability of error is called the channel capacity. To determine this measure of channel potential, a channel encoder receiving a source symbol every Ts second is assumed. If S represents the set of all source symbols and the entropy rate of the source
is written asH s , the channel encoder will receive H(s)/T( ) s information bits per second
on average. A channel codeword leaving the channel encoder every Tc second is also
assumed. In order to be able to transmit all the information from the source, there must be R information bits per channel symbol.
( ) c
H s T R
T
The number R is called the information rate of the channel encoder. The maximum information rate that can be used causing negligible probability of errors at the output is called the capacity of the channel. By transmitting information with the rate R, the channel is used every Tc seconds. The channel capacity is then measured in bits per
channel use. Assuming that the channel has bandwidth W, the input and output can be represented by samples taken Ts = 1/2W seconds apart. With a band-limited channel,
the capacity is measured in information bits per second. It is common to represent the channel capacity within a unit bandwidth of the channel, which means that the channel capacity is measured in bits/sec/Hz.
It is desirable to design transmission schemes that exploit the channel capacity as much as possible. Representing the input and output of a memoryless wireless channel with the random variables X and Y respectively, the channel capacity is defined as
( )
(
)
max ; bits/sec/Hz p x
C= I X Y (2.2)
where ( ; )I X Y represents the mutual information between X and Y. Equation (2.2)
states that the mutual information is maximized when considering all possible transmitter statistical distributions p(x). Mutual information is a measure of the amount of information that one random variable contains about another one. The mutual information between X and Y can also be written as
( ; )I X Y =H Y( )−H Y X( | ) (2.3)
where ( | )H Y X represents the conditional entropy between the random variables X
and Y. The entropy of a random variable can be described as a measure of the uncertainty of the random variable. It can also be described as a measure of the amount of information required on average to describe the random variable. Due to Equation (2.3), mutual information can be described as the reduction in the uncertainty of one random variable due to the knowledge of the other. Note that the mutual information
between X and Y depends on the properties of X (through the probability distribution of
X) and the properties of channel (through a channel matrix H). In the following, four
different kinds of channel capacities are introduced (single-input single-output (SISO), single-input multiple-output (SIMO), multiple-input single-output (MISO), and MIMO) to get the further concepts about the properties of the channel capacity.
2.2.1 SISO Channel Capacity
The ergodic (mean) capacity of a random channel (Nt = Nr = 1) with the average
transmit power constraint PT can be expressed as
( )
(
)
{
max: T ;}
bits/sec/Hz H p x P P C E I X Y ≤ = (2.4)where EH denotes the expectation over all channel realizations and P is the average
power of a single channel codeword transmitted over the channel. Compared to the definition in Equation (2.2), the capacity of the channel is now defined as the maximum of the mutual information between the input and output over all statistical distributions on the input that satisfy the power constraint. If each channel symbol at the transmitter is denoted by s, the average power constraint can be expressed as
2
T
P=E s⎡⎣ ⎤⎦≤P (2.5)
Using Equation (2.4), the ergodic (mean) capacity of a SISO system (Nt = Nr = 1) with a random complex channel gain h11 is given by
(
)
{
2}
2 11 log 1 bits/sec/Hz H C=E + ⋅ρ h (2.6)where ρ is the average signal-to-noise (SNR) ratio at the receive branch. If |h11| is
Rayleigh, |h11|2 follows a chi-squared distribution with two degrees of freedom.
(
)
{
2}
2 2 log 1 bits/sec/Hz H C=E + ⋅ρ χ (2.7)where χ22 is a chi-square distributed random variable with two degrees of freedom.
2.2.2 SIMO and MISO Channel Capacity
As more antennas deployed at the receiving end, the statistics of capacity improve. Then the ergodic (mean) capacity of a SIMO system with Nr receive antennas is given
by 2 2 1 1 log (1 ) bits/sec/Hz r N i i C ρ h = = +
∑
(2.8)where hi1 represents the gain for the receive antenna i. Note that the crucial feature in
Equation (2.8) is that increasing the number of receive antennas Nr only results in a
logarithmic increase in the ergodic (mean) capacity. Similarly, if transmit diversity is opted, in the common case, where the transmitter doesn’t have the channel knowledge, the ergodic (mean) capacity of a MISO system with Nt transmit antennas is given by
2 2 1 1 log (1 ) bits/sec/Hz t N i i t C h N ρ = = +
∑
(2.9)where the normalization by Nt ensures a fixed total transmit power and shows the
absence of array gain in that case (compared to the case in Equation (2.8), where the channel energy can be combined coherently). Again, the capacity has a logarithmic relationship with the number of transmit antennas Nt.
