1

ACKNOWLEDGEMENT

It is matter of great pleasure for me to submit this seminar report.

First and foremost, I would like to express my sincere gratitude to my supervisor, Pankaj Shukla sir

for his excellent guidance, outstanding support and constant encouragement throughout this seminar.

Without his exceptional contributions and invaluable advices, this seminar report would not have been

completed.

I want to show deep gratitude towards Mithilesh Kumar sir (Coordinator).

I am also very grateful and highly obliged to whole Electronics department for providing me this op-

portunity

I take this opportunity to express my deep sense of gratitude towards those, who have helped me in

various ways, for preparing my seminar.

1

ABSTRACT

Multiple-Input Multiple-Output (MIMO) systems can achieve high data-rate and high capacity trans-

mission. In MIMO systems, the scheme that weight substreams based on minimum mean square error

(MMSE) criterion at the transmitter using feedback channel state information (CSI) achieves good

performance ( as MMSE precoder).

Here we will discuss an MMSE precoder with mode selection for MIMO systems, which selects both

the number of substreams and the modulation scheme for each substream to minimize the average

BER at a fixed rate.

If we talk about the Standard MMSE receivers, they have various problems in tracking the rapidly

changing channel characteristics in fading environments. In addition, they require training sequence

for their adaptation. And Realistic direct sequence code division multiple access (DS/CDMA) chan-

nels are severely time-varying.

So we will also discuss a constrained MMSE receiver with a modified MMSE criterion. The con-

strained MMSE receiver is adaptively implemented using the orthogonal decomposition-based least

mean square (LMS) algorithm. In order to improve the accuracy of the reference signal and channel

estimates, pilot symbol-aided scheme is employed. It also makes the proposed MMSE receiver not

require any separate training sequence.

Regarding to MMSE ,We consider the design of linear precoding filters with respect to the MMSE

criterion for systems that employ an additional scalar gain next to a fixed receive filter. The precoding

filter and the scalar gain are to be jointly optimized. Currently, only the finite impulse response (FIR)

solution to this problem is known. So we will derive the infinite impulse response (IIR) MMSE pre-

coder both with and without causality constraint, i.e., finite and infinite latency time, respectively.

1

INDEX

INDEX 1

LIST OF FIGURES 5

LIST OF TABLES 6

CHAPTER I: INTRODUCTION 7

1.1 MMSE precoder for MIMO systems 7

1.2 Linear IIR-MMSE precoding for frequency selective MIMO channels 8

1.3 An Adaptive Pilot Symbol-Aided MMSE Receiver in Fading Channels 9

1.3.1 The proposed MMSE receiver 10

CHAPTER II: MIMO SYSTEM 13

2.1 Overview 13

2.2 What is MIMO system? 13

2.3 MIMO channel 15

2.4 Communication Channel Model 17

2.5 Channel state information (CSI) 18

2.6 CSI and MIMO 18

2.7 Precoding 19

2.8 Precoding and equalization for MIMO systems 19

1

2.8.1 Channel model 19

2.9 Precoding and equalization for SISO channels 21

2.10 Joint precoding and equalization for narrow- band MIMO systems 24

2.11 Linear precoding and equalization for broad-band MIMO systems 27

2.12 BSVD Based MIMO Precoding and Equalization 30

2.12.1 Introduction 30

2.12.2 Channel Model and System Set Up 32

2.12.2.1 MIMO System Model 32

2.12.2.2 Block Based Precoder and Equaliser 33

2.12.2.3 Proposed Design 34

2.13 Capacity of MIMO Channels 35

2.13.1 Single user MIMO capacity 35

A Constant MIMO Channel Capacity 35

B. Fading MIMO Channel Capacity 35

B.1 Capacity with Perfect CSIT and Perfect CSIR 36

B.2 Capacity with No CSIT and Perfect CSIR 36

B.3 Capacity with Partial CSIT and Partial CSIR 37

B.4 Capacity with No CSIT and No CSIR 37

B.5 Capacity with Partial CSIT and Partial CSIR 37

B.6 Frequency Selective Fading Channels 38

2.13.2 Multiuser MIMO 38

A. MIMO MAC 38

A.1 Constant Channel 39

1

A.2 Fading Channels 39

B. MIMO Broadcast Channel 40

2.14 Open Problems in Multiuser MIMO 40

CHAPTER-III: Theory of MMSE 41

3.1 Overview 41

3.2 What is MMSE? 41

3.3 Definition 41

3.4 Minimum mean square error estimation 42

3.4.1 Estimation of a continuous random variable 42

3.4.2 Orthogonality 44

3.4.3 Linear minimum mean square error estimation 45

3.5 MMSE equalization and different types of equalizers 46

3.5.1 Major types of MMSE equalizers algorithm 47

A. Algorithms Using Training (desired) Sequence 47

B. Blind Adaptive Methods 47

C. Different Matrix Inversion Methods 50

3.5.2 Applications of MMSE channel equalizers 51

A. Channel equalization for CDMA and WCDMA systems 51

B. Channel equalization for OFDMA systems 52

C. Channel equalization for MIMO systems 53

CHAPTER-IV: MMSE PRECODER FOR MIMO SYSTEMS 54

4.1 MMSE precoder with mode selection 54

1

A. Conventional MMSE Precoder 54

B. Mode Selection Algorithm 55

4.2 Method for reducing performance degradation due to feedback delay 57

A. Channel Prediction 58

B. Receive weight Robust to Feedback Delay 59

C. Combination of Channel Prediction and Receive weight Robust to Feedback Delay 59

4.3 Performance of MMSE precoder with mode selection 60

4.4 Linear IIR-MMSE precoding for frequency selective mimo channels 62

4.4.1 Optimal causal precoder 63

4.4.2 Optimal non-causal precoder 63

4.3.3 Precoder design 63

4.5 Precoding based on iterative LMMSE detection 65

A. Precoder Structure 65

B. Detection Principles 66

4.6 BER performance of the proposed MMSE receiver 68

CHAPTER V: CONCLUSION 71

REFERENCES 72

1

LIST OF FIGURES

1.1 System model 8

1.2 Modified MMSE receiver 10

1.3 Transient and steady state behavior of the proposed MMSE receiver as a function of number of

iterations for the case: a vehicle moving at 50 Km/h and the number of users K=20 (x-axis is scaled

by a factor of 100) 11

2.1 System with Precoding over Flat MIMO Channel 14

2.2 General MIMO channel model 19

2.3 Precoding and equalization for SISO frequency selective channel 21

2.4 Linear precoding and equalization for broadband MIMO 28

2.5 Linear precoding for broadband MIMO - simplified scheme 28

2.6 MIMO channel H(z) with precoder P(z) and equaliser E(z) including a multiplexing by P 33

3.1 (a) Simplified model of typical communication system, 46

(b) Equivalent composite channel model 46

3.2 General model of blind channel estimation 48

3.3 Classification of channel models 48

3.4 Simplified OFDM system model 52

4.1 System model of MMSE precoder with mode selection 56

4.2 The block format at the transmitter 57

4.3 Selection probability of each mode (12 bit/s/Hz) 61

4.4 BER of MMSE precoder with mode selection (12 bit/s/Hz) 61

4.5 BER of the proposed MMSE precoder with methods for reducing Performance degradation due to

feedback (Q = 2, 12 bit/s/Hz) 62

4.6 system model 62

4.7 The ―+‖ marks belong to the FIR-MMSE, the ―x‖ marks to the IIR-MMSE 64

1 4.8 (a) Transmission over a MIMO ISI channel 65

(b) The equivalent parallel subchannel model of the system in (a)

(c) The precoding scheme based on the equivalent model in (b)

4.9 Structure of an iterative receiver 66

4.10 Transient and steady state behavior of the proposed MMSE receiver as a function of number of

iterations for the case: a vehicle moving at 50 Km/h and the number of users K=20 (x-axis is scaled

by a factor of 100). 69

4.11 BER as a function of the number of users for an asynchronous DS/CDMA system in Rayleigh

fading environments: 69-70

(a) a vehicle moving at 50 Km/h and 69

(b) a vehicle moving at 100 Km/h 70

4.12 BER as a function of the pilot symbol insertion period M for the case:a vehicle moving at 50

Km/h and the number of users K=20. 70

LIST OF TABLES

Table 2.1 TABLE OF ABBREVIATION 35

Table 4.1 MODE PATTERN (12 BIT/S/HZ) 60

1

CHAPTER - I

INTRODUCTION

1.1 MMSE precoder for MIMO systems

Multiple-input multiple-output (MIMO) systems that realize high-speed data transmission with mul-

tiple antennas at both transmitter and receiver are drawing much attention. In MIMO systems, the ap-

proach of weighting substreams so as to minimize the symbol mean square error (MMSE) using

channel state information (CSI) has been studied (denoted hereafter as MMSE precoder). The CSI is

available at the transmitter through feedback channel between the receiver and the transmitter in fre-

quency division duplex (FDD) systems. Using an MMSE precoder and a decoder can diagonalize the

MIMO channels into eigen-subchannels, the system achieves good performance [1], [2].

Early studies on MMSE precoder have arbitrarily fixed the number of substreams. However, the op-

timum number of substreams differs with the channel. Thus, selecting the optimum number of sub-

streams based on CSI is needed. Moreover, in the presence of feedback delay, the CSI fed back to the

transmitter becomes outdated due to the time-varying nature of the channels. That is, there is a mis-

match between ―the configured precoder‖ and ―the optimum precoder when the signals are transmit-

ted‖; an outdated MMSE precoder cannot achieve the full performance possible. Therefore, various

methods for reducing this performance degradation have been considered. The performance degrada-

tion due to feedback delay can be reduced by predicting the CSI when the signals are transmitted and

configuring the MMSE precoder and decoder using the predicted CSI (denoted hereafter as channel

prediction)[3][4].The performance degradation due to feedback delay can be reduced by using the

receive weight determined from the actual CSI when the signals are transmitted instead of using the

outdated CSI fed back to the transmitter (denoted hereafter as receive weight robust to feedback de-

lay)[5].

Here we will propose an MMSE precoder with mode selection for MIMO systems, which selects both

the number of substreams and the modulation scheme for each substream to minimize the average

BER at a fixed rate. We evaluate the BER of the proposed MMSE precoder with two methods for re-

1 ducing performance degradation due to feedback delay, channel prediction and receive weight robust

to feedback delay. We also evaluate the BER of the proposed MMSE precoder with the method that

combines channel prediction with receive weight robust to feedback delay. Simulation results show

that the BER of the proposed MMSE precoder is improved compared to that of the conventional

MMSE precoder using the fixed number of substreams. We also show that the method that combines

channel prediction with receive weight robust to feedback delay can achieve good BER even when the

large feedback delay exists.

1.2 Linear IIR-MMSE precoding for frequency selective MIMO channels

Precoding is a well-researched method to migrate the computational burden in channel equalization

from the receiver to the transmitter. Nevertheless, in contrast to the MMSE equalizer which is given

by the classical Wiener filter, the MMSE precoder is a quite recent development ( [7, VI.A]).

Various MMSE precoders have been derived for quite a few different system models. Here , we con-

sider the system model depicted in Figure 1.1 . The channel H and the receive filter G are considered

to be fixed and known. The transmitter employs the precoding filter P. Additionally, we include the

scalar gain α as an additional simple receive filter. The filters P and α are to be jointly optimized sub-

ject to a transmit power constraint. The idea to include a scalar gain at the receiver can be traced back

at least to a paper of Karimi et. al. It offers two main advantages. The first advantage is of course an

additional degree of freedom which usually will improve the overall system performance. The second

advantage is that we can find closed-form solutions. The FIR-MMSE precoder for the model in Fig.

1.1 was derived by Choi and Murch [8] and, using a different technique, also by Joham et. al.[7].

However, the IIR counterpart is still unknown. In this , we are going to derive the IIRMMSE precod-

er, both with and without causality constraint.

Fig. 1.1 System model

We show that the IIR precoder has the same advantages compared to the FIR precoder as in the re-

ceive filter case.

First of all, the IIR precoder is the optimal linear precoder and therefore can never be outperformed

by the FIR precoder.

1 Second, improving the latency time always improves the performance of an IIR precoder. This is in

contrast to FIR precoding, where the optimal latency time is not obvious and has to be found by a

costly exhaustive search or by suboptimal approaches. IIR precoders do not require such optimization.

Third, IIR precoders often outperform FIR precoders of similar complexity.

1.3 An Adaptive Pilot Symbol-Aided MMSE Receiver in Fading Channels

Interference suppression in direct sequence code division multiple access (DWCDMA) system is cru-

cial in providing higher system capacity. Standard minimum mean squared error (MMSE) receivers

can suppress multiple access interference (MAI) in DS/CDMA systems. The standard MMSE receiv-

ers [9]-[11] offer better performance than the conventional matched filter (MF) receiver in fixed (or

static) channels and have simpler structures compared to other multi-user CDMA receivers. However,

the realistic radio channels are not static but time-varying due to fading. In fading environments, spe-

cifically under the low signal to interference plus noise ratio (SINR) during the deep fading, the per-

formance of the standard MMSE receivers is severely degraded. Recently, there have been various

efforts to overcome the problems of the standard MMSE receivers in fading channels [12]-[14]. The

modified MMSE receiver proposed in [12] removes the channel phase variations from the received

signal before it enters the adaptive filter. The phase estimation is accomplished at the output of the

adaptive filter where the desired SINR should be substantially higher than at the input. This mechan-

ism reduces the burden of the adaptive filter to track the channel phase variations and works well in

fading channels. However, since it compensates channel phase variations only, it is much likely to

suffer from the large variations of the channel amplitude.

On the other hand, the modified MMSE receiver proposed in [14] simultaneously tracks the phase and

amplitude changes of the desired signal in fading channels, and compensates the variations of the

channel amplitude as well as the channel phase. The channel coefficients are, however, estimated be-

fore the adaptive filter, so that the estimation error of channel coefficients has large variance. If the

channel estimation is accomplished at the output of the adaptive filter in the modified MMSE receiv-

er, the tap weights vector degenerates to zero. Here, we propose a new MMSE receiver with a con-

straint that makes the tap weights of adaptive filter not converge to zero. This constrained MMSE re-

ceiver is adaptively implemented using the orthogonal decomposition based least mean square (LMS)

algorithm. We employ a pilot symbol-aided scheme, where each pilot symbol is periodically inserted

into data symbol streams. These pilot symbols are used to estimate the channel coefficient and act as a

training sequence. The simulation results show that the proposed MMSE receiver gives significant

performance improvements in bit error rate (BER) over the conventional MF receiver and the mod-

ified MMSE receivers in an asynchronous Rayleigh fading channel.

