Antenna Selection and Space-Time Spreading Methods for Multiple-Antenna Systems

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SUDARSHAN, PALLAV. Antenna Selection and Space-Time Spreading Methods for Multiple-Antenna Systems. (Under the direction of Prof. Brian L. Hughes.)

The use of multiple antennas at the transmitter and receiver can significantly im-prove the performance of a wireless communication system. In recent years, there has been a lot of interest in deriving efficient receiver architectures and designing signalling and coding schemes that maximize the performance gains of a multi-antenna system. In this dissertation, we focus on two such issues: space-time spreading methods at the transmitter, and antenna selection techniques at the receiver.

For a synchronous code-division multiple-access (CDMA) system that employs multiple transmit antennas, we characterize the asymptotic spectral efficiency in terms of the number of users, processing gain, signal to noise ratio (SNR), array size, etc. Using this formula, we design the linear space-time spreading methods that maximize the spectral efficiency. The strategy for optimal spreading sequence allo-cation across antennas, and across users is also addressed. We show that the system capacity per chip is maximized when each user employs all the spreading sequences allocated to it on each transmit antenna.

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by

Pallav Sudarshan

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Electrical Engineering Raleigh, NC

2004

Approved By:

Dr. Hamid Krim Dr. Jack Silverstein

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Biography

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Acknowledgements

I would like to express my sincere gratitude to my advisor Prof. Brian Hughes, for giving me the opportunity to work with him. I have immensely benefitted from his intuitive approach to solving problems. His insightful thinking and invaluable guidance has made my graduate study experience a truly rewarding one.

I thank my committee members Prof. Alexandra Duel-Hallen, Dr. Hamid Krim, and Prof. Jack Silverstein for their useful comments and suggestions. I would also like to thank Dr. Huaiyu Dai for sharing and discussing many exciting ideas with me. I enjoyed the work done in collaboration with him very much.

I am thankful to Dr. Jinyun Zhang for providing me with the opportunity to work at Mitsubishi Electric Research Labs, where the final part of this work was done. I am immensely grateful to Dr. Neelesh Mehta for always motivating and encouraging me. I have learnt a lot from his unmitigated enthusiasm towards research. He has been a great guide and mentor to me ever since. I also extend my gratitude to Dr. Andreas Molisch for his valuable suggestions and rich discussions during my work at Mitsubishi.

I would like to thank my colleagues Ajith Kamath and Sandeep Krishnamurthy for their unflinching willingness to help. The technical and philosophical discussions with them are one of the most memorable moments of my graduate experience. Thanks are also due to Keyoor Gosalia, Chris Mary James, Xinying Yu and all other friends and colleagues at NCSU, for the nice time I had during my PhD. A special thanks to my close friends Aashish Mehra and Meeta Sharma, who were always there for me.

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List of Figures

1.1 Multiple-antenna system in a rich scattering environment . . . 4 1.2 Spreading in CDMA system . . . 9 2.1 Spectral efficiency of different space-time codes for t = 1,2,4 and

Eb/N0 = 10 dB . . . 34 2.2 Spectral efficiency as a function of number of spreading sequences per

user . . . 42 3.1 Block diagram for diversity transmission with RF-baseband design . . 49 3.2 Block diagram for spatial multiplexing with RF-baseband design . . . 51 3.3 Antenna arrangement for MEA with Nr receive antennas . . . 51

3.4 Beam pattern for FFT pre-processing for d/λ = 0.5,and Nr = 4 as a

function of the azimuth angle . . . 56 3.5 Beam pattern for MTI for d/λ = 0.5, Nr = 4, and θr = 45◦ as a

function of the azimuth angle . . . 60 3.6 Beam pattern for MTI for d/λ = 0.5, Nr = 4, and θr = 60◦ as a

function of the azimuth angle . . . 61 3.7 CDF of SNR for diversity system with Nt = 4, Nr = 4, L = 1, σr =

6◦_{, d}_{= 0}_.₅_λ,_T₌_I

Nt, and ρ= 10 dB. . . 74 3.8 Effect of spatial correlation σr on SNR for diversity system withNt =

4, Nr = 4, L= 1, θr = 60◦, d= 0.5λ,T=INt, and ρ= 10 dB. . . 75 3.9 CDF of capacity for spatial multiplexing system with Nt = 2, Nr =

4, θr= 45◦, σr= 6◦, d= 0.5λ,T=INt, and ρ= 10 dB. . . 77 3.10 Effect of spatial correlationσron CDF of capacity for spatial

multiplex-ing system withNt = 2, Nr = 4, L= 1, θr= 45◦,T=INt, andρ= 10 dB. 78 3.11 Beam pattern for 2-cluster TI pre-processing for d/λ = 0.5, Nr =

4, θr1 = 45◦, and θr2 = 75◦ as a function of the azimuth angle. . . 80 3.12 CDF of SNR for diversity system Nt = 4, Nr = 4, d = 0.5λ,T =

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3.13 CDF of capacity for spatial multiplexing system Nt = 2, Nr = 4, d =

0.5λ,T = INt, ρ = 10 dB, for two clusters with θr1 = 45

◦_{, θ}

r2 = 75◦, and σr = 6◦. . . 82

3.14 Impact of phase quantization and calibration error: Capacity CDF for

ρ= 6 dB, θ = 45◦_{, and} _σ

r = 6◦. . . 84

3.15 Impact of insertion loss: Capacity CDF for ρ = 6 dB, θ = 45◦ _and

σr = 6◦. . . 85

3.16 Impact of channel estimation error: Capacity CDF for ρ = 6 dB,

θ= 45◦ _and _σ

r= 15◦. . . 87

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List of Tables

3.1 List of abbreviations for different receiver architectures . . . 73 3.2 Average SNR (in dB) for different receiver architectures. . . 74 3.3 Ergodic capacity (in bits/s/Hz) for different receivers. . . 78 3.4 Ergodic capacity with phase quantization and calibration error for θ =

45◦ _and _σ

r = 6◦. . . 83

3.5 Ergodic capacity with insertion loss for θ= 45◦ _and _σ

r = 6◦. . . 85

3.6 Ergodic capacity with imperfect CSI for θ= 45◦ _and _σ

r= 15◦. . . 86

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Chapter 1

Introduction

The use of wireless links for communication of information began with the inven-tion of telegraph and radio in the late 19th _{century. While significant advances have} been made in the field of wireless communications in the last 100 years, it is in the last decade or so that the interest in wireless system design has really exploded. There is an exponentially increasing demand for broadband wireless access for applications such as multimedia information transfer on the move and high-speed wireless inter-net connectivity. These applications require reliable communication at very high data rates, which increases the bandwidth requirements of the system. However, wireless bandwidth is a scarce and expensive resource. Thus, bandwidth and power efficient schemes that can meet the ever increasing demand of high data rates need to be considered.