2.2.3 MIMO Channel Capacity
The capacity of a random MIMO channel with the power constraint PT can be
( ) ( )
( )
{
max: T ;}
bits/sec/Hz H p tr P C E I Φ ≤ = x x y (2.10)where Φ=E xx{ H}2 is the covariance matrix of the transmit signal vector x. By the relationship between mutual information and entropy and using Equation (2.1), Equation (2.10) can be expanded as follows for a given channel matrix H.
( )
( ) (
)
( ) (
)
( ) (
)
( ) ( )
; | | | I h h h h h h h h = − = − + = − = − x y y y x y Hx n x y n x y n (2.11)where h
( )
⋅ denotes the differential entropy of a continuous random variable. It is assumed that the transmit vector x and the noise vector n are independent.When y is Gaussian, Equation (2.11) is maximized. Since the normal distribution maximizes the entropy for a given variance. The differential entropy of a complex Gaussian vector y ∈ Cn
, the differential entropy is less than or equal to log det2
(
πeK)
,with equality if and only if y is a circularly symmetric complex Gaussian with { H}
E yy = K . For a real Gaussian vector y ∈ Rn with zero mean and covariance matrix,
K is equal tolog2
(
(2 ) det)
/ 2 ne
π K . Assuming the optimal Gaussian distribution for the transmit vector x, the covariance matrix of the received complex vector y is given by
{ }
{
(
)(
)
}
{
} { }
H H H H H H n d n E E E E = + + = + = + = + yy Hx n Hx n Hxx H nn HΦH K K K (2.12)The desired part and the noise part of Equation (2.12) denotes respectively by the superscript d and n. The maximum mutual information of a random MIMO channel is then given by
( ) ( )
(
)
(
)
(
)
(
)
( )
(
)( )
(
)
( )
(
)
( )
(
)
2 2 2 2 1 2 1 2 1 2log det log det
log det log det
log det log det log det d n n d n n d n n d n M H n M I h h e e π π − − − = − ⎡ ⎤ ⎡ ⎤ = ⎣ + ⎦− ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ = ⎣ + ⎦− ⎣ ⎦ ⎡ ⎤ = ⎢ + ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ + ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ + ⎥ ⎣ ⎦ y n K K K K K K K K K K K I HΦH K I (2.13)
When the transmitter has no knowledge about the channel, it is optimal to use a uniform power distribution. The transmit covariance matrix is then given by
/ t
T N t
P N
Φ = I . It is also common to assume uncorrelated noise in each receive antenna described by the covariance matrix 2
r
n
N
σ =
K I . The ergodic (mean) capacity for a complex additive white Gaussian noise (AWGN) MIMO channel can then be expressed as
2 2
log det bits/sec/Hz
r H T H N t P C E N σ ⎧ ⎡ ⎛ ⎞⎤⎫ ⎪ ⎪ = ⎨ ⎢ ⎜ + ⎟⎥⎬ ⎝ ⎠ ⎪ ⎣ ⎦⎪ ⎩ I HH ⎭ (2.14)
Equation (2.14) can also be written as
2
log det bits/sec/Hz
r H H N t C E N ρ ⎧ ⎡ ⎛ ⎞⎤⎫ ⎪ ⎪ = ⎨ ⎢ ⎜ + ⎟⎥⎬ ⎝ ⎠ ⎪ ⎣ ⎦⎪ ⎩ I HH ⎭ (2.15) where 2 / T P
ρ = σ is the average signal-to-noise (SNR) at each receive antenna. By the law of large numbers, the term /
r H
t N
N →
HH I as Nr is fixed and Nt gets large.
