MAC LAYER (WP4)

Internal report no.2

MAC layer: resource sharing principles and algorithms

Authors: Andrea Baiocchi, Fabrizio Capriotti, Francesca Cuomo, Cristina Martello

Date: 18-04-2001

MAC LAYER (WP4)

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Contents 1. Introduction ..................................................................................................................................2

2. Evaluation of the Signal-to-Noise Ratio ......................................................................................3

2.1 Introduction ..........................................................................................................................3

2.2 The Signal-to-Noise Ratio in the centralized scenario.........................................................3

2.3 The Signal-to-Noise Ratio in the distributed scenario .........................................................8

2.4 Interference from RF signals................................................................................................9

2.5 The adopted full-duplex mode ...........................................................................................12

2.6 Comparison among different variable bit rate allocation schemes ....................................13

3. Resource sharing ........................................................................................................................16

3.1 Resource sharing approach.................................................................................................18

3.2 Reservation Based traffic ...................................................................................................21

4. Simulations and performance results .........................................................................................27

5. References ..................................................................................................................................28

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1. Introduction

This document aims at laying out a modelling framework and a quantitative approach to the

resource sharing issue in an UWB air interface, where the main issues are: i) multiplexing of best

effort traffic with reservation based traffic with QoS (in terms of delay); ii) distributed algorithms.

Therefore, we assume that traffic offered to the UWB air interface (MAC layer) falls into two

categories: i) Reservation Based (RB) traffic; ii) Best Effort (BE) traffic.

The RB traffic serves those services requiring some guarantees as to throughput (hence transfer

delay). As seen by the MAC entities, the terminal to be addressed (among those that are

“connected”, i.e. reachable) and the bit rate of the MAC layer communication to be set up are given

as input, while the decision variables are the power levels to be used under the constraints on the

maximum transmittable average power and the communication quality (BER).

The BE traffic offers a transfer resource that adapts itself to the available bandwidth at a given

time and in a given point in space. The aim here is to achieve the maximum possible overall

throughput, possibly with fairness constraints, but no guarantees are offered as to achievable

throughput and transfer delay.

The resource sharing among the UWB terminodes requires both signalling and local

measurements.

The rest of the document is structured as follows.

Section 2 summarizes with some detail the derivation of the bit error probability for the UWB air

interface under the fundamental hypothesis of ideal channel with Gaussian additive noise (AWGN

channel). Moreover, unsynchronised randomly coded time hopping sequences are assumed for the

co-existing communications.

Section 3 lays down the principles of the resource sharing starting out from the expression of the

SNR and of the BER derived in the previous Section. A theoretical framework is assessed, and hints

are given as to how a distributed algorithm can exploit the theoretical results.

Section 4 presents a distributed algorithm based on the approach outlined in the previous

Section, by specifying signalling flows for RB and BE traffic.

Section 5 reports the analysis of the proposed algorithms based on simulations.

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2. Evaluation of the Signal-to-Noise Ratio

2.1 Introduction

In this section the UWB transmission technique performance are evaluated in terms of Signal-to

Noise Ratio and some strategies of managing and controlling the UWB parameters are considered

in order to explain how a variable throughput can be obtained. In 2.2 the Signal-to-Noise Ratio in a

centralized scenario is derived, while in 2.3 this result is generalized to a distributed context. The

section 2.4 provides an analysis of the effect of a RF signal in the UWB signal band. In the 2.5 a

full-duplex mode is defined and the relative impact on performance is analysed. Finally section 2.6

provide a comparison among different schemes for bit rate reconfiguration.

2.2 The Signal-to-Noise Ratio in the centralized scenario

We consider the case of multiple users transmitting to the same receiver, typically named Base

Station, by using a particular UWB transmission technique named Impulse Radio: the transmitted

signal is a sequence of very short-time (a nanosecond) pulses, named monocycles, whose

bandwidth is wide (a GigaHerz) [xx].

The transmitted signal relative to a single user k when time-hopping codes are used and without

modulation, is:

∑+∞

−∞=−−=

jc

kjfk

k TcjTtgAts )()( )()( (2.1)

where g(t) represents the monocycle, a normalized in amplitude base-band pulse not modulated by

any carrier and as for the used parameters their means are explained in the list below:

Ak is the pulse amplitude;

Tf is the pulse repetition period without the time-hopping code;

cj(k): is the j-th chip of the time-hopping code, it assumes values in the countable set [0, 1, 2, …,

Nh] and it determines the time shift of the relative pulse within a time interval lasting Tf so that it

does not overlap with the successive pulse;

Tc is the minimum time shift due to the time-hopping code.

A possible choice for g(t) is the Gaussian pulse (see Figure 1):

−

−=

22

2exp41)(mm

tttg

τπ

τπ (2.2)

where τm is a time constant that determines the pulse time duration.

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1 2 3 4 5 6 7 8 9

10-12

10-10

10-8

10-6

10-4

10

G(f)

f (GHz)0 1 2 3 4 5 6 7 8

-0.5

0

0.5

1

g(t)

t (ns)1 2 3 4 5 6 7 8 9

10-12

10-10

10-8

10-6

10-4

10

G(f)

f (GHz)1 2 3 4 5 6 7 8 9

10-12

10-10

10-8

10-6

10-4

101 2 3 4 5 6 7 8 9

10-12

10-10

10-8

10-6

10-4

10

G(f)

f (GHz)0 1 2 3 4 5 6 7 8

-0.5

0

0.5

1

g(t)

t (ns)0 1 2 3 4 5 6 7 8

-0.5

0

0.5

1

0 1 2 3 4 5 6 7 8-0.5

0

0.5

1

g(t)

t (ns)

Figure 1 – The Gaussian pulse both in the time and frequency domain

The time-hopping code is used in order to allow the contemporary access of multiple users; in fact

if the pulse sequence were periodic, two different signals could collide in a catastrophic way in an

asynchronous scenario, i.e. every pulse of a sequence overlapped in time with the relative pulse of

the other sequence. The time-hopping codes are periodic sequences with period Np (c(k)j=c(k)

j+Np)

and, in order to avoid two successive pulses to overlap, the relation 0≤ c(k)jTc≤ Tf must be satisfied.

The used modulation is PPM (Pulse Position Modulation) and is applied to the time-hopping coded

pulses sequence by introducing a time shift depending on the symbol to be transmitted. Referring to

a binary modulation, a sequence of NS not shifted pulses is transmitted for the symbol 0, while the

symbol 1 is transmitted by a sequence of NS pulses shifted of δ. The resulting expression of the

transmitted signal is:

∑+∞

−∞=−−−=

jS

kc

kjfk

k NjdTcjTtgAts /(()( )()()( δ (2.3)

fSk TNR /1= (2.4)

where:

d(k)(i) is the i-th symbol of the k-th user

Rk is the k-th user data-rate.