1 1.3.1 The proposed MMSE receiver

The modified MMSE criterion proposed in [14] can be implemented as Fig. 1.2.

Fig. 1.2 Modified MMSE receiver.

The cost function for the criterion is defined as

. (1.9)

Where d1 (m) is the product of the reference signal d,(m) and the estimate of the complex channel

coefficient denotes the Hermitian operation(complex conjugate and transpose opera-

tion). This modified MMSE criterion can simultaneously track the phase and amplitude changes of the

desired signal in fading channels. The criterion can be adaptively implemented using LMS algorithm

given by

. (1.10)

Where µ is the step size and (.)* denotes complex conjugate operation. The performance of this crite-

rion heavily depends on the accuracy of the reference signal and channel gain estimates. The output

signal of the adaptive filter gives less noisy signal (i.e. relatively less residual noise and interference)

than the input signal due to the MMSE criterion. Hence the output signal of the adaptive filter can be

used to give a fairly good estimate of the channel gain. The problem is that the channel estirnation at

the adaptive filter output gradually forces the tap weights of the modified MMSE receiver to be zero.

This fact comes from that the adaptive filter suppresses not only the interference due to other users but

1 also the desired signal. A simple example of this problem is shown in [15] and is experimentally

shown in Fig. 1.3.

Fig. 1.3 Transient and steady state behavior of the proposed MMSE receiver as a func-

tion of number of iterations for the case: a vehicle moving at 50 Km/h and the number

of users K=20 (x-axis is scaled by a factor of 100).

To solve this convergence problem, we propose a new constrained MMSE criterion given by

(1.11)

In (1.11), the constraint w1H(m)s1=1 prevents the convergence problem occurred when channel esti-

mation is accomplished at the output of the adaptive filter. The constrained MMSE criterion can be

adaptively implemented using the orthogonal decomposition-based LMS algorithm. From the ortho-

gonal decomposition method [9], the tap weights vector of adaptive filter is decomposed as

. (1.12)

Where it is assumed that spreading code vector s1 and adaptive component of the tap weights vector

x(m) are orthogonal, that is

. (1.13)

1 From ( 1.12 ) and (1.13), it follows that

(1.14)

Note that equation (1.14) is identical to the constraint given in (1.11). The received signal vector r1

(m) can also be decomposed as the sum of its projection vector on s, and its orthogonal vector, i.e.

(1.15)

Where

(1.16)

Then the orthogonal decomposition-based LMS algorithm is given by

(1.17)

From (1.14), it is clear that the orthogonal decomposition-based LMS algorithm keeps the tap weights

from approaching zero and works well with fading environment. In addition, we employ pilot symbol

aided scheme in order to improve the accuracy of the reference signal and channel estimate. Our

scheme does not require any separate training mode since each pilot symbol is periodically inserted

into data symbol streams during actual data transmission mode as proposed in [16]. That is, each pilot

symbol is inserted into every M data symbols. This scheme switches between the short pseudo-

training mode and decision-directed mode during actual data transmission without any separate train-

ing mode.

1

CHAPTER-II

MIMO SYSTEM

2.1 Overview

This chapter presents an overview of MIMO systems. It also highlights some of the key issues related

to MIMO systems.

2.2 What is MIMO system?

Fig. 2.1 shows a communication system employing Nt transmit antennas and Nr receive antennas,

which is called a Multiple-Input Multiple-Output (MIMO) system. In MIMO communication systems

[17, 18] the multiple data streams can be sent simultaneously from a transmitter employing multiple

antennas to a receiver that employs multiple receive antennas. The goal of a MIMO system is to in-

crease the data rate through spatial multiplexing and improving the error rate performance by increas-

ing signal diversity (this being achieved by increasing the number of transmit or receive antennas,

given that the probability of a fade at the same time in all the paths is reduced) to combat fading.

A MIMO system can be seen as a single-user point-to-point communication system. The special case

with Nt = Nr = 1 is called a Single-Input Single-Output (SISO) system. A second special case is when

Nt = 1 and Nr ≥ 2 and is called a Single-Input Multiple- Output (SIMO) system. Lastly, there exists

another special case if Nr = 1 and Nt ≥ 2, called a Multiple-Input Single-Output (MISO) system.

In MIMO systems with Nt transmit antennas and Nr receive antennas, we denote the equivalent low

pass channel impulse response between the j–th transmit antenna and the i– th receive antenna as

hi,j(τ, t). Thus, the randomly time-varying channel is characterized by the Nr × Nt matrix H(τ, t) de-

fined as

. (2.1)

1

Figure 2.1: System with Precoding over Flat MIMO Channel.

Suppose that the transmitted signal from the ith transmit antenna is xi(t). Then, the receive signal at

the jth receive antenna is given by

. (2.2)

Where ηj(t) is the additive noise. In matrix notation, this equation can be rewritten as

y(t) = H(τ, t) * x(t) + η(t) (2.3)

where x(t) = [x1(t), . . . , xNt(t)]T є C

Nt , y(t) = [y1(t), . . . , yNr(t)]

T є C

Nr , and η(t) = [η1(t), . . . , ηNr(t)]

T

є CNr

. For flat fading channels, the channel matrix H(τ, t) is transformed into the matrix H(t) given by

. (2.4)

and the received signal is now

. (2.5)

1 which can be expressed in matrix form as

(2.6)

In general, if we let f[n] = f(nTs + ∆) denote samples of f(t) every Ts seconds with ∆ being the sam-

pling delay and Ts the symbol time, then sampling y(t) every Ts seconds yields the discrete time signal

y[n] = y(nTs + ∆) given by

y[n] = H[q]x[n] + η(n) (2.7)

where n = 0, 1, 2, . . . corresponds to samples spaced with Ts and q denotes the slot time. The channel

remains unchanged during a block of NB symbols, i.e., over the data frame. Note that this discrete

time model is equivalent to the continuous time model in Eq. (2.6) only if ISI between samples is

avoided, i.e. if the Nyquist criterion is satisfied. In that case, we will be able to reconstruct the original

continuous signal from the samples by means of interpolation. This channel model is known as time-

varying flat block fading channels.

2.3 MIMO channel

How well can we know A MIMO channel?

Wireless channels are notorious for their spatiotemporal variation. So much so, that the usual recourse

is to assume stochastic models [19] and devise signaling methods which work well under uncertainty

[20, 21]. This state of affairs is especially frustrating for multiple input multiple output (MIMO)

channels because of the large potential gains possible were only the channel well known [20, 22]. Fur-

thermore, MIMO channels can be exploited for better inter-user isolation in multi-user systems [22,

23] and even, potentially, perfect wireless secrecy. A seeming oxymoron.. Precise channel knowledge

is essential for such applications. It is these tantalizing benefits which prompt us to ask the basic ques-

tion: how well can we know a MIMO channel?

The underlying physics of wireless channel composition is what makes these sorts of questions so

interesting. Stochastic variation of wireless channels is caused by the stochastic movement of scatter-

ers and transceivers. Scatterers could be vehicles, buildings, lampposts, trees, animals, people, some-

times even the atmosphere - Anything that interacts with radio waves and moves. And it is exactly

this dependence of the channel on the motion of macroscopic objects which suggests that knowing the

wireless channel and tracking its changes might not be the seeming fool's errand which has historical-

ly caused wireless theorists to collectively throw up our hands and invoke probability theory.

1 Specifically, the motion of the macroscopic objects which cause scattering is constrained by the ener-

gy available to move them. In particular, we can show that unpredictable motion is limited owing to

the energy necessary to change the momentum of an object. Thus, a skyscraper may sway, a vehicle

may speed and a person may shift position, but all will do so in a relatively predictable manner -

which implies that the information rate necessary to specify their positions could be relatively small.

The information and information rates which are necessary to specify relevant constituent parts of the

channel constitute exactly an upper bound on the amount of information that must be extracted from

measurements to specify the channel. Thus, by first deriving bounds on the information necessary to

specify and track relevant channel constituents and then comparing them to the amount of information

that can be extracted from channel interrogation, we can determine precisely whether knowing a fad-

ing MIMO channel well is possible or impossible. We suspect that because constituents of most scat-

tering environments are relatively massive objects and energy to move them stochastically is limited,

the information rates necessary to track changes and disseminate them appropriately could be mana-

geable. Establishing exactly when the channel can be known with an eye toward the benefits which

can accrue from such knowledge is our aim.

To do so, we must first quantify the effects of inaccurate channel knowledge and establish just how

well the channel must be known in order to realize the full benefits of MIMO systems. We will ex-

plore these issues somewhat anecdotally in this proposal to obtain a feel for the necessary precision,

but our ultimate goal is to determine how we may specify the channel state in bits - essentially a quan-

tization problem.

We must then consider the modeling of scattering channels at a physical level as an assortment of spa-

tially distributed scattering objects (perhaps grouped into ―scattering centers‖) whose ensemble prop-

erties and evolution we wish to track. We will argue that energy considerations constrain the stochas-

tic motion of scattering center constituents as well as transceivers to allow more directed estimation of

channel parameters from available probe measurements. Or more simply put, we will argue that our

approach, driven by energy constraints on objects in motion, can bound the channel tracking search

space and thus help with more efficient channel state evolution prediction.

Overall we seek to:

Quantify the necessary channel state information (CSI) for accuracy levels which enable ben-

efits such as increased rate, mutual interference mitigation and perfect secrecy in MIMO sys-

tems.

1

Understand channel variation as a function of energy bounds on the physical mobility of con-

stituent scattering objects and transceivers, thereby providing an upper bound on the entropy

rate of CSI.

Determine when and how such models might be acquired blindly (and/or under preexisting

generic classifications of scatterer types) from channel probes including potential augmenta-

tion with dedicated sensor measurements.

When channel state prediction is possible, demonstrate its feasibility through analysis and si-

mulation.

2.4 Communication Channel Model

We will use the generic signal space vector-channel model

r = Gu + w (2.8)

where received vector r and noise vector w are N-dimensional, and transmit vector u is M dimension-

al. The gain matrix G is thus of dimension N * M. The assumption with any such model is that all

transmitters and receiver waveforms lie in some common signal space defined by some common set

of orthonormal basis functions. Usually, owing to an assumption of channel linearity, time-invariance

and synchronization, these basis functions are sinusoids. However, the signal space description could

also be time-based with different dimensions corresponding to different time samples or whatever

convenient basis set is available. Here we will always assume w is Gaussian, though not necessarily

white, and that the gain matrix G is random in a way reflected by the physics of the particular channel

and constant over the signaling interval. That is, G is a specific ―channel instance‖ during each signal-

ing interval for which equation (2.8) applies. The sequence of channel instances is assumed to be

some ergodic though not necessarily stationary stochastic process. For a known channel G and Gaus-

sian noise w with covariance W, the channel capacity is given by [20, 24]

(2.9) .

Where Ru = E[uuH], P is the power available for signaling and the ui which comprise u are zero mean

and jointly Gaussian. The optimal Ru is the usual water-filling solution in the right-eigenspace of W-

1/2G, water filled over the inverse of it magnitude-squared non-zero singular values. The total capacity

is then C (G) [20] averaged over all channel instances

C = EG [C(G)] (2.10)

1

2.5 Channel state information (CSI)

In wireless communications, channel state information (CSI) refers to known channel properties of a

communication link. This information describes how a signal propagates from the transmitter to the

receiver and represents the combined effect of, for example, scattering, fading, and power decay with

distance. The CSI makes it possible to adapt transmissions to current channel conditions, which is

crucial for achieving reliable communication with high data rates in multiantenna systems.

CSI needs to be estimated at the receiver and usually quantized and fed back to the transmitter (al-

though reverse-link estimation is possible in TDD systems). Therefore, the transmitter and receiver

can have different CSI. The CSI at the transmitter and the CSI at the receiver are sometimes referred

to as CSIT and CSIR, respectively.

2.6 CSI and MIMO

Most current work on channel state information for MIMO systems asks questions about the rates

achievable, outage probabilities and other performance metrics when channel state is known to the

transmitter, to the receiver, both. or if neither, what performance is possible when the channel state

distribution is known (see [20] for a recent survey). Such studies provide powerful outer bounds, but

do not quite address on of the questions we hope to pose - how does one usefully quantify channel

state. This problem is a bit difficult in that the usual measures of accuracy. Such as variance bounds

on the entries gij of a gain matrix G - may not be as useful as in single input single output (SISO) cas-

es.

For instance, knowing the precise value of a nearly zero gain in a MIMO system is simply not that

important when channels with much stronger gains are available. More critical in this case would be

telling the difference between good and bad signaling dimensions quickly. Furthermore, although the

specific gains are important in a MIMO system, so is the structure of the vector space they express.

Finding ways to usefully measure and quantify CSI is therefore part of this proposed work, although

the usual approach is to find minimum mean square error estimates of channel matrices [25-27].

That notwithstanding, we will start our explorations with the usual model typically used for MIMO

capacity with estimation error problems [27]

. (2.11)

where Q is a zero mean matrix with independent identically distributed random entries of some ―error

variance‖ σe2 . G is renormalized so that Ĝ also has unit variance entries. This type of assumption is

reasonable for many channel interrogation methods.

1

2.7 Precoding

Precoding is a generalization of beamforming to support multi-layer transmission in multi-

antenna wireless communications. In conventional single-layer beamforming, the same signal is emit-

ted from each of the transmit antennas with appropriate weighting such that the signal power is max-

imized at the receiver output. When the receiver has multiple antennas, single-layer beamforming

cannot simultaneously maximize the signal level at all of the receive antennas. Thus, in order to max-

imize the throughput in multiple receive antenna systems, multi-layer beamforming is required.

In point-to-point systems, precoding means that multiple data streams are emitted from the transmit

antennas with independent and appropriate weightings such that the link throughput is maximized at

the receiver output. In multi-user MIMO, the data streams are intended for different users (known

as SDMA) and some measure of the total throughput (e.g., the sum performance) is maximized. In

point-to-point systems, some of the benefits of precoding can be realized without requiring channel

state information at the transmitter, while such information is essential to handle the co-user interfe-

rence in multi-user systems.

2.8 Precoding and equalization for MIMO systems

In this chapter, some precoding and equalization techniques are discussed. They include joint precod-

ing and equalization for single input single output frequency selective channels, joint precoding and

equalization for narrowband MIMO systems and linear precoding for broadband MIMO channels.

2.8.1 Channel model

In the following, a general discrete-time noise free model of baseband MIMO system with T transmit

and R receive antennas as shown in Figure 2.2 is considered.