Hence the capacity in the limit of large transmit antennas Nt can be written as
(
)
{
log 12}
bits/sec/HzH r
C=E N ⋅ +ρ (2.16)
Further analysis of the MIMO channel capacity given in Equation (2.15) is possible by diagonalizing the product matrix HHH either by eigenvalue decomposition or singular value decomposition (SVD). By using SVD, the matrix
product is written as
H
=
H UΣV (2.17)
where U and V are unitary matrices of left and right singular vectors respectively, and Σ is a triangular matrix with singular values on the main diagonal. All elements on the diagonal are zero except for the first k elements. The number of non-zero singular values k of Σ equals the rank of the channel matrix. Substituting Equation (2.17) into Equation (2.15), the MIMO channel capacity can be written as
2
log det bits/sec/Hz
r H H H N t C E I N ρ ⎧ ⎡ ⎛ ⎞⎤⎫ ⎪ ⎪ = ⎨ ⎢ ⎜ + ⎟⎥⎬ ⎝ ⎠ ⎪ ⎣ ⎦⎪ ⎩ UΣΣ U ⎭ (2.18)
The matrix product HHH can also be described by using eigenvalue decomposition on the channel matrix H written as
H = H
HH EΛE , (2.19)
where E is the eigenvector matrix with orthonormal columns and Λ is a diagonal matrix with the eigenvalues on the main diagonal. Using this notation, Equation (2.15) can be written as
2
log det bits/sec/Hz
r H H N t C E N ρ ⎧ ⎡ ⎛ ⎞⎤⎫ ⎪ ⎪ = ⎨ ⎢ ⎜ + ⎟⎥⎬ ⎝ ⎠ ⎪ ⎣ ⎦⎪ ⎩ I EΛE ⎭ (2.20)
After diagonalizing the product matrix HHH, the capacity formulas of the MIMO channel now includes unitary and diagonal matrices only. It is then easier to see that the total capacity of a MIMO channel is made up by the sum of parallel AWGN SISO subchannels. The number of parallel subchannels is determined by the rank of the channel matrix. In general, the rank of the channel matrix is given by
( )rank H = ≤k min(N Nt, r) (2.21)
Using the fact that the determinant of a unitary matrix is equal to 1 and Equation (2.21), Equations (2.18) and (2.20) can be expressed respectively as
2 2 1 log 1 bits/sec/Hz k H i i t C E Nρ σ = ⎧ ⎛ ⎞⎫ ⎪ ⎪ = ⎨ ⎜ + ⎟⎬ ⎪ ⎝ ⎠⎪ ⎩
∑
⎭ (2.22) 2 1 log 1 bits/sec/Hz k H i i t E Nρ λ = ⎧ ⎛ ⎞⎫ ⎪ ⎪ = ⎨ ⎜ + ⎟⎬ ⎪ ⎝ ⎠⎪ ⎩∑
⎭ (2.23)where σi2 are the squared singular values of the diagonal matrix Σ and λi are the
eigenvalues of the diagonal matrix Λ. The maximum capacity of a MIMO channel is achieved in the unrealistic situation when each of the Nt transmitted signals is received
by the same set of Nr antennas without interference. It can also be described as if each
transmitted signal is received by a separate set of receive antennas, giving a total number of N Nt⋅ r receive antennas.
With optimal combining at the receiver and receive diversity only (Nr = 1), the
channel capacity can be expressed as
(
)
{
2}
2 2 log 1 bits/sec/Hz r H N C=E + ⋅ρ χ (2.24) where 22 r Nχ is a chi-distributed random variable with 2Nr degrees of freedom. If there
are Nt transmit antennas and optimal combining between Nr antennas at the receiver,
the capacity can be written as
2 2 2 log 1 bits/sec/Hz r H t N t C E N Nρ χ ⎧ ⎛ ⎞⎫ ⎪ ⎪ = ⎨ ⋅ ⎜ + ⋅ ⎟⎬ ⎪ ⎝ ⎠⎪ ⎩ ⎭ (2.25)
Equation (2.25) represents the upper bound of a Rayleigh fading MIMO channel. When the channel is known at the transmitter, the maximum capacity of a MIMO channel can be achieved by using the water-filling (WF) principle on the transmit covariance matrix. The capacity is then given by
2 1 2 2 1 2 1 log 1 log 1 log ( ) bits/sec/Hz k H i i i t k H i i i t k i i C E N E N u ρ ε λ ρ ε σ λ = = + = ⎧ ⎛ ⎞⎫ ⎪ ⎪ = ⎨ ⎜ + ⎟⎬ ⎪ ⎝ ⎠⎪ ⎩ ⎭ ⎧ ⎛ ⎞⎫ ⎪ ⎪ = ⎨ ⎜ + ⎟⎬ ⎪ ⎝ ⎠⎪ ⎩ ⎭ =
∑
∑
∑
(2.26)where “+＂ denotes taking only those terms which are positive and u is a scalar, representing the portion of the available transmit power going into the ith subchannel which is chosen to satisfy
1 1 ( ) k i i u ρ λ− + = =
∑
− (2.27)Since u is a complicated nonlinear function of λ λ1, 2,…,λk, the distribution of the channel capacity appears intractable, even in the Wishart case when the joint distribution of λ λ1, 2,…,λk is known. Nevertheless, the channel capacity can be simulated using Equations (2.26) and (2.27) for any given HHH so that the optimal capacity can be computed numerically for any channel [29].