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by multiple paths to the receiver, each with possibly different path lengths, and hence different path delays, phase, amplitudes, and angles of arrival. The receiver thus sees a time-dispersed version of the transmitted signal, from which it can be difficult to recover the original signal. Also, as the signals from different paths superimpose, they might add constructively or destructively. Thus the amplitude of the received signal depends on the geometry of the scatterers. This phenomenon is calledmultipath fad-ing, and such channels are referred to as fading channels. Since the geometry of the scatterers changes in a random fashion, the fading channel gain is typically modelled as a random parameter. Due to these effects, multipath fading is one of the most prominent effects that limits the performance of a wireless channel.

One of the common methods to mitigate the ill-effects of fading is to use diversity. Diversity techniques reduce the fading-related fluctuations by repeated transmission of the same information. Conventional diversity methods include time diversity, where same information is transmitted repeatedly in time, or frequency diversity, where same information is transmitted on multiple frequencies. Although these diversity methods improve the performance of fading channels, they require additional resources, either in form of more transmission time for time diversity, or more bandwidth for frequency diversity.

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These diversity methods improve the reliability of a wireless link and can help achieve higher data rates. The reliability of a communication system is often mea-sured in terms of bit-error-rate (BER), defined as the probability that a transmitted bit of information will be received in error. A fundamental limit associated with the practically achievable data rates in a communication system is channel capacity. Shannon, in his seminal work in 1948 [1], proved that reliable, error-free communi-cation through a link is always possible as long as the rate of information transfer between the transmitter and the receiver is less than a certain threshold, known as channel capacity. An efficient communication system design maximizes the rate, while keeping the BER low.

For fading channels, since the fading gain is a random parameter, the maximum throughput will also be a random variable. Therefore, the performance of fading channels is measured in terms of ergodic capacity, defined as the maximum instan-taneous data rate averaged over all possible channel realizations. Ergodic capacity indicates, on average, how many bits of information can be transmitted per channel use.

In the next section, we give an overview of multiple-antenna systems, and their potential benefits in reducing the BER and increasing the capacity of a wireless system.

1.1

Multiple-Antenna Systems

Figure 1.1 shows a multiple-antenna system in a rich scattering wireless com-munication environment. The transmitter has Nt antennas and the receiver has Nr

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fad-Tx

1

Nt 1

Nt

Tx

1

Nr

Rx

Figure 1.1: Multiple-antenna system in a rich scattering environment

ing. If the delay between the signals from different paths between a pair of transmit and receive antennas is small compared to the inverse bandwidth of the signal, then it results in frequency non-selective fading, also known as frequency-flat fading. This occurs commonly in indoor wireless channels, where all the scatterers are located close to each other.

Consider a frequency flat fading wireless channel. Lethij denote the overall

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received symbols, y can be written as

y= ρ

Nt

Hx+n, (1.1)

where xdenotes the vector of transmitted symbols and n denotes the receiver noise. In a communication system, the transmitted signal is corrupted by additive thermal noise, generated from different sources like electronic devices, atmospheric effects, etc. The combined effect of all these sources is often modelled as additive white Gaussian noise (AWGN),i.e., the power spectral density of the noise is assumed to be flat. In (1.1), ρ denotes the signal-to-noise ratio (SNR) of the system. SNR is defined as the ratio of total power and noise spectral density.

MIMO systems have the potential to achieve enormous capacity gains, which are not possible using single antenna systems. Foschini and Gans [4] quantified the advantages of using multiple antennas. They showed that for uncorrelated fading,i.e., when the channel between each transmit and receive antenna pair fades independently, the capacity increases linearly with min(Nr, Nt) for a fixed SNR. This shows the

potential of achieving extremely high data rates by using multiple antennas at both ends of the communication system.

Motivated by these benefits, multiple-antenna systems have attracted much inter-est, both in the research community and in the industry. Several emerging commu-nication standards are based on MIMO architecture. For example, 802.11n wireless local area network (LAN) standard uses MIMO technology to increase capacity and reliability of indoor wireless channels. As another example, some third generation cellular standards, such as high speed downlink packet access (HSDPA) require the basestation and the handset to support multiple antennas, in order to meet their high-data-rate requirements.

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1.1.1

Gains in a Multiple-Antenna System

1. Multiplexing gain: The multiplexing gain [6] of a MIMO system is defined as

m= lim

ρ→∞

Ce log₂(ρ),

where Ce is the ergodic capacity of a multi-antenna system [4, 5]. Multiplexing gain gives an idea of the number of linearly independent signalling dimensions that the channel provides. The rank of the channel matrix indicates the spatial multiplexing gain.

2. Diversity gain: Diversity gain, d, [6] is defined as the number of degrees of freedom available to each data stream. Mathematically,

d=− lim

ρ→∞

log(BER) log(ρ) ,

i.e. the diversity gain is the negative of the exponent of the SNR in BER ex-pression. Diversity gain indicates how many independently fading copies of the same signal are available to the receiver.

3. Array gain: Also known as the beamforming gain, array gain refers to the gain in the SNR achieved by using multiple antennas. For example, for a system with one transmit antenna and two receive antennas, the beamforming gain is given by

B =|h1|2+|h2|2,

where h1 and h2 are the fading coefficients between the transmitter and the receive antennas.

1.1.2

Space-Time Codes

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space-time codes that maximize the data rate of a MIMO system. The broad categories of space-time codes include [7, 8] space-time trellis codes (STTC), space-time block codes (STBC), space-time turbo codes, quasi-orthogonal codes, layered designs, etc.

We will look at space-time block codes in more detail. STBC distribute a block of input data symbols over space and time. They are represented in terms of a matrix, whose rows represent time and the columns represent space. Depending upon their design, they provide diversity gain and/or multiplexing gain. The rate of block code is defined as the number of information symbols transmitted per channel use [33].

Next, we describe the structure of a few common STBC.

Alamouti Code

One of the first STBC was proposed by Alamouti [44], for the case of two transmit antennas. The code matrix is given by

X =



 x1 x2 −x∗

2 x∗1



_,

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BLAST

In Bell Labs Layered Space-Time (BLAST) transmission [22], independent data streams are transmitted from each antenna. The code matrix (vector) is given by

X =



       

x1

x2 ...

xNt



       

.

This scheme has rate Nt and diversity order 1. Unlike the Alamouti code, the

optimal receiver for BLAST is very complex because different data streams interfere with each other.

