1

First Year Probationary PhD Report Weisi Guo 01/02/2008

1. Research Outline

1.1 INTRODUCTION

The project I am working on has been awarded to Cambridge and Newcastle

Universities under an EPSRC Grant to investigate the potential benefits of

collaborative wireless networks when applied to Broadband fixed wireless access

(FWA). The report itself will commence with a short introduction, followed by a

literature review and some initial work undertaken on power allocation. Finally, a

section describing current and future work will be presented.

The potential advantages of a FWA network are rapid and low cost wide area

deployment, as well as high bit rates in both line-of-sight (LOS) and non-LOS

scenarios. FWA can deliver a complete range of network traffic, from high speed

multimedia services to telephony. FWA systems often have characteristics of near

LOS propagation between source and destination nodes, which gives rise to slow time

variation and limited frequency selectivity. Previously, research has been conducted

on FWA systems concerning aspects such as dynamic resource allocation,

propagation and throughput performance [23]. More recently, work presented in [24]

has incorporated spatial multiplexing and space time coding with the aim of

improving the throughput and quality of service. However, a persistent challenge

faced by FWA is the ever rising demand for high data rate services through the

wireless medium.

The concept of collaborative networks originated out of work conducted in the area of

Multiple Input Multiple Output (MIMO) antenna structures, whereby the use of

multiple antennas can improve the spectral efficiency through link diversity [1].

Whilst multiple antenna structures can improve capacity through path diversity, the

practical issues of using multiple antennas on a single device have led to the concept

of a distributed multiple antenna system known as a collaborative network, as shown

in Figure 1.1.

Figure 1.1 half duplex collaborative network with a single source, destination and two relays

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Therefore spatially distributed node deployment in a cooperative system potentially

allows path correlation to be reduced and thereby yielding a throughput improvement.

Essentially the multiple antennas are spread out over multiple devices and they

achieve the link diversity by cooperating with each other when transmitting data. The

primary propagation paths tend to be correlated, therefore decreasing the effectiveness

of space/time/frequency diversity.

The objective of this research is to investigate various approaches to collaborative

systems and to apply coding and power allocation schemes with the aim of further

improving the system throughput.

1.2 COLLABORATIVE NETWORKS

Basic collaborative networks involve two users and a destination. There exists an

inter-user (relay) channel between the users, and two uplink channels between the

users and the destination. First we look at the conventional no cooperation network or

otherwise referred to as a direct transmission scheme as shown in Figure 1.2.

• No Cooperation: Each user transmits its data (N blocks) independently to the

destination through uplink channels. Each channel is usually in orthogonal

frequencies employing frequency division multiplexing (FDM).

Figure 1.2 two users in direct transmission

• Cooperation: The nodes in a collaborative network may or may not have their

own information to send. If not, they are effectively acting as relay nodes.

Regardless of whether cooperative nodes have their own information to send,

we can divide collaborative networks into several types, though in general

they all operate in a half-duplex fashion. In the first time frame, the users each

transmit their own data to the desired destination, as with direct transmission

described previously. Owning to the broadcast nature of the medium, in that

same time frame, they also send their information to each other, as shown in

Figure 1.3.

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Figure 1.3 two users in the first stage of cooperative transmission

In the second time frame, they send each other’s data to the destination, as

shown in Figure 1.4. In so doing, they introduce path diversity.

Figure 1.4 two users in the second stage of cooperative transmission

1.3 COLLABORATIVE NETWORK STRUCTURES

There are numerous possible cooperation strategies, for example amplifying and

forwarding the received signals while others involve the application of coding.

Consequently, various tradeoffs in complexity and bit error rate (BER) performance

are available. Some basic cooperation techniques will now be introduced.

• Amplify and Forward (AF): The “amplify and forward” model is based on

sending the transmission data to both the destination and cooperating partners.

When a partner receives the data to be forwarded, it simply amplifies the

signal to negate any inter-user channel fading and sends it to the destination.

In the example in Figure 1.3 and 1.4, it can be seen that the relay transmission

is also delayed and occurs at a different frequency to that which it was

received. The amplify-and-forward scheme will be analyzed in detail in our

subsequent investigation.

• Decode and Forward: The “Decode and Forward” model is also based on

broadcasting the source transmission data to both the destination and

cooperating partners. To obtain reasonable performance it is vital to include

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forward error correction (FEC) coding on the transmitted data. When a

partner receives the data to be forwarded, it requires successful decoding to

take place before sending the data to the destination, either in coded or un-

coded form. “Detect and Forward” is a term often applied when the

transmitted data is not subject to FEC and is simply detected (not decoded) at

the relay before sending to the destination [19].

• Compress and Forward: The idea behind compress and forward in fact can be

weaved into other types of collaborative systems. Essentially the idea is to

only transmit a portion or to compress the relayed data to increase bandwidth

efficiency [19].

• Coded Cooperation: Coded cooperative systems decode the source-relay

transmission before re-encoding it using a different code (code diversity)

before transmission to the destination. The focus of the work in this approach

concerns finding suitable coding schemes and in general the degree of coded

cooperation can be adjusted.