As far as the receiver analysis is concerned, first we consider the case of a single user (user 1) and

an AWGN channel model (the multipath is not considered). In this context the received signal can

be written as:

)()()( 1)1(

11 tntsgAtr +−= τ (2.5)

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where:

g1 is the path gain from the transmitter to the receiver;

τ1 is the transmission delay.

The received signal relative to the i-th bit is r(t) considered in the time window within which that bit

is transmitted, that is:

)()()( 1)1(

11 tntsgAtr +−= τ with [ ]{ }fSfSi TNiTiNt 1)1(, 11 −+++=∈ τττ (2.6)

Another way to write the received signal relative to the i-th bit is by mean of the normalized

transmitted signal relative to the i-th bit:

)t(n)t(WgA)t(r bit += 11 with ∑−+

=−−−=

1)1(

1)1( )()(

s

S

Ni

iNjcjfbit TcjTtgtW τ (2.7)

In presence of an AWGN, the optimum receiver is a correlator with a correlation template signal

v(t) (see Figure 2):

)()()( δ−−= tgtgtv (2.8)

∑−+

=−−=−−−=

1)1(

1)1( )()()()(

s

s

Ni

iNjbitbitcjfbit tWtWTcjTtvtV δτ (2.9)

v(t)

0 0.2 0.4 0.6 0.8

0

0.5

1

1.5

-1.5

-1

-0.5

0.9

t (ns)

v(t)

0 0.2 0.4 0.6 0.8

0

0.5

1

1.5

-1.5

-1

-0.5

0.90 0.2 0.4 0.6 0.8

0

0.5

1

1.5

-1.5

-1

-0.5

0.9

t (ns)

Figure 2 – The correlator receiver

At the receiver a bit is decoded by correlating the received signal with the correlation template v(t).

Ideally, considering perfect synchronism with the received signal due to a perfect detection of the

time position pulse and assuming that a v(t)-pulse sequence according to the used time-hopping

code is generated at the receiver without errors, we have that:

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- if a symbol 0 was transmitted, then each of the Ns pulses is multiplied for the positive part of

v(t), so that a positive value after correlation is obtained;

- if a symbol 1 was transmitted, then the correlation is negative since the received pulses are

located in the negative part of v(t).

Thus the receiver makes Ns correlations per bit: given the result αj of each correlation, α represents

the variable on whose value the receiver bases the decision of the transmitted value:

∫++

+=

−−−=f

f

Tj

jTtcjfj dtTcjTtvtr

)1()1(

1

1

1

)()(

τ

τ

τα (2.10)

∑−+

⋅==

1)1( s

S

Ni

Nijjαα (2.11)

The Signal-to-Noise Ratio in presence of thermal noise only must be calculated after the correlator

as the ratio between the amplitude m of the desired signal risen to the second power and the noise

power. As for m, it represents the correlation value between the transmitted pulse and v(t):

pSt

bitbit mNgAdttVtWgAm

i

∫∈

==τ

1111 )()( (2.12)

dttvtgm p )()(∫= (2.13)

The noise power in output at the correlator is given by:

[ ]∑ ∫−

=

+∞

∞−

==1

00

22 )(2

SN

jSp

orec NmNdttv

Nσ (2.14)

Thus the Signal-to-Noise Ratio (where the multi access interference is not considered):

0

121

221

21

)1(N

NgA

NmN

mNgASNR S

Spo

pSOUT == (2.15)

Now we generalized the SNR expression to the case of multi-access interference. The received

signal is:

)()()()()( 1)1(

11)(

1

tNtsgAtntsgAtr totkk

kk

N

k

U

+−=+−= ∑=

ττ (2.16)

∑=

+−=uN

kk

kkktot tntsgAtN

2

)( )()()( τ (2.17)

In order to characterize Nout(t) some hypotheses can be stated:

1) the different users emitted signals are asynchronous and it results:

(τk -τ1)mod Tf ∈[0,Tf) (2.18)

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2) the time-hopping codes are pseudorandom, i.e. the code chips are statistically independent

random variables within a periodic interval lasting Np chips uniformly distributed in the set

[0,1…Nh], moreover different user select a code in a random and independent way;

3) during a bit transmission time the code chip are statistically independent due to the condition

Np>NS, where Ns is the number of chips per bit.

The hypotheses stated above allow to characterize the multi-access interference as a Gaussian white

stochastic process when the number of users is high; the mean value is zero because of the zero

average value of the transmitted pulses, due to the statistic independence of the single terms in

Nout(t), the variance is equal to the sum of the variance of the single terms themselves. Thus we

have:

{ } 0)( =tNE tot (2.19)

{ }∑=

+=UN

kkkkrectot nEgA

2

2222 )(σσ (2.20)

∫∈

−=it

bitkk

k dttVtsnτ

τ )()()( (2.21)

The term nk represents the k-th user interference and is computed in output at the user 1’s correlator

receiver. The interference is due to the asynchronism among different users signals and to the

pseudorandom codes, (as a consequence different users pulses can collide in time). The power of

the interference term is proportional to the number NS of the integrated pulses and it results:

{ } 22)( aSk NnE σ= with f

a T

dsdttvstg

2

2

)()(∫ ∫∞+

∞−

∞+

∞−

−

=σ (2.22)

Thus the Signal-to-Noise Ratio in presence of multiaccess interference in a centralized context and

assuming the same data rate for each user is:

∑=

+==

UN

kkkaSrec

ps

utotuOUT

gAN

)mNgA(

)N(

m)N(SNR

2

222

211

2

2

σσσ (2.23)

∑=

+=

uN

k

kk

pS

a

out

uOUT

gA

gA

mN)(SNR

)N(SNR

2 121

2

2

2

1

1

1

σ (2.24)

The specific case of perfect power control can be obtained from (2.24) as below:

)1()1(

1

1)(

2

2

−+

=

UpS

a

out

uOUT

NmNSNR

NSNRσ

(2.25)

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from which it can be shown that a performance degradation is experienced when the users number

NU increases or the number of pulses per bit NS decreases, which is equivalent to a decreasing bit

rate (see (2.4)).