1.2

Multiple-Access Methods

In a practical system, multiple users communicate simultaneously with a common node, for example, a basestation in the case of cellular networks, or an access point in the case of wireless LANs. This calls for a mechanism to multiplex the information from multiple users, so that the inter-user interference is minimized. Multiplexing can be achieved using time-division access (TDMA), frequency-division multiple-access (FDMA) or code-division multiple-multiple-access (CDMA), among other methods. In TDMA [2], each user communicates with the common basestation only during the time slot assigned to it, so that it does not interfere with other users. Similarly, in FDMA [2], each user is assigned a frequency slot for communication. In CDMA [23], each user spreads its information sequence over the entire time duration and the entire bandwidth using a unique pseudo-random spreading sequence. The spreading code sequences, known at the receiver, are used to separate the users.

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(b) Chipping Sequence

(c) Modulated Data (a) Data Sequence

Figure 1.2: Spreading in CDMA system

1.2.1

Code-Division Multiple-Access

CDMA systems use spreading sequences to spread the information symbols of each user. The spreading sequences can be designed using direct sequence (DS), frequency hopping, or time hopping methods. DS-CDMA involves modulation of the information signal by a digital, discrete-time, discrete-valued spreading sequence, also known as the chipping sequence. The individual chips are ±1, as shown in Fig. 1.2. The rate of chipping sequence is much higher than the symbol rate. This results in spreading of the symbol over a wider bandwidth.

Processing Gain

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Advantages of CDMA

CDMA offers some distinct advantages over other multiple access methods.

1. Multipath: As described earlier, in a wireless channel, the transmitted sig-nal travels through multiple paths, before reaching the receiver. This leads to time dispersion of the signal if each path has a different delay. CDMA can combat multipath interference if the chipping sequences are designed so that time-shifted versions of the sequences have low (or zero) correlation. Rake re-ceivers [24] exploit this property of DS-CDMA very effectively to combat multi-path interference. In fact, in a rich scattering environment, use of rake receivers provides a diversity gain to the system.

2. Anti-Jamming: Any narrowband interference that is added to a CDMA signal has a reduced effect after despreading.

3. Privacy: In order to recover the transmitted signal, the receiver needs to know the spreading sequence used.

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1.2.2

Multiuser Detection

In practice, spreading sequences do not have ideal autocorrelation and cross-correlation properties. After despreading, the interference from other users will not completely cancel out. A CDMA system is thus interference limited, i.e., adding an additional user deteriorates the performance of all the users. Various multiuser detec-tors have been proposed in the literature [9], that exploit the structure of spreading sequences to suppress the interference from other users. The simplest among them is matched filter receiver. It decorrelates the received signal of the user of interest and treats interference from other users as white noise. This receiver acts like a single-user receiver and does not use the knowledge of the spreading sequences of other users.

A decorrelating receiver completely cancels the interference caused by other users. This linear receiver projects the desired users received signal in the null space of the space spanned by interfering spreading sequences. Although the resulting signal is free of interference, the decorrelating receiver can lead to noise enhancement. The performance of this receiver degrades appreciably in the low SNR regime. The mini-mum mean square error (MMSE) receiver is a linear receiver that uses the information about the received SNRs of all the users. For additive white Gaussian noise, MMSE receiver is the optimal receiver in the mean-square-error sense. MMSE receiver also maximizes the signal-to-interference ratio of multiuser system.

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1.2.3

Capacity of Multiple-Access Systems

In the multiuser case, the capacity is defined by a region. For the purpose of illustration, consider a system with two users, communicating with a common receiver in a non-fading AWGN channel. Let R1 and R2 be the rates of user 1 and user 2, respectively. Then the achievable capacity region of this multiple-access channel is the closure of the convex hull of the set of points (R1, R2) satisfying [3]:

R1 ≤ log

µ

1 + P1

N0

¶

, (1.2)

R2 ≤ log

µ

1 + P2

N0

¶

, (1.3)

R1+R2 ≤ log

µ

1 + P1+P2

N0

¶

, (1.4)

where P1 and P2 is the power of user 1 and user 2, respectively, and N0 is the noise power spectral density.

1.3

Antenna Selection

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Often, the cost of an additional RF chain is much higher than that of an antenna element. This has prompted a lot of interest in the field of antenna selection, where a subset of available antennas are selected for baseband processing and up-conversion (down-conversion). Selecting a subset of antennas reduces the number of RF chains required, as well as simplifies the baseband signal processing, thereby considerably re-ducing the complexity of a MIMO system. This comes at the expense of performance loss, which under most conditions is usually small [46]. In fact, the diversity order of a judiciously selected antenna subset is same as that of a full complexity system using all the antennas [53].

The antenna selection algorithms proposed in the literature optimize different per-formance metrics, such as, information theoretic capacity [53–56], and probability of error [51, 52, 57]. Although antenna selection reduces the digital signal processing required by the MIMO system, finding the optimal set of antennas itself can be com-putationally intensive. This is especially true if the performance metric to maximize is capacity. Low-complexity, near-optimal algorithms to compute the antenna subset have also been proposed in the literature [53, 58].

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1.4

Dissertation Overview

In this dissertation, we derive the optimal space-time spreading methods that maximize the spectral efficiency of a multi-antenna CDMA system. We then design efficient, low-complexity receivers for MIMO systems. A pre-processing architecture is introduced, which reduces the number of RF chains required, while maintaining most of the benefits of multiple-antenna systems. Finally we consider antenna selection algorithms for multi-access MIMO systems, and derive the statistics-based optimal selection criteria.

In Chapter 2, we determine the fundamental information theoretic performance limits for a MIMO CDMA system. The performance of a CDMA system depends on the particular choice of spreading sequences. To remove this dependence we assume a random spreading model: the chips are randomly chosen to be ±1. Furthermore, we perform an asymptotic analysis, i.e., we let the number of chips, and the number of users in the system approach infinity. For such a system, we characterize the capacity using some recent results from random matrix theory. Such an analysis makes some of the properties of multi-antenna systems very transparent. Using this characterization, we then optimize the spectral efficiency over the class of all linear space-time spreading methods. Specifically, we design the codes for distributing the data over antennas and spreading sequences, that maximize the spectral efficiency. In the process, we also derive some interesting properties of the optimal space-time codes.

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Re-cent approaches [60, 61], involving the use of a pre-processing matrix at the carrier frequency before selection, have shown that most of the beamforming gain can be maintained, in spite of using reduced number of RF chains. These solutions propose the use of either a fixed, channel-independent pre-processing, or an instantaneous-channel-state-based pre-processing. While both these solutions yield performance benefits as compared to pure antenna selection (without pre-processing), they do not exploit the inherent correlation in the channel. In Chapter 3, we propose a channel-statistics-based pre-processing solution for antenna selection that fully exploits the channel correlation information. We also show the robustness of this architecture to RF and channel imperfections.