2.3 Diversity Based on MIMO Techniques
Antenna diversity, or spatial diversity, can be obtained by placing multiple antennas at the transmitter and/or the receiver. If the antennas are placed sufficiently far apart, the channel gains between different antennas pairs fade more independently, and independent signal paths are created. The required antenna separation depends on the local scattering environment as well as on the carrier frequency. For a mobile which is near the round with many scatterers around, the channel decorrelates over shorter spatial distances, and typical antenna separation of half to one carrier wavelength is sufficient. For base stations on high towers, larger antenna separation of several to 10’s of wavelengths may be required.
We will look at both receive diversity, using multiple receive antennas (single-input, multi-output SIMO channels), and transmit diversity, using multiple transmit antennas (multi-input, single-output MISO channels). Interesting coding problems arise in the latter and have led to recent excitement in space-time codes.
(I) (II)
Figure 2.2: (I) Receive diversity (II) Transmit diversity
2.3.1 Receive Diversity
In a flat fading channel with one transmit antenna and L receive antennas (Figure 2.2 (I)), the channel model is as follows:
[ ] [ ] [ ] [ ] 1,...,
l l l
y m =h m x m +w m l = L (2.28) where the noise w ml[ ] ~CN(0,N0) and independent across the antennas. We would
like to detect x[1] based on y1[1],...,yL[1]. This is exactly the same detection problem as in the use of a repetition cod over time, with L diversity branches now over space instead of over time. If the antennas are spaced sufficiently far apart, then we can assume that the gains hl[1]are independent Rayleigh, and we get a diversity gain of L. With receive diversity, there are actually two types of gain as we increase L. this can be seen for the error probability of BPSK conditioned on the channel gains:
2
( 2 SNR )
We can break up the total received SNR conditioned on the channel gains into a product of two terms:
2 1 2
SNR= SNRL L
⋅
h h . (2.30)
The first term corresponds to a power gain (also called array gain): by having multiple receive antennas and coherent combining at the receiver, the effective total received signal power increases linearly with L: doubling L yields a 3 dB power gain. The second term reflects the diversity gain: by averaging over multiple independent signal paths, the probability that the overall gain is small is decreased. The diversity gain L is reflected in the SNR exponent; the power gain affects the constant before the 1/ SNRL. Note that if the channel gains hl[1] are fully correlated across all branches, then we only get a power gain but no diversity gain as we increase L. on the other hand, even when all the h are independent there is a diminishing marginal return as L increases: l
due to the law of large numbers, the second term in (2.30),
2 2 1 1 1 [1] L l l h L L = =
∑
h , (2.31)Converges to 1 with increasing L (assuming each of the channel gains is normalized to have unit variance). The power gain, on the other hand, suffers from no such limitation: a 3 dB gain is obtained for every doubling of the number of antennas.
2.3.2 Transmit Diversity: Space-Time Codes
Now consider the case when there are L transmit antennas and one receive antenna, the MISO channel (Figure 2.2 (II)). This is common in the downlink of a cellular system since it is often cheaper to have multiple antennas at the base station than to having multiple antennas at every handset. It is easy to get a diversity gain of L: simply
one time, only one antenna is turned on and the rest are silent. This is simply a repetition code, and , as we have seen in the previous section, repetition codes are quite wasteful of degrees of freedom.
