PhD Seminar Bc

7/30/2019 PhD Seminar Bc

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PhD Seminar (ENCS8051)

New Power Allocation Strategy Among MIMO

Spatial Subchannels - Beam-Nulling

Mabruk Gheryani

Supervisor: Dr. Yousef R. Shayan

November 10, 2007


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Abstract

Since the discovery of multiple-input-multiple-output (MIMO) channel ca-

pacity, a lot of research efforts have been put into this field. It has been

recognized that adaptive techniques proposed for single-input-single-output

(SISO) channel, can also be applied to improve MIMO channel capacity.

The original MIMO channel can be converted to uncorrelated spatial

subchannels via singular value decomposition. Strategies of power allocationover these spatial subchannels for various channel state information scenarios

have been studied, such as equal power, water-filling, beamforming. A novel

scheme called beam-nulling has been proposed. Using the same feedback

bandwidth as beamforming, the new scheme discards the weakest spatial

subchannel but exploits the other subchannels. Hence, it can achieve signifi-

cant high capacity, which is near the optimal water-filling solution at medium

signal-to-noise ratio.Additionally, the capacities of equal power, beamforming and beam-

nulling are compared through theoretical analysis. Numerical results of the

three schemes and the optimal water-filling scheme are also compared. It is

shown that at low signal-to-noise ratio, beamforming nears the optimal, at

medium signal-to-noise ratio, beam-nulling nears the optimal, and at high

signal-to-noise ratio, equal power nears the optimal.

As future works, performance and application of beam-nulling and new

adaptation strategies will be further investigated. The remaining tasks have

been scheduled.


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Contents

List of Tables 3

List of Tables 4

Notations and Abbreviations 5

1 Introduction 7

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Problem Statement and Objective . . . . . . . . . . . . . . . . 10

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Preliminaries 12

2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Power Allocation Among Spatial Subchannels . . . . . . . . . 14

2.2.1 Equal Power . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Water-Filling . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Eigen-Beamforming . . . . . . . . . . . . . . . . . . . . 18

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3 Recent Research Results 22

3.1 Beam-Nulling . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Comparisons Among the Four Schemes . . . . . . . . . . . . . 26

3.2.1 At low SNR . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2 At medium SNR . . . . . . . . . . . . . . . . . . . . . 33

3.2.3 At high SNR . . . . . . . . . . . . . . . . . . . . . . . 34

4 Conclusions and Future Works 35

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.1 Performance and Application of Beam-Nulling . . . . . 36

4.2.2 Extended Adaptive Frameworks . . . . . . . . . . . . . 36

4.3 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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List of Tables

4.1 Schedule for the remaining tasks. . . . . . . . . . . . . . . . . 38

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List of Figures

2.1 MIMO channel model. . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Capacity for equal power allocation. . . . . . . . . . . . . . . . 15

2.3 Water-filling scheme. . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Capacity for water-filling. . . . . . . . . . . . . . . . . . . . . 18

2.5 Eigen-beamforming scheme. . . . . . . . . . . . . . . . . . . . 19

2.6 Capacity for eigen-beamforming. . . . . . . . . . . . . . . . . . 21

3.1 Beam-Nulling scheme. . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Capacity for beam-nulling. . . . . . . . . . . . . . . . . . . . . 263.3 2 2 Rayleigh fading channel. . . . . . . . . . . . . . . . . . . 293.4 3 3 Rayleigh fading channel. . . . . . . . . . . . . . . . . . . 303.5 4 4 Rayleigh fading channel. . . . . . . . . . . . . . . . . . . 313.6 5 5 Rayleigh fading channel. . . . . . . . . . . . . . . . . . . 32

4.1 Gantt chart for the remaining tasks. . . . . . . . . . . . . . . . 39

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Notations and Abbreviations

X: upper bold letter for matrix

x: lower bold letter for column vector

XH: hermitian of X XT: transpose of X diag[x]: a diagonal matrix with x on its main diagonal tr(X): trace ofX det(X): determinant of X {x}: a set ofx E(x): expectation of x AWGN: additive white Gaussian noise BF: beamforming BN: beam-nulling CSI: channel state information EQ: equal power

FDFR: Full Diversity Full Rate LDC: linear dispersion code MIMO: multiple-input-multiple-output

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SISO: single-input-single-output SNR: signal-to-noise ratio ST: space-time STBC: space-time block code SVD: singular value decomposition WF: water-filling i.i.d.: independently identically distributed

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Chapter 1

Introduction

1.1 Motivation

Since the discovery of multiple-input-multiple-output (MIMO) channel ca-

pacity, a lot of research efforts have been put into this field [1] [2].