In particular the maximum number of contemporary users can be determined with a given bit rate

and a desired SNRSPEC. In order to do that we define ∆P as the extra power – with respect to the case

of single user - that each user must transmit in order to reach the desired SNRSPEC when there are NU

–1 interference users:

SPEC

OUT

SNR

SNRP

)1(log10 10=∆ (2.26)

From (2.25) and (2.26) the number of users can be expressed as:

( ){ } 1101)( 10/2

2

+

−=∆ ∆− P

SPECa

pSU

SNR

mNPN

σ (2.27)

{ })10/(2

2

101)1(

)( P

Uaf

p

NT

mPR ∆−−

−=∆

σ (2.28)

When ∆P becomes infinite, the maximum number of contemporary users is obtained

12

2

+

=−

SPECa

pSMAXU

SNR

mNN

σ (2.29)

)1(2

2

−=

Uaf

pMAX

NT

mR

σ (2.30)

The bit error probability Pe can be obtained as a function of the SNR when the hypothesis of

Gaussian white stochastic process is assumed for the multiaccess interference process:

∫∞

−=

=

x

uUOUTe due)x(erfc,

)N(SNRerfcP

22

22

1

π (2.31)

2.3 The Signal-to-Noise Ratio in the distributed scenario

In the distributed context – as in an ad-hoc network – there does not exist a centralized node to

which the traffic is directed. (Each radio terminal must be considered as a terminode because it acts

both as a terminal and as a node.)

With respect to the generic link from the transmitter node i to the receiver one j the SNR can be

written as:

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ijSjilm imjm

mjmjmjmlapo

ijijSpijij

NgAgAmN

gNmASNR

−≠ ≠

−

++

=

∑ ∑),(),( ),,(

222

2)(

σ

(2.32)

ijfjilm imjm

mjmjmjmlapo

ijpij

RTgAgAmN

gmA

++

=

∑ ∑≠ ≠),(),( ),,(

222

2)(

σ

(2.33)

The expression (2.32) takes into account all the incoming transmissions to the node j (see the pairs

(m,j), m≠i) and all the transmissions directing to other nodes l≠j and contributing to the multi-access

interference level at the node j (see (m,l)≠(i,j).).

Thus the success in establishing a link depends on the values of the following quantities:

1) the multi-access interference level at the receiver;

2) the distance between the transmitter and receiver and the radio channel conditions

determining the path gain gij;

3) the desired bit rate of the link to be established Rij.

The (2.32) takes into account the thermal noise and the multi-access interference but does not

consider the extra interference due to other radio systems sharing the same (or part of) band of the

considered UWB system.

In the two following sections the interference from other systems and the SNR degradation for the

adopted full-duplex mode are studied so that a general expression for the SNR can be finally

obtained for the distributed case

2.4 Interference from RF signals

Referring to the distributed case, we consider the presence of other radio systems interfering signal

within the UWB system band. In a general way we can model it as a sinusoidal tone due to its very

narrow band with respect to the ultra wide band:

( ) )2( θπ += tfsinAtR or (2.34)

where the phase θ is a random variable uniformly distributed between 0 and 2π

In order to evaluate the SNR degradation due to a sinusoidal signal we can not assume the Gaussian

white model for the total interference process and thus the optimum receiver is not an adapted filter

any more; however we still consider it as receiver.

The output of the receiver due to the considered sinusoidal tone in a time interval lasting as a bit is:

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∑ ∑ ∫∫−

=

−

=

++++

+− ++=−−=

1

0

1

0 0

)()( )()()()(

)1(

)(

Sij Sij WWcjf

ci

jf

N

j

N

j

T

cijf

TTcjT

TcjT

fcijijr dttvTcjTtRdtjTTctvtRS

δδ

[ ]∑ ∫−

=

+

−−++=1

0 0

)( )()()(Sij wN

j

T

cijf dttgtgTcjTtR

δ

δ (2.35)

where TW is the monocycle duration.

From the (2.35) it can be obtained:

[ ] +

+++= ∑ ∫

−

=−

1

0 0

2ijNs

j

Tw)i(

jforijr dt)TccjTt(fsin)t(gAS θπ

[ ] =

++++− ∑ ∫

−

=

1

0 0

)( )(2)(Sij WN

j

T

cijfor dtTcjTtfsintgA θδπ

∑ ∫=

++++−=

Sij WN

J

T

cijfoor TcjTtftgfsinA

0 0

)( ))222(cos()()(2 θδπδπ (2.36)

We assume that the time shift due to the code is continuous rather than discrete; then we consider

interfering tones with To<Tf, and we define the random variable θj uniformly distributed in [0, 2π)

according to:

( ) jcjfofoo

o TcjTfTTT

f θθπ =++⇒<= 22,1

(2.37)

It can be noticed that the condition To<Tf restricts the set of sinusoidal tones under study to those

ones whose frequency – for example - is grater than 10 MHz when Tf is equal to 100 ns.

The random variable θj uniformly distributed in [0, 2π) means that for the pulses corresponding to

different code chips the interfering tone phases result random independent variables and as a

consequence the interference terms relative to the code sequence pulses do not sum in phase each

other. Thus the interference term can be rewritten as:

∑ ∑−

=

−

=− −=++−=

1

0

1

0

)(2))2(cos()()(2ij ijNs

j

Ns

jrjorjoorijr nfsinAtftgfsinAS δπθδπδπ (2.38)

and the interference power can now be obtained:

[ ]

=

=

==

∑ ∑

∑−

=

−

=Θ

−

=Θ−Θ−

1 1

0

22

21

0

222

)(4

)(4

ij ij

ij

Ns

ol

Ns

mrmrlor

Ns

krkorijrijr

nnEfsinA

nEfsinASEI

δπ

δπ

(2.39)

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Due to the hypotheses assumed for the random variables θj the terms with different indexes l and m

does not contribute to the sum since they are equal to zero as shown below:

[ ]

∫∫

∫ ∫

=++++=

=

++++=Θ

W

l

W

m

lm

T

lomo

T

Tw

o

Tw

lomorlrm

dttfEdttfE

dttfdttfEnnE

00

0

0))2(cos())2(cos(

))2(cos())2(cos(

θδπθδπ

θδπθδπ

θθ

θθ

(2.40)

Thus the (2.39) can be rewritten as follows:

imprjij

Tw

oijr

Ns

krkorijr

INsdttftgENsfsinA

nEfsinAIij

−

−

=Θ−

=

++=

=

=

∫

∑2

00

22

1

0

222

))2(cos()()(4

)(4

θδπδπ

δπ

θ

(2.41)

From (2.41) it can be noticed that the interference level due to a sinusoidal tone is proportional to

the number NS of pulses per bit while the desired term is proportional to NS risen to the second

power. So the quantity NS represents a processing gain even in contending with RF interference.

When we consider several interfering sinusoidal tones at different frequencies we have:

∑=

+=Nrj

kkkrk tfsinAtR

1

)2()( ϑπ (2.42)

where θk are statistically independent random variables uniformly distributed in [0, 2π).

