Asynchronous Iterative Waterfilling for Gaussian Frequency-Selective Interference Channels

0.42 MB·English

by  Gesualdo Scutari

Asyn
hronous Iterative Water(cid:28)lling for Gaussian
Frequen
y-Sele
tive Interferen
e Channels
1 2 1
Gesualdo S
utari , Daniel P. Palomar , and Sergio Barbarossa
{ }
E-mail: s
utari, sergio info
om.uniroma1.it,palomarust.hk
1
Dpt. INFOCOM, Univ. of Rome (cid:16)La Sapienza(cid:17), Via Eudossiana 18, 00184Rome, Italy.
8 2
0 Dpt. of Ele
troni
and Computer Eng., Hong Kong Univ. of S
ien
e and Te
hnology, Hong Kong.
0
2 Submitted to IEEE Transa
tions on Information Theory, August 22, 2006.
∗
n
Revised September 25, 2007. A

epted January 14, 2008.
a
J

6 Abstra
t
1
Thispaper
onsidersthe maximizationofinformationratesfortheGaussianfrequen
y-sele
tive

]
T interferen
e
hannel, subje
tto powerand spe
tral mask
onstraintson ea
hlink. Toderivede
en-
I
tralized solutions that do not require any
ooperation among the users, the optimization problem
.
s
c is formulated as a stati
non
ooperative game of
omplete information. To a
hieve the so-
alled
[
Nash equilibria of the game, we propose a new distributed algorithm
alled asyn
hronous itera-

1
tive water(cid:28)lling algorithm. In this algorithm, the users update their power spe
tral density in a
v
0
ompletely distributed and asyn
hronousway: someusers may update their power allo
ationmore
8
frequently than others and they may even use outdated measurements of the re
eived interferen
e.
4
2 Theproposedalgorithmrepresentsauni(cid:28)edframeworkthaten
ompassesandgeneralizesallknown
.
1 iterative water(cid:28)lling algorithms, e.g., sequential and simultaneous versions. The main result of the
0
8 paper
onsists of a uni(cid:28)ed set of
onditions that guarantee the global
onverge of the proposed
0
algorithm to the (unique) Nash equilibrium of the game.
:
v
i
X Index Terms: Game theory, Gaussian frequen
y-sele
tive interferen
e
hannel, Nash equilib-
r rium, totally asyn
hronous algorithm, iterative water(cid:28)lling algorithm.
a
1 Introdu
tion and Motivation
Inthispaperwefo
usonthefrequen
ysele
tiveinterferen
e
hannelwithGaussiannoise. The
apa
ity
region of the interferen
e
hannel is still unknown. Only some bounds are available (see, e.g., [1, 2℄ for
a summary of the known results about the Gaussian interferen
e
hannel). A pragmati
approa
h that
leads to an a
hievable region or inner bound of the
apa
ity region is to restri
t the system to operate
as a set of independent units, i.e., not allowing multiuser en
oding/de
oding or the use of interferen
e

an
elation te
hniques. This a
hievable region is very relevant in pra
ti
al systems with limitations on
∗
Part of this work was presented in IEEE Workshop on Signal Pro
essing Advan
es in Wireless Communi
ations,
(SPAWC-2006), July 2-5, 2006, and in Information Theory and Appli
ations (ITA) Workshop, Jan. 29 - Feb. 2, 2007.
This work was supportedbythe SURFACEproje
t fundedbytheEuropean Communityunder Contra
t IST-4-027187-
STP-SURFACE.
1
the de
oder
omplexity and simpli
ity of the system. With this assumption, multiuser interferen
e is
treated as noise and the transmission strategy for ea
h user is simply his power allo
ation. The system
design redu
es then to (cid:28)nding the optimum Power Spe
tral Density (PSD) for ea
h user, a

ording to
a spe
i(cid:28)ed performan
e metri
.
−
Within this
ontext, existing works [4℄ [15℄
onsidered the maximization of the information rates
of all the links, subje
t to transmit power and (possibly) spe
tral mask
onstraints on ea
h link. The
latter
onstraints are espe
ially motivated in adaptive s
enarios, e.g.,
ognitive radio, where previously
allo
ated spe
tral bands may be reused, but provided that the generated interferen
e falls below spe
i-
(cid:28)ed thresholds [3℄. In [4, 5℄, a
entralized approa
h based on duality theory was proposed to
ompute,
under te
hni
al
onditions, the largest a
hievable rate region of the system (i.e., the Pareto-optimal set
of the a
hievable rates). Our interest, in this paper, is fo
used on (cid:28)nding distributed algorithms with
no
entralized
ontrol andno
ooperation among theusers. Hen
e, we
astthe systemdesign under the