To combat channel quality variation and thus further improve system

performance such as power efficiency, error rate and average data rate, it has

been recognized that adaptive techniques proposed for single-input-single-

output (SISO) channel [3] [4], can also be applied to improve MIMO chan-

nel capacity. In this method, a feedback channel is utilized to provide CSI

from the receiver to the transmitter. According to the feedback of CSI, the

transmitter will adjust transmission parameters, such as power allocation,

modulation, coding rate, etc. This is conditioned by the fact that the chan-

nel keeps relatively constant until the transmitter receives the CSI and then

transmits the next data block accordingly.

The ideal scenario is that the transmitter has full knowledge of channel

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state information (CSI) fed back from the receiver and the CSI keeps con-

stant before the transmitter sends information to the receiver. With such a

perfect CSI feedback, the original MIMO channel can be converted to multi-

ple uncoupled SISO channels via singular value decomposition (SVD) at the

transmitter and the receiver [1]. In other words, the original MIMO chan-

nel can be decomposed into several orthogonal spatial subchannels with

various propagation gains. To achieve better performance, various strategies

to allocate constrained power to these subchannels can be implemented de-

pending on the availability of CSI at the transmitter [6]- [8]. In this study,

we propose to develop new scheme with limited CSI feedback.

1.2 Literature Survey

If the transmitter has full knowledge about channel matrix, i.e., full CSI,

the so-called water-filling (WF) principle is performed on each spatial sub-

channel to maximize the channel capacity. This scheme is optimal in thiscase. Note that in practice, water-filling power allocation has to cooper-

ate with the other adaptive parameters to improve performance or/and data

rate, such as constellation and coding rate. For example, over time-invariant

MIMO channels, it is known that the optimal performance (ergodic capac-

ity) is attained by power water-filling across channel eigenvalues with the

total power constraint [1]. Also, for time-varying MIMO channels, the op-

timal performance is obtained through power water-filling over both spaceand time domains with the average power constraint [11]. The space-time

WF-based scheme and the spatial WF-based scheme for MIMO fading chan-

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nels were compared in [12]. The comparison shows that for Rayleigh channels

without shadowing, space-time WF-based scheme gains little in capacity over

spatial WF-based scheme. However, for Rayleigh channels with shadowing,

space-time WF-based scheme achieves higher spectral efficiency per antenna

over spatial WF-based scheme. A WF-based scheme using imperfect CSI in

MIMO systems was studied in [13].

For the WF-based scheme, the feedback bandwidth for the full CSI grows

with respect to the number of transmit and receive antennas and the perfor-

mance is often very sensitive to channel estimation errors. To overcome these

disadvantages, various beamforming techniques for MIMO channels are also

investigated intensively. Beamforming is a linear signal processing technique

that control the complex weights of the transmit and receive antennas jointly

to optimize the signal-to-noise ratio (SNR) in one direction [10]. In other

words, beamforming can increase the sensitivity in the direction of wanted

signals but decrease the sensitivity in the direction of interfering signals.

In an adaptive beamforming scheme, complex weights of the transmit

antennas are fed back from the receiver. If only partial CSI is available at

the transmitter such as the eigenvector associated with the strongest spa-

tial subchannel, eigen-beamforming [7] is optimal in this case. The eigen-

beamforming scheme only allocates power to the strongest spatial subchan-

nel but can achieve full diversity and high signal-to-noise ratio (SNR). Also,

in practice, the eigen-beamforming scheme has to cooperate with the other

adaptive parameters to improve performance or/and data rate, such as con-

stellation and coding rate.

There are also other beamforming schemes based on various criteria.