None of the above schemes have specified what type of power allocation they use.

Effectively equal power allocation is usually assumed, where each user allocates the

same power to transmit its own and its partner’s data. Power allocation and coding

for cooperative networks depends on both the channel environment and the objective

function that we are trying to optimise. The aim is to find appropriate methods to

optimise the power allocation and coding in any given condition. In the next section

will look at the previous work regarding collaborative and relay networks.

2. Literature Review

The principle of collaborative networking is relatively new in the sense that the first

publications concerning coded cooperation were published in 2002 by Hunter and

Nosratinia [17]. This came some while later than Alamouti’s practical conventional

MIMO system presented in 1998 [20]. The field can generally be split into five major

areas, each looking different aspects of collaborative networks.

2.1 SYSTEM CAPACITY AND CHANNEL STATE INFORMATION

Optimizing system capacity is often the end goal of much of the research work in

cooperative networks. In the case of collaborative networks, adapting the transmit

power has been shown to be necessary in order to maximize the instantaneous

capacity. Goldsmith et al. [13] proved that Channel State Information (CSI)

availability at the transmitter (required for adaptive power allocation) influences both

transmitter and receiver collaboration to varying degrees. In a given Decode and

Forward collaborative network the received signal Y can be written in the form:

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NHAEXY += , (2.1)

where H is the channel fading matrix, A is the degree of cooperation, N is the additive

noise distributed as N(0, N0), and E is the power of transmitted bits X. It can be then

shown [3] that the instantaneous capacity C is:

)]1

[det(log2

1max

0

2

TTTHAHAEE

NIC += . (2.2)

The absence of CSI at the transmitter in a collaborative relay network will have a

detrimental effect on the capacity. If the CSI is absent, the system can either use

equal power allocation or attempt to allocate power via various other methods. In any

event a node may reserve a potentially un-necessary percentage of its power for

relaying. Blind power allocation has been shown to cause poor performance in

collaborative networks. Ibars et al. [3] showed that without CSI and with a good

signal-to-noise ratio (SNR) in the uplink, it is often better not to cooperate when

transmitting.

Figure 2.1 without CSI, it is better to not cooperate even with good SNR links [3].

Figure 2.1 shows how the network capacity can be degraded as the number of

collaborative users increase, in the absence of CSI for transmit power allocation. It is

also observed that in [3], a node with a high SNR should co-operate more often than

one with a low SNR. Intuitively the better relay channels (those with a lower

attenuation) offer a higher probability of passing on the message without errors than

do channels having a higher attenuation. However, these relay channels can become

saturated if all the data is allocated to low attenuation i.e., high SNR relay paths, after

which it is better to divert the power elsewhere.

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2.2 DIVERSITY AND MULTIPLEXING TRADE-OFF

The diversity and multiplexing trade-off (DMT) is a fundamental consideration in

MIMO systems. A trade-off exists between multiplexing different data to achieve a

high rate, and eventually sending the same data in order to achieve a lower error rate

through spatial diversity. Both have the aim to achieve superior system capacity. In

collaborative networks, the available multiplexing gain is limited by the finite

capacity links between single antenna relay nodes. Erkip et al. [11] state that the

multiplexing gain and the diversity of any coded collaborative system can be written

as:

(2.3)

for a channel rate R, and bit error probability P. So for any coding scheme, this cut

set represents the lower bound of the probability of outage (probability an outage will

occur in a specified time frame), which is an upper bound for the diversity order of

the system (number of independent propagation paths). Simulations presented by

Prasad et al. in [12] show that for a system of M collaborative nodes and N

destinations, the non-clustered diversity order is limited by the term: N+M-1. It is

also shown that for any collaborative network, the diversity order can never exceed

that of a MIMO equivalent due to the introduction of a noisy/imperfect relay channel.

Therefore conventional MIMO systems, for example: the Alamouti [20] based perfect

relay (noiseless relay channel) system; can be seen as an upper performance bound to

a similar collaborative system.

The distributed multiplexing trade off is a popular research area for MIMO systems,

but it is less applicable for a distributed networks due to each user transmitting

independent data.

2.3 RELAY NETWORKS

A relay network can be generalised into a source and destination pair with a number

of relays in between, as shown in Figure 2.2. As before, CSI and power allocation are

important to maximise instantaneous throughput efficiency [13].

Figure 2.2 a single hop relay network

)log(

)(lim)(

SNR

SNRRrgain

SNR ∞→=

)log(

)](log[lim)(

SNR

SNRPddiversity

SNR ∞→−=

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Del Coso et al. [3] have published work on the capacity of high node population relay

networks, whereby a large number of single hop relays are available for cooperation.

Using a relay selection process that maximised the instantaneous network capacity, he

showed how increasing the number of relays available changes network capacity

(Figure 2.3) and how many of those relays are actually utilised (Figure 2.4). The

curves also show the variation due to uplink channel SNR. Figure 2.3 shows how the

network capacity increases with the number of available relays. However, the

increase becomes asymptotic as the relay population increases. The capacity benefit

of further cooperation becomes linear beyond a certain number of relays.