Fig. 2.2 General MIMO channel model

1 In general case, the channel can be considered to be frequency selective with finite impulse response

(FIR). The channel transfer function C(z) є CR×T

(z) can be written as

(2.12)

The maximum support length of the channel impulse responses (CIRs) between each pair of transmit

and receive antennas is L+1. The matrix C[l] є CR×T

contains the l th time slice of these CIRs. In gen-

eral case, it can be further assumed that each of the T MIMO inputs emerges from a time multiplex of

P input lines. Similarly, each of the R outputs can be demultiplexed into P signals. With the input

symbol vector u[n] є CPT

and output symbol vector r[n] є CPR at discrete time instance n defined as

where ui[n], (i = {1, . . . , T}) is the signal sent to the ith transmit antenna and rj [n], (j = {1, . . . ,R}) is

the signal received on the jth receive antenna, the resulting spatio-temporal MIMO system can be

written as

(2.13)

Whereby and the spatio-temporal MIMO matrix takes the block-pseudo-circulant form

1

. (2.14) .

The matrices Cp(z), p = 0, 1, . . .P − 1, are the P polyphase components of C(z) such that

(2.15)

or alternatively

In the following, based on the above general channel model, several joint precoding and equalization

schemes will be reviewed.

2.9 Precoding and equalization for SISO channels

In this section, a linear joint optimal precoding and equalization design for block transmission over

frequency selective channel, which was proposed in [28] by Scaglione et al. is discussed. Similar to

other block transmission schemes, this approach also utilises transmit redundancy in the form of zero

padding intervals to mitigate the inter-block interference (IBI) caused by the channel frequency selec-

tivity, then the joint optimal precoder and equaliser filter banks are designed to remove the intrablock

interference and combat noise. The optimal criteria are maximum output SNR (MaxSNR), minimum

mean square error under constrained transmit power (MMSE/CP) [28] and maximum information rate

(MaxIR) [29].

The discrete-time system model is illustrated in Figure 2.3. The channel in this case is a SISO fre-

quency selective channel, corresponding to the model presented in Section 2.8.1 with T = R = 1.

Figure 2.3: Precoding and equalization for SISO frequency selective channel

1 The input symbol stream is converted into a sequence of blocks of size M. A guard interval is inserted

to mitigate IBI through the upsamplers by P where P > M. Thus, the input blocks of size M are

mapped into blocks of size P by the precoder filter bank. After being transmitted through frequency

selective channel

with impulse response c[n], the received blocks of size P are mapped back to a sequence of blocks of

size M by the equaliser filter bank. The input and output signal vectors are defined as

the output vector of the precoder and the noise-free output of the channel as

and the received vector with noise and the corresponding noise vector as

(2.16)

In the general case where there is no constraint on the length of precoding filters, channel and equalis-

ers, the relation between input and output vectors of precoder, equaliser and channel can be written as

(2.17)

1

(2.18)

Where

(2.19)

(2.20)

(2.21)

The optimal precoder and equaliser under MaxSNR criteria are designed to maximize the SNR at the

equaliser output subject to zero-forcing (ZF) constraint GHF = I. The equaliser is derived as a func-

tion of the precoder matrix from the condition of maximizing the output SNR, the precoder matrix is

then calculated from the ZF constraint.

1 Based on the following eigendecompositions

(2.22)

where U, V, Vn are unitary matrices and ∆, Λ are diagonal matrices, the optimal precoder and equalis-

er under the MaxSNR criterion are given by [28]:

(2.23)

where K is defined as transmit-amplification gain and depends on the transmit power and σv2 is the

variance of noise.

The optimal precoder and equaliser pairs mentioned above render a frequency selective SISO channel

into a number of flat sub channels. One should note that the water-filling algorithm for MMSE/CP

design differs from the water-filling algorithm for MaxIR design. In the former, depending on the val-

ue of the water level, the transmit power allocated for each flat subchannel can be a convex function

or monotonically decreasing function of _ii while in the later the transmit power allocated for each

flat subchannel is always a monotonically increasing function of λii.

2.10 Joint precoding and equalization for narrow- band MIMO systems

This section will discuss a joint precoding and equalization design proposed in [30. 31] to mitigate

multiuser interference (MUI) while maximizing the system capacity or allowing power control in

multiuser narrowband MIMO systems. The algorithm, which is referred to as block diagonalisation, is

based on the SVD of the channel matrices, the precoder is designed to remove the inter-user interfe-

rence and achieve maximum capacity or allow power control and the equaliser is designed to separate

the individual data streams.

Consider a multiuser flat-fading MIMO system with a base station which has T transmit antennas and

K users, each with Ri receive antennas. Let the total number of antennas of all receivers to be

. The MIMO channel between the transmitter and the ith user is represented by matrix

1 Hi є C

Ri×T . One can see that this MIMO system is a specific case of the system described in Section

2.8.1 with L = 0, P = 1 and H(z) = C0 is actually a stacked version of all matrices Hi. For simplicity,

the matrix H(z) is written as H(z) = H. Let the precoding matrix associated with the ith user be de-

noted by Bi є CT×mi

where mi is the length of the vector of symbols si[n] which is destined to

this user. The input vector si[n] is linearly mapped by the precoder Bi to vector ui[n] which is actually

broadcast from the transmit antennas. At the input of the ith receiver, the received signal includes the

contributions from the signals for all users as well as the noise and can be written as

(2.24)

where vi[n] denotes the spatially white noise and interference with covariance matrix

To capture the operation of the whole system in matrix form, the received data from all of the receiv-

ers are stacked together and represented as

(2.25)

where the definitions of y[n],H,B, s[n] are clear from the equation

In order to remove inter-user interference, the precoders are designed so that HiBi ≠ 0 and HjBi = 0 for

j≠i. In other words, HB is a block-diagonal matrix. Define matrix Ĥi є C(R−Ri)×T

as follow

. . (2.26)

Assume T > R and let rank (Ĥi) = Li ≤(R − Ri), consider the following SVD

(2.27)

where i,1 contains the first Li rows that correspond to the non-zero singular values of Ĥi and

contains the last T − Li rows. It is clear that the columns of span the null space of

. Thus, can help to eliminate MUI and therefore let , the precoder matrix B for all users

can be written as

1

(2.28)

The overall channel transfer matrix preprocessed by B now becomes

(2.29)

Further, let L΄i = rank(Hi ) and consider the SVD

(2.30)

where Vi,1 holds the first L΄i right singular vectors that correspond to the non-zero singular values in

diagonal matrix ∑i є CL΄i×L΄i

, one can see that

(2.31)

This motivates the use of an equaliser Wi = UiH and setting the precoder as

(2.32)

So that the product

(2.33)

is a completely diagonal matrix, which means the MUI and the co-channel interference caused by

MIMO components is completely eliminated.

In order to maximise the system capacity, a water-filling algorithm with single water level is per-

formed on the diagonal elements of ∑ so that the transmit power will be allocated accordingly. Thus

the precoder B now becomes

1

(2.34)

where Λ is a diagonal matrix with diagonal elements λjj obtained from the above mentioned water-

filling algorithm.

The system capacity is given by

. (2.35)

With the precoder matrix defined as in equation (2.34), it can be seen that mi = L΄i and therefore, it is

necessary that L΄i ≥ 1 so that the transmission for the ith user can take place

In the case of the power control problem where one have to minimize the transmit power subject to

achieving a desired transmission rate for each user, the precoder and equaliser matrices can be derived

in similar steps as mentioned above, except that the matrix _ is defined by performing water-filling

separately for each user, where the constrained transmit power for each user is scaled to achieve the

required transmission rate.

2.11 Linear precoding and equalization for broad-band MIMO systems

In this section, an approach for joint precoding and equalization for frequency selective MIMO sys-

tems is discussed. This approach was proposed in [32] and similar to the one in [33], it is also based

on block transmission and utilises redundancy in the form of guard intervals to mitigate inter-block

interference and exploits the channel eigendecompositions to design the optimal precoders and equa-

lisers. Several design criteria were proposed in [32] which targeted minimum MSE and BER under

constraints on the transmit average power or peak power. This section will focus on the optimal de-

signs under constrained transmit average power.

Consider the MIMO channel model in Section 2.8.1, assume that the channel is stationary or slowly

time-varying, the channel with linear precoder and equaliser is illustrated in Figure 2.4.

The blocks of input symbols s[n] є CN are mapped into vectors u[n] є C

PT by the precoder F so that

u[n] = Fs[n] (2.36)

1

Figure 2.4: Linear precoding and equalization for broadband MIMO

Through the parallel-to-serial converter, each vector u[n] is divided into P blocks of length T, which

will be transmitted through T transmit antennas after pulse shaping. At the receiver, the received sym-

bol blocks of length R from the receive antennas are stacked together by the serial-to-parallel conver-

ter to form the vector y[n] є CPR

. The equaliser G є CN×PR

will perform the inverse mapping on y[n] to

give the estimated output symbol blocks ŝ[n].

The system can be equivalently illustrated as in Figure 2.5 where the MIMO channel is now

represented by pseudo-circulant matrix H(z) as in (2.4). With P > L, polynomial matrix H(z) has unit

order and the relation between the input and output blocks can be written as

. (2.37)

whereby H0 and H1 are two coefficient matrices of H(z), v[n] is the block of noise samples.

Figure 2.5: Linear precoding for broadband MIMO - simplified scheme

Assume that the channel is stationary or slowly time-varying, one can write H0 and H1 in the follow-

ing form

1

,

(2.38)

. (2.39)

From equations (2.37) and (2.39), one can see that similar to the case of SISO dispersive channels in

Section 2.9, the IBI here can be eliminated by setting the term GH1F to zero, which also leads to either

the TZ approach where the last LT rows of precoder matrix are forced to be zero or LZ approach

where the first LR columns of equaliser matrix are set to zero. These two approaches are equivalent to

the setting of the last L blocks among P transmitted blocks of length T to zero (TZ method) or dis-

carding the first L blocks among P received blocks of length R (LZ method) as mentioned in [32]

With the IBI eliminated and P = M + L, equation (2.37) can be simplified as

. (2.40)

whereby in the TZ case G0 є CN×PR, F0 є C

MT×N and H є C

PR×MT

. (2.41)

1

and in the LZ case G0 є CN×MR

, F0 є CPT×N

and H є CMR×PT

,

(2.42)

In the case when the channel is time-varying, H is a block-banded matrix.

Note that in order for the output symbols to be recovered by linear equaliser G, it is necessary that

N ≤ rank (H). Therefore in the TZ case, it is required that N ≤ min (PR,MT) and in the LZ case,

N ≤ min(MR, PT) is required.

2.12 BSVD Based MIMO Precoding and Equalization

The use of guard intervals to eliminate IBI in block transmission systems reduces the spectral effi-

ciency of the system. This chapter will discuss a new approach to precoding and equalization for fre-

quency selective MIMO channels by applying a recently proposed broadband singular value decom-

position (BSVD) to decouple the MIMO channel matrix into approximately independent frequency

selective SISO sub channels. In a second step, the remaining ISI in the sub channels is eliminated.

This first step helps to remove not only co-channel interference (CCI) but can also eliminate part of

the inter-symbol interference with a very small loss in channel power gain as will be demonstrated.

The proposed method can provide better bit error rate performance than that of a benchmark design.

Under a quality of service constraint, the proposed design can achieve higher data throughput and mu-

tual information than that of the benchmark design while maintaining a similar symbol error perfor-

mance.

2.12.1 Introduction

In wireless communications MIMO systems arise when multiple antennas are used at both the trans-

mitter and receiver sides. Such systems can offer transmission with increased capacity over SISO

channels provided that the transmission paths are uncorrelated [34, 35, 36] and at the same time pro-

vide an increase in range and reliability without consuming additional bandwidth.

In many cases, it is assumed that the channel state information (CSI) is available only at the receiver

(CSIR). In such cases, either space-time coding [37, 38, 39], the V-BLAST [34, 40] or equalization

techniques [41, 42, 43] can be applied. However, in some scenarios, such as frequency division dup-

lex (FDD) or time division duplex (TDD) systems, the CSI can be made available at the transmitter

1 (CSIT) either through a feedback channel or through the reciprocity of the channel. In such cases, the

problem of joint transmit and receive processing or joint precoder and equaliser design [44, 45, 46,

47,1,32] becomes very appealing as it can achieve much higher performance than systems with iso-

lated designs.

A large number of research publications focus on the case of narrowband or frequency flat fading

MIMO [30, 48,49], where the channel can be represented by a matrix and the standard singular value

decomposition (SVD) plays a central role in the joint design process in order to decouple the MIMO

channel into independent flat sub channels.

With the demand for higher transmission rates, the transmission channel can no longer be considered

as narrowband and designs for the resulting broadband MIMO systems have attracted attention. In

broadband MIMO systems, apart from the elimination of CCI caused by the MIMO components, ad-

ditionally the elimination of ISI caused by the channel frequency selectivity is required. A widely ap-

plied approach is based on block transmission. Firstly, multicarrier modulation can be utilized to de-

compose the broadband problem into a number of narrowband ones, where the above mentioned po-

werful SVD-based designs can decouple the MIMO system. Secondly, single carrier broadband ap-

proaches have been formulated for the single-input single output case in [28, 29], which can be easily

extended to the MIMO case [32]. In [46], Palomar et al. also assume that the IBI has been eliminated

by the use of guard intervals and then generalize the results on joint design of linear precoding and

equalization for flat fading MIMO systems to several criteria using convex optimization functions.

There, the equaliser is first derived as a Wiener filter, then under different optimization criteria that

have been unified in form of Schur-concave or Schur-convex functions, the precoder is determined

via the SVD of the whitened channel.

In [47], an iterative algorithm to design joint optimal precoder and equaliser for broadband MIMO

channels was proposed. There, the optimal precoder and equaliser in the form of FIR MIMO filters

are designed to minimize the system MSE under constrained transmit power and in the presence of

near-end crosstalk.

The designs in [28, 46, 2] generally rely on a block-transmission approach, which requires a certain

amount of redundancy to eliminate inter-block interference (IBI). This redundancy limits the spectral

efficiency of the system. The loss in spectral efficiency can be reduced by increasing transmit block

size, however due to the constrained transmit power, the energy per symbol will be decreased and

therefore the bit error rate performance becomes poorer. Also, one can see that the use of guard inter-

vals always requires an amount of degrees of freedom (DOF) equal to the channel order to be invested

into IBI cancellation only and therefore it cannot be traded-off against ISI and noise amplification

1 unless channel shortening is used. In [47], it is also shown that the use of zero-padding intervals as

proposed in [28,46] is not optimal in terms of SNR performance.