More generally, any time diversity code of block length L can be used on this transmit diversity system: simply use one antenna at a time and transmit the coded symbols of the time diversity code successively over the different antennas. This provides a coding gain over the repetition code. One can also design a code specifically for the transmit diversity system. There have been a lot of research activities in this area under the rubric of space-time coding and here we discuss the simplest, and yet one of the most elegant, space-time code which is called Alamouti scheme. This is the transmit diversity scheme proposed in several third-generation cellular standards. Alamouti scheme is designed for two transmit antennas; generalization to more than two antennas is possible, to some extent.
2.3.2.1 Alamouti Scheme
With flat fading, the two transmit, single receive channel is written as
1 1 2 2
[ ] [ ] [ ] [ ] [ ] [ ]
y m =h m x m +h m x m +w m (2.32) where hi is the channel gain from transmit antennas i. the Alamouti scheme transmits
two complex symbols u1 and u2 over two symbol times: at time 1, x1[1]=u1, x2[1]=u2; at
time 2, x1[2]= -u2*, x2[1]= -u1. If we assume that the channel remains constant over the
two symbol times and set h1=h1[1]= h1[2], h2=h2[1]= h2[2], then we can write the
matrix form:
[
] [
]
1 *2[
]
1 2 2 1 [1] [2] u u [1] [2] y y h h w w u u ⎡ − ⎤ = ⎢ ⎥+ − ⎣ ⎦ . (2.33)1 2 1 * * * 2 1 2 [1] [1] [2]* [2] h h u y w h h u y w ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ ⎢ ⎥+ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (2.34)
We observe that the columns of the square matrix are orthogonal. Hence, the detection problem for u1,u2 decomposes into two separate, orthogonal, scalar problems. We
project y onto each of the two columns to obtain the sufficient statistics 1, 2 i i i r = h u +w i= (2.35) where [ ,1 2]t h h =
h and wi ~CN(0,N0) and w1, w2 are independent. Thus, the
diversity gain is 2 for the detection of each symbol. Compared to the repetition code, 2 symbols are now transmitted over two symbol times instead of 1 symbol, but with half the power in each symbol (assuming that the total transmit power is the same in both cases).
The Alamouti scheme works for any constellation for the symbols u1, u2, but suppose now they are BPSK symbols, thus conveying a total of two bits over two symbol times. In the repetition scheme, we need to use 4-PAM symbols to achieve the same data rate. To achieve the same minimum distance as the BPSK symbols in the Alamouti scheme, we need 5 times the energy per symbol. Take into account the factor of 2 energy saving since we are only transmitting one symbol at a time in the repetition scheme, we see that the repetition scheme requires a factor of 2.5 (4dB) more power than the Alamouti scheme. Again, the repetition scheme suffers from an inefficient utilization of the available degrees of freedom in the channel: over the two symbol times, bits are packed into only one dimension of the received signal space, namely along the direction [ ,1 2]t
h h . In contrast, the Alamouti scheme spreads the information onto two dimensions- along the orthogonal directions *
1 2 [ , ]t h h and * 2 1 [ , ]t h −h .
2.4 Spatial Multiplexing Based on MIMO
Techniques
The use of multiple antennas at both ends of a wireless link has recently been shown to have the potential of achieving extraordinary data rate. The corresponding technology is known as spatial multiplexing [30]-[34]. It allows a data rate enhancement in a wireless radio link without additional power or bandwidth consumption. In spatial multiplexing systems, different data streams are transmitted from different transmit antennas simultaneously or sequentially and these data streams are separated and demultiplexed to yield the original transmitted signals according to their unique spatial signatures at the receiver. An illustration of the spatial multiplexing system is shown in Figure. 2.3. The separation step is made possible by the fact that the rich scattering multipath contributes to lower correlation between MIMO channel coefficients, and creates a desirable full rank and low condition number coefficient matrix condition to resolve Nt unknowns from a linear system of Nr equations. In the
following, two typical spatial multiplexing schemes, D-BLAST [20] and V-BLAST [32], [33], are introduced.