Figure 2.3 – network capacity changes with relay availability [3].

Figure 2.4 shows the optimal number of relays to utilize as the number of available

relays increases. A higher uplink SNR, leads to a much lower number of relays

required to reach optimal cooperation.

Figure 2.4 - number of relays used for optimal performance [3].

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2.4 CODED COLLABORATION

It is important to mention coded cooperation, because it forms a large and important

research area concerning collaborative systems. Collaborative coded cooperation was

first presented by Hunter and Nosratinia [17], where it is shown that both users

experience significant gains in performance when slow fading is considered; this was

then extended to fast fading channels [17]. They also presented results concerning

turbo-coded cooperation [23].

Figure 2.5 - BER performances for No Cooperation, Amplify and forward, Detect and forward and

Coded Cooperation systems [17].

Figure 2.5 demonstrates that at high SNRs, cooperation is better than no cooperation

for a range of channel conditions (to be discussed later). Nosratinia et al. [17] also

show that coded cooperation performs better than schemes using simple

Amplify/Detect and Forward relays.

2.5 POWER ALLOCATION

Power allocation in conventional MIMO systems utilizes a method called

“waterfilling”. In this approach the transmitter allocates more power to the channels

with high SNR, and less or no power to weaker, i.e., low SNR, channels [10]. In

conventional MIMO there exists a power budget constraint:

+

−

=

∑

−=

1

02

011 cN

n nC

budget

h

N

NP

λ, (2.5)

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where NC is the number of sub-channels, N0 is the additive noise power density, and

hn is the fading variance of each channel. Note, x+ means max(x, 0), since power can

be allocated only positively or not at all, i.e. not negatively. From the constraint in

(2.5), the system capacity given by:

∑−

=−

+=

1

0 0

2

1...1logmax

0

C

CNC

N

n

nn

PPN

N

hPC , (2.6)

can be maximized using a Lagrange multiplier (λ) based approach. Effectively, the

1/λ term sets a maximum level for pouring water (power), and N0/|hn|2 is the noisiness

of the channel.

Figure 2.6 an example of waterfilling power allocation scheme for MIMO channels

As Figure 2.6 illustrates, the noisier the channel the less water is poured (power

allocated) into the channel. If the noise level exceeds the threshold level, then no

power is allocated.

Goldsmith et al. [13] showed how transmit power allocation using available CSI is

important to optimise capacity through power allocation schemes such as waterfilling.

Clearly the availability of CSI in a centralised control hub would be the ideal way to

implement power allocation. An alternative to maximising capacity for a centralised

power allocation system could be to minimize the overall BER after receiving each

node’s instantaneous CSI data. However, in an ad-hoc network such a centralised

control system may not be feasible. Adeane et al. [8] explored a distributed power

allocation system that reduces the volume of control data exchange and computational

complexity. A node allocates power based on its partner’s CSI information, and then

relays this decision to its partners. This in turn can fine tune the power allocation

from its neighbours. It is demonstrated that a distributed system can match the

performance of a centralised one after a number of iterations as depicted in Figure 2.7.

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The overall CSI knowledge of non-adjacent nodes can be further improved by

estimating it from the mean SNR.

Figure 2.7 shows that even with only partial CSI, a distributed power allocation

system can approach the performance of a centralised power allocation scheme.

Figure 2.7 BER performance of direct transmission, equal power allocation, distributed power

allocation and centralised power allocation [8].

3. Work Undertaken During First Year

3.1 SYSTEM MODELING

Collaborative networks are commonly modelled as a 2x1 (two source transmitters and

one destination) with half-duplex transmission [1]. This is because transmitter

cooperation is shown to be more beneficial than receiver cooperation when CSI is

available [13]. There have also been papers where a single source and multiple relays

are used (either single hop or multi hop) to forward the data in an AF fashion.

In our case, we have created two models in Matlab to simulate a collaborative

network, as shown in Figure 3.1. A simple duplex 2x1 AF model and a more

elaborate multiple relay AF system are used for all the initial work on power

allocation and relay selection. The latter relay network is merely an extension of the

existing 2x1 system, into an Nx1 system. The former is tested with basic parameters

against text book and published results to validate correct operation. It is also worth

noting that we use a duplex system that had adapted its power so that it performs the

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same as standard half-duplex systems. This is so that we can allocate power

adaptively and still compare our performance fairly with other half duplex-systems.

SR

RD

SD

SR

RD

SD

Figure 3.1 – 2x1 AF system model with a source transmitter, a relay and a transmitter

Specifically, the 2x1 AF model compromises of a single source (S), an AF relay (R),

and a destination (D). The transmitted signal is a stream of random bits modulated

using a Binary Phase Shift Keying (BPSK) scheme. Slow Rayleigh fading gain is

assumed for the source to destination uplink labelled F and is modelled as an

independently identically distributed circularly complex Gaussian N(0, f2), where f

2 is

the fading variance. Similarly the source to relay channel fading gain is labelled G,

and the relay to destination is labelled H, as seen in Figure 3.1. The additive noise

(zSD, zSR, and zRD) for each respective channel is distributed as N(0, N0).