In this chapter, a method for precoding and equalization for point-to-point broadband MIMO channels

is proposed. Different from block transmission based approaches in [32, 46], in the first step a recent-

ly proposed broadband singular value decomposition (BSVD) [50, 51] is utilized to decompose the

broadband MIMO channel matrix, which is polynomial, into two paraunitary matrices and a poly-

nomial diagonal matrix and thus the broadband MIMO channel can be decomposed into a number of

nearly independent frequency selective (FS) SISO subchannels whose transfer functions correspond to

the main diagonal elements of the diagonal polynomial matrix mentioned above. The use of BSVD

helps to eliminate not only the CCI in the MIMO channel but also a part of ISI when combined with a

water-filling algorithm in the second step. In the second step, the decoupled FS SISO subchannels are

precoded and equalized using standard methods . Since ISI has been eliminated partly with the help of

the BSVD, this approach loosens the constraint of ISI elimination and provides a possibility to

achieve a better spectral efficiency for the decoupled FS SISO subchannels and thus leads to an im-

proved system performance.

2.12.2 Channel Model and System Set Up

2.12.2.1 MIMO System Model

Here the precoding and equalization design for the stationary broadband MIMO channel, whose mod-

el has been described in Section 2.8.1 will be considered.

In the general case, when one assumes that the signal on each of the T inputs has resulted from a time-

multiplexing of P input signals, and each of R outputs is demultiplexed into P signals, the MIMO

channel can be represented by the pseudocirculant matrix H(z) є CRP×TP

(z) as explained in Section

2.8.1. Recall that H(z) is given by

(2.43)

where Cp(z) are the polyphase components of C(z),

(2.44)

1 In the following a generic model with precoder P(z) є C+ (z) and equaliser E(z) єC

PR×K(z) and the

channel H(z) as illustrated in Figure 2.6 is considered.

Figure 2.6 MIMO channel H(z) with precoder P(z) and equaliser E(z) including

a multiplexing by P.

2.12.2.2 Block Based Precoder and Equalizer

When P = 1, H(z) = C(z) is an R × T polynomial matrix of order L. As the number of polyphase com-

ponents P increases, the size of H(z) becomes larger, but its polynomial order reduces in accordance

with the shortening polyphase responses. Once P = L is reached, the polyphase components Cp(z) are

constants with no dependency on z. However, the block-pseudo-circulant form of H(z) in (2.14) en-

sures that for all P > L, the spatio-temporal MIMO system matrix H(z) will be a first order polynomi-

al, which means that IBI always exists.

As it has been reviewed in Sections 2.9 and 2.11, to overcome the polynomial order and therefore to

eliminate the ISI or IBI, the block transmission based system in [28] for T = R = 1 and in [1,2] for

arbitrary T and R rely on a time multiplex that is chosen longer than the channel order, i.e. P > L. As a

result, H(z) now becomes a sparse block-pseudo-circulant matrix of only first order in z, as noted ear-

lier. Specifically

H(z) = H0 + H1z−1 (2.45)

whereby H0 and H1 are given in (2.38) and (2.39) for the MIMO case.

The polynomial order of the MIMO system matrix H(z) can be eliminated by suppressing H1 through

either TZ or LZ approaches as mentioned in Sections 2.9 and 2.11. Thus the polynomial nature of

H(z) has been eliminated and the precoder and equalizer can be selected as non-polynomial matrices.

However, one can see again that these TZ or LZ approaches or even the multicarrier approach which

uses cyclic prefix always require at least the first L degrees of freedom (DOF) to be used only for the

ISI elimination.

1

2.12.2.3 Proposed Design

The proposed design has two components. Firstly, the MIMO system matrix H(z) is decoupled into a

number of independent FS subchannels based on a recently proposed broadband singular value de-

composition [50, 51]. This allows to factorize H(z) as

(2.46)

whereby S(z) = diag{S00(z), S11(z), · · · SK−1,K−1(z)} and U(z) and V(z) are paraunitary matrices.

The factorization (2.46) motivates the use of a precoder P(z) containing the first K columns of V(z)

and an equaliser E(z) containing the first K rows of Ũ(z) such that the broadband MIMO channel ma-

trix is decomposed into K _≤min(RP, TP) independent FS subchannels. One can write

(2.47)

(2.48)

where is the signal vector at the input of the precoder P(z), CK(z) is the signal

vector at the output of the equaliser E(z), and V (z) єCRP

(z) characterizes additive white Gaussian

noise as illustrated in Figure 2.5.

Although the CCI has been eliminated with the help of P(z) and E(z), the decoupled subchannels are

still dispersive and cause ISI. Therefore in a second step a precoder and equaliser are designed for

each decoupled subchannel so that the remaining ISI is eliminated. These precoders and equalisers

can be the linear optimal precoders and equalisers in [28, 29] which are also reviewed in Section 2.9

or nonlinear optimal precoders and equalissers proposed in [52]. One can see that depending on the

precoding and equalization method applied for the SISO FS subchannels, the block transmission

might be invoked in the second step, but only for a small portion of the system design. In addition, the

second stage design of precoders and equalisers can take the individual properties of each subchannel

— such as its SNR — into account. Note that P is both the number of polyphase components and the

block size. For block transmission systems in [2, 28], P >> 1, while for the designs proposed here, P is

generally very small and can be equal to 1.

1

2.13 Capacity of MIMO Channels

2.1 Table of abbreviation

First we will discuss the single user MIMO capacity results which can be summarized under various

assumptions on CSI.

2.13.1 Single user MIMO capacity

A Constant MIMO Channel Capacity

When the channel is constant and known perfectly at the transmitter and the receiver, MIMO channel

can be converted to parallel, non-interfering single input single output (SISO) channels through a sin-

gular value decomposition of the channel matrix.

Interestingly, the worst-case additive noise for this scenario is where the worst case noise is also given

explicitly for low SNR.

Although the constant channel model is relatively easy to analyze, wireless channels in practice typi-

cally change over time due to multipath fading. The capacity of fading channels is investigated next.

B. Fading MIMO Channel Capacity

With slow fading, the channel may remain approximately constant long enough to allow reliable esti-

mation at the receiver (perfect CSIR) and timely feedback of the channel state to the transmitter (per-

fect CSIT). However, in systems with moderate to high user mobility, the system designer is inevita-

bly faced with channels that change rapidly. Fading models with partial CSIT or CSIR are more ap-

plicable to such channels. Capacity results under various assumptions regarding CSI are summarized

in this section.

1

B.1 Capacity with Perfect CSIT and Perfect CSIR

Perfect CSIT and perfect CSIR model a fading channel that changes slow enough to be reliably meas-

ured by the receiver and fed back to the transmitter without significant delay. The ergodic capacity of

a flat-fading channel with perfect CSIT and CSIR is simply the average of the capacities achieved

with each channel realization. The capacity for each channel realization is given by the constant chan-

nel capacity expression. Since each realization exhibits a capacity gain of min(M,N) due to the mul-

tiple antennas, the ergodic capacity will also exhibit this gain.

B.2 Capacity with No CSIT and Perfect CSIR

The two relevant definitions in this case are capacity versus outage (capacity CDF) and ergodic capac-

ity. For any given input covariance matrix the input distribution that achieves the ergodic capacity is

to be complex vector Gaussian, mainly because the vector Gaussian distribution maximizes the entro-

py for a given covariance matrix. This leads to the transmitter optimization problem - i.e., finding the

optimum input covariance matrix to maximize ergodic capacity subject to a transmit power (trace of

the input covariance matrix) constraint. Mathematically, the problem is to characterize the optimum Q

to maximize capacity

(2.49)

Where

(2.50)

is the capacity with the input covariance matrix and the expectation is with respect to

the channel matrix H. The capacity C(Q) is achieved by transmitting independent complex circular

Gaussian symbols along the eigenvectors of Q. The powers allocated to each eigenvector are given by

the eigenvalues of Q.

While the channel models assume uncorrelated and frequency flat fading, practical channels exhibit

both correlated fading as well as frequency selectivity. The need to estimate the capacity gains of

BLAST for practical systems in the presence of channel fade correlations and frequency selective fad-

ing sparked off measurement campaigns. The measured capacities are found to be about 30% smaller

than would be anticipated from an idealized model. However, the capacity gains over single antenna

systems are still overwhelming.

1 B.3 Capacity with Partial CSIT and Partial CSIR

There has been much interest in the capacity of multiple antenna systems with perfect CSIR but only

partial CSIT. It has been found that unlike single antenna systems where exploiting CSIT does not

significantly enhance the Shannon capacity, for multiple antenna systems the capacity improvement

through even partial CSIT can be substantial. Key work on capacity of such systems has provided

many interesting results.

B.4 Capacity with No CSIT and No CSIR

With perfect CSIR, channel capacity grows linearly with the minimum of the number of transmit and

receive antennas. However, reliable channel estimation may not be possible for a mobile receiver that

experiences rapid fluctuations of the channel coefficients. Since user mobility is the principal driving

force for wireless communication systems, the capacity behavior with partial CSIT and CSIR is of

particular interest.

MIMO capacity in the absence of CSIT and CSIR is in which the channel matrix components are

modeled as i.i.d. complex Gaussian random variables that remain constant for a coherence interval of

T symbol periods after which they change to another independent realization. Capacity is achieved

when the T x M transmitted signal matrix is equal to the product of two statistically independent ma-

trices: a T x T isotropically distributed unitary matrix times a certain T x M random matrix that is di-

agonal, real, and nonnegative. This result enables them to determine capacity for many interesting

cases. Marzetta and Hochwald show that, for a fixed number of antennas, as the length of the cohe-

rence interval increases, the capacity approaches the capacity obtained as if the receiver knew the

propagation coefficients. However, perhaps the most surprising result is the following: In contrast to

the reported linear growth of capacity with min(M,N).under the perfect CSIR assumption, showed

that in the absence of CSI, capacity does not increase at all as the number of transmitter antennas is

increased beyond the length of the coherence interval T. The MIMO capacity in the absence of CSIT

and CSIR was further explored by Zheng and Tse. They show that at high SNRs the optimal strategy

is to use no more than M* = min(M,N[T/2]) transmit antennas. In particular, having more transmit an-

tennas than receive antennas does not provide any capacity increase at high SNR.

B.5 Capacity with Partial CSIT and Partial CSIR

In some results little hope of achieving the high capacity gains predicted for MIMO systems is left

when users are highly mobile. However, before resigning ourselves to these less than optimistic re-

sults we note that these results assume no CSIT or CSIR. Even for a rapidly fluctuating channel where

1 reliable channel estimation is not possible, it might be much easier to estimate the distribution of the

channel fades instead of the channel realizations themselves. This is because the channel distribution

changes much more slowly than the channel itself. The estimated distribution can be made available

to the transmitter through a feedback channel. This brings us to the realm of MIMO capacity with par-

tial CSI.

B.6 Frequency Selective Fading Channels

While flat fading is a realistic assumption for narrowband systems where the signal bandwidth is

smaller than the channel coherence bandwidth, broadband communications involve channels that ex-

perience frequency selective fading. Research on the capacity of MIMO systems with frequency se-

lective fading typically takes the approach of dividing the channel bandwidth into parallel flat fading

channels, and constructing an overall block diagonal channel matrix with the diagonal blocks given by

the channel matrices corresponding to each of these sub channels. Under perfect CSIR and CSIT, the

total power constraint then leads to the usual closed-form waterfilling solution. Note that the waterfill

is done simultaneously over both space and frequency. Even SISO frequency selective fading chan-

nels can be represented by the MIMO system model in this manner.

2.13.2 Multiuser MIMO

We consider the two basic multi-user MIMO channel models: the MIMO multiple-access channel

(MAC), and the MIMO broadcast channel (BC). Since the capacity region of a general MAC has been

known for quite a while, there are quite a few results on the MIMO MAC for both constant channels

and fading channels with different degrees of channel knowledge at the transmitters and receivers.

The MIMO BC, however, is a relatively new problem for which capacity results have only recently

been found. As a result, the field is much less developed.

A. MIMO MAC

In this section we summarize capacity results on the multiple-antenna MAC. We first consider the

constant channel scenario, and then look at the fading channel. Since the capacity region of a general

MAC is known, the expressions for the capacity of a constant MAC are quite straightforward. For the

fading case, one must consider the CSI (channel knowledge) available at the transmitter and receiver.

We consider four cases: perfect CSIR and perfect CSIT, perfect CSIR and partial CSIT, perfect CSIR

and no CSIT, and finally the case where there is neither CSIT nor CSIR.

1 A.1 Constant Channel

The capacity of any MAC can be written as the convex closure of the union of pentagon regions for

every product input distribution satisfying the user-by-user power constraints. For the Gaussian MI-

MO MAC, however, it has been shown that it is sufficient to consider only Gaussian inputs and that

the convex hull operation is not needed . For any set of powers (P1,……,PK), the capacity of the MI-

MO MAC is:

The i-th user transmits a zero-mean Gaussian with spatial covariance matrix rI. Each set of covariance

Matrices (Q1…..,QK,)corresponds to a K dimension polyhedron (i.e.{(R1,….,RK)}:

and the capacity region is equal to the union of all such

polyhedrons.

A.2 Fading Channels

Channel capacity for fading channels has many more parameters than for constant channels due to the

time variation of the channel. As in the single-user case, the capacity of the MIMO MAC where the

channel is time-varying depends on the definition of capacity and the availability of channel know-

ledge at the transmitters and the receiver. The capacity with perfect CSIT and CSIR is very well stu-

died, as is the problem of capacity with perfect CSIR only. However, little is known about the capaci-

ty of the MIMO MAC (or the MIMO BC for that matter) with partial CSI at either the transmitter or

receiver.

With perfect CSIR and perfect CSIT, the system can be viewed as a set of parallel non interfering

MIMO MACs (one for each fading state) sharing a common power constraint. Thus, the ergodic ca-

pacity region can be obtained as an average of these parallel MIMO MAC capacity regions, where the

averaging is done with respect to the channel statistics. The iterative waterfilling algorithm of easily

extends to this case, with joint space and time waterfilling.

In the capacity region of a MAC with perfect CSIR but no CSIT Gaussian inputs are optimal, and the

ergodic capacity region is equal to the time average of the capacity obtained at each fading instant

with a constant transmit policy (i.e. a constant covariance matrix for each user). Thus, the ergodic ca-

pacity region is given by

1

If the channel matrices Hi have i.i.d. complex Gaussian entries and each user has the same power con-

straint, then the optimal covariances are scaled versions of the identity matrix.

B. MIMO Broadcast Channel

When the transmitter has only one antenna, the Gaussian broadcast channel is a degraded broadcast

channel, for which the capacity region is known. However, when the transmitter has more than one

antenna, the Gaussian broadcast channel is generally non-degraded. The capacity region of general

non-degraded broadcast channels is unknown, but the seminal work of Caire and Shamai and subse-

quent research on this problem has shed a great deal of light on this channel and the sum capacity of

the MIMO BC has been found.