Transmitter Receiver
2.4.1 Diagonal Bell Lab’s Layered Space-Time
(D-BLAST)
Space-time coding (STC) performs channel coding across space and time to exploit the spatial diversity offered by MIMO systems to increase system capacity. However, the decoding complexity of the space-time codes is exponentially increased with the number of transmit antennas, which makes it hard to implement real-time decoding as the number of antennas grows. To reduce the complexity of space-time based MIMO systems, D-BLAST architecture has been proposed in [30]. Rather than try to achieve optimal channel coding scheme, in D-BLAST architecture, the input data stream is divided into several substreams. Each substream is encoded independently using an elegant diagonally-layered coding structure in which code blocks are dispersed across diagonals in space-time and the association of corresponding output stream with transmit antenna is periodically cycled to explore spatial diversity. To decode each layer, channel parameters are used to cancel interference from undetected signals to make the desired signal as “clean” as possible.
Figure 2.4 shows the typical encoding steps in D-BLAST. Considering a system with Nt transmit and Nr receive antennas, the high rate information data stream is first
demultiplexed into Nt subsequences. Each subsequence is encoded by a conventional
1-D constituent code with low decoding complexity. The encoders apply these coded symbols to generate a semi-infinite matrix C of Nt rows to be transmitted. The element
in the pth row and tth column of C, ctp, is transmitted by the pth transmit antenna at time t . As illustrated in Figure. 2.3, c c c c c c11, 22, 33, 41, 52, 63 are encoded by encoder α,
1 2 3 1 2 3
2, 3, 4, ,5 6, 7
c c c c c c are encoded by encoder β, and c c c c c c13, 42, 53, ,61 72, 83 are encoded by encoder γ.
1 : 3 DEMUX Input Bits Encoder α Encoder β Encoder γ
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
=
C
1 1c
2 2c
3 3c
1 2c
2 3c
3 4c
1 3c
2 4c
3 5c
1 4c
2 5c
3 6c
0
0
0
Encoder α Encoder β Encoder γ 1 5c
c
16 2 6c
c
72 3 7c
c
83 Space Time Layer Decision ( Decoding ) Data Rate N∝
1 : 3 DEMUX Input Bits Encoder α Encoder β Encoder γ
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
=
C
1 1c
2 2c
3 3c
1 2c
2 3c
3 4c
1 3c
2 4c
3 5c
1 4c
2 5c
3 6c
0
0
0
Encoder α Encoder β Encoder γ 1 5c
c
16 2 6c
c
72 3 7c
c
83 Space Time Layer Decision ( Decoding ) Data Rate N∝
Figure 2.4: Diagonal Bell Labs’ Layered Space-Time encoding procedure
Figure 2.5 shows the typical decoding steps of interference suppression, symbol detection, decoding, and interference cancellation performed in D-BLAST. The receiver generates decisions for the first diagonal of C. Based on these decisions, the diagonal is decoded and fed back to remove the contribution of this diagonal from the received data. The receiver continues to decode the next diagonal and so on. The encoded substreams share a balanced presence over all paths to the receiver, so none of the individual substreams is subject to the worst path. Therefore, the data received at time t by the qth receive antenna is rtq, which contains a superposition of ctp,
1, 2, , t
p= … N , and an AWGN noise component. Then, the received data vector can be expressed as rt =H ct t+ξt at any time instance t . The D-BLAST method uses a repeated process of interference suppression, symbol detection, and interference
cancellation to decode all symbols, Nt, Nt 1,..., 1
t t t
c c − c . This decoding process can be expressed in a general form described in the following.
1.
2.
3.
4.
5.
6.
Nulled Detected Detected Nulled Detected Cancelled Decode1.
2.
3.
4.
5.
6.
Nulled Detected Detected Nulled Detected Cancelled DecodeFigure 2.5: Diagonal Bell Labs’ Layered Space-Time decoding procedure.