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For example, an optimal eigen-beamforming space-time block code (STBC)

scheme based on channel mean feedback was proposed in [7]. A MIMO sys-

tem based on transmit beamforming and adaptive modulation was proposed

in [8], where the transmit power, the signal constellation, the beamforming

direction, and the feedback strategy were considered jointly. The analysis of

MIMO beamforming systems with quantized CSI for uncorrelated Rayleigh

fading channels was provided in [9].

1.3 Problem Statement and Objective

Note that the conventional beamforming is optimal in terms of maximizing

the SNR at the receiver. However, it is sub-optimal from a MIMO capac-

ity point of view, since only one data stream, instead of parallel streams,

is transmitted through the MIMO channel [17]. Inspired by existing beam-

forming schemes, we will propose a new beamforming-like technique called

minimum eigenvector beam-nulling (BN). This scheme uses the same feed-back bandwidth as beamforming. That is, only one eigenvector is fed back

to the transmitter. Unlike the eigen-beamforming scheme in which only the

best spatial subchannel is considered, in the beam-nulling scheme, only the

worst spatial subchannel is discarded. Hence, the loss of channel capacity as

compared to the optimal water-filling scheme can be reduced. In this scheme,

power is only allocated to the other good spatial subchannels. As compared

to the beamforming scheme, this scheme outperforms significantly in termsof channel capacity.

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1.4 Organization

The rest of the seminar is organized as follows. In Chapter 2, preliminaries

are presented. In Chapter 3, current research results are provided. Finally

in Chapter 4, we will conclude the seminar and present the research schedule

for the remaining tasks.

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Chapter 2

Preliminaries

2.1 Channel Model

In this study, the channel is assumed to be a Rayleigh flat fading channel

with Nt transmit and Nr (Nr Nt) receive antennas. Lets denote thecomplex gain from transmit antenna n to receiver antenna m by hmn and

collect them to form an Nr Nt channel matrix H = [hmn]. The channelis known perfectly at the receiver but partially informed to the transmitter.

The entries in H are assumed to be independently identically distributed

(i.i.d.) symmetrical complex Gaussian random variables with zero mean and

unit variance. The MIMO channel is shown in Figure 2.1.

The symbol vector at the Nt transmit antennas is denoted by

x = [x1, x2, . . . , xNt ]T. During the duration of x, H keeps constant but vary

for next x. According to information theory [5], the optimal distribution

of the transmitted symbols is Gaussian. Thus, the elements {xi} of x areassumed to be i.i.d. Gaussian variables with zero mean and unit variance,

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hmnxn

x1

ym

y1

yNrxNt

Figure 2.1: MIMO channel model.

i.e., E(xi) = 0 and E|xi|2 = 1. By using the linear model, the receivedsignals can be written as

y = Hx + z (2.1)

z is the additive white Gaussian noise (AWGN) vector with i.i.d. symmetrical

complex Gaussian elements of zero mean and variance 2z .

The singular-value decomposition of H can be written as

H = UVH (2.2)

where U is an Nr

Nr unitary matrix, is an Nr

Nt matrix with singular

values {i} on the diagonal and zeros off the diagonal, and V is an Nt Nt unitary matrix. For convenience, we assume 1 2 . . . Nt, U =[u1u2 . . . uNr ] and V = [v1v2 . . . vNt ]. {ui} and {vi} are column vectors. We

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assume that the rank of H is r (r Nt). That is, the number of non-zerosingular values is r.

From (2.2), the original channel can be considered as consisting of r

uncoupled parallel subchannels. Each subchannel corresponds to a singular

value of H. In the following context, the subchannel is also referred to as

spatial subchannel. For instance, one spatial subchannel corresponds to

i, ui and vi.

2.2 Power Allocation Among Spatial Subchan-

nels

We assume that the total transmitted power is constrained to P. Given the

power constraint, different power allocation among spatial subchannels can

affect the channel capacity tremendously. In the following context, depending

on the power allocation among spatial subchannels, several popular schemes

are presented.