onvenient framework of game theory. In parti
ular, we formulate the rate maximization problem as
a strategi
non-
ooperative game of
omplete information, where every link is a player that
ompetes
against the others by
hoosing the spe
tral power allo
ation that maximizes his own information rate.
An equilibrium for the whole system is rea
hed when every player is unilaterally optimum, i.e., when,
given the
urrent strategies of the others, any
hange in his own strategy would result in a rate loss.
This equilibrium
onstitutes the
elebrated notion of Nash Equilibrium (NE) [17℄.
The Nash equilibria of the rate maximization game
an be rea
hed using Gaussian signaling and
−
a proper PSD from ea
h user [9℄ [12℄. To obtain the optimal PSD of the users, Yu, Ginis, and Cio(cid:30)
proposed the sequential Iterative WaterFilling Algorithm (IWFA) [6℄ in the
ontext of DSL systems,
modeled as a Gaussian frequen
y-sele
tive interferen
e
hannel. The algorithm is an instan
e of the
Gauss-Seidel s
heme [18℄: the users maximize their own information rates sequentially (one after the
other), a

ording to a (cid:28)xed updating order. Ea
h user performs the single-user water(cid:28)lling solution
giventheinterferen
egeneratedbytheothersasadditive(
olored)noise. Themostappealingfeaturesof
thesequential IWFAare itslow-
omplexity and distributed nature. In fa
t, to
ompute the water(cid:28)lling
solution, ea
h user only needs to measure the noise-plus-interferen
e PSD, without requiring spe
i(cid:28)
knowledgeofthepowerallo
ationsandthe
hanneltransferfun
tionsofallotherusers. The
onvergen
e
−
ofthesequential IWFAhasbeenstudiedinanumber ofworks [7℄, [11℄ [15℄, ea
htimeobtaining milder

onvergen
e
onditions. However, despiteitsappealingproperties,thesequentialIWFAmaysu(cid:27)erfrom
slow
onvergen
e if the number of users in the network is large, just be
ause of the sequential updating
strategy. In addition, the algorithm requires some form of
entral s
heduling to determine the order in
whi
h users update their PSD.
To over
ome the drawba
k of slow speed of
onvergen
e, the simultaneous IWFA was proposed in
[9,11,12℄. Thesimultaneous IWFAisaninstan
eoftheJa
obis
heme [18℄: atea
h iteration, theusers
update their own strategies simultaneously, still a

ording to the water(cid:28)lling solution, but using the
interferen
e generated by the others in the previous iteration. The simultaneous IWFA was shown to
2
onverge to the unique NE of the rate maximization game faster than the sequential IWFA and under
−
weaker
onditions on the multiuser interferen
e than those given in [6, 7℄, [13℄ [15℄ for the sequential
IWFA. Furthermore, di(cid:27)erently from [6, 7, 13, 15℄, the algorithm as proposed in [11℄ takes expli
itly
into a

ount the spe
tral masks
onstraints. However, the simultaneous IWFA still requires some form
of syn
hronization, as all the users need to be updated simultaneously. Clearly, in a real network with
many users, the syn
hronization requirement of both sequential and simultaneous IWFAs goes against
the non-
ooperation prin
iple and it might be una

eptable.
This paper generalizes the existing results for the sequential and simultaneous IWFAs and develops
a uni(cid:28)ed framework based on the so-
alled asyn
hronous IWFA, that falls within the
lass of totally
asyn
hronous s
hemes of [18℄. In this more general algorithm, all users still update their power allo
a-
tions a

ording to the water(cid:28)lling solution, but the updates
an be performed in a totally asyn
hronous
way (in the sense of [18℄). This means that some users may update their power allo
ations more fre-
quently than others and they may even use an outdated measurement of the interferen
e
aused from
the others. These features make the asyn
hronous IWFA appealing for all pra
ti
al s
enarios, either
wired or wireless, as it strongly relaxes the need for
oordinating the users' updating s
hedule.
Themain
ontributionofthispaperistoderivesu(cid:30)
ient
onditionsfortheglobal
onvergen
eofthe
asyn
hronous IWFA to the (unique) NE of the rate maximization game. Interestingly, our
onvergen
e