2.14 Open Problems in Multiuser MIMO

Multiuser MIMO has been the primary focus of research in recent years, mainly due to the large num-

ber of open problems in this area. Some of these are:

1. No CSI: The capacity region of MIMO MAC and BC systems when neither the transmitter(s) nor

the receiver(s) know the channel.

2. BC with receiver CSI alone: The Broadcast channel capacity is only known when both the transmit-

ter and the receivers have perfect knowledge of the channel.

3. Imperfect CSI: Since perfect CSI is never possible, a study of capacity with imperfect CSI for both

MAC and BC is of great practical relevance.

4. Non-DPC techniques for BC: Dirty-paper coding is a very powerful capacity-achieving scheme, but

it appears quite difficult to implement in practice. Thus, non-DPC multi-user transmissions schemes

for the downlink (such as downlink beamforming) are also of practical relevance.

1

CHAPTER-III

THEORY OF MMSE

3.1 Overview

This chapter presents all about MMSE, i.e. MMSE equalization, detection, filtering etc.

3.2 What is MMSE?

In statistics and signal processing, a minimum mean square error (MMSE) estimator describes the

approach which minimizes the mean square error (MSE), which is a common measure of estimator

quality.

The term MMSE specifically refers to estimation in a Bayesian setting, since in the alternative fre-

quentist setting there does not exist a single estimator having minimal MSE. A somewhat similar con-

cept can be obtained within the frequentist point of view if one requires unbiasedness, since an esti-

mator may exist that minimizes the variance (and hence the MSE) among unbiased estimators. Such

an estimator is then called the minimum-variance unbiased estimator (MVUE).

3.3 Definition

Let X be an unknown random variable, and let Y be a known random variable (the measurement). An

estimator is any function of the measurement Y, and its MSE is given by

where the expectation is taken over both X and Y.

The MMSE estimator is then defined as the estimator achieving minimal MSE.

1

3.4 Minimum mean square error estimation

A recurring theme in this chapter and in much of communication, control and signal processing is that

of making systematic estimates, predictions or decisions about some set of quantities, based on infor-

mation obtained from measurements of other quantities. This process is commonly referred to as infe-

rence. Typically, inferring the desired information from the measurements involves incorporating

models that represent our prior knowledge or beliefs about how the measurements relate to the quanti-

ties of interest.

Our discussion in this chapter, we focus primarily on choosing our estimate to minimize the expected

or mean value of the square of the error, referred to as a minimum mean-square-error (MMSE) crite-

rion. In this we consider the MMSE estimate without imposing any constraint on the form that the

estimator takes. In next section we restrict the estimate to be a linear combination of the measure-

ments, a form of estimation that we refer to as linear minimum mean-square-error (LMMSE) estima-

tion.

3.4.1 Estimation of a continuous random variable

To begin the discussion, let us assume that we are interested in a random variable Y and we would

like to estimate its value, knowing only its probability density function. We will then broaden the dis-

cussion to estimation when we have a measurement or observation of another random variable X, to-

gether with the joint probability density function of X and Y .

Based only on knowledge of the PDF of Y, we wish to obtain an estimate of Y — which we denote as

ŷ— so as to minimize the mean square error between the actual outcome of the experiment and our

estimate ŷ. Specifically, we choose ŷ to minimize

. (3.1)

Differentiating (3.1) with respect to ŷ and equating the result to zero, we obtain

(3.2)

1 Or

(3.3)

From which

(3.4)

The second derivative of E[(Y- ŷ)2

] with respect to ŷ is

(3.5)

which is positive, so (3.4) does indeed define the minimizing value of ŷ. Hence the MMSE estimate

of Yin this case is simply its mean value, E[Y ].

The associated error — the actual MMSE — is found by evaluating the expression in (3.1) with ŷ=

E[Y ]. We conclude that the MMSE is just the variance of Y, namely σY

2

:

(3.6)

The associated MMSE is the variance σ2

Y|X of the conditional density fY|X(y|x), i.e., the MMSE is the

conditional variance. Going a further step, if we have multiple measurements, say X1 = x1, X2 = x2,

,XL = xL, then we work with the a posteriori density

(3.7)

Apart from this modification, there is no change in the structure of the solutions. Thus, without further

calculation, we can state the following:

The MMSE estimation of Y given X1=x1,….XL=xl, is the conditional expectation of Y:

ŷ(x1…..xl) = E[Y|X1=x1,….XL=xl] (3.8)

For notational convenience, we can arrange the measured random variables into a column vector X,

and the corresponding measurements into the column vector x. The dependence of the MMSE esti-

mate on the measurements can now be indicated by the notation ŷ(x), with

1

(3.9)

The minimum mean square error (or MMSE) for the given value of X is again the conditional va-

riance, i.e., the variance σ2

Y|X of the conditional density fY|X(y|

x).

3.4.2 Orthogonality

A further important property of the MMSE estimator is that the residual error Y-ŷ(X) is orthogonal to

any function h(X) of the measured random variables:

(3.10)

where the expectation is computed over the joint density of Y and X. Rearranging this, we have the

equivalent condition

(3.11)

i.e the MMSE estimator has the same correlation as Y does with any function of X. In particular,

choosing h(X)=1 ,we find that,

(3.12)

The latter property results in the estimator being referred to as unbiased: its expected value equals the

expected value of the random variable being estimated. We can invoked the unbiasedness property to

interpret (3.10) as stating that the estimation error of the MMSE estimator is uncorrelated with any

function of the random variables used to construct the estimator.

The proof of the correlation matching property in (3.11) is in the following sequence of equalities:

(3.13)

Rearranging the final result here, we obtain the orthogonality condition in (3.10).

1 3.4.3 Linear minimum mean square error estimation

In general, the conditional expectation E(Y|X) required for the MMSE estimator developed in the pre-

ceding sections is difficult to determine, because the conditional density fY|X(y|x) is not easily deter-

mined.

A useful and widely used compromise is to restrict the estimator to be a fixed linear (or actually af-

fine, i.e., linear plus a constant) function of the measured random variables, and to choose the linear

relationship so as to minimize the mean square error. The resulting estimator is called the linear min-

imum mean square error (LMMSE) estimator. We begin with the simplest case.

Suppose we wish to construct an estimator for the random variable Y in terms of another random va-

riable X, restricting our estimator to be of the form

(3.14)

where a and b are to be determined so as to minimize the mean square error

(3.15)

Note that the expectation is taken over the joint density of Y and X ; the linear estimator is picked to

be optimum when averaged over all possible combinations of Y and X that may occur. We have ac-

cordingly used subscripts on the expectation operations in (3.15) to make explicit for now the va-

riables whose joint density the expectation is being computed over; we shall eventually drop the sub-

scripts.

Once the optimum a and b have been chosen in this manner, the estimate of Y, given a particular x, is

just ŷℓ(x)= ax + b, computed with the already designed values of a and b. Thus, in the LMMSE case

we construct an optimal linear estimator, and for any particular x this estimator generates an estimate

that is not claimed to have any individual optimality property. This is in contrast to the MMSE case

considered in the previous sections, where we obtained an optimal MMSE estimate for each x, namely

E[Y|X = x], that minimized the mean square error conditioned on X = x. The distinction can be sum-

marized as follows: in the unrestricted MMSE case, the optimal estimator is obtained by joining to-

gether all the individual optimal estimates, whereas in the LMMSE case the (generally non-optimal)

individual estimates are obtained by simply evaluating the optimal linear estimator.

1

3.5 MMSE equalization and different types of equalizers

A generic communication system model is shown in Fig. 3.1(a). Where s(t), is the data sequence to be

transmitted, w(t) is AWGN, and y(t) is the signal at the output of the receiver filters. The simplified

equivalent model is provided in Fig. 3.1(b), where n(t) = w(t) * gR(t) is the noise at the output of the

receiver filter. The transmitter filter, propagation channel, and the receiver filters are represented by

the composite channel with transfer function

(3.16)

As a result, the problem becomes that of deconvolution.

Fig. 3.1 (a) Simplified model of typical communication system,

(b) Equivalent composite channel model.

From here on, we shall refer to the composite channel model as simply channel. According to Fig.

3.1, the transmitted signal undergoes amplitude and phase distortion given by |H(f)| and Ө(f), respec-

tively. The goal of the receiver is to estimate the transmitted signal s(t) based on the distorted noisy

signal y(t). This can be facilitated by applying the received signal to the filter with the transfer func-

tion equal to the inverse of the transfer function of the channel, H−1

(f). The algorithms performing

estimation of H(f) and applying its inverse to the received signal, y(t), fall under the category of chan-

nel equalizers. The scope of this is limited to subclass of channel equalization algorithms that minim-

ize the mean square error between the transmitted signal s(t) and its estimate ŝ(t):

(3.17)

1

In its sampled form the model of Fig. 3.1(b) may be represented as Bayesian linear model [54]:

y=Hs+n (3.18)

If we assume that s(t) is a Gaussian process, then the sampled signal s[n] becomes discrete-time Gaus-

sian process and the LMMSE estimator is given by [54]

(3.19)

where the matrix A = CsHT(HCsH

T + σn

2I)

−1 represents the Wiener filter.

3.5.1 Major types of MMSE equalizers algorithm

In this section, the MMSE equalizers are classified according to algorithm type.

A. Algorithms Using Training (desired) Sequence

The use of training sequence in channel equalization allows to simplify receiver architecture, while

improving the reliability of the data reception. This comes at the expense of reduced maximum data

rates, since part of the bandwidth is must be allocated to the training sequence. Most cellular commu-

nication standards incorporate some kind of training signal. In the case of CDMA-based systems this

role is played by the code division multiplexed pilot sequence. It has been shown that the pilot PN

sequence can be used as a desired signal for chip-level equalization [55, 56]. In the case of GSM, the

time-multiplexed midamble is used for the same purpose. In the context of Wiener equation such

training sequence is typically used as the desired signal. That is the training sequence is used to esti-

mate the impulse response of the channel. The distortion introduced by the channel is then corrected

by convolving the received signal with the inverse of the estimated impulse response. In time-varying

channels such as those encountered in wireless communications, the channel estimates must be up-

dated fast enough to track the changes in the channel transfer function.

B. Blind Adaptive Methods

The blind channel equalization techniques are characterized by the fact that the channel estimation is

based solely on the received signal with no access to the transmitted signal. The blind estimation

techniques find application in the scenarios where the training sequence is not possible or where sig-

nificant advantage can be gained by reducing overhead associated with training sequence. Wireless

transmission over time varying channel is a typical application. Combination of blind and training

sequence-based methods has been explored by [57]. There, it has been assumed that part of the trans-

mitted sequence is known by the receiver, and the rest is unknown. The blind estimation techniques

1 exploit various properties of the channel and the input signal. One such example would be input signal

with known statistical properties such as distribution or moments. The general model of the blind es-

timation scenario is given in Fig. 3.2.

Fig. 3.2 General model of blind channel estimation [58]

The two main types of channel model used in blind estimation can be classified as multichannel and

multirate systems [58]. The multichannel model of Fig. 3.3 (a) represents single-input P-output chan-

nel model. The channel diversity is often assumed, which implies that the FIR models of different

channels have different zeros. This ensures unique identifiability of each channel as well as ability to

construct inverse of the FIR model. The multichannel blind equalization for SIMO and MIMO sys-

tems was covered in [59] and [60], respectively, where sequential algorithms were proposed.

In the case of multirate model of Fig. 3.3(b), if the input sequence sk is wide sense stationary, then yk

is wide sense cyclostationary with period P. This property allows to estimate the channel using the

second order statistics.

Fig. 3.3 Classification of channel models [24].

1 The multirate model of Fig. 3.3(b) basically translates to the requirement for fractional spacing of the

equalizer taps. This means that the delay between the equalizer taps must be less than the symbol rate

of the system. According to the model of Fig. 3.1, the received complex symbol y(t) is given by

(3.20)

where sk is an information symbol in a signal constellation with symbol interval T, h(t) is the compo-

site channel impulse response representing transmitter pulse shaping filter, the wireless multipath

channel, and receiver filters. n(t) represents the additive white noise. If the received signal is sampled

at then equation (3.20) becomes

(3.21) .

Where

(3.22)

And

(3.23)

It can be shown that the power spectral density of x(i) is given by

(3.24)

If is evaluated at , k= 0, 1, ... we obtain

(3.25)

As can be seen from equation (3.25), the information regarding the phase of the channel is preserved.

However, if Ts = T, the noiseless signal x(i) is wide-sense stationary with power spectral density given

by equation (3.25) with k = 0:

1

(3.26)

Hence, the information about the phase characteristic of the channel is lost and cannot be recovered

unless it is known a priori that the channel is either minimum phase or maximum phase [61].

C. Different Matrix Inversion Methods

Since the Wiener solution represents the MMSE solution to the channel estimation problem. The ne-

cessary part of deriving the tap weights of the equalizer is the computation of the inverse of the auto-

correlation matrix of the received signal. The direct inverse computation in seldom appropriate due to

its complexity. Hence, variety of indirect methods have been proposed and successfully employed in

channel equalization applications. This section shall describe some of the most popular matrix inver-

sion techniques. Cholesky Factorization allows to factor the matrix inverse R-1

=(HCsHT+σn

2I)

-1 from

equation (3.19) into the product of an upper triangular and a lower triangular matrices (that are Hermi-

tian transpose of each other):

(3.27)

where D = diag(P0, P1, ..., PM) and Pi = E[|bi(k)|2] = the average power of the ith backward prediction

error bi(k), which is obtained via Gram-Schmidt orthogonalization given by

(3.28)

where L is the lower triangular matrix with 1‘s along the main diagonal. Hence, the determinant of L

is unity, which guarantees that this matrix is nonsingular. The nonzero elements of each row represent

the weights of backward prediction-error filter of order corresponding to the row number. The predic-

tion error filter coefficients may be found recursively using Levinson-Durbin algorithm as was shown

in [62]. Another example of the adaptive method based on Cholesky factorization was proposed in

[63], where the Hermitian, Toeplitz properties of the channel correlation matrix were utilized to re-

duce complexity of the factorization method. These two methods were derived in the context of sin-

gle-antenna GSM receiver. Yet, the Choleskybased adaptive algorithms were shown to be applicable

for channel equalization in MIMO HSDPA systems as well [64], which will be covered in more detail

in Section 3.5.2.

The matrix inversion techniques used in solving the Wiener equations are not limited to Cholesky fac-

torization.For example, the use of Conjugate Gradient (CG) method in the context of MMSE equaliz-

ers has been investigated in [65] and [55]. One of the advantages of the CG algorithm is that it is

guaranteed to converge in N steps (where N is the number of rows of the autocorrelation matrix, Ryy).