By a similar proceeding with respect to that in the previous case and thanks to the linearity of the

receiver, the interference term results:

∑ ∑

∑∑

=

−

=−

−

=−

=−

−=

=++−=

rj Sij

ij

N

k

N

lkrlkrk

Ns

lklkkkr

Nrj

kijr

nfsinA

tftgfsinAS

1

1

0

1

01

)(2

))2(cos()()(2

δπ

θδπδπ (2.43)

From (2.43) the interference power is:

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[ ]

∑ ∫

∑ ∑

∑ ∑

=

=

−

=−Θ

=

−

=−Θ−Θ−

++=

=

=

=

==

Nrj

k

Tw

kkrkij

Nrj

K

Ns

lkrlkrk

Nrj

k

Ns

lkrlkrkijrijr

dttftgEfsinANs

nfsinAE

nfsinAESEI

ij

ij

1

2

0

22

1

1

0

222

2

1

1

0

2

))2(cos()()(4

)(4

)(4

θδπδπ

δπ

δπ

θ

(2.44)

Thus the obtained SNR is:

( )( ) ( )ijfimprj

jilm imjmmjmjmjmlapo

pijijij

RTIgAgAmN

mgASNR

+

++

=

−≠ ≠

∑ ∑., ,,

222

22

σ

(2.45)

2.5 The adopted full-duplex mode

We consider a node j with several both out coming and incoming established links. Let nTj be the

number of contemporary out coming links. The relative transmissions are synchronous, moreover

we assume that they use orthogonal codes, while the multicode mode is examined in the next

section.

As for the received signals, they must be considered asynchronous both each other both with respect

to the node j transmissions, since they income from different users.

The adopted full-duplex mode consists in turn off the receiver while transmitting as long as the

pulse plus the time shift of the PPM lasts. In this way the error causes due to contemporary

transmission and reception are avoided, in fact since an error would occur if a transmitted pulse

were decoded as a received one, the receiver must turned off during the transmission. Of course that

has an impact on the SNR in terms of a degradation of its. In order to explain that the Figure 3

shows the case of an incoming pulse lost due to the turned off receiver. For sake of simplicity the

case of synchronism in the TC but not in the Tf is shown.

Tx node j

Rx node i

“LOST” PULSE

Tf

Tx node j

Rx node i

“LOST” PULSE

Tf

Figure 3 - The case of an incoming pulse lost due to the turned off receiver

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In order to evaluate the SNR degradation due to the described full-duplex scheme, we must consider

that the decision variable α is obtained from NS transmitted pulses whereas a number NSij < NS is

received: the variables NSij result random variables statistically independent and equidistributed, so

that for the decision variable α it can be written:

ijh

TjNs

jj

Ns

jj Ns

N

nE

ijij

)1(11

−=

⇒ ∑∑

==αα (2.46)

From (2.45) and (2.46) the SNR in the considered general case of a distributed network, of the

described full-duplex mode and of both multi-access and RF signal interference, is:

( )( ) ( )

)1(

., ,,

222

22

h

tj

ijfimprjjilm imjm

mjmjmjmlapo

pijijij N

n

RTIgAgAmN

mgASNR −

+

++

=

−≠ ≠

∑ ∑σ

(2.47)

2.6 Comparison among different variable bit rate allocation schemes

The aim of this section is evaluating different variable bit rate allocation schemes. Thus we consider

the centralized case when several users are transmits to a Base Station with different bit rate; then

from (2.23) the SNR of the k-th user is:

∑≠=

⋅+

=UN

kjjjjapo

Skkpkuk

gAmN

NgmANSNR

,1

22

2)()(

σ (2.48)

Let suppose the user 1’s bit rate must be reconfigured and in particular – but without loss of

generality – it must be increased by an integer factor n1. There are three ways to do that:

1) by decreasing by the factor n1 the number of pulses per bit;

2) by using a number of n1 pseudorandom codes (even with the previous NS);

3) by using a number of n1 codes (even with the previous NS) orthogonal and synchronized

each other.

Let consider the first way.

In this case the SNR relative to the user 1 is:

1

12

22

112*

11

)()( γ

σ

=

+

=

∑=

ngAmN

NgmANSNR

UN

jjjapO

Spu (2.49)

In order to obtain the same previous value of the SNR, the user 1 must increase the transmitting

power by a factor n1 so that it interferes with the other users more. The level power is chosen

according to:

MAC LAYER (WP4)

14

211

2*1 )()( AnA = (2.50)

1

21

2

221

21)(

Sp

N

jjjapo

Nmg

gAmN

A

U

+

=∑=

σγ

(2.51)

UN

kjjjjapo

Skkpkuk Nk

AngAmN

NgmANSNR

U

,2,)(

)(

,2

211

22

2

=

++

=

∑≠=

σ

(2.52)

Now we consider the second case of using n1 random codes, so that the transmitted signal is:

∑∑=

−− −−−=1

11

)1()1(1

)1( ))/(()(n

i jS

ic

ijf NjdTcjTtgAts δ (2.53)

This multi-rate mode corresponds to assign n1 channels each of bit rate 1/NS1Tf; since the codes are

pseudorandom, they interfere each other, thus resulting in a degradation of the SNR that must be

compensated by increasing the transmitting power. Below the expressions for the SNR and for the

needed transmitting power are given:

1

12

211

22

112

11

)1(

)()( γ

σ

=

−++

=

∑=

−

gAngAmN

NgmANSNR

UN

jjjapo

Spui (2.54)

[ ]2*

1211

21

112

12

1

2

221

21 )()(,

)1()( AAnA

nNmg

gAmN

A RMCaSp

N

jjjapo

U

>=−−

+

= −=∑

γσ

σγ

(2.55)

The needed power results higher with respect the previous case (see (2.50)) because of the mutual

interference of the n1 codes.

The idea of the third case is to assign a set of n1 codes chosen such that they do not interfere each

other (i.e. in an orthogonal set and by using a code time shift at least equal to Tc =TW +δ ). In this

case we obtain the same SNR expression of (2.53), here obtained with the assumption given below:

imNjcc pm

ji

j ≠=≠ −− ,...1,)1()1( (2.56)

1

2

22

112

11

)()( γ

σ

=

+

=

∑=

−UN

jjjapO

Spui

gAmN

NgmANSNR (2.57)

2*1

211

21 )()()( AAnA OMC ==− (2.58)

MAC LAYER (WP4)

15

From (2.58) it can be noticed that the node 1 transmitting power is equal to that of the first case. As

for the interference on other users, we must compare the one due to the transmission of a single

code to the one due to the transmission of n1 codes at a power level less by a factor n1.

We consider signals synchronous with respect to the code word but not with respect to the single

chip. In the single code case the collision probability with other users pulses is 1/Nh and the

transmitting power is (A1)2n1; in the multi-orthogonal codes case the collision probability increases

by a factor n1 while the transmitting power decreases by the same factor. As a consequence the

interference caused to other users is the same in the two cases.