onditions are shown to be independent of the users' update s
hedule. Hen
e, they represent a uni(cid:28)ed
set of
onditions en
ompassing all existing algorithms, either syn
hronous or asyn
hronous, that
an
be seen asspe
ial
ases of our asyn
hronous IWFA. Our
onditions also imply that both sequential and
simultaneous algorithms are robust to situations where some users may fail to follow their updating
s
hedule. Finally, we show that our su(cid:30)
ient
onditions for the
onvergen
e of theasyn
hronous IWFA

oin
ide with those given re
ently in [11℄ for the
onvergen
e of the (syn
hronous) sequential and
−
simultaneous IWFAs, and are larger than
onditions obtained in [6, 7℄, [13℄ [15℄ for the
onvergen
e of
the sequential IWFA in the absen
e of spe
tral mask
onstraints.
The paper is organized as follows. Se
tion 2 provides the system model and formulates the opti-
mization problem as a strategi
non-
ooperative game. Se
tion 3
ontains the main result of the paper:
the des
ription of the proposed asyn
hronous IWFA along with its
onvergen
e properties. Se
tion 4
re
overs the sequential and simultaneous IWFAs as spe
ial
ases of the asyn
hronous IWFA and then,
asaby produ
t, itprovides auni(cid:28)ed setof
onvergen
e
onditions forboth algorithms. Finally, Se
tion
5 draws some
on
lusions.
2 System Model and Problem Formulation
In this se
tion we
larify the assumptions and the
onstraints underlying the system model and we
formulate the optimization problem addressed in this paper expli
itly.
3
2.1 System model
We
onsider a Gaussian frequen
y-sele
tive interferen
e
hannel
omposed by multiple links. Sin
e
our goal is to (cid:28)nd distributed algorithms that require neither a
entralized
ontrol nor a
oordination
among the users, we fo
us on transmission te
hniques where interferen
e
an
elation is not possible
and multiuser interferen
e is treated by ea
h re
eiver as additive
olored noise. The
hannel frequen
y-
1
sele
tivity is handled, with no loss of optimality, adopting a multi
arrier transmission strategy. Given
the above system model, we make the following assumptions:
A.1 Ea
h
hannel
hanges su(cid:30)
iently slowly to be
onsidered (cid:28)xed during the whole transmission, so
that the information theoreti
results are meaningful;
A.2 The
hannel from ea
h sour
e to its own destination is known to the intended re
eiver, but not
to the other terminals; ea
h re
eiver is also assumed to measure with no errors the overall PSD of the
noiseplusinterferen
es generated bytheotherusers. Basedonthisinformation, ea
hre
eiver
omputes
the optimal power allo
ation a
ross the frequen
y bins for its own transmitter and feeds it ba
k to its
2
transmitter through a low bit rate (error-free) feedba
k
hannel.
A.3 All the users are blo
k-syn
hronized with an un
ertainty at most equal to the
y
li
pre(cid:28)x length.
This imposes a minimum length of the
y
li
pre(cid:28)x that will depend on the maximum
hannel delay
spread.
We
onsider the following
onstraints:
Co.1 Maximum overall transmit power for ea
h user:
N
E ks k2 = p¯ (k) ≤ NP , q = 1,...,Q,
q 2 q q (1)
n o Xk=1
s N q N p¯ (k) , E |s (k)|2
q q q
where
ontains the symbolstransmitted by user onthe
arriers, denotes
q k, P n o
q
the power allo
ated by user over
arrier and ispower in units of energy per transmitted symbol.
Co.2 Spe
tral mask
onstraints:
E |s (k)|2 = p¯ (k) ≤ p¯max(k), k = 1,...,N, q = 1,...,Q,
q q q
(2)
n o
p¯max(k) k
q
where represents the maximum power that is allowed to be allo
ated on the -th frequen
y bin
q
from the -th user. Constraints like (2) are imposed to limit the amount of interferen
e generated by
ea
h transmitter over pre-spe
i(cid:28)ed bands.
p , (p (1),...,p (N))
q q q
Themaingoalofthispaperistoobtaintheoptimalve
torpowerallo
ation ,
for ea
h user, a