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Chapter 2

Optimal Space-Time Spreading

Codes for CDMA Systems

This chapter considers the asymptotic spectral efficiency of a general linear space-time spreading system for CDMA with random spreading sequences, in the limit of large user populations. The resulting parametric formulas for spectral efficiency provide a design criterion for space-time coded CDMA. Using this criterion, we derive optimal space-time codes.

2.1

Introduction

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Several transmit diversity methods have been proposed in the literature for im-proving the performance of CDMA using multiple antennas at the base station and the handset. Transmit diversity methods are usually grouped into two categories, depending on the transmitter’s knowledge of current channel conditions [15]. Closed-loop methods, such as transmit beamforming [27] and selection transmit diversity [34], require channel information to be fed back from the receiver to the transmitter. On the other hand, open-loop methods, such as orthogonal transmit diversity [32] and space-time spreading (STS) [29], require no such feedback. Both types of methods play a role in future wideband CDMA systems under different conditions, which de-pend, in part, on the quality of the channel estimates.

Most work on transmit diversity for CDMA has focussed on the downlink [10–14], where spreading codes are orthogonal, and on particular transmit diversity methods. Huang, et al. [10] describe some interesting variants of BLAST for a CDMA system, and calculate the capacity gain for downlink channel. In [11], the spectral efficiency for a CDMA system using orthogonal spreading sequence is calculated for STS and compared with the performance of BLAST. While most of the previous work has focused on orthogonal CDMA systems, relatively little attention has been paid to the use of multiple-antennas in non-orthogonal CDMA systems. It is not clear what is the impact of non-orthogonal spreading sequences, and whether it is possible to sig-nificantly improve on these simple transmit diversity methods. Thus, it is natural to examine the fundamental information-theoretic limits of space-time coding methods and to examine the performance of a larger class of transmit diversity methods.

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more transparent [19, 20, 35, 38, 39]. In particular, the spectral efficiency (capacity per chip) of randomly-spread CDMA is characterized in [39] when all users are re-ceived with equal power, and in [35], when users are plagued by frequency-flat fading. Multiple receive antennas are also considered in [35]. The ergodic and outage spec-tral efficiency of a randomly-spread multi-antenna CDMA system with a BLAST-like transmission was derived in [18]. The authors compared two extreme cases for the allocation of spreading sequences: using same sequences across all the antennas, and using a different sequence on each antenna. However, it is not clear that the best strategy to allocate spreading sequences across the antennas is one of these simple techniques.

In this chapter, we characterize the asymptotic spectral efficiency for a general, synchronous CDMA transmission scheme, in which spreading sequences are not tied to a specific transmit antenna. The asymptotic spectral efficiency is derived as a function of the array size, SNR, and the number of users per chip, for large user populations, in presence of frequency-flat fading. Then we derive the conditions that must be satisfied by the space-time codes to maximize the asymptotic spectral efficiency. Using these conditions, we examine some interesting properties of optimal space-time codes and the maximum achievable spectral efficiency.

The rest of the chapter is organized as follows. In Section 2.2 we introduce the transmission model and the channel model. The spectral efficiency of a CDMA system is defined in Section 2.3. Section 2.4 gives a brief background on random matrix theory, followed by the derivation of asymptotic spectral efficiency in Section 2.5. The optimal space-time spreading codes are derived in Section 2.6, followed by conclusions in Section 2.7.

The following notation is used in the chapter: (.)† _{for the Hermitian transpose}

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a matrix, Tr(.) for the trace of a matrix, EX[.] for the expectation with respect to

X, and Nc(m, σ2) denotes a complex Gaussian random variable with mean m and

variance σ2_.

2.2

Transmission Model

2.2.1

A General Model of Linear Modulation

We consider a synchronous direct-sequence CDMA system in which K equal-energy users transmit to a common receiver. Each user transmitsdindependent data streams over t antennas using m spreading sequences. The spreading sequences are not necessarily tied to specific transmit antennas. The only requirement is that the transmitted signal must be linear in the data symbols. The signal transmitted by user k is given by

Xk =

1

√

tmN

d X

l=1

(xklAl+x∗klBl)Ck, (2.1)

where N is the number of chips per spreading sequence, xk1, . . . , xkd are the data

symbols, Ck is an m× N matrix of spreading sequences, and Al and Bl are t ×

m complex modulation matrices, which distribute the data over the spreading se-quences and antennas. Assuming that the data symbols are independent, identically-distributed (i.i.d.) Nc(0,1) complex random variables, and independent of Ck, where

E

n

CkC†k o

=NIm, the matrices are required to satisfy an average energy constraint:

E

n

Tr

h

XkX†_k io

= 1

tm

d X

l=1 Tr

h

AlA†_l +BlB†_l i

≤ 1. (2.2)

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Example 1. Orthogonal Transmit Diversity (OTD) [29]

In OTD, each user transmits different data streams on each antenna, using dif-ferent spreading sequences. Fort transmit antennas, this scheme can be expressed in the form of (2.1), by choosing d=t, m=t, Bl =0 and

Al =

√

t diag(δ1l, . . . , δtl), l = 1,· · · , d.

Here 0 is a matrix with all-zero elements, diag(a, . . . , z) refers to a diagonal matrix with elements a . . . z along the main diagonal, and

δij =  



1 if i=j

0 otherwise

.

Example 2. Space-Time Spreading (STS) [29]

In STS, two data symbols xk1 and xk2 are spread over two antennas as follows:

Xk =

1

√

4N



 xk1c1 +xk2c2

x∗

k2c1−x∗k1c2





where c1 and c2 are spreading sequences of length N. This can be modelled as in (2.1) with m=d=t= 2 and

A1 =



 1 0

0 0



_, _B1 ₌ 

 0 0

0 −1



_, _A2 ₌ 

 0 1

0 0



_, _B2 ₌ 

 0 0

1 0



_. _(2.3)

Example 3. Independent data, same codes [18] (BLAST)

When different data streams are transmitted on each antenna using the same spreading sequence, the transmitted matrix can be written as

Xk =

1 √ tN         

xk1c

xk2c ...

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To express it in the form of (2.1), choose d=t, m= 1, Bl =0, and

Al = [δ1l, . . . , δtl]†, l = 1,· · · , d.

This scheme can be considered as a wideband representation of the Bell Labs layered space-time (BLAST) transmission [22]. This scheme (along with a variant that uses adaptive rate control) has also been included in Lucent’s 3GPP proposal for high speed downlink packet access (HSDPA) [25].