2.2.1 Equal Power

If the transmitter has no knowledge about the channel, the most judicious

strategy is to allocate the power to each transmit antenna equally. In this

case, the received signals can be written as

y = PNt

Hx + z (2.3)

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The associated instantaneous channel capacity with respect to H can be

written as [1]

Ceq =Nti=1

log

1 +

P

Nt2z2i

(2.4)

In the following figure, numerical results of ergodic (average) channel

capacity for 2 2, 3 3 and 4 4 Rayleigh flat fading channels are shownin Figure 2.2.

0 5 10 15 20 250

5

10

15

20

25

SNR (dB)

Capacity(bit/s/Hz)

Equal power

2x23x3

4x4

5x5

Figure 2.2: Capacity for equal power allocation.

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2.2.2 Water-Filling

If the transmitter has full knowledge about the channel, the most judicious

strategy is to allocate the power to each spatial subchannel by water-filling

principle [1]. It allocates more power when a spatial subchannel has larger

gain (i.e. {i}) and less when a subchannel gets worse. With V at thetransmitter and U at the receiver, the original MIMO channel is converted to

r uncoupled parallel SISO channels. The WF scheme is shown in Figure 2.3.

Constellation

Mapper

Ant-1

Ant-Nt

Ant-1

Ant-Nr

~y

Binary

Info.source Binary

Info.

Out

V U

Channel

Estimation

Feedback

P1x1

S/PDetector

PowerAllocation

Nt-r0's

H

Prxr

Figure 2.3: Water-filling scheme.

For spatial subchannel i, i = 1, 2, . . . , r, the received signal isyi =

Piixi + zi (2.5)

wherer

i=1Pi = P as a constraint and zi the is AWGN variable with zero mean

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and 2z variance. Following the method of Lagrange multipliers, the optimal

Pi can be found as [1]

Pi = max

1

L ln 2

2z

i, 0

, i = 1, 2, . . . , r (2.6)

where L is the Lagrange multiplier. The instantaneous channel capacity is

for this spatial subchannel is

Cwf,i = log

1 +

Pi2z

2i

(2.7)

Then the total channel capacity with respect to H is

Cwf =r

i=1

Cwf,i (2.8)

The WF scheme maximizes the channel capacity by power allocation over

spatial subchannels. Since 1 2 . . . r, we have P1 P2 . . . Pr andCwf,1 Cwf,2 . . . Cwf,r. In practice, if each spatial subchannel requiresthe same error rate performance, the spatial subchannel with larger gain

(i.e.

Pii) can have higher rate; while if each spatial subchannel has the

same rate, the spatial with larger gain will have better performance. Theapplication of these two approaches will depend on the type of service. For

example, the important data needs high quality but the voice can tolerate

low quality.

In the following figure, numerical results of ergodic (average) channel

capacity for 2 2, 3 3 and 4 4 Rayleigh flat fading channels are shownin Figure 2.4.

In the WF scheme, the feedback bandwidth for the perfect CSI grows

with respect to the number of transmit and receive antennas. In practice,

the bandwidth for CSI feedback is often very limited. The large feedback

bandwidth also affects the transmission efficiency.

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0 5 10 15 20 250

5

10

15

20

25

SNR (dB)

C

apacity(bit/s/Hz)

Waterfilling

2x2

3x3

4x4

5x5

Figure 2.4: Capacity for water-filling.

2.2.3 Eigen-Beamforming

To save feedback bandwidth, beamforming can be considered. Beamform-

ing is a linear signal processing technique that control the complex weights

of the transmit and receive antennas jointly to optimize the signal-to-noise

ratio (SNR) in one direction [7] [10]. For the MIMO model, the optimal

beamforming is called eigen-beamforming. In the following context, this

optimal solution is referred to as beamforming.

In this scheme, the eigenvector associated with the maximum singular

value 1 from the transmitter side, i.e., v1, is feedback to the transmitter as

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the transmission beamformer. The Eigen-beamforming scheme is shown in

Figure 2.5.

Constellation

MapperDetector

Ant-1

Ant-Nt

Ant-1

Ant-Nr

~y1

Binary

Info.

source

Binary

Info.

Out

v1 u1

Channel

Estimation

Feedback

H

Px1

Figure 2.5: Eigen-beamforming scheme.