ording to the optimality
riterion introdu
ed in the next se
tion.
1
Multi
arrier transmission is a
apa
ity-lossless strategy for su(cid:30)
iently large blo
klength [21, 22℄.
2
In pra
ti
e, both measurements and feedba
k are inevitably a(cid:27)e
ted by errors. This s
enario
an be studied by
extending our formulation to games with partial information [23, 24℄, but this goes beyond the s
ope of the present
paper.
4
2.2 Problem formulation as a game
We
onsider a strategi
non-
ooperative game [23, 24℄, in whi
h the players are the links and the payo(cid:27)
3
fun
tions are the information rates on ea
h link: Ea
h player
ompetes rationally against the others
by
hoosing the strategy that maximizes his own rate, given
onstraints Co.1 and Co.2. A NE of the
game is rea
hed when every user, given the strategy pro(cid:28)le of the others, does not get any rate in
rease
by
hanging his own strategy.
q
Using the signal model des
ribed in Se
tion 2.1, the a
hievable rate for ea
h player is
omputed
q
as the maximum information rate on the -th link, assuming all the other re
eived signals as additive

olored noise. It is straightforward to see that a (pure or mixed strategy) NE is obtained if ea
h user
transmits using Gaussian signaling, with a proper PSD. In fa
t, for ea
h user, given that all other users
4
use Gaussian
odebooks, the optimal
odebook maximizing mutual information is also Gaussian [21℄.
q
Hen
e, the maximum a
hievable rate for the -th user is given by [21℄
N
1
R = log(1+sinr (k)),
q q
N (3)
k=1
X
sinr (k) k
q
with denoting the Signal-to-Interferen
e plus Noise Ratio (SINR) on the -th
arrier for the
q
-th link:
sinr (k) , H¯qq(k) 2p¯q(k)/dγqq , |Hqq(k)|2pq(k) ,
q σq2(k)+(cid:12) r6=q (cid:12)H¯qr(k) 2p¯r(k)/dγrq σq2(k)+ r6=q|Hqr(k)|2pr(k) (4)
(cid:12) (cid:12)
H¯qr(k) P (cid:12)(cid:12) (cid:12)(cid:12) P r q
where denotes the frequen
y-response of the
hannel between sour
e and destination ex-
γ
d γ d r
qr qr

luding the path-loss with exponent and is the distan
e between sour
e and destination
q σ2(k) q
q
; is the varian
e of the zero mean
ir
ularly symmetri

omplex Gaussian noise at re
eiver
k
over the
arrier ; and for the
onvenien
e of notation, we have introdu
ed the normalized quantities
H (k) , H¯ (k) P /dγ p (k) , p¯ (k)/P
qr qr r qr q q q
and .
p
Observe that in the
ase of pra
ti
al
oding s
hemes, where only (cid:28)nite order
onstellations
an
be used, we