In conclusion the multi-code mode can be used in order to support links with different SNR

requirements or to establish link with different users, so that the general expression of the

transmitted signal is:

∑ ∑=

−−−

− −−−=KN

i jiSk

ikc

ikjfiK

k NjdTcjTtgAts1

)()()( ))/(()( δ (2.59)

)max( iSkpk NN −> (2.60)

where the (2.60) represents a necessary condition to obtain that a chip occurs only one time within a

bit period in each of the Nk used channels.

MAC LAYER (WP4)

16

3. Resource sharing

By assuming an AWGN channel model with perfect timing, the SNR of the communication

between a transmitter i and a receiver j is

SNRij =Ai

2mp2N s,ijgij

N0mp + IRF , j + σa2

Ak2gkj

k =1k≠ i, j

NTX

∑1 −

n j

Nh

j =1,…, NRX (3.1)

where NRX and NTX are the number of receivers and transmitters respectively. The index i of the

transmitter is a function of j, at least as long as we consider unicast transmissions.

For the remaining of this Section we make the following simplification: we assume that a

receiving terminal has no simultaneous transmissions. Thus, also each transmitter does not have any

reception going on and we can label the transmitter-receiver pair of each communication with the

same index, say i. Let N = NTX = NRX be the number of such pairs.

The expression of SNR can be simplified also from a notational point of view, by introducing the

transmission power. Let

Pi =1

TfA

i2W 2 (t )dt

0

Tf

∫ =1

TfA

i2EW (3.2)

with

EW = W 2 (t )dt−∞

∞

∫

mp = W(t ) W (t ) −W (t − δ)[ ]dt−∞

∞

∫

σa2 = 1

TfW(t − s) W( t ) −W (t − δ)[ ]dt

−∞

∞

∫

2

ds−∞

∞

∫

(3.3)

Note that EW, mp and σa all have the same dimension (energy, e.g. measured in Volt2.s; to have

Joule we have to apply the signal to a load).

Then the SNR in (3.1) can be rewritten as

SNRi =Pigii

Ri ηi + Tf σ2 Pkgkik =1,k ≠i

N

∑

i =1,…, N (3.4)

where we let the following definitions

Tf the pulse repetition time interval;

Ri the binary bit rate of the i-th communication;

Pi the average power emitted by the i-th transmitter;

gij the path gain from the i-th transmitter to the j-th receiver;

MAC LAYER (WP4)

17

ηi the background noise energy; with the model in (3.1), this is N0 + IRF , j mp( )EW m p ;

σ2 an adimensional parameter equal to σ

a2 m

p2 .

Typical values of the above parameters are as follows [3]: the pulse duration is 0.75 ns and Tf =

100 ns; σ2 = 1.9966e–3; in case of thermal background noise, ηi = 2.568e-21 V2.s. Note that powers

are measured in V2.

We can also write (3.4) by using the energy per bit instead of the power level at the transmitter.

Let Ebi be the energy per bit used by the i-th transmitter; then

SNRi =Ebigii

ηi + Tf σ2 Rk Ebk gkik =1,k≠ i

N

∑=

Ebigii

ηi + σ2 Ebk Ns,kgkik=1,k ≠i

N

∑i =1,…, N (3.5)

With the same hypotheses leading to (3.4) and assuming further that UWB self-interference can

be modelled as a Gaussian process, the probability of error per bit can be calculated leading to:

Pe(SNR) = Q SNR( ), Q( x) =1

2πe−u2 / 2du

x

∞

∫ (3.6)

The denominator of the SNR in (3.4) and (3.5) highlights the two sources of noise: internal

UWB self-interference and external noise and interference (e.g. RF). The first contribution arises

because random codes are assumed (no orthogonality): it can be controlled by means of admission

control, possibly working differently for a reservation based traffic and for a best effort traffic.

Along with admission control, transmission level measurements and adaptation of power

levels/rates are crucial to the working of the UWB terminals. External interference and noise are to

be taken as given and not modifiable.

The minimum allowable energy per bit to contrast external interference can be obtained by the

requirement SNRi ��γi, by letting the internal UWB interference tend to 0; so

Ebi ≥ Ebmin,i =

ηiγ i

g ii (3.7)

By accounting for the upper limit on the power level at the transmitter available for data

transmission (not signalling), Pmax, we get an upper limit for the bit rate achievable by the i-th

communication:

Ri ≤

Pi

Ebi≤

Pmaxgii

ηiγ i≡ Rmax,i (3.8)

In the rest of this Section we deal with power levels and bit rates as if they were continuously

variable quantities. They are not. Power is usually adjusted in fixed steps (e.g. by 1 dB step). As for

bit rates, remind that Ri = 1/(NsTf); Tf being fixed, the bit rate can be varied by either adjusting Ns,

which is an integer, (variable “spreading factor” code) or by using more than one code per

communication (multicode). The first solution suffers from very gross quantization for high rates.

The second one requires several impulse shifting chains, orthogonalization of the codes of a same

user, and distribution of the overall transmitted power among the codes.

The rest of this Section aims at laying out a sort of “theoretical approach” to the resource sharing

MAC LAYER (WP4)

18

issue. To make it work, at least in a world where AWGN channel model can be assumed (!), many

aspects need be investigated. Section 3.1 lays out the guidelines of the resource sharing approach.

Section 3.2 deals with resource sharing for best effort traffic (BE traffic). Section 3.3 is devoted to

reservation based (RB) traffic. Section 3.4 draws some conclusions and outlines the lines of action

to develop a practical approach. Throughout this report mobility is marginally considered. Its full

impact should be investigated further. However, power/bit rate adaptation mechanism here defined

should be carried over rather straightforwardly.

3.1 Resource sharing approach

In general, a basic requirement of the physical layer is to offer communication links with an error

probability no greater than a given threshold value. This in turn means that the SNR of a

communication link be no less than a specified threshold γ. Moreover, a power limiting constraint

adds usually either with respect to the average value of the power level or to the peak power. We

consider the former one in the following.

Therefore, from (3.4) we infer that the power levels and bit rates should meet the following

constraints

Pigii − Ri γ iTf σ2 Pkgkik =1,k≠ i

N

∑ ≥ ηi Ri γ i i =1,… , N

0 < Pi ≤ Pmax Ri > 0

(3.9)

The assignment problem splits into two subproblems depending on the required service

guarantees. In case of RB MAC flows, the target value of Ri comes as a requirement from the upper

layer; hence the point is to check if a feasible power level can be set in all transmitters so that the

required bit rates be supported. This in turn is equivalent to the existence of a solution to the

inequalities system (3.9) with a given set of values of the bit rates.