We assume one symbol, saying x1, is transmitted. At the receiver, the

received vector can be written as

y1 =

PHv1x1 + z1 (2.9)

where z1 is the additive white Gaussian noise vector with i.i.d. symmetricalcomplex Gaussian elements of zero mean and variance 2z .

The eigenvector associated with the maximum singular value from the

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receiver side, i.e., u1, is applied as the receiver beamformer. Then we have

y1 = uH1 y1 =

P 1x1 + z1 (2.10)

where z1 is Gaussian with zero mean and variance 2z . As can be seen from

(2.10), only the spatial subchannel associated with the maximum singular

value 1 is applied.

The associated instantaneous channel capacity with respect to H can be

written as

Cbf = log1 + P2z

21 (2.11)In the following figure, numerical results of ergodic (average) channel capacity

for 22, 33 and 44 Rayleigh flat fading channels are shown in Figure 2.6.The eigen-beamforming scheme can save feedback bandwidth and is op-

timized in terms of SNR [10]. However, since only one spatial subchannel is

considered, this scheme suffers from loss of channel capacity [17], especially

when the number of antennas grows. Also, as can be seen from the figure,

the gap of capacity between different numbers of antennas is nearly the same

for all the SNR region.

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0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

SNR (dB)

Capacity(bit/s/Hz)

Beamforming

2x2

3x3

4x4

5x5

Figure 2.6: Capacity for eigen-beamforming.

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Chapter 3

Recent Research Results

In this chapter, the current achievements will be presented as follows.

3.1 Beam-Nulling

Inspired by the eigen-beamforming scheme, we will propose a new beamforming-

like scheme called beam-nulling (BN). This scheme uses the same feedback

bandwidth as beamforming. That is, only one eigenvector is fed back to the

transmitter. Unlike the eigen-beamforming scheme in which only the best

spatial subchannel is considered, in the beam-nulling scheme, only the worst

spatial subchannel is discarded. Hence, the loss of channel capacity as com-

pared to the optimal water-filling scheme can be reduced. The Beam-Nulling

scheme is shown in Figure 3.1.

In this scheme, the eigenvector associated with the minimum singular

value from the transmitter side, i.e., vNt, is feedback to the transmitter.

By a certain rule, a subspace orthogonal to the weakest spatial channel is

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Constellation

Mapper

Ant-1

Ant-Nt

Ant-1

Ant-Nr

~y

Binary

Info.

source

BinaryInfo.

Out

U

ChannelEstimation

Feedback

x1

S/P

xNt-1

Detector

H

Figure 3.1: Beam-Nulling scheme.

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constructed. That is, the following condition should be satisfied.

HvNt = 0 (3.1)

The Nt (Nt 1) matrix = [g1g2 . . . gNt1] spans the subspace. Note thatthe rule to construct the subspace should also be known to the receiver.

An example to construct the orthogonal subspace is presented as follows.

We construct an Nt Nt matrix

A = [vNtI] (3.2)

where I = [I(Nt1)(Nt1)0(Nt1)1]T. Applying QR decomposition to A, we

have

A = [vNt] R (3.3)

where R is an upper triangular matrix with the (1,1)-th entry equal to 1.

is the subspace orthogonal to vNt.

At the transmitter, Nt 1 symbols denoted as x are transmitted only

over the orthogonal subspace . The received signals at the receiver can be

written as

y =

P

Nt 1Hx + z (3.4)

where z is additive white Gaussian noise vector with i.i.d. symmetrical

complex Gaussian elements of zero mean and variance 2z .

Substituting (2.2) into (3.4) and multiplying y by UH, we have

y = PNt 1B

0Tx + z (3.5)

where z is additive white Gaussian noise vector with i.i.d. symmetrical com-plex Gaussian elements of zero mean and variance 2z . With the condition in

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(3.1),

VH = B

0T (3.6)

where

B =

vH1 g1 vH1 g2 . . . v

H1 gNt1

vH2 g1. . . . . .

......

.... . .