an use the gap approximation analysis [25, 26℄ and write the number of bits transmit-
N q |H (k)|2
qq
ted over the substreams from the -th sour
e still as in (3) but repla
ing in (4) with
|H (k)|2/Γ , Γ ≥ 1
qq q q
where is the gap. The gap depends only on the family of
onstellation and on
P M
e,q
; for -QAM
onstellations, for example, if the symbol error probability is approximated by
P (sinr (k)) ≈ 4Q 3sinr (k)/(M −1) , Γ = (Q−1(P /4))2/3
e,q q q q e,q
the resulting gap is [25, 26℄.
(cid:0)p (cid:1)
In summary, we have a game with the following stru
ture:
G = {Ω,{P } ,{R } },
q q∈Ω q q∈Ω
(5)
3
The rationality assumption means that ea
h user will never
hose a stri
tly dominated strategy. A strategy pro(cid:28)le
xq zq Φq(xq,y−q)<Φq(zq,y−q), y−q ,(y1,...,yq−1,yq+1,...,yQ),
is stri
tly dominated by if for a given admissible
Φq q.
where denotes thepayo(cid:27) fun
tion of player
4
Observethat,ingeneral, Nashequilibriaa
hievableusingarbitrarynon-Gaussian
odesmayexist. Inthispaper,we
fo
us only ontransmission using Gaussian
odebooks.
5
Ω , {1,2,...,Q} Q P
q
where denotes the set of the a
tive links, is the set of admissible (normalized)
N q
5
power allo
ation strategies, a
ross the available
arriers, for the -th player, de(cid:28)ned as
N
1
P , p ∈ RN : p (k) = 1, 0 ≤ p (k) ≤ pmax(k), k = 1,...,N ,
q ( q N q q q ) (6)
k=1
X
pmax(k) , pmax(k)/P R q
q q q q
with and is the payo(cid:27) fun
tion of the -th player, de(cid:28)ned in (3).
q
The optimal strategy for the -th player, given the power allo
ation of the others, is then the
6
solution to the following maximization problem
N
1
maximize log(1+ sinr (k))
q
pq N , ∀q ∈ Ω
Xk=1 (7)
subject to p ∈ P
q q
sinr (k) P q
q q
where and and are given in (4) and (6), respe
tively. Note that, for ea
h , the maximum
p , p , (p ,...,p ,p ,...,p ).
q −q 1 q−1 q+1 Q
in (7) is taken over for a (cid:28)xed
The solutions of (7) are the well-known Nash Equilibria, whi
h are formally de(cid:28)ned as follows.
p⋆ = p∗,...,p∗ ∈ P ×...×P
1 Q 1 Q
De(cid:28)nition 1 A (pure) strategy pro(cid:28)le is a Nash Equilibrium of
G ( ) (cid:16) (cid:17)
the game in 5 if R (p⋆,p⋆ )≥ R (p ,p⋆ ), ∀p ∈ P , ∀q ∈ Ω.
q q −q q q −q q q
(8)
Observe that, for the payo(cid:27) fun
tions de(cid:28)ned in (3), we
an indeed limit ourselves to adopt pure
strategies w.l.o.g., as we did in (5), sin
e every NE of the game is proved to be a
hievable using pure
strategies in [10℄.
A