Example 4. Same data, independent codes [18]

In this scheme, the same data symbol is sent on each of the transmit antennas using different spreading sequences. This can be modelled using (2.1) by setting

d= 1, m =t,B1 =0, and A1 =

√

tIt, where It is t×t identity matrix.

2.2.2

MISO Channel Model

For the sake of simplicity, we assume that all users employ the same linear modu-lation scheme. The arguments are easily generalized to unequal powers and different modulation. We assume that perfect instantaneous channel state information (CSI) is available at the receiver.

We consider a multiple-input single-output (MISO) channel, i.e. the transmitter has multiple antennas, while the receiver has a single antenna. At the receive antenna, the output of a filter matched to the chip waveform is synchronously sampled. In frequency-flat fading, we can write the samples in vector form as

Y = √ρ

K X

k=1

HkXk+W, (2.4)

where Y is an 1×N vector of received chip samples, Hk is a 1×t vector of fading

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i.i.d.complex Gaussian zero-mean, unit-variance random variables. Hk are

indepen-dent (not necessarily Gaussian) random vectors such that E

n

HkH†k o

=t. Note that the results are not changed by allowingHk and Ck to vary with time.

We can write the channel in a unified matrix form by collecting the data symbols of all users into one Kd-dimensional vector x= [x1, . . . ,xK], xk = [xk1, . . . , xkd]:

Y =

r

ρ

N (xHA+x

∗_H

B)C+W

where HA = diag{A1, . . . ,AK} and HB = diag{B1, . . . ,BK} are dK ×mK

block-diagonal matrices, and

Ak =

1 √ mt         

HkA1 HkA2

... HkAd

        

, Bk=

1 √ mt         

HkB1 HkB2

... HkBd

        

, C =

     C1 ... CK    

. (2.5)

Recall that the algebra of n×m complex matrices can be expressed in terms of 2n×2m real matrices via the correspondence

C = <(C) +j=(C)←→C¯ =



 <(C) −=(C) =(C) <(C)



_,

where <(C) and =(C) are the real and imaginary parts of C, respectively. Thus, in real form xkAk+x∗kBk can be written as



 <(xk) −=(xk) =(xk) <(xk)



 

 <(Ak) −=(Ak) =(Ak) <(Ak)



₊ 

 <(xk) =(xk) −=(xk) <(xk)



 

 <(Bk) −=(Bk) =(Bk) <(Bk)



_.

The two rows of the resulting matrix are redundant; dropping the second row gives (1/√2)¯xkM¯k where (with a slight abuse of notation) ¯xk =

√

2[<(xk) − =(xk)] and

¯ Mk =



 <(Ak+Bk) −=(Ak+Bk) =(Ak− Bk) <(Ak− Bk)



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Thus, the channel can be written in real form as ¯

Y =

r

ρ

2Nx¯H¯C¯ + ¯W, (2.7)

where ¯x= [¯x1, . . . ,x¯K], W¯ =

√

2[<(W) − =(W)],

¯ H =          ¯

M1 O · · · O

O M2¯ · · · O

... ... ... ...

O O · · · M¯K          ¯

C = √2

     ¯ C1 ... ¯ CK    

. (2.8)

2.3

Spectral Efficiency

Once we have the channel in a composite form, we can calculate the spectral efficiency of (2.7). The capacity region of a multi-access system (2.7) is given by the convex hull of the maximum rate of each user [3]. Thus the maximum achievable sum-rate spectral efficiency in bits/chip is given by

η(N, K, m, t, ρ) = 1 2N log

¯ ¯ ¯I+ ρ

2NH¯C¯C¯

†_H¯†¯¯_¯ _(2.9)

= 1

2N log

¯ ¯ ¯I+ ρ

2NC¯

†_H¯†_H¯_C¯¯¯_¯_, _(2.10)

where (2.10) follows from the matrix identity |I+AB| = |I+BA|. The formula for spectral efficiency above does not provide much insight into the behavior of the multi-antenna CDMA system. For example, the impact of varying the number of users, SNR, and number of antennas is not very clear from (2.10). Furthermore, it does not provide a mechanism to design the codes that maximize the spectral efficiency.

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behavior of the spectral efficiency of a multiple-transmit-antenna system for large-user populations. In particular, we consider the asymptotic behavior of (2.10) under the following assumptions:

1. The number of users, K, and the number of chips per bit, N, approach infinity, but their ratio is bounded, i.e.

K, N → ∞,

0< β= K

N <∞

2. The spreading sequence, ck, has i.i.d. complex elements, and each element has

i.i.d. real and imaginary parts with mean zero and variance 1/2.

Verdu et al. show in [39], [35] that many fundamental properties of a CDMA system become apparent under the assumptions mentioned above. The performance of a system with random spreading sequences serves as a lower bound to the per-formance of a CDMA system with optimally designed deterministic spreading se-quences. Furthermore, random spreading accurately models CDMA systems where the pseudonoise sequences span many symbol periods [39].

2.4

Random Matrices in the Large System Limit

In this section, we give a brief overview of the main results in the theory of random matrices. In [26], Tulino and Verdu provide an excellent overview of the applications of random matrices to communications. We review some definitions and results from [26] in this section.

2.4.1

Empirical Eigenvalue Distribution

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Definition: Empirical Eigenvalue Distribution (edf)

For any n×n Hermitian matrix A, the empirical eigenvalue distribution, FA, is

defined as

FA(λ) =

1

n

n X

i=1

1{λi ≤λ}

where λ1, . . . , λn are the eigenvalues of A and 1{.} is the indicator function.

The edf of a matrix gives a measure of the fraction of eigenvalues that are less than a certain number. Intuitively, this can be thought of as the matrix equivalent of cumulative distribution function (cdf) of a random variable.

2.4.2

Stieltjes Transform

It is often convenient to represent the edf of a matrix in terms of its Stieltjes transform.

Definition: Stieltjes Transform

LetX be real-valued random variable with distributionFX. Its Stieltjes transform

is defined as

SF(z) = Z _∞

−∞

1

λ+zdFX(λ).

From the definition above, we can see that the Stieltjes transform of a matrix can be written as

S(z) = E

½

1

λ+z

¾

,

where the expectation is taken with respect to (w.r.t.) the empirical eigenvalue distri-bution of the matrix. The reader is referred to [26] for examples of Stieltjes transforms of some common class of matrices, and their applications to communications.

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Lemma 1. [36, Thrm 1.1]: Consider the n×n random matrix

Qn=

1

nC

†

nPpCn.