3.2 INVESTIGATING POWER ALLOCATION

3.2.1 Background

A great deal of current research on various aspects of collaborative networks assumes

an equal allocation of power at the transmitter between source data and relayed data.

To date research concerning power allocation can be found indirectly in work

conducted to obtain BER expressions for AF systems. BER expressions for both

equal and maximum ratio combining 2x1 systems with a perfect relay channel can be

found in Tse [10], and an approximate one has been found by Ribeiro et al. [2] for a

noisy relay collaborative Nx1 network. A channel with transmit energy per bit (E),

fading variance (f2), and noise variance N0/2, has a SNR that can be defined as:

0

2

N

Ef=γ . (3.1)

Tse et al. [10] uses the combined receive SNR (each channel’s SNR combined at the

destination) in statistically similar channels to produce an exact BER expression for a

perfect relay 2x1 (extended also to Nx1) system, where γ is the average channel SNR

(since all channels are assumed to have the same SNR). The bit error probability of a

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2x1 system for equal gain combining (EGC) and maximum ratio combining (MRC)

are given respectively by:

+−−=

2)1(

111

2

1

γEGCP , (3.2)

2

11

2

1

12

+−

++=

γ

γ

γ

γMRCP . (3.3)

As one can observe, the bit error probability is a function of a single variable, namely

the average received SNR γ, since the channels are pre-conditioned to be statistically

identical, therefore leaving no scope for realistic systems with different channel

fading and adapting power accordingly. Ribeiro [2] on the other hand produced an

expression with more scope for power allocation purposes, albeit an approximation,

for Nx1 systems with unequal channels and noisy relays. In particular he showed that:

RDSDSR

AFk

Pγγγ

111

*4

32

+= , (3.4)

where k=2 for binary phase-shift keying (BPSK) modulation and each of the channels

has its own average SNR (γ): source – relay (γSR), source – destination (γSD) and relay

– destination (γRD). Note that the quality of each channel is described by a different

SNR value. Whilst Ribeiro does not propose a power allocation scheme, if we

differentiate (3.4), it offers a potential power allocation strategy. Independently we

have determined our own expression for calculating BER and in the next section we

will go onto show how this expression can also provide a power allocation scheme,

namely the “biased theoretical scheme”.

3.2.2 Initial Work on BER Expression and Power Allocation

Our approach to this problem utilises an expression for BER that has been derived

using a slightly different method of system analysis. Traditionally, BER expressions

involve the error function having received SNR as part of the argument. In our

approach, we consider the exact error probability of a given bit of information. This

approach yields a BER expression for a 1x1 system, which is identical to the Tse’s

exact BER result:

+−=

γ

γ

11

2

111xP . (3.5)

Please refer to Appendix 1 for the detailed derivation of this equation. Using the

same idea with an approximation during integration, an expression for the BER of a

2x1 system has also been derived:

13

++

−+

−

+−=

RDSD

RDSDRDSD

RD

SD

SD

eP

γγ

γγγγ

γ

γ

γ

1

11

2

1

11

2

1. (3.6)

This result can be seen in figure 3.2 and plotted for comparison are appropriate text

book exact BER plots and Ribeiro’s result. This equation (3.6) enables power

allocation problem to be addressed in the presence of different channel fading

conditions.

Figure 3.2 – 2x1 AF BER: Tse’s Text Book Equal and MRC, Ribeiro and our Biased BER

Figure 3.2 shows the exact BER performance plots obtained from (3.2) and (3.3).

Also plotted are Ribeiro’s approximate BER results and the BER result for a system

using our “biased theoretical” BER expression (3.6). All systems assume a noiseless

relay AF structure. As we can see the exact BER curves for EGC and MRC are

parallel, whilst our biased estimation eventually converges towards them. Due to

approximations in our derivation, it can be seen that our BER is not accurate at low

SNRs. However, it has the advantage of being closer to the exact BER curve than

Ribeiro’s expression, whilst having the scope to express the system in different

channel fading conditions, thus allowing further work concerning power allocation.

3.2.3 Deriving Power Allocation

From the BER expression given in (3.6), it can be seen that we have separate SNR

terms for each individual channel. We introduce the term α which is a power

allocation factor that ranges in value from 0 to 1. Therefore in the equal power

allocation scheme, α = 0.5 for both channels. We define:

,0

2

N

EfSD

αγ = and ,

)1(

0

2

N

EhSD

αγ

−= (3.7)

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where f2 and h

2 are the fading variances for source-destination and relay-destination

channels respectively, and N0 is the noise variance. Differentiation of the BER

expression (3.6) with respect to α gives three possible power allocation factors, in

particular, when:

• )/( hff +=α , power is allocated based on the ratio of the fading gains of each

uplink channel.

• α = 1, the source allocates all its power to transmit to the destination, because

the relay-to-destination path has a low SNR (i.e., h = 0).

• α = 0, the source allocates all its power to transmit to the relay, because the

source-to-destination path has a low SNR (i.e., f = 0).