ordingto(7),allthe(pure)Nashequilibriaofthegame,iftheyexist,mustsatisfythewater(cid:28)lling
solution for ea
h user, i.e., the following system of nonlinear equations:
p⋆ =WF p⋆,...,p⋆ ,p⋆ ,...,p⋆ = WF (p⋆ ) , ∀q ∈ Ω,
q q 1 q−1 q+1 Q q −q (9)
(cid:16) (cid:17)
WF (·)
q
with the water(cid:28)lling operator de(cid:28)ned as
σ2(k)+ |H (k)|2p (k) pmqax(k)
[WF (p )] , µ − q r6=q qr r , k = 1,...,N,
q −q k " q P|Hqq(k)|2 # (10)
0
b
[x] x [a,b] µ
a 7 q
where denotes the Eu
lidean proje
tion of onto the interval . The water-level is
hosen
(1/N) N p⋆(k) = 1.
to satisfy the power
onstraint k=1 q
5In order to avoid the trivial solution pP⋆q(k) = pmqax(k) for all k, PNk=1pmqax(k) > N is assumed for all q ∈ Ω.
Furthermore,in thefeasible strategy set of ea
h player, we
an repla
e, without loss of generality, theoriginal inequality
power
onstraint (1/N) PNk=1pq(k) ≤ 1, with equality, sin
e, at the optimum, this
onstraint must be satis(cid:28)ed with
equality.
6
Intheoptimizationproblemin(7),any
on
avestri
tlyin
reasingfun
tionoftherate
anbeequivalently
onsidered
as payo(cid:27) fun
tion of ea
h player. The optimalsolutions of this new set of problems
oin
ide with those of (7).
7The Eu
lidean proje
tion [x]ba, with a ≤ b, is de(cid:28)ned as follows: [x]ba = a, if x ≤ a, [x]ba = x, if a < x < b, and
[x]b =b x≥b
a , if .
6
pmax(k) = +∞ ∀q, ∀k
q
Observe that in the absen
e of spe
tral mask
onstraints (i.e., when , ), the
G
Nash equilibria of game are given by the
lassi
al simultaneous water(cid:28)lling solutions [6, 7℄, where
WF (·) pmax(k) = +∞ ∀q, ∀k.
q q
in (9) is still obtained from (10) simply setting , Interestingly, the
G
presen
e of spe
tral mask
onstraints does not a(cid:27)e
t the existen
e of a pure-strategies NE of game ,
as stated in the following.
G
Proposition 1 The game in (5) always admits at least one pure-strategy NE, for any set of
hannel
realizations, power and spe
tral mask
onstraints.
Proof. The proof
omes from standard results of game theory [23, 24℄ and it is given in [10℄.
G
In general, the game may admit multiple equilibria, depending on the level of multiuser interfer-
en
e [10℄. In the forth
oming se
tions, we provide su(cid:30)
ient
onditions ensuring the uniqueness of the
NE and we address the problem of how to rea
h su
h an equilibrium in a totally distributed way.
3 Asyn
hronous Iterative Water(cid:28)lling
G
To rea
h the NE of game , we propose a totally asyn
hronous distributed iterative water(cid:28)lling pro
e-
dure, whi
h we name asyn
hronous Iterative WaterFilling Algorithm. The proposed algorithm
an be
seen as an instan
e of the totally asyn
hronous s
heme of [18℄: all the users maximize their own rate
in a totally asyn
hronous way. More spe
i(cid:28)
ally, some users are allowed to update their strategy more
frequently than the others, and they might perform their updates using outdated information about
the interferen
e
aused from the others. What we show is that the asyn
hronous IWFA
onverges to a
G
stable NE of game , whi
hever the updating s
hedule is, under rather mild
onditions onthe multiuser
interferen
e. Interestingly, these
onditions are also su(cid:30)
ient to guarantee the uniqueness of the NE.
To provide a formal des
ription of the asyn
hronous IWFA, we need to introdu
e some preliminary
de(cid:28)nitions. We assume, without any loss of generality, that the set of times at whi
h one or more users
T = N = {0,1,2,...}. p(n)
+ q
update their strategies is the dis
rete set Let denote the ve
tor power
q n T ⊆ T n q
q
allo
ation of user at the -th iteration, and let denote the set of times at whi
h user
p(n) n ∈/ T , p(n) τq(n)
q q q r
updates his power ve
tor (thus, implying that, at time is left un
hanged). Let
r q n
denote the most re
ent time at whi
h the interferen
e from user is per
eived by user at the -th
q q
τ (n) 0 ≤ τ (n) ≤ n q
r r
iteration (observe that satis(cid:28)es ). Hen
e, if user updates its power allo
ation
n
at the -th iteration, then it water(cid:28)lls, a