In case of best effort traffic, the problem as stated in (3.9) is undetermined, since the bit rates can

take in principle any positive value. It is clear from the form of (3.9) that a feasible solution for the

power levels always exists, provided that the bit rates are sufficiently small.

Then, we can pose an optimisation problem by stating a function of the Pi‘s and the Ri‘s and

requiring that this function be maximized, under the constraints (3.9). An obvious target is to

maximized the overall rate of the active communications. To account for binary errors, one could

maximize the “capacity” of the ongoing communications. This leads to the following target

function:

H (r,p) = Ri ⋅C Pe(SNRi )( )

i=1

N

∑ , C( p) =1 + p log2 ( p) + (1− p) log2 (1 − p) (3.10)

If the constraints in the first line of (3.9) were all satisfied with the equality, the above function

becomes just a linear combination of the bit rates. If all threshold values γi are equal to a same value

γ, the target function (3.10) can be restated as just the sum of the bit rates. In general, we can find

the value of Ri such that Ri ⋅C(Pe(SNRi )) = Ri ⋅C(Q( ai Ri )) is maximized, with SNRi = ai/Ri. It is

MAC LAYER (WP4)

19

easy to see that this function has a unique maximum for Ri ranging in (0,���� ���� �� �� �� ���

solving the equation ϕ(u)=uϕ’(u), where ϕ(u) = C(Q( u )). Let the unique positive solution be

denoted by u*. Then the optimum Ri is Ri,opt = ai/u* and the target function (3.10) becomes

H (r,p) ≡ H (p) = Ri,opt ⋅C Pe u*( )

i =1

N

∑ =C Pe u*( )

u*

Pi gii

ηi + Tf σ2Pkgki

k =1,k≠ i

N

∑⋅

i =1

N

∑ (3.11)

A modified form of the target function (3.10) represents the sum of capacities per unit of energy

spent by each transmitting terminal. Then, we might want to maximize:

G(r,p) =Ri ⋅C Pe(SNRi )( )

Ebii=1

N

∑ =R

i2

PiC Pe Pigii Ri ηi + Tf σ2 Pk gki

k=1,k ≠i

N

∑

i=1

N

∑ (3.12)

With either function (3.10) or (3.12) we can proceed in two ways. If one requires that SNRi = γi

(with equality), the bit rates can be easily found out from these constraints and plugged into the

target function expression, so that it becomes only a function of the power levels. Alternatively, one

can choose the optimum rate for each communication, by assuming all power levels are fixed. The

existence of these optimal rates follows from the form of the bit error probability function and from

the target function expressions. This last approach does not constrain the SNR in any way.

The above formulation highlights a form of duality between RB and BE traffic. In the former

case, power levels are adjusted (optimised) for given values of the bit rates. In the latter case, we

optimise the bit rates and calculate the power level thereof, based on the SNR constraint.

Summing up, the RB traffic problem can be stated as follows

given r > 0, find p such that

Pigii − Ri γ iTf σ2 Pkgkik =1,k≠ i

N

∑ ≥ ηi Ri γ i i =1,… , N

0 < Pi ≤ Pmax

(3.13)

Let

��

M =

g11 γ1σ2Tf R1( ) −g12 � −g1N

−g21 g22 γ2σ2T f R2( )� −g2N

� � � �

−gN 1 −g N 2 � gNN γ N σ2Tf RN( )

(3.14)

The matrix M is not symmetric in general, since gij is the path gain from the i-th transmitter to

the j-th receiver. The above problem can be compactly stated in matrix form as

pM ≥ h ≡1

Tf σ2η1,…,ηN[ ] (3.15)

A solution exists (actually with equality in each of the first N inequalities of (3.13)) iff the matrix

M (with positive diagonal entries and nonpositive off-diagonal entries) is such that the spectral

MAC LAYER (WP4)

20

radius of (D–M)D–1 is less than 1, where D is the diagonal part of M. Remind that the spectral

radius of a matrix is the maximum of the modulus of its eigenvalues.

A sufficient condition for M–1 to exist and to be a non negative matrix, is that the matrix M be

strictly (or irreducibly) diagonally dominant. As an example, dominance by columns means that at

the i-th receiver it must be

gii > γ i σ2Ns,i gki

k=1,k≠i

N

∑ i =1,…, N (3.16)

The difference between the left and the right hand sides can be taken as a definition of the i-th

receiver margin. Conversely, this could be used as a criterion to select an intermediate terminode in

a multihop path, i.e. by choosing the terminode i that maximizes the above margin. This is because

the spectral radius of a matrix is upper bounded by any matrix norm. So, we have

ρ (D − M)D−1( )≤ (D − M)D−1

1= max

1≤ i≤N

γ iσ2 Ns,i

giigki

k =1,k≠ i

N

∑

(3.17)

The amount by which the rightmost side of (3.17) is less than 1 represents a margin, since it

upper bounds the rate of convergence of the power levels from their initial values to the optimum

values, according to

p(n +1) = p(n)(D − M)D−1 + hD−1, n ≥ 0 (3.18)

Let us assume that the solution of the problem (3.15) exists. Since M–1 is a non negative matrix,

the power vector is a monotonously increasing function of the right-hand side vector. Hence we can

evaluate the maximum additional interference that can be tolerated by each communication

(receiver) under that maximum average power constraint. If e denotes a row vector of 1’s, the

maximum additional interference a is found as

Pmaxe = hM−1 + aM−1 ⇒ a = PmaxeM − h (3.19)

Provided that no more than that interference is added to any of the receiver, it can safely make

the transmitter adjust its power (and hence trigger power adjustments in other communications),

without violating the maximum power constraint. The “power margin” of the i-th transmitter is

∆Pi =Pmax

γ i Rigii − γ iσ

2 Ns,i gkik =1,k ≠i

N

∑

− ηi (3.20)

This is the amount of additional interference that can be loaded to the i-th receiver, without

violating the SNR and the maximum average power constraints.

As for BE traffic with SNR constraints, the problem formulation can be as follows:

MAC LAYER (WP4)

21

maximize H (p) =C Pe max{u* ,γ i }( )

max{u*,γ i}

Pi gii

ηi + Tf σ2 Pk gkik=1,k ≠i

N

∑i =1

N

∑

0 ≤ Pi ≤ Pmax i =1,… , N

Ri = 1

max{u*, γ i}

Pi gii

ηi + Tf σ 2 Pkgkik=1,k ≠i

N

∑i =1,…, N

(3.21)

Under the SNR constraint in the first line of (3.21), the assignment problem for BE traffic

reduces to the maximization of a function of the vector P in the hypercube [0,Pmax]N.