What we have produced is a statistical power allocation scheme that is updated over

time. We refer to it as Parameter Biased AF. This scheme can be implemented

practically, where a time-varying power allocation is slowly updated based on

previous channel conditions. If we replace the statistical parameters with the

instantaneous fading coefficients of the channels and include the correction factor β

(to compensate for the fading G between the source and relay channel) we obtain:

GHF

F

βα

+= . (3.8)

This approach, which we call Instantaneous Biased AF or AF Lower Bound, requires

a constant updating of the power allocation factor for each transmitted block of data.

Although more complex, this scheme is optimal in terms of bit error rate. Therefore,

its performance can be used as a lower bound on the bit error rate of various AF

schemes and is plotted in Figure 3.3.

Figure 3.3 – comparing equal power allocation, parameter variance based power allocation and

instantaneous power allocation (AF lower bound)

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Figure 3.3 shows the performance difference between collaborative networks

employing biased-AF with statistical parameters and biased-AF with instantaneous

coefficients. It is shown, whilst biased-AF with statistical parameters offers a better

performance than equal power allocation, biased-AF with instantaneous coefficients

achieves a markedly better performance than both. This can be seen as a lower bound

to the defined AF scenario. Having found a power allocation method that gives

improvements over an equal power allocation system, we go on to compare it with

Ribeiro’s system, had he proposed a power allocation scheme based on his BER

expression of (3.4). To this end we differentiated his expression (3.4) and in a similar

manner to that described previously yielded the results shown in figure 3.4.

Figure 3.4 – power allocation comparison of no cooperation, equal power AF, Ribeiro power AF and

our own biased power AF lower bound

Having found a power allocation method that gives improvements over an equal

power allocation system, we go on to compare it with Ribeiro’s system, had he

proposed a power allocation scheme based on his BER expression of (3.4). To this

end we differentiated his expression (3.4) and in a similar manner to that described

previously yielded the results shown in Figure 3.4. The non-cooperative case is

compared with the previous AF power allocation strategies. As expected, the equal

power allocation and Ribeiro et al. [2] error minimization scheme performs better

than no cooperation. However, our proposed scheme, achieves a further performance

gain of ~2.5dB over equal power allocation at a bit error rate of 10-3

.

3.2.4 Discrepancies and Further Power Allocation Work

As we have just described, the work carried out yielded a possible approach for power

allocation based on the approximate BER expression presented in (3.6). However as

noted previously, this model doesn’t quite match the exact BER curve under similar

channel conditions (as seen previously in Fig 3.2), and unfortunately we were unable

to explain the reasons for the differences, especially at low SNR values. At high SNR

and equal channel conditions, when γSD = γRD= γ, our BER expression (3.6) reduces to:

16

++−=

121

4

12

2

γγ

γEGCP , (3.9)

whereas the exact expression (3.2) reduces to:

++

+−=

12

21

2

12

2

γγ

γγEGCP . (3.10)

Owing to these differences we have decided to abandon power allocation based on

expression (3.6) and to turn our attention to obtaining power allocation schemes

through the use of Moment Generating Functions (MGF). This approach follows that

of Alouini et al. [21] in their work on relay systems and will be presented in more

detail in section 4.1.

3.3 INVESTIGATING RELAY SELECTION

The work done in power allocation was extended to address relay selection and

management because in a realistic system, we are likely to have multiple partners

acting as potential relays. The system model now has a single source transmitter and

a number of relays scattered randomly. This is achieved by assigning random source

to relay channel fading (i.e., SNR values). All relays are single hop AF nodes, as

shown in Figure 3.5. Every relay is potentially a collaborative partner for the source;

however, some without a doubt will have better channels than others. In a situation

where we have N potential relay partners, the question is which relays to use and how

much power should be allocated to them given a fixed power budget. Potentially all

the relays can be used with various power allocation factors. Some relays will be

extremely poor and yield negative benefits to the overall system if used. A threshold

value Ω is introduced to dissuade using all the relays in a selfish manner, thus it

determines how many potential relays to cooperate with, and can be seen as a “relay

usage cost”.

Figure 3.5 – randomly scattered one hop relays

Source

Destination

ith

Relay

G

H

F

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The relay selection solution can adopt a criterion similar to that used previously in

power allocation (equation 3.8) that is based upon the channel fading coefficients:

Ωβ

β≥

+∑ =

N

i iii

kkk

HGF

HG

0

. (3.11)

The numerator βkGkHk is the fading gain of the kth

relay channel while the

denominator is the sum of all possible data routes to the transmitter. A high cost will

select relays with a strong fading path, whereas a low cost will allow a larger variety

of relays to be incorporated. Simulation is performed with a single source, a single

destination and N relays. The source to destination channel is set to an SNR range of

0-40 dB to allow comparison with [3]. We keep the previous system model and use a

constant power budget for the entire system. To simulate randomly located relays,

their channels’ fading variance is randomly distributed between ranges which depend

on the number of relays. Consequently as we increase the number of available relays

we are simulating a larger cooperative cell.