Let Q R be the QR decomposition of t t H , where t Q is an Nt r × Nr unitary
matrix and R is an Nt r × Nt upper triangular matrix. Multiplying the received signal
by Q , we can get tH N N Nr t H H H H H t t t t t t t t t t t t t t t t t = = + = + = + I ξ y Q r Q H c Q ξ Q Q R c Q ξ R c ξ (2.36) where
1, 1,1 1,2 2, 2,2 1 1 2 2 , 0 0 0 , , 0 0 0 0 0 0 0 0 t t t t r r N t t t N t t t t t N N t t t t t N N t t r r r r r y y r y ξ ξ ξ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = =⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ y R ξ " " % # # % # # % % % # " (2.37)
Since R is a upper triangular matrix, the elements in the vector t y can be t expressed as
{
}
, 1 2 contribution from c , ,..., t p N p p p p p p t t t t t t t y =r c +ξ + + c + c (2.38)Hence, the interference from ctq, q< ≤p Nt , is first suppressed in ytk and the residual interference terms in Equation (2.38) can be cancelled by the available decisions 1
ˆp
cτ+ , 2
ˆp
cτ+ ,…, ˆcNt
τ . Assuming all these decisions are correct, then the present
decision variable is , , 1, 2, , p p p p p t t t t t c =r c +ξ p= … N (2.39)
The relation between p
c and p
c in Equation (2.39) can be interpreted as the input
and output of a SISO channel with the channel power gain rp p, 2 and AWGN. The channel power gain rp p, 2 is independently chi-squared distributed with 2×(Nr−p+1)
degrees of freedom. Moreover, if there are no decision feedback errors, the pth row of the C matrix can be treated as transmitted over a (Nt, Nr)=(1, Nr−p+1) system without
2.4.2 Vertical Bell Lab’s Layered Space-Time
(V-BLAST)
The D-BLAST algorithm has been proposed by Foschini for achieving a substantial part of the MIMO capacity. In an independent Rayleigh scattering environment, this processing structure leads to theoretical rates which grow linearly with the number of antennas (assuming equal number of transmit and receive antennas) with these rates approaching ninety percents of Shannon capacity. However, the diagonal approach suffers from certain implementation complexities which make it inappropriate for practical implementation. Therefore, a simplified version of the BLAST algorithm is known as V-BLAST (vertical BLAST) [32], [33]. It is capable of achieving high spectral efficiency while being relatively simple to implement. The essential difference between D-BLAST and V-BLAST lies in the vector encoding process. In D-BLAST, redundancy between the substreams is introduced through the use of specialized intersubstream block coding. In V-BLAST, however, the vector encoding process is simply a demultiplex operation followed by independent bit-to-symbol mapping of each substream. No intersubstream coding, or coding of any kind, is required, though conventional coding of the individual substreams will certainly be applied.
Figure 2.6 shows the typical encoding steps in V-BLAST. The coding procedure can be viewed as there is an encoder on each transmit antenna. The output coded symbols of one encoder are transmitted from the corresponding transmit antenna. The output coded symbol of the pth encoder is used to fill the pth row of C.
1 : 3 DEMUX Input Bits Encoder α Encoder β Encoder γ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
=
C
3 1c
2 1c
3 2c
1 1c
2 2c
3 3c
1 2c
2 3c
3 4c
Encoder α Encoder β Encoder γ 1 3c
c
14 2 4c
Space Time Layer Decision ( Decoding ) Data Rate N∝⎫
⎬
⎭
1 : 3 DEMUX Input Bits Encoder α Encoder β Encoder γ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦=
C
3 1c
2 1c
3 2c
1 1c
2 2c
3 3c
1 2c
2 3c
3 4c
Encoder α Encoder β Encoder γ 1 3c
c
14 2 4c
Space Time Layer Decision ( Decoding ) Data Rate N∝⎫
⎬
⎭
Figure 2.6: Vertical Bell Labs’ Layered Space-Time encoding procedure.
Decode
1.
3.
2.
4.
6.
5.
Detected Detected Detected Nulled Nulled Nulled Cancelled Detected Nulled Decode1.
3.
2.
4.
6.
5.
Detected Detected Detected Nulled Nulled Nulled Cancelled Detected NulledFigure 2.7 shows the typical decoding steps in V-BLAST. The detection procedure is to extract the strongest substream from the signals received by the all receive antennas simultaneously. Then, the procedure proceeds with the remaining weaker signals, which are easier to recover when the strongest signals have been removed as a source of interference. Following the data model in D-BLAST, let Hl=1=H and t
1 l
t = =
r r at the first decoding step at a given time instant t . In each step l, the
pseudo-inverse of H is calculated to be the nulling matrix l G . l
(
)
1 ( ) ( ) ( ) l l l H l l H + − = = G H H H H (2.40)Each row of G can be used to null all but the lth desired signal. The layer shows the l biggest post-processing SNR suggested to be detected first to reduce the error propagation effect efficiently [24]. At this step, the row of G with the minimum l norm is chosen and the corresponding row is defined as the nulling vector
l T k w . {1 1} 2 ,..., arg min || ( ) || l l l j j k k k − ∉ = G (2.41) ( ) l l l T k = k w G (2.42)
The post-processing SNR for the klth detected component of c can be defined as
2 2 2 | | || || l l l k k k c ρ σ < > = w (2.43) Then, using l k
w to suppresses all layers but the one transmitted from antenna kl and a soft decision value is obtained
l l
k T l
t k t
c = w r (2.44)
Therefore, the klth layer can be detected within the constellation set S
2 ˆkl arg min || kl || t t x S c c c ∈ = − (2.45)
As soon as one layer is detected, the part of the detected signal can be subtracted from the received vector to improve the detection performance for the later layers.