...

vHNt1g1 . . . . . . vHNt1

gNt1

(3.7)

B is an (Nt 1) (Nt 1) unitary matrix. As can be seen from (3.5), theavailable spatial channels are Nt 1. Since the weakest spatial subchannel isnulled in this scheme, power can be allocated equally among the left better

Nt 1 subchannels. Equation (3.5) can be rewritten as

y =

P

Nt 1Bx + z (3.8)

where

y and

z are column vectors with the first (Nr 1) elements of

y and

z, respectively, and = diag[1, 2, . . . , (Nt1)]. From (3.8), the associatedinstantaneous channel capacity with respect to H can be found as

Cbn =Nt1i=1

log

1 +

P

(Nt 1)2z2i

(3.9)

In the following figure, numerical results of ergodic (average) channel capacity

for 22, 33 and 44 Rayleigh flat fading channels are shown in Figure 3.2.

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0 5 10 15 20 250

5

10

15

20

25

SNR (dB)

C

apacity(bit/s/Hz)

Beamnulling

2x2

3x3

4x4

5x5

Figure 3.2: Capacity for beam-nulling.

As can be seen, the beam-nulling scheme only needs one eigenvector to

be fed back. However, since only the worst spatial subchannel is discarded,

this scheme can increase channel capacity significantly as compared to the

conventional beamforming scheme.

3.2 Comparisons Among the Four Schemes

In this section, we compare the new proposed beam-nulling scheme with the

other schemes. Water-filling is the optimal solution among the four schemes

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for any SNR.

Let Ceq, Cbf, and Cbn denote the ergodic channel capacities of equal

power, beamforming and beam-nulling, respectively. That is,

Ceq = E(Ceq) (3.10)

Cbf = E(Cbf) (3.11)

Cbn = E(Cbn) (3.12)

where E() denotes expectation with respect to H. Let = P/2z denote SNR.

Differentiating the above ergodic capacities with respect to respectively, we

have

Ceq

= E

Nti=1

1

+ Nt2i

(3.13)Cbf

= E

1

+ 121

(3.14)

Cbn

= ENt1

i=1

1

+ Nt12i

(3.15)

The differential will also be referred to as slope. The second order differ-

entials are listed as follows.

2Ceq

2

= E

Nt

i=1 1

+ Nt2i

2 (3.16)

2Cbf2

= E

1 + 1

21

2 (3.17)

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2Cbn2 = E

Nt1i=1

1

+ Nt12i

2 (3.18)

As can be seen, the above ergodic capacities are concave and monotonically

increasing with respect to . With the fact that 1 2 . . . Nt, it canbe readily checked that the slopes of ergodic capacities associate with equal

power and beam-nulling are bounded as follows.

E Nt + Nt

1

Ceq

E Nt + Nt

Nt

(3.19)E

Nt 1 + Nt1

1

Cbn

E Nt 1

+ Nt1(Nt1)

(3.20)

For the case of Nt = 2, beamforming and beam-nulling have the same

capacity for any SNR as can be seen from equations of capacity and slope.

If 0, i.e., at low SNR, it can be easily found thatCbf

Cbn

Ceq

, 0 (3.21)

If , i.e., at high SNR, it can be easily found thatCeq

Cbn

Cbf

, (3.22)

Hence, at medium SNR, Cbn

has the largest value. That is, among equal

power, beamforming and beam-nulling, for low SNR beamforming has largest

capacity, for high SNR equal power has largest capacity, for medium SNR,

beam-nulling has largest capacity. Numerical results will be provided to

demonstrate our analysis.

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0 5 10 15 20 250

2

4

6

8

10

12

SNR (dB)

C

apacity(bit/s/Hz)

2x2

EQ

WF

BF

BN

Figure 3.3: 2

2 Rayleigh fading channel.

From Figure 3.3 to 3.6, capacities of water-filling, beamforming, beam-

nulling and equal power are compared over 22, 33, 44 and 55 Rayleighfading channels, respectively. Note that since SNR is measured in dB, the

curves become convex. In these figures, EQ stands for equal power, WF

stands for water-filling, BF stands for beamforming and BN stands for

beam-nulling. In the figures with subfigures, subfigure (a) is for all region

SNR, subfigure (b) is detailed for SNR region from low to medium, and

subfigure (c) is detailed for SNR region from medium to high. Note that for

2 2 channel, beamforming and beam-nulling have the same capacity.