As the number of available relays is increased, we expect the overall collaborative

system to perform better. As we vary the relay usage cost Ω, we affect the number of

relays utilized for cooperation. In particular, at zero cost (Ω=0) all relays are allowed

to be used for cooperation; however at a heavy cost (Ω>>0) relaying will not be

allowed, so we will revert to direct transmission, whilst an optimized cost will

minimize the bit error rate by using a percentage of the available relays. Figure 3.6.

shows the BER improvement as we increase the number of available relays. It also

demonstrates how adjusting the relay usage cost can change the BER. A relay usage

of 70% achieves a superior performance to that of full relay usage. However, further

reduction to 40% sees a worse BER performance than for 70% relay usage.

Consequently, we conclude that there is a tradeoff between the number of relays

utilized and the performance.

Figure 3.6 – BER performance with increasing relays

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We go one step further and investigate the effects that the relay usage cost has on a

given set of relays with random fading values. As we increase the value of Ω, we can

see a drop in the number of relays utilized, as shown in Figure 3.7. This leads to an

elimination of the relays with weak fading paths and usage of only those relays with

strong paths. Biased power allocation is then given to these relays under a fixed

budget.

Figure 3.7 – threshold effect on BER and percentage of relays utilized

For a given threshold, there exists an optimum number of relays to use. For example

in Figure 3.7, it can be seen that the best BER performance is achieved at a threshold

value of about 3 to 4. This in turn relates to using approximately 35% to 20% of the

10 total available relays.

Figure 3.8– percentage of relays used as number of available relays increase

As shown in Figure 3.8: when we increase the number of relays that percentage falls.

Allocating biased power to the chosen relays gives a better performance especially in

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a relatively small relay node population. This is consistent with the observation in [3],

where the number of optimum relays, which maximize capacity, is about 30%, at an

SNR of 0dB and a relay population of about 10.

4. Project Plan and Future Work

4.1 POWER ALLOCATION

A key element in our intended thesis is determining how to allocate power in

collaborative network having non-equal channel conditions subject to various

optimisation criteria, specifically to maximise capacity or to minimize BER.

Previously, we obtained our own expression for the probability of a bit error in order

to determine how to allocate power in non-equal channel conditions. This led to a

power allocation scheme that gave better performance in the range of SNRs we

investigated. Owning to the discrepancies between our BER expression and the exact

expression, we have decided to pursue other possible other approaches to investigate

power allocation on statistically dissimilar channels.

Alouini et al. [21] have already shown that by using an MGF based approach, BER

expressions for single hop non-cooperative relay systems can be found:

++

+=

)1)(1(2

1

21

21

γγ

γγeP , (4.1)

where γ1 is the source – relay SNR, and γ2 is the relay – destination SNR. This

approach can perhaps be extended to our collaborative networks, with the goal of

producing a power allocation system.

The collaborative 2x1 system’s BER can be expressed as:

∫= γγγ dpPPe )()( , (4.2)

where P(γ) is the probability of error in additive noise, and p(γ) is the PDF of the

combined SNR, γ, of the two uplink channels at the destination. Note that each uplink

channel’s SNR, follows a Chi-square probability distribution. In order to find the

PDF of the receive SNR, we must combine together two independent Chi-Squared

distributed random variables (i.e., γSD and γRD) having different variances, i.e.,

RDSD γγγ += . (4.3)

20

We can potentially use MGFs to find the distribution of p(γ). The MGF of p(γ) is the

product of the MGF of γSD and γRD. The sum of any number of independent

distributions is the product of their Fourier transforms (MGF):

∏=

Ψ=ΨN

i

i

0

. (4.4)

Another line of investigation would be to apply a waterfilling-based power allocation

method to collaborative networks by including the relay path. Firstly a capacity

expression must be found in order to determine the Lagrange multiplier (water level)

as described by (2.5). After this has been done, the work can be extended to include

work done by Verdu et al. [14] on a power allocation scheme suited to all modulation

schemes.

4.2 NOISY RELAY CHANNEL AND PATH LOSS MODELING

So far the work has focused on a simplified version of a 2x1 collaborative network

whereby the relay channel is assumed to be noiseless or perfect. Currently the BER

equation devised will become less accurate as the relay channel noise increases. In

reality, relays will be noisy and power allocation needs to be changed accordingly to

accommodate the fading and the noise of a relay channel.

We also need to improve the simulation model in order that it can more accurately

represent real systems. Consequently accurate models of propagation path loss will

be included in the simulation. In the first instance a simple path loss will be

calculated using the formula:

CdnL += )(log10 10, (4.5)

where L is the path loss in decibels, n is the path loss exponent, d is the distance

between the transmitter and the destination, usually measured in meters, and C is the

path loss at a defined reference distance. The path loss exponent is environment

dependent, for example it has a value of 2 for free space and 4 for propagation over a

flat reflecting earth. Further models for path loss can be considered for certain

environments and terrains such as the Stanford University Interim (SUI) 3 model,

which is useful for rural area modeling. We will then see what the resulting impact is

on our relay network and compare the results with those obtained in [3].

4.3 COMBING POWER ALLOCATION WITH CODING AND CHANNEL ACCESS SCHEMES

The goal of the project is to eventually combine the individual efforts of project

members. Novel ideas in channel coding, access schemes and power allocation need

to be integrated into a system architecture that is coherent and realistic for FWA.