(a) If Cn is an p ×n random matrix with i.i.d. elements satisfying E{cij} = 0,

E{|cij|2}= 1;

(b) Pp is a (possibly random) p×p nonnegative-definite Hermitian matrix,

indepen-dent of C, such that FPp converges in distribution, almost surely (a.s.), to a fixed distribution G on [0,∞) as p→ ∞;

then, as n → ∞ and p/n → c > 0, FQn also converges in distribution (a.s.) to a fixed distribution F. This distribution is most easily given in terms of its Stieltjes Transform

SF(z) = Z

1

λ+zdF(λ) (2.11)

by the implicit relation

z = 1

s −c

Z

λ

1 +sλdG(λ), (2.12)

where for =(z) >0, SF(z) is the unique solution of (2.12) satisfying =(SF(z)) < 0. 2

2.4.3

R-Transform

The R-transform is closely related to the inverse function of SF(z) and plays a

central role in the theory of free random variables [40, Sec. 3.2]. For the problem at hand, we will see that RF(s) is easier to calculate than SF(z), and will prove useful

in deriving codes that maximize the spectral efficiency.

1_{The Stieltjes Transform is defined differently than in [36]. The relationship is}_S

F(−z) =mF(z) =

R

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Observe thatSF(z) is strictly decreasing forz >0, and so has an inverse function,

say z =f(s), defined for 0 < s < sF = R

λ−1_dF₍_λ_{). For 0}_{< s < s}

F, the R-transform

is defined as

SF µ

1

s −RF(s)

¶

=s (2.13)

i.e. 1

s −RF(s) =z. (2.14)

Comparing Eq. (2.14) with (2.12), we see that for a matrix of the formQndefined

in Lemma 1, RF(z) is given by

RF(z) = c Z

λ

1 +zλdG(λ). (2.15)

The normalized R-Transform RF(z) is defined as

RF(z) = 1

cRF(z) =

Z

λ

1 +zλdG(λ).

2.5

Asymptotic Spectral Efficiency for Optimal

De-tector

In this section, we characterize the asymptotic spectral efficiency of the CDMA system described in (2.7) that uses an optimal detector at the receiver. The spectral efficiency (2.10) can be written as

η = 1 2N log2

¯ ¯ ¯I+ ρ

2NC¯

†_H_¯†_H_¯_C_¯¯¯_¯ _(2.16)

= 1

2N

2N X

i=1

log (1 +ρλi(ΣN)) (2.17)

2_{The Stieltjes and R-transforms are defined slightly differently than in [40, Sec. 3.2], where}

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whereΣN = (1/2N) ¯C†H¯†H¯C¯ and λi(ΣN) is the ith largest eigenvalue of ΣN. Under

the asymptotic assumptions of section 2.3, (2.17) can be expressed as

η=

Z _∞

0

log (1 +ρλ)dFΣN(λ), (2.18)

where FΣN(λ) is the edf of ΣN. The eigenvalue distribution of ΣN is generally a very complex function of the modulation, the fading statistics, and the spreading sequences. When random spreading sequences are used, however, Verdu-Shamai [39] and Tse-Hanly [38] have shown that FΣN converges to a deterministic limit for the particular case of one data stream and one transmit antenna. In this work, we extend these results to arbitrary linear space-time spreading, fading statistics, and an arbitrary number of transmit antennas.

2.5.1

Parametric Formula for Asymptotic Spectral Efficiency

To apply Lemma 1 to (1/2N) ¯C†_H¯†_H¯_{C, we need only prove that}¯ _F_¯

H†_H¯ converges almost surely in distribution as K → ∞. To this end, observe that ¯H†_H¯ _{is block}

diagonal, with the i.i.d. Hermitian random matrices ¯M†_kM¯k running down the

diag-onal. The empirical eigenvalue distribution of ¯H†_H¯ _{is therefore the average of the}

empirical distributions of the blocks:

F_H¯†_H¯(λ) = 1

K

K X

k=1

F_M¯†

kM¯k(λ).

For eachλ, F_H¯†_H¯(λ) is a sum of i.i.d. random variables that take values in [0,1], and thus converges almost surely to the fixed limit

G(λ) = E

n

F_M¯† kM¯k(λ)

o

. (2.19)

We conclude that ΣN = (1/2N) ¯C†H¯†H¯C¯ satisfies the hypotheses of Theorem 1.

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distributionF that is given by (2.11), (2.12), and (2.19), wherec=mβ, and that the spectral efficiency converges almost surely to the fixed limit

η(β, m, t, ρ) = lim

N→∞,K/N→βη(N, K, m, t, ρ)

=

Z _∞

0

log (1 +ρλ)dF(λ). (2.20)

This result is completely general, and applies to any kind of linear modulation, any channel fading statistics, and any number of transmit antennas and spreading se-quences.

Our main goal is to design space-time spreading methods (2.1) that maximize the asymptotic spectral efficiency η(β, m, t, ρ) for every choice of parameters. In principle, the result above allows us to calculate η(β, m, t, ρ) for each choice of d

and the modulation matricesA1, . . . ,AdandB1, . . . ,Bd. However, the equations are

quite cumbersome and optimizing them seems far from straightforward. Fortunately, the code design problem can be made more transparent by re-parameterizing it in terms of the R-transform of F, RF(s). In this section, we will show that RF(s) is

easier to calculate than SF(z), and η(β, m, t, ρ) can be expressed directly in terms

of RF(s). Furthermore, we will see later that a given code optimizes the spectral

efficiency for all SNR if and only if it optimizes RF(s) for all s.

The system of equations (2.11), (2.12), and (2.19) is easier to solve forRF(s) than

SF(z). From (2.15), we have

RF(s) = c Z

λ

1 +λsdG(λ)

= mβE_M¯_k

½

1 2mTr

h

¯

M_k†M¯k(I+sM¯†kM¯k)−1 i¾

(2.21)

where ¯Mk is given in (2.6) and (2.21) follows from (2.19).

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Theorem 1. The spectral efficiency for the general linear modulation system (2.1) satisfies the following parametric equations

¯

ηF(s) = Z _s

0

RF(s)dσ−ln[1−sRF(s)]−sRF(s) (2.22)

ρ = s

1−sRF(s)

(2.23)

for all s=SF(z) such that 0< sRF(s)<1 where RF is defined as in (2.21).

Proof. First rewrite the spectral efficiency (2.18) in nats/chip as a function of the inverse SNR z =ρ−1_:

ηF(z) = Z _∞

0 ln

µ

1 + λ

z

¶

dF(λ), z >0.

Observe that

η_F0 (z) =

Z _∞

0 1

λ+zdF(λ)−

1

z

= SF(z)−1

z.