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0 5 10 15 20 250

2

4

6

8

10

12

14

16

SNR (dB)

Ca

pacity(bit/s/Hz)

3x3

EQ

WF

BF

BN

(a) 0 25 dB

0 2 4 6 8 100

1

2

3

4

5

6

SNR (dB)

Capacity(bit/s/Hz)

3x3

EQWF

BF

BN

(b) 0

10 dB

12 14 16 18 20 226

7

8

9

10

11

12

13

14

15

16

SNR (dB)

Capacity(bit/s/Hz)

3x3

EQWF

BF

BN

(c) 12

22 dB

Figure 3.4: 3 3 Rayleigh fading channel.

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0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

22

SNR (dB)

Ca

pacity(bit/s/Hz)

4x4

EQ

WF

BF

BN

(a) 0 25 dB

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

SNR (dB)

Capacity(bit/s/Hz)

4x4

EQWF

BF

BN

(b) 0

8 dB

16 17 18 19 20 21 22 23 2410

11

12

13

14

15

16

17

18

19

20

SNR (dB)

Capacity(bit/s/Hz)

4x4

EQWF

BF

BN

(c) 16

24 dB


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0 5 10 15 20 250

5

10

15

20

25

SNR (dB)

Ca

pacity(bit/s/Hz)

5x5

EQ

WF

BF

BN

(a) 0 25 dB

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

SNR (dB)

Capacity(bit/s/Hz)

5x5

EQWF

BF

BN

(b) 0

8 dB

22 22.5 23 23.5 24 24.5 25 25.5 2620

20.5

21

21.5

22

22.5

23

23.5

24

24.5

25

SNR (dB)

Capacity(bit/s/Hz)

5x5

EQWF

BF

BN

(c) 22

26 dB


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As can be seen from all these figures, the water-filling has the best capac-

ity at any SNR region. The other schemes perform differently at different

SNR regions as discussed in the following context.

3.2.1 At low SNR

At low SNR, the beamforming is the closest to the optimal water-filling, e.g.,

the SNR region below 3 dB for 3 3 fading channel, the SNR region below3.2 dB for 4

4 fading channel, and the SNR region below 3.5 dB for 5

5

fading channel. At low SNR, the water-filling scheme can only allocate power

to one or two spatial subchannels. Especially, if only one spatial subchannel

can be use, the beamforming scheme is just the optimal water-filling, which

can be seen from the capacity of 2 2 fading channel in Figure 3.3.

3.2.2 At medium SNR

At medium SNR, the proposed beam-nulling is the closest to the optimalwater-filling, e.g., the SNR region from 3 dB to 16 dB for 3 3 fading channel,the SNR region from 3.2 dB to 20.5 dB for 44 fading channel, and the SNRregion from 3.5 dB to 23.5 dB for 5 5 fading channel. The beam-nullingscheme only discards the weakest spatial subchannel and allocates power to

the other spatial subchannels. As can be seen from the numerical results, the

beam-nulling scheme performs better than the other schemes in this case.

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3.2.3 At high SNR

At high SNR, the equal power scheme is the closest to the optimal water-

filling, e.g., the SNR region over 16 dB for 3 3 fading channel, the SNRregion over 20.5 dB for 4 4 fading channel, and the SNR region over 23.5dB for 5 5 fading channel. As can be seen from the figures, at high SNR,the equal power scheme will converge to the water-filling scheme.

In summary, the application of the above four schemes shall depend on the

SNR region and the availability of CSI. At medium SNR, the proposed beam-

nulling scheme can achieve larger capacity than the beamforming scheme with

the same feedback bandwidth.

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Chapter 4

Conclusions and Future Works

In this chapter, we will conclude the study and present our remaining tasks

together with their schedule.

4.1 Conclusions

Via singular-value decomposition, the original MIMO channel is converted

to uncorrelated spatial subchannels. Based on the concept of spatial sub-

channels, we studied various power allocation strategies for various channel

state information scenarios, such as equal power, water-filling, beamforming.