1 Even faster convergence is attained when eigenvalues of Ryy are clustered together. The computation-

al efficiency of the CG method based on the fact that the algorithm does not need to compute the es-

timate of Ryy−1

) at every iteration but requires computation of only one matrix-vector product per ite-

ration [65]. It uses the estimates of autocorrelation matrix and cross correlation vector as its inputs.

The method lands itself well to both sample-by-sample processing (using forgetting factor, ') as well

as block processing. In the case of sample-by-sample processing, the inputs at the time of reception of

kth data sample are

(3.29)

If the block processing (of block size P is desired, the input to the algorithm at the m-th block are giv-

en by

(3.30)

(3.31)

The asymptotic performance of the CG algorithm was investigated in [66]. The results indicate that

the this method is numerically stable and in steady-state behaves similarly to method of steepest des-

cent.

3.5.2 APPLICATIONS OF MMSE CHANNEL EQUALIZERS

This section covers the classification of the MMSE equalizers according to their intended application.

A. Channel equalization for CDMA and WCDMA systems

List of reference: [67], [68], [69], [55] Terminology: the chip sequence in the context of a CDMA sys-

tem is the data correlated with the PN spreading sequence. The symbol sequence is the data prior to

being spread (at the transmitter) or after beoing depsread (at the receiver) via the PN spreading se-

quence. The rate of the chip sequence is M times higher than that of the symbol sequence, where M is

the spreading factor.

1 With increase in available computing power in the mobile receivers, the chip-level equalization has

been gaining popularity. Its main advantage is that it allows to resolve multipath rays with relative

arrival delays less than the chip interval. This is accomplished by oversampling the received signal at

the rates equal to multiple of the chip rate of the system and using the fractionally-spaced equalizer

taps [55], [70]. From the statistical perspective, the received data sequences at symbol level (after de-

spreading), are non-cyclostationarity due to application of the long scrambling code [71]. Finally,

another advantage of chip-level equalization is that it allows to model the received sampled signal as

i.i.d. process [65]. Under this assumption, the LMMSE equalizer expression (3.19) reduces to

(3.32)

With increase in the number of user applications requiring the wireless data transfers, the optimization

of the mobile receiver performance becomes more important. It has been shown that in most multi-

path fading channel conditions the MMSE equalizer-based receiver outperforms the rake-based re-

ceiver [55], [67], [68]. The gain due to channel equalization is especially significant at high SNR le-

vels.

B. Channel equalization for OFDMA systems

The OFDM-based systems are widely used for high-speed wireless communications. Some their at-

tractive features include its spectral efficiency and its robustness against multi-path fading due to eli-

mination of ISI. The application of channel equalization for OFDM systems was investigated in [72],

[73], and [74]. Denoting the received OFDM symbol yi = [y0,i, ..., yN−1,i]T and corresponding transmit-

ted symbol as si = [s0,i, ..., sN−1,i]T , we can obtain the following channel model:

(3.33)

The corresponding MMSE equalization matrix is then given by [74]

(3.34)

where F represents the inverse FFT. The simplified form of OFDM channel model is also depicted in

Fig. 3.4.

Fig. 3.4 Simplified OFDM system model [73].

1

C. Channel equalization for MIMO systems

Recently, some of the existing commercial wireless data standards have included MIMO enhance-

ments. This spurred more research devoted to finding computationally efficient equalization methods

for MIMO channels. In the context of CDMA-based MIMO systems, the equalization is needed in

order to restore the orthogonality of the spreading codes, which is degraded due to multipath propaga-

tion and code reuse by multiple transmit antennas. In the case of MIMO system with NT transmit an-

tennas and NR receiver antennas, the convolution matrix of the channel is given by [64]

(3.35)

where, assuming the impulse response of the length of individual propagation channel equal to Lh and

equalizer length equal to Lf , the channel matrix between kth transmit and the mth receive antenna is

given by

(3.36)

Defining the stacked received signal vector at time instant i as it is possible represent the overall mod-

el of the MIMO transmission channel in the form of Bayesian linear model similar to that of equation

(3.18):

(3.37)

Hence, the LMMSE estimator of same form as that of equation (3.19) is required, where the matrix

inversion can be facilitated by an iterative method based on the Singular Value Decomposition that

was proposed in (3.32).

1

CHAPTER-IV

MMSE precoder for MIMO systems

4.1 MMSE precoder with mode selection

In this section, we first describe the conventional MMSE precoder that has the perfect CSI at the

transmitter, and then the proposed mode selection algorithm that selects both the number of sub-

streams and the modulation scheme for each substream to minimize the average BER at a fixed rate.

A. Conventional MMSE Precoder

The precoder and decoder are jointly configured to minimize the symbol mean square error subject to

the total transmit power constraint given by

subject to trace

. (4.1)

where trace(・) denotes the trace of matrix and is the total transmit power available. Let us de-

fine the eigenvalue decomposition (EVD) as

. (4.2)

where V denotes an Nt ×B unitary matrix that forms a basis for the range space of , Λ de-

notes a diagonal matrix containing the B nonzero eigenvalues arranged in a descending order from

top-left to bottom-right, ˜Λcontains the zero eigenvalues, ˜V denotes an Nt ×(Nt −B) unitary matrix

that constitutes a basis for the null space of . Based on (2), the optimal MMSE precoder and

decoder are given by

F=VP (4.3)

1

. (4.4)

where Γ denotes B × B diagonal matrix and P denotes the B×B diagonal power allocation matrix that

satisfies the power constraint with the following (i, i) entry given by

. (4.5)

where λii denotes the (i, i) entry of Λ, (x)+ denotes the operator that replaces the negative element by

zero, andB ≤B is such that |pnn|2 > 0 for n є [1, B ], and |φnn|

2 = 0 for all n.

Let ŝ denote the B × 1 received signal vector given by

ŝ = GHFs + Gn (4.6)

Substituting (4.2) -(4.4) into (4.6)

Yields

. (4.7)

This result shows that MMSE precoder F and decoder G can diagonalize H into B eigen-subchannels

and each substream can be detected without inter-substream interference because ΓΛP is a diagonal

matrix.

B. Mode Selection Algorithm

The MMSE precoder F and the decoder G can diagonalize H into the B eigen-subchannels. However,

error of some substreams increases compared to that of the other. As a result, using a fixed number of

substreams may lead to poor BER performance. Here , we propose an MMSE precoder with mode

selection for MIMO systems to improve the performance of the MMSE precoder, which selects a

combination of the number of substreams and the modulation scheme for each substream to minimize

the average BER at a fixed rate shown in Fig. 4.1 .

1

Fig. 4.1 System model of MMSE precoder with mode selection

We refer to the combination as mode. The average BER of MIMO system using MMSE precoder

is upper-bounded by

. (4.8)

where mi denotes the number of bits assigned to the ith substream, pii and λii denote the (i, i) entry of P

and Λ, respectively, and αi and βi denote constants determined by the modulation scheme [6].

We explain the mode selection algorithm of the proposed scheme. First, a combination (m1. . . mBopt )

of the number of transmitted substreams and the modulation scheme to minimize the right-hand side

of (1.7) at a fixed rate is selected. Then, the optimal MMSE precoder and decoder are designed using

(1.3) based on the number of selected substreams Bopt .

1

4.2 Method for reducing performance degradation due to feedback delay

The block at the transmitter is shown in Fig. 4.2. Each data stream is split into blocks of length Lb. To

estimate MIMO channels, the symbol known to the ith antenna is inserted in the kth block, denoted by

si(k), and is spread by an antenna-specific signature code ci = [ci(0), . . . , ci(Nt − 1)]T which are de-

signed to be orthogonal to each other. Specifically, at time index n = kLb+l, where l = 0, . . . , Nt−1,

the known symbols si(k)ci(l) are transmitted as shown in Fig. 4.2. The Nt received signals yj(k) =

[yj(kLb), . . . , yj(kLb + Nt − 1)]T corresponding to the pilots of the kth block at the jth receive antenna

are given by

. (4.9)

Fig. 4.2 The block format at the transmitter

where nj(k) is defined similar to yj(k). Owing to orthogonality among the signature codes, we acquire

the estimated channel values for each frame as

. (4.10)

Where

Here, feedback delay is assumed to be Q blocks. To utilize MMSE precoder, CSI estimated in the kth

block at the receiver is fed back to the transmitter. However, CSI fed back to the transmitter becomes

outdated for the (k + Q)th block where it is used owing to the time-varying nature of the channels.

That is, there is a mismatch between ―the configured precoder‖ and ―the optimum precoder when the

1 signals are transmitted‖; an outdated MMSE precoder cannot achieve the full performance possible.

Here, as methods for reducing the performance degradation due to feedback delay, we consider two

methods, channel prediction [4] and receive weight robust to feedback delay [5]. We also consider the

method that combines channel prediction with receive weight robust to feedback delay.

A. Channel Prediction

Having obtained the estimated channel response hji(kLb) for each block, the receiver then obtains the

predicted channel response ĥji(n) at any time index n, using optimal Wiener filtering that exploits the

time-domain correlation. Given this assumption, at the block k, the receiver predicts the MIMO chan-

nel response Q frames ahead, using the Pth-order filter as

. (4.11)

Where

wji =[ wji(0,. . . wji(P-1) ]T

,

=

We seek the filter wji that minimizes the mean square error (MSE) as

. (4.12)

Based on (1), the optimal wji is given by

. (4.13)

The entries of matrix R and vector u, rji and ui respectively, are given by

1

(4.14)

(4.15)

where j, i ε [0, P] and Ep = |si(k)|2 denotes the energy per pilot symbol. From (4.14), (4.15), it is seen

that channel prediction needs estimation of maximum Doppler frequency fd. We assume the Ideal es-

timation of fd.

In the following, at the block k, denotes the estimated channel matrix whose entries are

and Ĥk denotes the predicted channel matrix whose entries are . The proposed MMSE

precoder with channel prediction is denoted by Prediction-MMSE. In Prediction- MMSE at the block

k, we conduct its weight design and mode selection based on the predicted channel matrix Ĥk+Q.

B. Receive weight Robust to Feedback Delay

For receive weight robust to feedback delay, MMSE decoder is designed based on the channel matrix

estimated at the present (k +Q)th block instead of the channel matrix used for configuring

MMSE precoder as

(4.16)

where denotes the precoder matrix configured based on . The proposed MMSE precoder with

receive weight robust to feedback delay is denoted by Robust-MMSE.

C. Combination of Channel Prediction and Receive weight Robust to Feedback Delay

We propose a method that combines channel prediction with receive weight robust to feedback delay.

In the method, we configure its precoder based on the predicted channel matrix Ĥ k+Q. On the con-

trary, we configure its decoder based on the channel matrix estimated at the block k + Q as

(4.17)

1

where denotes precoder matrix configured based on Ĥk+Q. The proposed MMSE precoder with

method that combines channel prediction with receive weight robust to feedback delay is denoted by

Combination-MMSE.

4.3 Performance of MMSE precoder with mode selection

We evaluate the proposed scheme by simulation. The simulation parameters are set as follows-

The number of transmit antennas is Nt = 4, and that of receive antennas is Nr = 4. The flat Rayleigh

fading channel model is employed. The modulation scheme is QPSK, 16QAM, and 64QAM. The

fixed rate is 12 bit/s/Hz. The mode patterns are shown in Tables 4.1. The normalized Doppler fre-

quency is fdTs = 1.0×10−4

. The block length is Lb = 200 symbols. The filter order in the channel pre-

diction is P = 50. The number of feedback delay blocks is Q = 2, 5.

Table 4.1

Figs. 4.3 and 4.4 show the selection probability of each mode and BER of MMSE precoder with mode

selection (12 bit/s/Hz), respectively. We assume that MMSE precoder has the perfect CSI at the

transmitter. That is, we assume the environment that can neglect feedback delay. From Fig. 4.3, we

can see that when SNR is low, the selection probability for Mode 5 is high. We can also see that when

SNR is low, the selection probability for Mode 3 is high. From Fig. 4.4, we can see that the BER of

the proposed MMSE precoder that adaptively selects the mode is slightly improved compared to that

of the conventional MMSE precoder with the fixed mode. This is because the mode to mi-

nimize BER is selected by mode selection.

1

Fig. 4.3 Selection probability of each mode (12 bit/s/Hz)

Fig. 4.4 BER of MMSE precoder with mode selection (12 bit/s/Hz)

Fig. 4.5 shows BER of the proposed MMSE precoder with methods for reducing performance degra-

dation due to feedback delay for Q = 2. As a comparison of the proposed MMSE precoder with three

methods, we consider the proposed MMSE precoder that can use perfect CSI at the transmitter (de-

noted by Ideal MMSE). We also consider the proposed MMSE precoder in the presence of feedback

delay without using methods reducing performance degradation due to feedback delay (denoted by

MMSE). We set the fixed rate 12 bit/s/Hz for mode selection. From Fig. 4.5, we can see that the BER

of MMSE is largely degraded and the error floor is generated owing to feedback delay. On the other

hand, Prediction-MMSE and Robust-MMSE reduce the BER degradation due to feedback delay and

1 almost achieve the identical BER each other. We can also see that BER of Combination-MMSE is

slightly improved compared to those of Prediction-MMSE and Robust-MMSE. This is because the

transmit weight of Combination-MMSE becomes close to that of Ideal-SVD by using channel predic-

tion.

Fig. 4.5 BER of the proposed MMSE precoder with methods for reducing

Performance degradation due to feedback (Q = 2, 12 bit/s/Hz)

In addition, the receive weight of Combination-MMSE also becomes close to that of Ideal-SVD, be-

cause that of Combination-MMSE is configured by both transmit weight that is close to that of Ideal-

SVD and the channel matrix for the actual transmission. Thus, Combination-MMSE can achieve the

BER close to that of Ideal-SVD.

4.4 Linear IIR-MMSE precoding for frequency selective MIMO channels

The system model is shown in figure 4.6

Fig. 4.6 system model

1 In formula, we have

y(z) = H(z)P(z)s(z) + η(z)

ŝ(z) = αG(z)y(z) = αG(z)H(z)P(z)s(z) + αG(z)η(z)

where H ∈ RH∞q×p

is the channel, G ∈ RH∞r×q

is a fixed receive filter, P ∈ RH∞p×r

is the precoder,

and α ≥ 0 is a scalar gain. The signals s, y, η denote transmitted and received signals, and noise. We

assume that both s and η are mutually independent white random sequences with zero mean and cov

riance matrices I andση2I > 0, respectively. We want to minimize the mean square error

2.