ording to (10), the interferen
e level
aused by the power
allo
ations of the others:
p(τq(n)) , p(τ1q(n)),...,p(τqq−1(n)),p(τqq+1(n)),...,p(τQq(n)) .
−q 1 q−1 q+1 Q (11)
(cid:18) (cid:19)
The overall system issaid to be totally asyn
hronous if the following weak assumptions are satis(cid:28)ed
q q
q 0 ≤ τ (n) ≤ n lim τ (n ) = +∞ |T | = ∞ {n }
r k→∞ r k q k
for ea
h [18℄: A1) ; A2) ; and A3) ; where is a
T −
q
sequen
e of elements in that tends to in(cid:28)nity. Assumptions A1 A3 are standard in asyn
hronous

onvergen
e theory [18℄, and they are ful(cid:28)lled in any pra
ti
al implementation. In fa
t, A1 simply
7
(τq(n))
n q p
−q
indi
ates that, at any given iteration , ea
h user
an use only the interferen
e ve
tors
allo
ated by the other users in the previous iterations (to preserve
ausality). Assumption A2 states
(τq(n))
n , p
k −q
that, for any given iteration index the values of the
omponents of in (11) generated prior
(n)
n , p n n
k q k
to arenotusedintheupdatesof ,when be
omessu(cid:30)
iently larger than ; whi
h guarantees
that old information is eventually purged from the system. Finally, assumption A3 indi
ates that no
n
user fails to update its own strategy as time goes on.
G, Dmin ⊆ {1,··· ,N} {1,...,N}
q
Given game let denote the set (possibly) deprived of the
arrier
q
indi
es that user would never use as the best response set to any strategies adopted by the other
users [10℄:
Dmin , k ∈ {1,...,N} :∃ p ∈ P [WF (p )] 6= 0 ,
q −q −q su
h that q −q k (12)
WF (·) (cid:8) P , P ×···×P ×P ×···×P (cid:9)
q −q 1 q−1 q+1 Q
with de(cid:28)ned in (10) and . In [10℄, an iterative
D Dmin ⊆ D ⊆ {1,··· ,N} Smax ∈RQ×Q
q q q +
pro
edure to obtain a set su
h that isgiven. Let the matrix
be de(cid:28)ned as |H¯ (k)|2 dγ P
qr qq r
max , r 6= q,
[Smax]qr , k∈Dq∩Dr |H¯qq(k)|2 dγqr Pq if
(13)
 0,
otherwise,
 Dq ∩Dr Dq
with the
onvention that the maximum in (13) is zero if is empty. In (13), ea
h set
an
{1,··· ,N} Dmin ⊆ D ⊆ {1,··· ,N} Dmin
q q q
be
hosen as any subset of su
h that , with de(cid:28)ned in (12).
N
it
Using the above notation, the asyn
hronous IWFA is des
ribed in Algorithm 1 (where denotes the
number of iterations).
Algorithm 1: Asyn
hronous Iterative Water(cid:28)lling Algorithm
(0)
n = 0 p =
q
Set and any feasible power allo
ation;
n = 0 :N ,
it
for
WF p(τ1q(n)),...,p(τqq−1(n)),p(τqq+1(n)),...,p(τQq(n)) , n ∈ T ,
p(n+1) = q 1 q−1 q+1 Q if q ∀q ∈ Ω
q  (cid:18) (cid:19)
(n)
 p , ;
 q
otherwise
 (14)
end
The
onvergen
e of the algorithm is guaranteed under the following su(cid:30)
ient
onditions.
Theorem 1 Assume that the following
ondition is satis(cid:28)ed:
ρ(Smax) < 1,
(C1)
Smax ρ(Smax) Smax N → ∞
8 it
where is de(cid:28)ned in (13) and denotes the spe
tral radius of . Then, as ,
8 ρ(S) S ρ(S)=max{|λ|:λ ∈eig(S)} eig(S)
The spe
tral radius of the matrix is de(cid:28)ned as , with denoting the set
S
of eigenvalues of [27℄.
8
G
the asyn
hronous IWFA des
ribed in Algorithm 1
onverges to the unique NE of game , for any set
of feasible initial
onditions and updating s
hedule.
Proof. The proof
onsists in showing that, under (C1),
onditions of the Asyn
hronous Convergen
e
Theoremin[18℄aresatis(cid:28)ed. Akeypointintheproofisgivenbythefollowingpropertyofthemultiuser
WF(p) = (WF (p ))
water(cid:28)lling mapping q −q q∈Ω based on the interpretation of the water(cid:28)lling solution
(10) as a proper proje
tor [11℄:
WF(p(1))−WF(p(2)) ≤ β p(1)−p(2) , ∀p(1), p(2) ∈ P ,
(15)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
k·k (cid:13) β (cid:13) (cid:13) (cid:13) 1
where is a proper ve
tor norm and is a positive
onstant, whi
h is less than if
ondition (C1)
is satis(cid:28)ed. See Appendix A for the details.
Togiveadditionalinsightintothephysi
alinterpretationofthe
onvergen
e
onditionsofAlgorithm
1, we provide the following
orollary of Theorem 1.
Corollary 1 A su(cid:30)
ient
ondition for (C1) in Theorem 1 is given by one of the two following set of

onditions: 1 |H¯ (k)|2 dγ P
qr qq r
max w < 1, ∀q ∈ Ω,
wqr6=qk∈Dr∩Dq(cid:26)|H¯qq(k)|2(cid:27) dγqr Pq r (C2)
X
1 |H¯ (k)|2 dγ P
qr qq r
max w < 1, ∀r ∈Ω,
wrq6=rk∈Dr∩Dq(cid:26)|H¯qq(k)|2(cid:27)dγqr Pq q (C3)
X
w , [w ,...,w ]T
1 Q 9
where is any positive ve
tor.
D D = {1,...,N}
q q
Note that, a