( )
1 ˆkl kl l l l t t ct + = − r r H (2.46)where
( )
Hl kl denotes the klth column of H . Then, the channel matrix is deflated to laccount for its removal.
( )
1 kl
l+ = l
H H (2.47)
where the notation
( )
Hl kl denotes the matrix obtained by zeroing columns1, 2,..., l
k k k of H . Therefore, the diversity gain is increased by one at each step l when we decrease the number of layers to be nulled out in the next step by one.
The Zero-Forcing (ZF) V-BLAST detection algorithm can be summarized as follows: Initialization:
( )
1 1 1 2 1 1 arg min || ( ) ||j j i k + ← = = G H G Recursion:( )
( )
{1 } 1 1 +1 1 1 2 1 , ( ) ˆ ( ) ˆ ( ) arg min || ( ) || 1 l l l l l l l l l l l l l T k k k T l k k T l k k k k k l l l k l l l l l l j j k k c c c Q c c k i i + + + + + + ∉ = = = = = − = = = ← + w G w r w r r r H H H G H GBy using the minimum mean square error (MMSE) nulling matrix instead of the ZF one, it can improve the detection performance especially for the mid-range SNR values [32]. 1 1 ( ) ( ) l l H l l H SNR − ⎛ ⎞ =⎜ + ⎟ ⎝ ⎠ G H H I H (2.48)
In the MMSE detection case, the noise level on the channel is taken into account besides nulling out the interference. Thus, the SNR has to be estimated at the receiver
2.5 Beamforming Based on MIMO Techniques
Traditionally, the intelligence of the multiantenna system is located in the weight selection algorithm. Simple linear combining can offer a more reliable communications link in the presence of adverse propagation conditions such as multipath fading and interference. Beamforming is a key technique in smart antenna and increases the average SNR. In the following, we will introduce two schemes: beamforming and eigenbeamforming.2.5.1 Generic Beamforming
As a feedback channel from the receiver to the transmitter can be obtained, beamforming can be utilized to maximize the receiver SNR and provide array gain [33]. With beamforming technique applied to both transmitter and receiver, the beamformer output of the receiver is given by
(
)
ˆ H H H
T T
R R R
s =w Hw s+n =w Hw s+w n (2.49)
where wR and wT denote the weight vectors of transmitter and receiver, respectively,
matrix H, the beamformer output can be rewritten as
1 2 *
1
ˆ R
s =λ s + w n (2.50)
where λ1 is the largest eigenvalue of the matrix HHH. According to Equation (2.50),
the corresponding SNR is 2 1 BF SNR T n P λ σ = (2.51)
where PT is the total transmitted power.
2.5.2 Eigenbeamforming Technique
The eigenbeamforming is an attractive method for downlink [34]. It can provide good diversity gains with less amount of feedbacks due to the short-term selection of eigenmode. Eigenbeamforming is particularly suitable for spatially correlation channels, and is easily applied to the spatial downlink channel. This is because that the spatial downlink channel possesses a higher spatial correlation and few dominant eigenmodes, in which each eigenmode can be considered as an uncorrelated path to the mobile station [35]. In a cellular system, the signal can be spatially selectively transmitted with only few directions due to the vanish of local scatters around the antenna array of the base station. This makes the eigenbeamforming effectively. Consider a system with NT transmit antennas at the base station and NR receive
antennas at the mobile station. The received signal vector at the mobile station is given by
( )t = γ ( ) ( ) ( )t t s t + ( )t
y H w n (2.52)
where γ is the input SNR, H(t) is the channel matrix, and w(t) is the beamforming