. (4.18)

Where L ∈ N is the latency time, subject to the power constraint

. (4.19)

4.4.1 Optimal causal precoder

A well-known issue with IIR filters is that the optimal filters usually are non-causal and thus cannot

be implemented in real-time. Therefore, a causality constraint in form of a finite latency time has to be

incorporated explicitly.

4.4.2 Optimal non-causal precoder

The optimal precoder without explicit causality constraint usually is non-causal and therefore cannot

be implemented in real-time. Nevertheless, it still often is of theoretical interest because it constitutes

the limit of the optimal causal precoder as the latency time goes to infinity. Formally, the optimal pre-

coder without causality constraint is the solution to the following problem.

Problem - Find P ∈ LH∞p×r

and α ≥ 0 such that (4.18) is minimized for L = 0 subject to (4.19).

4.4.3 Precoder design

We compare the FIR-MMSE precoder for various filter lengths Nf with the IIR-MMSE precoder from

Theorem 1.

1 Theorem1: (optimal causal precoder). Let and define

Then,

. (4.20) .

The latency time of the FIR-MMSE precoder was chosen such that the MSE is minimized [75]. In

contrast, as the MSE of (4.20) decreases monotonically with L, so we should make the latency time L

of the IIR precoder as large as possible. In order to ensure a fair comparison, we chose L such that the

number KIIR of parameters necessary to describe to the resulting IIR precoder is not larger than the

number of parameters necessary to describe the FIR precoder, i.e., KFIR = Nf qp = 4Nf . The IIR part of

the precoder, i.e., (S ξopt )

−1 , has the same McMillan degree as the channel, i.e., κ = 4. Therefore, when

we use a state space realization in block controllable companion form of (S ξopt)−1

, κp−1

+ κq + qp =

14 parameters are required to describe it. The FIR part of the IIR-MMSE precoder has length L.

Therefore, the total number of parameters necessary in the IIR case is KIIR = 14+Lqp = 14+4L. In our

simulations, we chose L = Nf − 4, which gives KIIR = KFIR − 2 < KFIR.

Consider an uniform power profile, i.e., c0 =c1 = c2 = 1. We considered the filter lengths Nf ∈{8, 12,

16}.Fig. 4.7 show the simulation results. We observe that the IIR-MMSE outperforms the FIR-MMSE

for all three filter lengths with improvements up to an order of magnitude in the high SNR regime.

Also note that there are error floors. The reason is that we consider square channels. These are likely

to have unstable zeros, i.e., zeros outside the unit circle. Such channels cannot be causally inverted,

which is why there are errors no matter how high the SNR

Fig. 4.7 the “+” marks belong to the FIR-MMSE,

the “x” marks to the IIR-MMSE.

1

4.5 Precoding based on iterative LMMSE detection

A. Precoder Structure

Let χ be a coded vector generated by a generic encoder (consisting of an FEC encoder followed by

random interleaving). Without loss of generality, the average power per entry of χ is normalized to 1.

We will treat χ as a sequence of independent and identically distributed (i.i.d.) symbols, which is ap-

proximately ensured by random interleaving [53,76,77]. Assume that the channel has been diagona-

lized as in Fig. 4.8(a).We adopt the precoding scheme illustrated in Fig. 4.8(c) defined by linear trans-

form

(4.21)

where is the transmitted signal vector over the parallel channel in and

(4.22)

that is used for power allocation among sub-channels.

Fig. 4.8 (a) Transmission over a MIMO ISI channel

(b) The equivalent parallel subchannel model of the system in (a)

(c) The precoding scheme based on the equivalent model in (b).

1 The matrix ‎P in (4.21) is the rotation matrix which can be either a DFT matrix or a Hadamard matrix.

In either case P, has the following two properties

1) P is unitary.

2) The entries of P have a constant modulus of .

Incidentally, using a Hadamard matrix as P is more cost-effective as then P χ involves additions only.

B. Detection Principles

The use of the precoder in (4.21) will reintroduce inter-symbol interference. We now consider

applying iterative LMMSE detection [78] to suppress this interference. A key in the discus-

sion below is to take into account the feedback information. This brings about significant per-

formance improvement. We have

(4.23)

where

(4.24)

and is a vector of AWGN samples with variance σ2. The optimal estimation for χ usually involves

prohibitively high complexity. We will take a suboptimal alternative shown in Fig. 4.9. The receiver

consists of two local operators, namely, the elementary signal estimator (ESE) and the decoder

(DEC). The coding constraint is ignored in the ESE and the impact of the channel matrix A is ignored

in the DEC, which implies reduced complexity at the cost of certain performance loss. To compensate

for the performance loss, an iterative procedure is applied to the two local operators.

Fig. 4.9 Structure of an iterative receiver

1 For the ESE, even after ignoring the coding constraint, the optimal detection of is still a very diffi-

cult task when is discrete (as in most practical systems). To overcome the difficulty, we take a

two-step approach. In the first step, we treat as approximately Gaussian distributed. We assume

that the a priori mean (denoted by ) and a priori auto-covariance of are available. To simplify

the problem, we further assume that the a priori autocorrelation of can be written as . In prac-

tice, the a priori information can be initialized to and updated using the

feedbacks from the DEC during the iterative process. (We will return to this later in Section III-E.)

Then the standard LMMSE estimation of .

(4.25)

Where . In the second step, we write the ith entry of in the following

form

(4.26)

In (4.26)

(4.27)

(4.28)

where a(i) is the ith row of A, and Ω (i, i) is the ith diagonal entry in the following matrix:

(4.29)

Here, represents an unknown interference-and-noise term that is independent of x(i). we treat

as an AWGN sample. Its variance is given by

(4.30)

Then we can detect x(i) based on (4.26) in a symbol-by-symbol manner using the true distribution of

x(i) .

The rationale behind the above two-step approach is that is statistically close to x(i) after

the LMME operation in (4.25). Thus, the symbol-by-symbol detection in (4.26) is relatively reliable.

1

4.6 BER performance of the proposed MMSE receiver

We compare the BER performance of the proposed MMSE receiver with that of other receivers: the

conventional MF receiver, the standard MMSE receiver, the modified MMSE receivers. We use the

following acronyms for the receivers in all plots: MF stands for the conventional MF receiver, MMSE

for the standard MMSE receiver, MMSE-PC for the modified MMSE receiver with phase compensa-

tion, MMSEAPC (A) and (B) both for the MMSE receivers with amplitude and phase compensation.

MMSE-PC is implemented based on the design concept proposed in [12] and [79]. In this scheme, the

channel phase compensation is conducted in front of the adaptive filter and the channel phase estima-

tion is made from the output signal of the adaptive filter. In MMSEAPC (A), the channel coefficient

estimation is made from the input (more noisy) signal of the adaptive filter. This scheme is similar to

the work proposed in [14]. The proposed receiver is denoted by MMSE-APC (B). In MMSE-APC

(B), the channel coefficient estimate is obtained from the output (less noisy) signal of the adaptive

filter. The BER performance is evaluated by Monte-Carlo simulation. We consider an asynchronous

BPSK DS/CDMA system for a reverse link with K users in the presence of Rayleigh fading and

AWGN. We use Gold code sets with chip length N = 31 as the spreading code. The channel band-

width is 3.968 MHz and the carrier frequency is 2.0 GHz. For the desired user, E,, / No is set to 20 dB.

Transmitter powers of all active users are set to be equal. The tap weights vector for adaptive imple-

mentations is initialized as s1. Adaptive implementations for MMSE, MMSE-PC, and MMSE-APC

(A) and (B) are based on the corresponding LMS algorithms. To make a fair comparison of the BER

performance in a steady state, all receivers are operated in pilot symbol-aided mode.

Fig. 4.10 shows the transient and steady state behavior of the proposed MMSE receiver, denoted by

MMSE-APC (B), for the case: a vehicle moving at 50 Km/h and the number of users K = 20. The

proposed MMSE receiver is adaptively implemented using the orthogonal decomposition-based LMS

algorithm and the conventional LMS algorithm. It is observed that, for the conventional LMS algo-

rithm, one realization of the tap weights of the adaptive filter goes to zero as the number of iterations

increases. However, the orthogonal decomposition-based LMS algorithm does not go to zero and

works well in this fading environment. This effectiveness is caused by the constraint w(m)H s, = 1 .

1

Fig. 4.10 Transient and steady state behavior of the proposed MMSE receiver

as a function of number of iterations for the case: a vehicle moving at 50

Km/h and the number of users K=20 (x-axis is scaled by a factor of 100).

Fig. 4.11 (a) and Fig. 4.11 (b) show the BER performance as a function of the number of users for two

different fading environments: (a) a vehicle moving at 50 Km/h and (b) a vehicle moving at 100

Km/h. Pilot symbol insertion period M is set to 8. The standard MMSE receiver has severely inferior

performance to MF and the modified MMSE receivers in both environments, and is apparently not

working in fading environments. MMSE-PC shows slightly different BER performances to the chang-

ing environments, specifically exposing some degree of performance degradation for the mobile speed

of 50 Km as the number of users increases. The BER performance of MMSE-APC (A) is better than

MF, but worst among the modified receivers. However, the proposed MMSE receiver offers outstand-

ing performance improvement in BER over MF, MMSE, MMSE-PC, and MMSE-APC (A). The BER

performance remains nearly identical for both mobile environments. This means that the proposed

receiver is nearly insensitive to the speed of the fading.

1

Fig. 4.11 BER as a function of the number of users for an asynchronous DS/CDMA sys-

tem in Rayleigh fading environments:

(a) a vehicle moving at 50 Km/h and (b) a vehicle moving at 100 Km/h

Fig. 4.12 shows the BER performance as a function of the pilot symbol insertion period M for the

case: a vehicle moving at 50 Km/h and the number of users K = 20. Estimation interval is selected by

trial-and-error so that it can give the best channel coefficient estimate for each M. As M increases, the

BER performance is gradually degraded. The proposed MMSE receiver gives significant performance

improvements over MF and MMSE-APC: (A) for most values of M. The performance of the proposed

MMSE receiver approaches that of the conventional MF receiver as M goes to 512.

Fig. 4.12 BER as a function of the pilot symbol insertion period M for the case:a vehicle

moving at 50 Km/h and the number of users K=20.

1

CHAPTER V

CONCLUSION

We proposed an MMSE precoder with mode selection for MIMO systems, which selects both the

number of substreams and the modulation scheme for each substream to minimize the average BER at

a fixed rate and evaluated the BER of the proposed MMSE precoder with two methods for reducing

performance degradation due to feedback delay.

Then we discussed about a new, constrained MMSE receiver with the modified MMSE criterion in

the presence of Rayleigh fading and AWGN. To overcome the intrinsic problem of the modified

MMSE criterion, we imposed a Constraint on the filter tap weights. For implementations of the con-

strained MMSE criterion, we used the orthogonal decomposition-based LMS algorithm. This ortho-

gonal decomposition is actually analogous to the constrained MMSE optimization with a constraint

w(m)Hsl = 1. Also, we employed pilot symbol-aided scheme. The inserted pilot symbols periodically

act as a pseudo-training sequence and improve the accuracy of the channel estimates.

We also discussed the IIR-MMSE precoder with and without causality constraint.

We also evaluated the BER of the proposed MMSE precoder with method that combines channel pre-

diction with receive weight robust to feedback delay. Simulation results showed that the BER of the

proposed MMSE precoder is improved compared to that of the conventional MMSE precoder using

the fixed number of substreams. We also showed that method that combines channel prediction with

receive weight robust to feedback delay can achieve good BER even when the large feedback delay

exists.

1

REFERENCES

[1] H. Sampath, P. Stoica, and A. J. Paulraj, ―Generalized linear precoder and decoder design for

MIMO channels using the weighted MMSE criterion,‖ IEEE Trans. Commun., vol.49, no.12,

pp.2198–2206, Dec. 2001

[2] A. Scaglione, P. Stoica, and S. Barbarossa, G. B. Giannakis, and H. Sampath, ―Optimal designs

for space-time linear precoders and decoders,‖ IEEE Trans. Signal Processing, vol.50, pp.1051–1064,

May 2002.

[3] G. E. Oien, H. Holm, and K. J. Hole, ―Impact of channel prediction on adaptive coded modulation

performance in Rayleigh Fading,‖IEEE Trans. Veh. Techno., pp.758–769, May 2004.

[4] S. Zhou and G. B. Giannakis, ―How accurate channel prediction needs to be for transmit-

beamforming with adaptive modulation over Rayleigh MIMO channels?,‖IEEE Trans. Wireless

Commun., vol.3, no.4, pp.1285– 1294, July 2004

[5] G. Lebrun, J. Gao, and M. Faulkner, ―MIMO transmission over a time-varying channel using

SVD,‖ IEEE Trans. Commun., vol.4, no.2, pp.757–764, March 2005

[6] K. Miyashita, T. Nishimura, T. Ohgane, Y. Ogawa, Y. Takatori, and K. Cho, ―High data-rate

transmission with eigenbeam-space division multiplexing (E-SDM) in a MIMO channel,‖ IEEE VTC

2002 fall, vol.3, pp.24–28, Sept. 2004.

[7] M. Joham, W. Utschick, and J. A. Nossek, ―Linear transmit processing in mimo communications

systems,‖ IEEE Trans. Signal Process., vol. 53, pp. 2700–2712, Aug. 2005.

[8] R. L.-U. Choi and R. D. Murch, ―New transmit schemes and simplified receivers for mimo wire-

less communication systems,‖ IEEE Trans. Wireless Commun., vol. 2, pp. 1217– 1230, Nov. 2003.

[9] U. Madhow and M. L. Honig, ―MMSE interference suppression for direct-sequence spread-

spectrum CDMA,‖ IEEE Trans. Commun., vol. 42, no. 12, pp. 3178-3188, Dec. 1994.

[10] P. B. Rapajic and B. S. Vucetic, ―Adaptive receiver structures for asynchronous CDMA sys-

tems,‖ IEEE J. Select. Areas Commun. , vol. 12, no. 4, pp. 685-697, May 1994.

[11] S. L. Miller, ―An adaptive direct-sequence code-division multiple access receiver for multiuser

interference rejection,‘‗ IEEE Trans. Commun., vol. 43, no. 21314, pp. 1746-1755, Feb./Mar./Apr.

1995.

[12] S. L. Miller and A. N. Barbosa, ―A modified MMSE receiver for detection of DS-CDMA signals

in fading channels,‖ in Proc IEEE Military Communications Conf, 1996, pp. 898-902.

[13] L. J. Zhu and U. Madhow, ― Adaptive interference suppression for direct sequence CDMA over

severely time-varying channels,‖ in Proc. IEEE Global Telecommunications Conf, 1997, pp. 91 7-

922.