Since limz→∞ηF(z) = 0, we have

ηF(z) = − Z _∞

z

η_F0 (ζ)dζ (2.24)

=

Z _∞ z

µ

1

ζ −SF(ζ)

¶

dζ. (2.25)

Making the change of variablesζ =f(σ), we obtain

ηF(z) = ¯ηF(s) = Z _s

0

µ

σ− 1

f(σ)

¶

f0(σ) dσ. (2.26) where s=SF(z). For any 0< ² < s, consider the integral

Z _s

² µ

σ− 1

f(σ)

¶

f0(σ)dσ =

Z _s

²

σf0(σ)dσ−lnf(σ)|s_²

= sf(s)−²f(²) +

Z _s

² ·

1

σ −f(σ)

¸

dσ

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Set z² = f(²) and observe that z² → ∞ as ² → 0. It follows from (2.11) that

²f(²) =z²SF(z²) → 1 as ² → 0. Recalling that f(z) = 1/z−RF(z) and ρ = 1_z and

using (2.12), we obtain (2.22).

The theorem above gives a parametric formula for the spectral efficiency of a general linear space-time model in terms of the R-transform. Now we compute the R-Transform for a few well known space-time modulation methods discussed before, and compare their spectral efficiencies.

Example 5. Orthogonal Transmit Diversity (OTD)

For OTD we have m=d=t,

Al = diag(δ1l, . . . , δtl)

and Bl =0, so that the complex channel formulas can be applied.

¯

M†_kM¯k =

1

d

d X

l=1

A†_lH†_kHkAl =

1

tdiag(kHk1k

2_{, . . . ,}_k_H

ktk2)

where the diagonal elements are i.i.d. and exponentially distributed. Thus, we have

SG(z) = E_Hˆ_k

½

1

tTr

h

(zI+ ¯M†_kM¯k)−1 i¾

= E_Hˆ_k

½

1

z+ (1/t)kHk1k2

¾

=

Z _∞

0

t tz+λe

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Thus, the normalized R-transform is

R(s) = 1

s

·

1− 1

sSG

µ 1 s ¶¸ = 1 s · 1− Z _∞ 0 t t+sλe

−λ_dλ ¸

=

Z _∞

0

λ t+sλe

−λ_dλ

= 1

s

Z _∞

0

λ t/s+λe

−λ_dλ

= 1 s µ t s ¶

et/s_Γ(2)Γ(₋₁_{, t/s}₎ _(2.27)

= t

s2e

t/s_Γ(₋₁_{, t/s}₎_. _(2.28)

Eqn. (2.27) follows from the identity [28, 3.383-10, pg. 319]

Z _∞

0

τn−1_e−τ

τ +α dτ = α

n−1_eα_Γ(_n_)Γ(1₋_{n, α}₎_,

where Γ is the incomplete Gamma function Γ(n, x) =

Z _∞ x

xn−1e−x dx.

Example 6. Space-Time Spreading (STS)

As shown in Example 2, for STS we have

A1 = 1

√

2



 1 0

0 0



_, _B₁ ₌ _√1

2



 0 0

0 −1



_,

A2 = 1

√

2



 0 1

0 0



_, _B₂ ₌ _√1

2



 0 0

1 0



_.

Thus, we have

¯ Mk =

1 2         

<(h1) −<(h2) −=(h1) =(h2)

=(h1) =(h2) <(h1) <(h2)

<(h2) <(h1) −=(h2) −=(h1)

−=(h2) =(h1) −<(h2) <(h1)

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For i.i.d. path gains, it follows that ¯

M†_kM¯k =

1 4

£

|h1|2+|h2|2

¤

I4.

Thus,

E_M¯_k

½

1 2mTr

h

(zI+ ¯M†_kM¯k)−1 i¾

= E_M¯_k

½

4

4z+|h1|2+|h2|2

¾

=

Z _∞

0

4λ

4z+λe

−λ_dλ.

The normalized R-transform is

R(s) = 1

s

·

1− 1

sSG

µ 1 s ¶¸ = 1 s · 1− Z _∞ 0 4λ

4 +sλe

−λ_dλ ¸

=

Z _∞

0

λ2 4 +sλe

−λ_dλ

= 1

s

Z _∞

0

λ2 4/s+λe

−λ_dλ

= 1 s µ 4 s ¶₂

e4/s_Γ(3)Γ(₋₂_,₄_/s₎

= 32

s3e

4/s_Γ(₋₂_,₄_/s₎_.

Example 7. BLAST transmission

Proceeding as above, we can show that for BLAST transmission (Example 3), the R-Transform can be written as

R(s) =

µ

t s

¶_t₊₁

et/sΓ(−t, t/s). (2.29) For consistency with previous results on spectral efficiency of CDMA systems [39], [35], we assume that SNR denotes the energy per N chips. Then the SNR can be expressed in terms of the energy per bit (Eb) and noise variance (N0/2) using the relationship

ρ = N

K Eb

N0

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

1 2 3 4 5 6

users per chip (K/N)

spectral effciency (bits/chip)

BLAST (t=1) BLAST (t=2) STS (t=2) OTD (t=2) OTD (t=4) BLAST (t=4) No fading

Figure 2.1: Spectral efficiency of different space-time codes fort = 1,2,4 andEb/N0 =

10 dB

where η is the spectral efficiency in bits/s/Hz.

The spectral efficiency versus β of various schemes discussed above is plotted in Fig. 2.1 for Eb/N0 = 10 dB for different number of transmit antennas. For OTD, at low loads, increasing the number of transmit antennas gives a considerable per-formance improvement. However, as the load increases, we see that all the schemes seem to converge to a common limit, independent of the number of antenna and the number of spreading sequences m.

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ap-proaches that of no-fading case. Also note that the performance of OTD is better than that of BLAST. However, OTD requires t spreading sequences per user, compared to a single spreading sequence required for BLAST. In a spreading-sequence-limited system (e.g.one using orthogonal Walsh codes), the sum capacity per chip of BLAST will be higher than that of OTD, while the link level performance of BLAST will be worse [10]. STS outperforms both BLAST and OTD fort= 2. In fact, we shall show later that fort =m = 2, STS is indeed the optimal code that maximizes the spectral efficiency.

2.5.2

Performance in the Wideband Regime

Now we analyze the behavior of spectral efficiency in the wideband regime, i.e.as

β →0. Recall that

ρ= Eb

N0

η β.