Inspired by the beamforming scheme, we proposed a novel scheme called

beam-nulling. Using the same feedback bandwidth as beamforming, the

new scheme exploits all spatial subchannels except the weakest one and thus

achieves significant high capacity near the optimal water-filling scheme at

medium signal-to-noise ratio. Additionally, the capacities of equal power,

beamforming and beam-nulling were compared through theoretical analysis

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first and then numerical results of these three schemes are also compared

with the optimal water-filling scheme. The comparison showed that at low

signal-to-noise ratio, beamforming is the closest to the optimal water-filling,

at medium signal-to-noise ratio, beam-nulling is the closest to the optimal

solution, and at high signal-to-noise ratio, equal power is the closest to the

optimal solution.

4.2 Future Works

In the future, based on our current achievement, the remaining two works

are identified and will be presented together with their schedule as follows.

4.2.1 Performance and Application of Beam-Nulling

Currently, the capacity of beam-nulling was studied in the seminar. Based on

this knowledge, we will study how to achieve the promised capacity of beam-

nulling in practice. Additionally, we will study how to cooperate with the

other schemes and thus improve the performance further. For example, the

proposed scheme can concatenate with the linear dispersion code to achieve

better performance with more flexibility.

4.2.2 Extended Adaptive Frameworks

The power allocation strategy of beam-nulling and beamforming can be ex-

tended to improve capacity further if more feedback bandwidth is available.

That is, if more than one eigenvector, e.g. k eigenvectors, can be avail-

able at the transmitter, the existing beamforming scheme and the proposed

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beam-nulling scheme can be further extended, respectively. The extended

schemes will implement or discard k spatial subchannels and are referred to

as multi-dimensional (MD) beamforming and multi-dimensional beam-

nulling, respectively. The theoretical analysis and numeric results in terms

of capacity will be provided to evaluate the new extended schemes. We will

study the performance of the extended scheme. Similarly, we will study

how to cooperate with the other schemes and thus improve the performance

further.

4.3 List of Publications

Conferences

[1] Mabruk Gheryani, Z. Wu and Y. Shayan, SINR Analysis for Full-Rate

Linear Dispersion Code Using Linear MMSE, the 4th IEEE International

Symposium on Wireless Communication Systems, Oct. 2007.

[2] Mabruk Gheryani, Z. Wu and Y. Shayan, Design of Adaptive MIMOSystem Using Linear Dispersion Code, Submitted to IEEE International

Conference on Communications (ICC2008), 2008.

[3] Mabruk Gheryani, Y. Shayan, X. Wang and Z. Wu, Error Perfor-

mance of Linear Dispersion Codes, Submitted to IEEE International Con-

ference on Communications (ICC2008), 2008.

[4] Mabruk Gheryani, Z. Wu and Y. Shayan, Power Allocation Strategy

for MIMO System Based on Beam Nulling, Submitted to IEEE Interna-tional Conference on Communications (ICC2008), 2008.

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Journals

[1] Mabruk Gheryani, Z. Wu and Y. Shayan, Design of Adaptive MIMO

System Using Linear Dispersion Code, A transaction paper has been sub-

mitted to IEEE Trans. on Wireless Communications.

[2] Mabruk Gheryani, Y. Shayan, X. Wang and Z. Wu, Error Perfor-

mance of Linear Dispersion Codes, A transaction letter has been submitted

to IEEE Trans. on Wireless Communications.

4.4 Schedule

The remaining tasks are scheduled in Table 4.1 and the associated Gantt

chart is also shown in Figure. 4.1.

Table 4.1: Schedule for the remaining tasks.

ID Task Name Schedule

1. Performance and Application of Beam-Nulling 2007/6 - 2007/11

2. Extended Adaptive Frameworks 2007/7 - 2007/12

3. Wrap-ups 2008/1 - 2008/3

4. Thesis Writing 2008/3 - 2008/5

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ID Task Name

2007

Q1

1Performance and Application of

Beam-Nulling

2 Extended Adaptive Frameworks

2008

Q2 Q3 Q4 Q1 Q2

3

4 Thesis Writing

Wrap-ups

Figure 4.1: Gantt chart for the remaining tasks.

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