21

Coding is an additional layer of cooperation in collaborative networks which is being

investigated by team members in both Newcastle and Cambridge. Turbo, LDPC (low

density parity check) and other codes are being actively investigated with positive

results.

We propose to integrate the existing collaboration structure with coding and a channel

access scheme to create a complete system model. Numerous channel access schemes

are possible such as TDMA (time division multiple access), FDMA (frequency

division multiple access), SDMA (space division multiple access) or CDMA (code

division multiple access) by Sendonaris et al. [9]. The general half duplex model [1]

is a form of TDMA-FDMA access scheme.

All these are yet to be investigated and their combined effects seen, this is proposed

for the later stages of the project. Performance comparisons will be made between

potential systems and information theoretic bounds.

4.4 OTHER ISSUES

Wireless channels suffer frequency selective fading, and even in FWA where

selectivity is mild, the problem is yet to be fully addressed in a collaborative network.

The use of orthogonal frequency division multiplexing (OFDM) is an area worth

investigating to tackle this problem.

Power allocation needs to be practically assigned in either a centralised or a

distributed manner. How to effectively allocate power in a distributed node network

and pass on CSI accordingly needs to be investigated perhaps based on the work done

in [8].

For uplink cooperation, where the control of a user is independent and selfish,

cooperation can be difficult to encourage. Whether nodes wish to cooperate for

overall system benefit but personal loss is an important question to consider. Game

theory is an area which may provide answers into tackling such a problem. The

addition of hidden penalties and incentives can persuade selfish nodes to join the

cooperative system.

Further work includes possibly extending the network topology to mesh or multi-hop

relays, which would considerably complicate the fading and coding model. Creating

hybrid MIMO networks with the addition of extra antennas on each collaborative

node is also another area to consider, given that FWA is unlikely to employ small

mobile devices. This would give rise to additional spatial diversity and potentially

further improve performance.

22

5. Conclusion

The goal of our research is to investigate the benefits of wireless collaborative

networks over conventional systems and establish ways of realistically producing

such gains through power allocation, coding and protocol design.

In the literature review, we found that the main areas of research are in coding,

diversity and multiplexing trade-offs; relay selection, capacity optimisation and power

allocation. The majority of existing work is focused in coding and capacity modelling.

From the latter, relay selection strategies and power allocation schemes have been

devised. Most of the original work assumed equal channel conditions, which have

yielded exact BER expressions. When work was done on un-equal channel

conditions, they have produced expressions approximating the BER performance.

We looked at the problem of how to allocate power effectively in un-equal channel

conditions. We first tackled the problem from an approach based on expressions for

the exact bit error probability. This work yielded interesting results and power

allocation methods. However, the error model was not accurate at low SNR values,

but did converge to the correct result at high SNR. We are currently trying to tackle

the same problem utilising moment generating functions. We aim to obtain a more

solid power allocation frame work, upon which we can build further work.

Our work will eventually be extended to address larger relay networks that

incorporate various more realistic path loss models. In the future it is important to

integrate the power allocation framework with coding and modulation schemes to

devise an overall system model. Problems such as frequency selective fading and

user cooperation willingness are also possible areas for investigation in cooperative

networks.

REFERENCES

[1] Laneman J., Wornell G., “Cooperative Diversity in Wireless Networks: Efficient protocols and

outage behaviour” IEEE Transactions on Information Theory, vol. 50, pp. 3062-3080, Dec 2004

[2] Ribeiro A., Cai X., Giannakis G.B., “Symbol Error Probabilities for general Cooperative Links”,

IEEE Transactions on Wireless Communications vol. 4, no. 3, May 2005

[3] del Coso A., Ibars C., “Capacity of Decode and Forward Cooperative Links with Full Channel

State Information” Proceedings of 39th

Asilomar Conference on Signals, Systems and Computers,

pp. 1514-1518, Nov 2005

[4] Proakis J.G., Digital Communications, McGraw-Hill New York, 1989.

[5] Croft A., Mathematics for Engineers, Prentice-Hall, 2003.

[6] Cover T.M., El Gamal A.A, “Capacity Theorems for the Relay Channel”, IEEE Transactions on

Information Theory, vol.25, no. 5, pp 572-584, September 1979

23

[7] Laneman J., Wornell G., “Cooperative Diversity in Wireless Networks; Efficient Protocols and

Outage Behaviour”, IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062-3080,

Dec. 2004.

[8] Adeane J., Rodrigues M., Wassell I., “Optimum Power Allocation in Cooperative Networks”,

Proceedings of the Postgraduate Research Conference in Electronics, Apr. 2005

[9] Sendonaris A., Erkip E., Aazhang B., “User Cooperation Diversity Part II. Implementation

aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp.

1939-1948, Nov. 2003

[10] Tse D., Viswanath E., “Fundamentals of Wireless Communication”, Cambridge University Press,

2005.

[11] Erkip E, Yuksel M, “Diversity and Multiplexing Tradeoffs in Cooperative Wireless Systems”,

Proceedings of 40th

Annual Conference on Information and Sciences, pp. 1062-1067, Mar. 2006.