For a fixed Eb

N0, as β → ∞, ρ → 0. Thus the wideband regime is also the low-SNR

regime. Define z =s/ρ. Substituting this in (2.23), we get

z+mβzρRF(ρz) = 1

Since limρ→0RF(ρz) = _m1E n

Tr( ¯Mk†M¯k) o

= 1

m and limρ→0z = ₁₊_η1Eb N₀

, after some simple manipulations, we can show that

lim

β→∞η = log(1 +η

Eb

N0

). (2.30)

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2.6

Optimal Space-Time Modulation

As noted earlier, our aim is to characterize those space-time spreading methods that optimize the spectral efficiency. Unfortunately, the spectral efficiency is a rel-atively complex function of the underlying code structure. However, the following theorem shows that optimizing RF(s) for all s > 0 optimizes the spectral efficiency

for all SNR. Thus, the R-transform can be used as a design metric for linear space-time codes.

Theorem 2. For any distributionF letηF(z)andRF(s)denote the spectral efficiency

and R-transform, respectively. Then RF(z) ≤ RG(z) for all z > 0 if and only if

ηF(z)≤ηG(z) for all z >0.

Proof. By definition of RF(s), we have

SF µ

1

s −RF(s)

¶

=s

for all 0 < s < sF = R

λ−1_dF₍_λ_{). Moreover, as} _s _{varies from 0 to} _s

F, zF(s) = 1/s−

RF(s) varies from 0 to ∞. For all s < min{sF, sG}, we therefore have SF(zF(s)) =

SG(zG(s)) = s. Since RF(s) ≤ RG(s), we have zF(s) ≥ zG(s). From (2.11), SF(z)

is strictly decreasing and hence SF(zG(s))≥ SG(zG(s)) for all s > 0, or equivalently

SF(z) ≥ SG(z) for all z > 0. Since ηF(z) = R_∞

z [1/ζ − SF(ζ)]dζ it follows that

ηF(z)≤ηG(z) for all z >0.

2.6.1

Open-Loop Upper Bound

We now characterize those linear space-time spreading methods (2.1) that optimize

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Theorem 3. For a channelHksuch that its elementsHkj, j= 1,· · · , tare i.i.d.

Gaus-sian distributed, for all linear space-time spreading methods, we have

RF(s)≤β µ

mt s

¶_t₊₁

emt/s_Γ(₋_{t, mt/s}₎_. _(2.31)

with equality if and only if ¯

Mk†M¯k= (1/mt)kHkk2I,a.s. (2.32) Proof. Letλ1, . . . , λ2m denote the (random) eigenvalues of ¯M†_kM¯k, and let

¯

λ = 1 2m

2m X

i=1

λi =

1 2mTr

h

¯ M†_kM¯k

i

.

Since λ(1 +λs)−1 _{is strictly concave for}_{s >}_{0, we have}

RF(s) = E ½

1 2mTr

h

¯

M_k†M¯k(I+sM¯†kM¯k)−1 i¾ = E ( 1 2m 2m X i=1 λi

1 +λis )

≤ E

½ _¯

λ

1 + ¯λs

¾

.

Letting G¯_λ denote the cdf ofλ, we have

RF(s) ≤ Z _∞

0

λ

1 +λsdG¯λ(λ) (2.33)

with equality if and only λ1 =· · ·=λ2m = ¯λ, almost surely, or equivalently

¯

M†_kM¯k= ¯λI (a.s.).

Using (2.6) and (2.5), and the identities Tr[A†_{A] = Tr[}_<_(A)T_<_(A)+₌_(A)T₌_(A)]

and Tr[AB] = Tr[BA], we can express ¯λ in a simpler form: ¯

λ = 1 2mTr

£

(Ak+Bk)†(Ak+Bk) + (Ak− Bk)†(Ak− Bk) ¤

= 1

mTr

h

A†_kAk+Bk†Bk i = 1 mTr " 1 mt d X l=1 ³

A†_lH†_kHkAl+B†lH

†

kHkBl ´#

= 1

m

h

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where

¯

A = 1

mt

d X

l=1

³

AlA†l +BlB†l ´

.

Since ¯A is Hermitian, it can be diagonalized by a unitary transformation, say UAU¯ † _{= diag(}_σ

1, . . . , σt). If Hk is Gaussian, then Hk and HkUhave the same pdf,

and hence we can assume without loss of generality that ¯A = diag(σ1, . . . , σt). For

alls >0, observe that

Z _∞

0

λ

1 +λsdGλ¯(λ) = E

( _P_t

j=1σj|Hkj|2 1 +sPt_j₌₁σj|Hkj|2

)

is symmetric and strictly concave inσ1, . . . , σt, and is therefore maximized if and only

if all of the eigenvalues are equal: ¯A = (1/t)Tr[ ¯A]It. Moreover, since σ(1 +sσ)−1

is strictly increasing and the energy constraint (2.2) gives Tr[ ¯A] ≤ 1, we have the further upper bound

Z _∞

0

λ

1 +λsdGλ¯(λ)≤

Z _∞

0

λ

1 +λsdGmt1 kHkk2(λ) with equality if and only if

¯

M†_kM¯k =

1

mtkHkk

2_I _a.s. _(2.34)

The bound above can be evaluated in closed form. Recall that the pdf of kHkk2

is

p(λ) = 1 Γ(t)λ

t−1_e−λ_{, λ}_≥₀_. _(2.35)

Thus the optimal normalized R-transform is

Z _∞

0

λ

1 +λsdGmt1 kHkk2(λ) = 1 Γ(t)

Z _∞

0

λ/mt

1 +λs/mtλ

t−1_e−λ_dλ

= 1

sΓ(t)

Z _∞

0

λt

mt/s+λe

−λ_dλ

= 1

m

µ

mt s

¶_t₊₁

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which follows from the identity [28, 3.383-10, pg. 319]

Z _∞

0

τn−1_e−τ

τ +α dτ = α

n−1_eα_Γ(_n_)Γ(1₋_{n, α}₎_,

where Γ is the incomplete Gamma function

Γ(n, x) =

Z _∞ x

xn−1_e−x _dx.

We obtain

Antenna Selection and Space-Time Spreading Methods for Multiple-Antenna Systems

Biography

Contents

List of Figures

List of Tables

Chapter 1

Multiple-Antenna Systems

Gains in a Multiple-Antenna System

Space-Time Codes

BLAST

Multiple-Access Methods

Code-Division Multiple-Access

Multiuser Detection

Antenna Selection

Chapter 2

Introduction

Transmission Model

MISO Channel Model

Spectral Efficiency

Random Matrices in the Large System Limit

Stieltjes Transform

R-Transform

Asymptotic Spectral Efficiency for Optimal

Parametric Formula for Asymptotic Spectral Efficiency

Performance in the Wideband Regime

Optimal Space-Time Modulation

Open-Loop Upper Bound