ording to the de(cid:28)nition of in (13), one
an always
hoose in (C1)
and (C2)-(C3). However, less stringent
onditions are obtained by removing unne
essary
arriers, i.e.,
the
arriers that, for the given power budget and interferen
e levels, are never going to be used.
Re
allthat,if(cid:28)niteorder
onstellationsareused,Theorem1isstillvalidusingthegap-approximation
|H (k)|2
qq
method [25, 26℄ as pointed out in Se
tion 2.2. It is su(cid:30)
ient to repla
e ea
h in (C1) with
|H (k)|2/Γ .
qq q
Remark 1 - Global
onvergen
e and robustness of the algorithm: Even though the rate
maximization game in (7) and the
onsequent water(cid:28)lling mapping (10) are nonlinear,
ondition (C1)
guarantees the global
onvergen
e of the asyn
hronous IWFA. Observe that Algorithm 1
ontains as
spe
ial
ases a plethora of algorithms, ea
h one obtained by a possible
hoi
e of the s
heduling of
q
{τ (n)} {T }
r q
the users in the updating pro
edure (i.e., the parameters and ). The important result
stated in Theorem 1 is that all the algorithms resulting as spe
ial
ases of the asyn
hronous IWFA are
guaranteed torea
htheuniqueNEofthegame, underthesamesetof
onvergen
e
onditions(provided
w
9
The optimal positive ve
tor in (C2)-(C3) is given by the solution of a geometri
programming, as shown in [11,
Corollary 5℄
9
− {T }
q
that A1 A3 are satis(cid:28)ed), sin
e
ondition (C1) does not depend on the parti
ular
hoi
e of and
q
{τ (n)}.
r
Remark2-Physi
alinterpretationof
onvergen
e
onditions: Asexpe
ted,the
onvergen
eof
theasyn
hronousIWFAandtheuniquenessofNEareensurediftheinterferers aresu(cid:30)
iently farapart
from the destinations. In fa
t, from (C2)-(C3) one infers that, for any given set of
hannel realizations
andpower
onstraints, thereexistsadistan
ebeyondwhi
h the
onvergen
e oftheasyn
hronousIWFA
(andtheuniquenessofNE)isguaranteed,
orresponding tothemaximumlevelofinterferen
e thatmay
be tolerated by ea
h re
eiver [as quanti(cid:28)ed, e.g., in (C2)℄ or that may be generated by ea
h transmitter
[as quanti(cid:28)ed, e.g., in (C3)℄. But the most interesting result
oming from (C1) and (C2)-(C3) is that
|H (k)|2/
qr
the
onvergen
e of the asyn
hronous IWFA is robust against the worst normalized
hannels
|H (k)|2; |H (k)|2/|H (k)|2
qq qr qq
in fa
t, the sub
hannels
orresponding to the highest ratios (and, in
|H (k)|2
qq
parti
ular, the sub
hannels where is vanishing) do not ne
essarily a(cid:27)e
t the
onvergen
e of
D
q
the algorithm, as their
arrier indi
es may not belong to the set .
Remark 3 - Distributed nature of the algorithm: Sin
e the asyn
hronous IWFA is based on the
water(cid:28)lling solution (10), it
an be implemented in a distributed way, where ea
h user, to maximize
his own rate, only needs to measure the PSD of the overall interferen
e-plus-noise and water(cid:28)ll over it.
More interestingly, a

ording to the asyn
hronous s
heme, the users may update their strategies using
a potentially outdated version of the interferen
e. Furthermore, some users are even allowed to update
their power allo
ation more often than others, without a(cid:27)e
ting the
onvergen
e of the algorithm.
These features strongly relax the
onstraints on the syn
hronization of the users' updates with respe
t
to those imposed, for example, by the simultaneous or sequential updating s
hemes.
We
an generalize the asyn
hronous IWFA given in Algorithm 1 by introdu
ing a memory in the
updating pro
ess, as given in Algorithm 2. We
all this new algorithm smoothed asyn
hronous IWFA.
Algorithm 2: Smoothed Asyn
hronous Iterative Water(cid:28)lling Algorithm
(0)
n = 0 p = α ∈ [0,1), ∀q ∈ Ω
q q
Set and any feasible power allo
ation and ;
n = 0 :N ,
it
for
α p(n)+(1−α )WF p(τq(n)) , n ∈ T ,
p(n+1) = q q q q −q if q ∀q ∈ Ω;
q  p(n), (cid:16) (cid:17) , (16)
q
 otherwise

end
α ∈ [0,1) α
q q
Ea
h fa
tor in Algorithm 2
an be interpreted as a forgetting fa
tor: The larger
α
10 q
is, the longer the memory of the algorithm is. Interestingly the
hoi
e of 's does not a(cid:27)e
t the
10
Inthispaper,weareonly
onsidering
onstant
hannels. Nevertheless,inatime-varyings
enario, (16)
ouldbeused
to smooththe(cid:29)u
tuations dueto the
hannelvariations. Insu
h a
ase, if the
hannelis (cid:28)xedor highlystationary, it is
αq 1 αq

onvenientto take
lose to ;
onversely,if the
hannelis rapidly varying,it is better to take a small .
10