These results generalize quite straightforwardly to the case of multi transmission/reception for

each terminal. As a matter of fact, the reduction of the SNR indicated by (3.1) can be easily

incorporated into the constant γi. Moreover, the practically interesting cases are those with one or

two transmission going on at the same time a reception is active (bi-directional link or two bi-

directional link, when a terminode is acting as a relay). So the reduction of the SNR is generally

negligible (Nh is in the order of 100).

A major point requiring an in-depth analysis is the impact of terminal mobility. This essentially

implies a variability of the path gain (provided the assumed AWGN model still holds!). This in turn

entails that power levels (for RB traffic) and bit rates (for BE traffic) should be adjusted

accordingly. In the former case, the constraints (3.13) could become no more sustainable. In that

case, the forced termination of some communication is necessary (or possibly a modification from

RB to BE of some traffic).

As stated above, the assignment problems are “centralized” optimisation problems. To make

them practical in a distributed UWB environment, a possibly approximate approach should be taken

based on local measurements and signalling, with a trade-off between signalling load and accuracy.

3.2 Reservation Based traffic

The basic principle a practical solution of the problem (3.13) can be based on is to define

margins. To this end, each pair of terminals wishing to establish a RB connection, will choose a

power level such that the SNR constraint is met with a margin of β dB.

We address the issue of a new UWB terminal pair wishing to start its communication. Let this

pair be labelled with 0. We assume that the constraints (3.13) are met for the on-going N RB traffic

communications. The rate R0 associated to the new connection is constrained by the QoS

requirements (e.g. limited transfer delay for real time services, time deadlines for given blocks of

data). The power level P0 has to be chosen so that

P0g00

η0 + Tf σ2 Pk gk 0k =1

N

∑= ξ0γ 0 (3.22)

MAC LAYER (WP4)

22

where ξ0 is a margin and η0 includes the UWB self interference due to BE traffic.

A number of constraint must be met: i) ξ0 must be not less than 1; ii) the interference due to the

new RB connection on the on-going RB connections must be limited within their respective SNR

safety margins, denoted by ∆k for the k-th receiving terminal (for a formal definition of the margin

see later on); iii) the power level P0 cannot exceed Pmax.

The SNR safety margin is so defined: let the SNR of the j-th terminal pair be equal to ξjγj. Then,

the margin is the amount of additional UWB interference that can be tolerated by maintaining that

the SNR be no less than γj. Formally,

Pjg jj

Rj η j +U j( )= ξ jγ j andPj g jj

R j η j +U j + ∆ j( )= γ j (3.23)

From (3.23) it is easily derived that:

∆ j =Pj g jj

γ j Rj1 −

1

ξ j

(3.24)

The constraints on (3.22) above can be so formulated

P0 =ξ0γ 0R0 η0 +U0( )

g00≤ min

1≤ j≤N

∆ j

T f σ2g0 j

≡ Pallowed

1 ≤ ξ0 ≤ β

P0 =ξ0γ 0R0 η0 +U0( )

g00≤ Pmax

(3.25)

If there exist a ξ0 such that all inequalities in (3.25) hold, then the new connection can be

admitted, otherwise it is rejected. In the former case, ξ0 is set according to:

ξ0 = min β,g00Pmax

γ0 R0 η0 +U 0( ),g00Pallowed

γ 0 R0 η0 +U0( )

≥1 (3.26)

and the margin of the 0-th receiving terminal can be computed by means of (3.24).

To reduce the signalling load due to the update of the margin values as the terminals move

and/or the radio channel propagation changes the SNR and hence the adopted margins, we could

consider a range of acceptable values of the SNR instead of a specific value γ.

3.3. Best Effort traffic

First, we address the issue of a new UWB terminal pair starting its communication. Let this pair

be labelled with 0. In this case, the terms ηi include all the external (non UWB) interference and

background noise at the receiver, plus the UWB interference of communications having higher

priority of best effort traffic (i.e. reserved bandwidth communications).

Then, we require the following constraints to be satisfied (and we assume that they are with Eb0

= 0):

MAC LAYER (WP4)

23

Ebigii

ηi + Tf σ2 Rk Ebk gkik =1,k≠ i

N

∑ + Tf σ2 R0Eb0g0i

≥ γ i i = 1,…, N

Eb0g00

η0 + Tf σ2 RkEbk gk 0k =1

N

∑≥ γ 0

(3.27)

In (3.27) the bit rates and energy per bit of each communication except the 0-th one are already

fixed. The point is what are the feasible values of the new communication. For a positive bit rate to

be allowed for the new communications it is necessary to either reconfigure the assignable

quantities of the other communications or to have set them so that the SNR constraints are satisfied

with a margin. If the latter is the case, we have to answer the question as to which is the most

appropriate value of the margin for the active communications.

To illustrate an example, let us assume only best effort traffic and let us take a TCP-like

approach to congestion control. The concept is as follows: a terminal pair wishing to initialise a

communication, start transmitting at a rate depending on the interference level measured by the

receiver and on the path gain from the transmitter to the receiver. Each other pair of communicating

terminals (i.e. already active communications) senses a possibly increased interference level and

adjusts its rate consistently, so as to maintain a target SNR value. The point to answer is whether it

is convenient that the new communication be activated. This depends obviously on the objective

function to be optimised. To this end, we choose the problem formulation as in (3.21).

Specific to this type of optimisation there is the step-by-step approach: given that N

communications are active, what are the best rate and power level for a new (N+1)-th

communication?

Let us assume that N communications involving N pairs of terminals are active with values of Pi

and Ri chosen so as to maximize the objective function and be compatible with the following

constraints

SNRi =Pigii

Ri ηi + Tf σ2 Pkgkik=1,k ≠i

N

∑

=Pi gii

Ri Ii= γ i i =1,… , N

Pi ≤ Pmax

(3.28)

When the new communication is activated, the above constraints become as follows:

MAC LAYER (WP4)

24

SNRi =Pi gii

Ri* ηi + Tf σ2 Pkgki

k =1,k≠ i

N

∑ + Tf σ 2P0g0i

=

= Pi gii

Ri* Ii + T f σ2 γ 0I0

g00g0i R0

= Pigii

Ri* I i + ci R0( )= γ i i =1,…, N

SNR0 =P0g00

R0 η0 + Tf σ2Pk gk0

k =1

N

∑

=P0g00

R0I0= γ0

Pi ≤ Pmax i = 0,1,… , N

(3.29)

where the rates of the already on going communications have been adjusted to the new value Ri* to

maintain the same SNR as before the 0-th communication started. Moreover, the notation

ci = Tf σ2 γ 0 I0 g00( )g0i is used.