[14] M. Latva-aho and M. Juntti, ―Modified adaptive LMMSE receiver for DS-CDMA systems in

fading channels,‖ in Proc. PIMRC‗97, 1997, pp.

1 [15] S . R. Kim, Y. G. Jeong, and 1. K. Choi, ―A Constrained MMSE Receiver for DS/CDMA Sys-

tems in Fading Channels,‖ IEEE Trans. Commun., 1998, submitted. 5 54-558.

[16] F. Ling, ―Coherent detection with reference-symbol based channel estimation for direct sequence

CDMA uplink communications,‖ in Proc. IEEE Vehicular Technology Con$. 1993, pp. 400-403.

[17] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005

[18] J. G. Proakis and J. Salehi, Digital Communications. McGraw Hill, 1983

[19] Andrea Goldsmith. Wireless Communications. Cambridge University Press, 2005

[20] A. Goldsmith, S. A. Jafar, N. Jindan, and S. Vishwanath, ―Capacity limits of MIMO channels,‖

IEEE Journal on Selected Areas in Communications, vol. 21, no. 5, pp. 684–702, 2003.

[21] A.O. Hero III. Secure Space Time Communication. IEEE Trans. Info. Theory, 49(12):3235., De-

cember 2003

[22] J. Winters. On the Capacity of radio communications systems with diversity in a Rayleigh Fading

Environment. IEEE J. Sel. Areas Commun., 5(5):871.878, June 1987.

[23] P.B. Rapajic and D. Popescu. Information Capacity of a Random Signature Multiple- Input Mul-

tiple-Output Channel. IEEE Trans. Commun., 48(8):1245.1248, August 2000.

[24] G. J. Foschini, D. Chizhik, M. J. Gans, C. Papdias, and R.A Valenzuela. Analysis and Perfor-

mance of Some Basic Space-Time Architectures. IEEE JSAC, 21(3):303 . 320, April 2003.

[25] Honglei Miao and M.J. Juntti. Space-time channel estimation and performance analysis for wire-

less MIMO-OFDM systems with spatial correlation. IEEE Trans. on Vehic. Tech., 54(6):2003.2016,

November 2005.

[26] D. Kotoulas and N. Koukoulas, P.and Kalouptsidis. Subspace projection based blind channel or-

der estimation of MIMO systems. IEEE Trans. on Acoustics, Speech and Signal Processing,

54(4):1351.1363, April 2006.

[27] T. Yoo and A. Goldsmith. Capacity and Power Allocation for Fading MIMO Channels With

Channel Estimation Error . IEEE Trans. Info. Theory, 52(5):2203.2214, May 2006.

[28] A. Scaglione, G. B. Giannakis, and S. Barbarossa, ―Redundant Filterbank Precoders and Equaliz-

ers Part I: Unification and Optimal Designs,‖ IEEE Transactions on Signal Processing, vol. 47, no. 7,

pp. 1988–2006, 1999.

[29] A. Scaglione, S. Barbarossa, and G. B. Giannakis, ―Filterbank transceivers optimizing informa-

tion rate in block transmissions over dispersive channels,‖ IEEE Transactions on Information Theory,

vol. 45, no. 3, pp. 1019–1032, 1999.

[30] C. Peel, Q. Spencer, A.L. Swindlehurst, and B. Hochwald, ―Downlink transmit beamforming in

multi-user MIMO systems,‖ in IEEE Sensor Array and Multichannel Signal Processing Workshop

Proceedings, Sitges, Spain, July 2004,vol. 1, pp. 43–51.

[31] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, ―Zero-forcing methods for downlink spatial

multiplexing in multiuser MIMO channels,‖ IEEE Transactions on Signal Processing, vol. 52, no. 2,

pp. 461–471, February 2004.

1 [32] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, and H. Sampath, ―Optimal designs for

space-time linear precoders and decoders,‖ IEEE Transactions on Signal Processing, vol. 50, no. 5,

pp. 1051–1064, 2002.

[33] A. Scaglione, G. B. Giannakis, and S. Barbarossa, ―Redundant Filterbank Precoders and Equaliz-

ers Part I: Unification and Optimal Designs,‖ IEEE Transactions on Signal Processing, vol. 47, no. 7,

pp. 1988–2006, 1999

[34] G. J. Foschini and M. Gans, ―On Limits of Wireless Communications in a Fading Environment

when Using Multiple Antennas,‖ Wireless Personal Communications, vol. 6, pp. 311–335, 1998.

[35] I. E. Telatar, ―Capacity of Multi-Antenna Gaussian Channels,‖ European Transactions on Tele-

communications, vol. 10, no. 6, pp. 585–595, November 1999.

[36] A. Goldsmith, S. A. Jafar, N. Jindan, and S. Vishwanath, ―Capacity limits of MIMO channels,‖

IEEE Journal on Selected Areas in Communications, vol. 21, no. 5, pp. 684–702, 2003.

[37] V. Tarokh, N. Shesadri, and A. R. Calderbank, ―Space-time Codes for High Data Rate Wireless

Communications: performance criterion and code construction ,‖ IEEE Transactions on Information

Theory,vol. 44 Issue:2,pp. 744 – 765, Mar 1998

[38] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, ―Space-Time Block Codes from Orthogonal

Designs,‖ IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, July 1999.

[39] H. Jafarkhani, Space-Time Coding: Theory and Practice, Cambridge University Press, New

York, 2005.

[40] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, ―VBLAST: An Architec-

ture for Realizing Very High Data Rates over The Rich Scattering Wireless Channel,‖ in Proc. URSI

International Symposium on Signals, Systems and Electronics Conference, New York, 1998, pp. 295–

300.

[41] N. Al-Dhahir and A.H. Sayed, ―The Finite-Length Multi-Input Multi-Output MMSE-DFE,‖

IEEE Transactions on Signal Processing, vol. 48, no. 10, pp. 2921–2936, October 2000.

[42] X. Zhu and R. D. Murch, ―Layered Space-Time Equalization for Wireless MIMO Systems,‖

IEEE Transactions on Wireless Communications, vol. 2, no. 6, pp. 1189–1203, November 2003.

[43] K. D. Rao, ―Equalization of a MIMO Channel Using FIR Inverses,‖ IEEE Transactions on Signal

Processing, vol. 54, no. 7, pp. 2844 – 2848, July 2006.

[44] J. Yang and S. Roy, ―Joint Transmitter-Receiver Optimization for Multi-Input Multi-Output Sys-

tems with Decision Feedback,‖ IEEE Transactions on Information Theory, vol. 40, no. 5, pp. 1334–

1347, September 1994.

[45] J. Yang and S. Roy, ―On Joint Transmitter and Receiver Optimization for Multiple-Input Mul-

tiple-Output (MIMO) Transmission Systems,‖ IEEE Transactions on Communications, vol. 42, no.

12, pp. 3221–3231, December 1994.

1 [46] D.P. Palomar, J.M. Cioffi, and M.A. Lagunas, ―Joint Tx-Rx beamforming design for multicarrier

MIMO channels: a unified framework for convex optimization,‖ IEEE Transactions on Signal

Processing, vol. 51, no. 9, pp. 2381 –2401, September 2003.

[47] A. Hjørungnes, Marcello L. R. de Campos, and Paulo S. R. Diniz, ―Jointly Optimized Transmit-

ter and Receiver FIR MIMO Filters in the Presence of Near-End Crosstalk,‖ IEEE Transactions on

Signal Processing, vol. 53, no. 1, pp. 346–359, January 2005.

[48] K. C. Hwang and K. B. Lee, ―Joint Transmit and Receiver Filters Design for Multiple-Input Mul-

tiple-Output (MIMO) Systems,‖ IEEE Transactions on Wireless Communications, vol. 4, no. 4, pp.

1635–1649, July 2005.

[49] A. Yasotharan, ―Multirate Zero-Forcing Tx-Rx Design for MIMO Channel Under BER Con-

straints,‖ IEEE Transactions on Signal Processing, vol. 54, no. 6, pp. 2288–2301, June 2006.

[50] J. G. McWhirter, P. D. Baxter, T. Cooper, S. Redif, and J. Foster, ―An EVD Algorithm for Para-

Hermitian Polynomial Matrices,‖ IEEE Transactions on Signal Processing, vol. 55, no. 5, pp. 2158–

2169, May 2007.

[51] J. G. McWhirter and P. D. Baxter, ―A Novel Technique for Broadband SVD,‖ in 12th Annual

Workshop on Adaptive Sensor Array Processing, MIT Lincoln Labs, Cambridge, MA, March 2004.

[52] F. Xu, N. T. Davidson, J. K. Zhang, and K. M. Wong, ―Design of Block Transceivers With Deci-

sion Feedback Detection,‖ IEEE Transactions on Signal Processing, vol. 54, no. 3, pp. 965–978,

March 2006.

[53] M. T¨uchler, R. Koetter, and A. Singer, ―Turbo equalization: principles and new results,‖ IEEE

Trans. Comm.,vol. 50, pp. 754-767, May 2002.

[54] S. Kay, Fundamentals of statistical signal processing: estimation theory. Prentice Hall.

[55] K. Hooli, M. Latva-aho, and M. Juntti, ―Performance evaluation of adaptive chip-level channel

qualizers in WCDMA downlink,‖ Communications, ICC 2001.IEEE International Conference on,

2001.

[56] Z. Rong and L. Guangqiu, ― Low complexity code-multiplexed pilot aided adaptive MMSE equa-

lizer for CDMA systems,‖ 4th International Conference on Wireless Communications, Networking

and Mobile Computing, 2008. WiCOM ‘08, 2008.

[57] E. Carvalho and D. Slock, ―Deterministic quadratic semi-blind FIR multichannel estimation: Al-

gorithms and performance,‖ In Proc. ICASSP., 2000.

[58] L. Tong and S. Perreau, ―Multichannel blind identification: From subspace to maximum likelih-

ood methods,‖ ProcIEEE, 1998.

[59] I. Kacha, K. Abed-Meraim, and A. Belouchrani, ―A fast adaptive blind equalization algorithm

robust to channel order over-estimation errors,‖ IEEE Sensor Array and Multichannel Signal

Processing Workshop Proceedings, 2004

[60] ―A new blind adaptive MMSE equalizer for MIMO systems,‖ IEEE 16th International Sympo-

sium on Personal, Indoor and Mobile Radio Communications, 2005.

1 [61] L. Tong, G. Xu, and T. Kailath, ― Blind identification and equalization of multipath channels,‖

Communications, 1992. ICC ‘92, Conference record, SUPERCOMM/ICC ‘92, Discovering a New

World of Communications., IEEE International Conference on,1992.

[62] ―Implementation of the levinson algorithm for MMSE equalizer,‖ 2008 International SoC Design

Conference, 2008.

[63] S. Kim, J. Lee, and Y. Kim, ―Adaptive cholesky based MMSE equalizer in GSM,‖ Circuits and

Systems, 2008.APCCAS 2008. IEEE Asia Pacific Conference on, 2008.

[64] C. Mehlfuhrer and M. Rupp, ― A robust MMSE equalizer for MIMO enhanced HSDPA,‖ Sig-

nals, Systems and Computers, 2006. ACSSC ‘06. Fortieth Asilomar Conference on, 2006.

[65] S. Chowdhury and M. Zoltowski, ― Application of conjugate gradient methods in MMSE equali-

zation for the forward link of DSCDMA,‖ Vehicular Technology Conference. VTC 2001 Fall. IEEE

VTS 54th, 2001.

[66] P. Chang and A. Willson, ―Analysis of conjugate gradient algorithms for adaptive filtering,‖

IEEE transactions on Signal Processing,2000.

[67] T. P. Krauss, M. D. Zoltowski, and G. Leus, ― Simple MMSE equalizers for CDMA downlink to

restore chip sequence: Comparison to zero-forcing and rake,‖ Proceedings.2000 IEEE International

Conference on Acoustics, Speech, and Signal Processing, 2000. ICASSP ‘00., 2000

[68] Z. Rong and L. Guangqiu, ― Low complexity code-multiplexed pilot aided adaptive MMSE equa-

lizer for CDMA systems,‖ 4th International Conference on Wireless Communications, Networking

and Mobile Computing, 2008. WiCOM ‘08, 2008.

[69] B. H. Kim, X. Zhang, and M. Flury, ―Linear MMSE spacetime equalizer for MIMO multicode

CDMA systems,‖ IEEE Transactions on Communications, 2006.

[70] B. Mouhouche, P. Loubaton, and W. Hachem, ―Asymptotic analysis of reduced-rank chip-level

MMSE equalizers in the downlink of CDMA systems,‖ IEEE TRANSACTIONS ON SIGNAL

PROCESSING, 2007

[71] M. J. Heikkila, K. Ruotsalainen, and J. Lilleberg, ―Space-time equalization using conjugate-

gradient algorithm in wcdma downlink,‖

2002.

[72] A. Ikhlef and J. Louveaux, ― An enhanced MMSE per subchannel equalizer for highly frequency

selective channels for FBMC/OQAM systems,‖ 2009. SPAWC ‘09. IEEE 10th Workshop on Signal

Processing Advances in Wireless Communications, 2009.

[73] A. Scaglione, G. B. Giannakis, and S. Barbarossa, ― An improved MMSE equalizer for one-

dimensional modulation OFDM systems,‖ IEEE TRANSACTIONS ON SIGNAL PROCESSING,

1999.

[74] P. Tan and N. Beaulieu, ―An improved MMSE equalizer for one-dimensional modulation OFDM

systems,‖ VTC-2005-Fall.2005 IEEE 62nd Vehicular Technology Conference, 2005.

1 [75] M. Joham, W. Utschick, and J. A. Nossek, ―Latency time optimization for FIR and block trans-

mit filters,‖ in Proc. ISSPA, (Paris, France), pp. 273–276, 2003.

[76] X. Wang and H. V. Poor, ― Iterative (turbo) soft interference cancellation and decoding for coded

CDMA,‖ IEEE Transaction on Commun. , vol. 47, no. 7, pp. 1046–1061, Jul. 1999.

[77] M. Tüchler, R. Koetter, and A. C. Singer, ―Turbo equalization: Principles and new results,‖ IEEE

Trans. Commun., vol. 50, no. 5, pp. 754–767, May 2002.

[78] X. Yuan, Q. Guo, X. Wang, and L. Ping, ―Evolution analysis of lowcost iterative equalization in

coded linear systems with cyclic prefixes,‖ IEEE J. Sel. Areas Commun., vol. 26, no. 2, pp. 301–310,

Feb. 2008.

[79] N. Barbosa and S. L. Miller, ―Adaptive detection of DSICDh4A signals in fading channels,‖

IEEE Transaction on Commun., vol. 46, no 1, pp. 115-124, Jan. 1998.