[12] Prasad N & Varanasi MK, “Diversity and Multiplexing Trade-off Bounds for Cooperative

Diversity Protocols”, International Symposium on Information Theory, pp.268-274, Oct. 2004.

[13] Goldsmith AJ and Ng CTK, “Capacity Gain from Transmitter and Receiver Cooperation”,

International Symposium on Information Theory, Sept. pp. 397-401, 2005.

[14] Verdu S, “Optimum Power Allocation for Parallel Gaussian Channels with Arbitrary Inputs”,

IEEE Transactions on Information Theory, vol. 7, pp. 3033-3051, July 2006.

[15] Wittneben A, Kuhn M & Hammerstrom I, “Impact of Relay Gain Allocation on the Performance

of Cooperative Diversity Networks”, IEEE Vehicular Technology Conference, vol. 3, pp. 1815-

1819, Sept. 2004.

[16] Hunter T. E., Nosratinia A., “Cooperation Diversity through Coding”, International Symposium

on Information Theory, pp. 220, July 2002.

[17] Nosratinia A., Hunter T. E., “Coded Cooperation under Slow Fading, Fast Fading and Power

Control”, Proceedings of 36th Asilomar Conference on Signals, Systems and Computers, vol. 1,

pp. 118-121, Nov. 2002

[18] Zhao B., Valenti M. C., “Distributed Turbo Coded Diversity for Relay Channel”, Electronics

Letters, pp. 786-787, Mar. 2003

[19] Hunter T. E., Nosratinia A., “Performance Analysis of Coded Cooperation Diversity”, IEEE

International Conference on Communications, vol. 4, pp. 2688-2692, May 2003

[20] Alamouti S.M, “A simple transmit diversity system for wireless communications”, IEEE

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[21] Hasna M, Alouini SM., “Performance Analysis of Two Hop Relayed Transmissions over Rayleigh

Fading Channels”, IEEE Vehicular Technology Conference, vol. 4, pp. 1992-1996, May 2002.

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vol. 53, pp. 494-503, Sep 2005

24

APPENDIX 1 – PROBABILITY ANALYSIS OF BER AND POWER ALLOCATION

Figure A.1 shows a 2x1 AF system model. The following equations give the received

signal at the destination for a 1x1 direct transmission and an equal gain combining

2x1 systems, where Y is the received signal, X is the transmitted signal, F and H are

Rayleigh fading gains distributed as N(0, f2) and N(0, h

2) respectively, and Z is the

additive noise for a respective channel distributed as N(0, N0):

1x1 System: ZFXY += , (1)

2x1 System: RDSD ZZHXFXY +++= . (2)

SR

RD

SD

SR

RD

SD

Figure A.1 – 2x1 AF system model with a source transmitter, a relay and a transmitter

When transmitting a BPSK signal, we transmit either a positive or a negative symbol

bit. Each symbol is faded (by a random positive multiplicative factor having a

Rayleigh probability distribution function) and then additive white Gaussian noise

(AWGN) added to it. Therefore, errors can only occur when enough counter-

productive noise is added to override the initial signal’s magnitude combined with the

fading magnitude. Therefore the probability of an error is:

)1()1|1()1()1|1( −=−=+=++=+=−== XPXYPXPXYPPe . (3)

Given the fact that there is an equal number of positive and negative bits transmitted,

P(X = +1) = P(X = -1) = 0.5, and since the noise is symmetrically distributed, we

only need to evaluate one of the possibilities of error, hence:

)1|1()1|1(2

1)1|1(

2

1+=−==−=+=++=−== XYPXYPXYPPe . (4)

Consequently, the probability of obtaining an incorrect decoding is given by:

25

)()|()1|1( noisePnoiseerrorPXYPPe =+=−== . (5)

The probability distribution of the noise is simply the negative part of the Gaussian

noise that can possibly produce an error on a positive symbol (i.e. X = +1). The

probability of error given this negative noise depends only on the fading of the

channel for our 1x1 direct transmission example:

dZdFf

FXP

Zz

fF

e ∫∫ ∞−

−−

−=0

2/

0

2/

2

2222

exp2

1exp σ

πσ. (6)

We are only concerned with the case when the fading is insufficiently great and the

noise sufficiently high to produce an error, P(error | noise). Therefore, the limit FX

comes from the inequality when Y<0:

zFX −< . (7)

This gives us the exact BER expression for a 1x1 system, which is identical to that in

text books [10]:

+−=

γ

γ

11

2

111xP . (8)

Using the same idea with an approximation during integration, an expression for the

BER of a 2x1 system has been derived:

++

−+

−

+−=

RDSD

RDSDRDSD

RD

SD

SD

eP

γγ

γγγγ

γ

γ

γ

1

11

11

2

1. (9)

The equation has been plotted in the main text of the report (Figure 3.2) and also

compared against exact BER plots. This expression offers the ability to handle

different channel fading conditions and so allow power allocation expression to be

developed. Unfortunately, as explained in the main text, it displays deviations from

the expected BER results at low SNR values.