The value of P0 is fixed by the second of (3.29). There remains to choose the rate R0. The

objective functions can be as follows

G( R0 ) = R0 f (SNR0 )Eb0

+R

i* f (SNRi )

Ebi*

i =1

N

∑ =

= R0g00

γ 0 I0f (γ 0 ) + Ri

Iigii

γ i Ii + ci R0( )2f (γ i )

i =1

N

∑

H ( R0 ) = R0 f (SNR0 ) + Ri* f (SNRi )

i =1

N

∑ = R0 f (γ0 ) + RiI i

Ii + ci R0f (γ i )

i=1

N

∑

R0 ∈ 0, R0max[ ], R0max = Pmaxg00

γ 0 I0

(3.30)

In 3.12 we use the notation f(.) = C(Pe(.)).

Since they lead to the same result, we confine ourselves to the second one. As a minor

simplification (but trivial modifications only would be needed to generalize) we assume that all

SNR target values are equal to the same γ. Then, we can redefine the second objective function in

(3.30) by cancelling the common positive factor f(γ), thus considering the function (with a slight

abuse of notation)

H ( R0 ) = R0 + RiIi

Ii + ci R0i =1

N

∑

R0 ∈ 0, R0max[ ], R0max = Pmaxg00

γI0

(3.31)

It can be easily checked that the second derivative of this function is always positive, hence it is

convex; the first derivative at R0 = 0 can be of either sign. In any case, the maximum of H(R0) is

achieved at the border of the interval where R0 varies, i.e. Hmax = max{H(0), H(R0)}.

The meaning of this result is as follows. If H(R0) > H(0), the new communication can increase

MAC LAYER (WP4)

25

the overall net throughput of the air interface, and its best rate is the maximum one, compatible with

the power constraint. This implies that the transmitter should use the maximum power level. If the

reverse is true, the new communication should not be initialised, since the benefit it brings about is

outweighted by the rate penalty induced in all other affected communications. The condition

H(R0max) > H(0) is easily checked out to be equivalent to

ci Ri

Ii + ci R0maxi =1

N

∑ <1 (3.32)

This is intuitively appealing, since the term ci contains the path gain from the 0-th transmitter to

the i-th receiver, hence a measure of how much “disturbing” the new communication is to the

already established i-th one.

As for the possibility to measure the quantities in (3.32), we can write ciR0max = diPmax. The i-th

MS can measure both Ii and Ii+ciR0max hence, by knowledge of Pmax, it can evaluate di. If the ratios

diRi/(Ii+ciR0max) are signalled to the 0-th terminal, after it probes the air by sending a signal at Pmax,

that terminal can decide whether its communication can actually start by comparing the sum of

those ratios to the threshold g00/(γ0I0). This comparison can be carried out by the receiving terminal.

This requires a pair of terminals wishing to initialise a communication to cooperate, so that the

transmitting terminal sends a sort of paging signal with the identity of the receiving terminal at a

power level Pmax. Then, all the terminals acting as the receiving side of an already active

communication will start measuring a modified level of interference and finally will signal back the

ratio diRi/(Ii+ciR0max) to the terminal paged by the initial signalling. So, the receiver of the 0-th pair

can verify the inequality in (3.32) and decide whether the communication can actually take place.

The optimisation of the overall net rate with a step-by-step approach (i.e., as new

communications come up) turns out to translate into a dyadic choice: either give up to transmitting,

or transmit at peak power (hence the maximum allowed rate). That peak power transmission be

optimal is not intuitive in a self-interference prone environment. However, it must be stressed that a

one step optimisation of the overall system could lead to a different power and rate assignment.

Interestingly enough, it can be shown that also the overall optimisation of the best effort traffic in a

single step is attained by either shutting up a transmitting terminal or letting it transmit at peak

power.

To show this result, we consider N terminal pairs, each having a communication. We wish to

solve the constrained optimisation problem (3.21), which can be easily reduced to

maximize H (P) = Pigii f (γ i )

γ i ηi + Tf σ2 Pk gkik=1,k≠ i

N

∑

i =1

N

∑

0 ≤ Pi ≤ Pmax

(3.33)

By some tedious algebra, it can be checked that

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26

∂2H (P)

∂Pi2

=2 Tf σ2gij( )2 Pj g jj f (γ j )

γ j η j + Tf σ2 Pk gkjk=1,k≠ j

N

∑

3j=1j≠ i

N

∑ ≥ 0 (3.34)

for any vector P in the hypercube [0,Pmax]N. The inequality (3.34) imply that the target function

H(P) is convex with respect to each scalar variable, whatever the values of the others.

So, we can write:

H (P) = H (P1, P2 ,…, PN )

≤ max H (0,P2,… ,PN ), H (Pmax ,P2,…,PN ){ }≤ max H (0,0,…,PN ), H (0,Pmax ,…, PN ), H (Pmax ,0,… ,PN ), H (Pmax ,Pmax ,…, PN ){ }≤ max

ε∈ 0,1{ }NH (ε1Pmax ,ε2Pmax ,…,ε N Pmax ){ }

(3.35)

where the last max operator is taken over all the permutation of N binary values belonging to {0,1}.

From (3.35) it is clear that the maximum of H(P) is achieved in one of the vertices of the hypercube

[0,Pmax]N. This corresponds to the extreme choice of either 0 or maximum power level transmission

of the N pairs of terminals.

Therefore, a main result is that the maximization of the overall net rate for BE traffic leads to

maximum power transmission for some terminals while the others cannot set up their

communications. The Pmax/0 pattern of the N pairs of terminals need not be the same as that

obtained by step-by-step optimisation (think of starting from a pair that would have to stay silent on

the overall optimisation). The interesting result is that there is no point is using any intermediate

level of the average power.

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4. Simulations and performance results

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5. References

[1] Sanjay Lal, Elvino S. Sousa “Distributed Resource Allocation for DS-CDMA-Based

Multimedia ad hoc Wireless LAN’s”, IEEE JSAC, vol. 17, No. 5, pp. 947-967, May 1999.

[2] J.-P. Hubaux, T. Gross, J.-Y. Le Boudec, and M. Vetterli "Toward Self-Organized Mobile Ad

Hoc Networks: The Terminodes Project", IEEE Communications Magazine, January 2001.

[3] T. A. Elbatt, S. V. Krishnamurthy, D. Connors, S. Dao, “Power Management for Throughput

Enhancement in Wireless Ad Hoc Networks,” IEEE 2000, pp. 1506-1513.

[4] D. Kim, “Rate-Regulated Power Control for Supporting Flexible Transmission in Future

CDMA Mobile Networks,” IEEE J. Select. Areas Commun., vol. 17, pp. 968-977, May 1999.

[5] S. J. Oh, T. L. Olsen, K. M. Wasserman, “Distributed Power Control and Spreading Gain

Allocation in CDMA Data Networks,” in Proc. INFOCOM ’00, 2000, pp. 379-385.