Bounds on the Lambert function and their application to the outage analysis of user cooperation

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by  Ioannis Chatzigeorgiou

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Bounds on the Lambert function and their application
to the outage analysis of user cooperation
Ioannis Chatzigeorgiou, Member, IEEE
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Abstract—Problems formulated in terms of logarithmic or
exponentialequationsoftenusetheLambertW functionintheir
solutions. Expansions, approximations and bounds on W have 1
been derived in an effort to gain a better understanding of the
relationshipbetweenequationparameters.Inthispaper,wefocus 0
6
ononeofthebranchesofW,denotedasW ,wederivetractable
1 −1
upper and lower bounds and we illustrate their usefulness in
0 identifying conditions under which user cooperation can yield a z)−1
2 lower outage probability than non-cooperative transmission. W(
−2
n
Index Terms—Lambert function, bounds, cooperation, decode
a
and forward, outage probability, Rayleigh fading.
J −3

19 I. INTRODUCTION −4 W0(z)
THE Lambert function, denoted as W(z) for z ∈ C, is W−1(z)
]
T defined as the function that satisfies −−51 0 1 z2 3 4 5
.I W(z)eW(z) =z. (1)
s
Fig. 1. The two real-valued branches of the Lambert W function. Both
c Taking logarithms of both sides of (1) and rearranging terms,
[ branchesaredefinedforz≥−1/e,whereW−1(z)≤−1andW0(z)≥−1.
we obtain the equivalent expression [1] Thetwobranchesmeetatthepoint(−1/e,−1).

1
v W(z)=ln(z)−ln(W(z)). (2)
5 If z is real, that is z ∈ R, W(z) can take two possible real solutions involving the Lambert function were derived in [6],
9
8 valuesfor−1/e≤z <0.ValuessatisfyingW(z)≥−1belong [7] but their focus was mainly on the primary branch W0.
4 to the principal branch, which is denoted as W0(z), while Algorithmic approaches were used in [8] to derive accurate
0 values satisfying W(z) ≤ −1 belong to the W−1(z) branch. approximations to the W−1 branch, for example
1. The two branches meet at the branch point for z = −1/e, W (z)≈ln(−z)−2α−1
−1
60 wfohrezre≥W00(−be1/loe)ng=toWt−he1(−pr1i/nec)ip=al−br1a.ncAhllWva0l(uze)s. oTfheWt(wzo)  (cid:34) (cid:18) 1+ln(−z)(cid:19)12(cid:35)−1 (3)
1 × 1− 1+α −
branches of the W(z) are depicted in Fig. 1. 2
:  
v The Lambert W function has been recognised in the solu-
Xi tion of scientific and engineering problems and it has been where α=0.3205.
introduced as a special function in numerical computation In this paper, we use functional analysis methods to derive
r
a software packages, such as Maple and MATLAB. It has upper and lower bounds on the W branch. Our objective is
−1
also appeared in recent research in communications, such toobtainexpressionswhichmightnotbeasaccurateas(3)but
as relaying strategies [2], moment generating functions for are simpler, more tractable and can shed light on the essence
modelling signal fading [3] and long-haul cooperative power of solutions that involve the W function.
−1
allocation methods [4]. Numerical evaluation of the Lambert The remainder of this paper has been organised as follows.
W function makes use of recursive schemes. Non-recursive Section II presents bounds on the natural logarithm which
expansionshavealsobeenobtainedforsmallandlargevalues are used as a stepping stone for the derivation of bounds on
of z [1], [5]. For instance, the first few terms of the series the W function. Section III considers a simple cooperative
−1
expansion of W(z) about the branch point are network and assesses the tightness and usefulness of the
1 11 proposed upper and lower bounds. Section IV summarises the
W(z)=−1+p− p2+ p3+...
3 72 main points of the paper and discusses future work.
(cid:112) (cid:112)
where p= 2(e·z+1) for W and p=− 2(e·z+1) for
0
W . Tractable bounds that provide clearer interpretations of II. DERIVATIONOFBOUNDSONTHEW−1 FUNCTION
−1
We will now introduce two lemmata which will help us
I. Chatzigeorgiou is with the School of Computing and Communica-
derive lower and upper bounds on the Lambert function
tions,InfoLab21,LancasterUniversity,LA14WA,UnitedKingdom,(e-mail:
[email protected]) W−1(z) for z =−e−u−1 and u>0.
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Lemma 1. For x≥0, the natural logarithm is bounded from
above by
−1.9
x2 (cid:18) 1 (cid:19)2
ln(1+x) ≤ x− (4) −2
2 1+x/3
where equality holds for x=0. −2.1
Proof: Let
−2.2
x2 (cid:18) 1 (cid:19)2 LambertW−1(−e−u−1)
f(x)=ln(1+x)−x+ . −2.3
2 1+x/3 Upperboundforu∈(0,∞)
Iff(x)≤0forx≥0,thelemmaistrue.Differentiatingf(x) −2.4 Lowerboundforu∈(0,∞)
with respect to x, we obtain Lowerboundforu∈(0,1)
−2.5
df(x) x3(x+9) 0.2 0.25 0.3 0.35 0.4 0.45 0.5
=− . u
dx (x+1)(x+3)3
We observe that d f(x) = 0 for x = 0, while d f(x) < 0 cFoigrr.es2p.ondsTthoethueprpiegrht-bhoaunnddsiodne etxhpereLssaimonbeortf (f7u)ncatnidon(8W),−w1h(ic−hei−suv−al1id)
dx dx
for x>0. Therefore, f(x) is a decreasing function which has for every u > 0. The left-hand side of (7) generates a lower bound which
canbemadetighterforu∈(0,1),usingtheleft-handsideof(8).
a maximum at x=0, implying that f(x)≤0.
As a lower bound on the natural logarithm, we use the first
two terms of the Taylor series expansion of ln(1+x), that is
approaches the minimum value of zero for x→0. Therefore
x2 f(x,1)>0 for every x>0.
ln(1+x) ≥ x− . (5)
2 Theorem 1. The Lambert function W (−e−u−1) for u > 0
−1
Lemma 2. For x>0 and g(x)=x−ln(1+x), we have is bounded as follows
2 (cid:112) √ √ 2
g(x) < x− 2g(x) < g(x). (6) −1− 2u−u<W (−e−u−1)<−1− 2u− u. (7)
3 −1 3
Proof: For c∈R, let Proof: Using (2), we express W (−e−u−1) as
−1
f(x,c)=cg(x)+(cid:112)2g(x)−x. W (−e−u−1)=ln(−e−u−1)−ln(cid:0)W (−e−u−1)(cid:1)
−1 −1
=−u−1+ln(−1)−ln(cid:0)W (−e−u−1)(cid:1)
To prove the left-hand side and right-hand side of (6), we −1
need to show that f(x,2/3)<0 and f(x,1)>0, respectively. =−u−1+ln(1)−ln(cid:0)−W−1(−e−u−1)(cid:1)
Substituting g(x) in f(x,c) and taking the derivative, we =−u−1−ln(cid:0)−W (−e−u−1)(cid:1).
−1
obtain
(cid:113) From the definition of the Lambert function, we have
(cid:0) (cid:1) (cid:0) (cid:1)
df(x,c) x− 1+x(1−c) 2 x−ln(1+x) W−1(−e−u−1)<−1, thus −W−1(−e−u−1)−1>0. We now
= (cid:113) . invoke (6), set x=−W (−e−u−1)−1 and write g(x) as
dx (cid:0) (cid:1) −1
(1+x) 2 x−ln(1+x)
g(x)=x−ln(x+1)
Weobservethatlimx→0(cid:0)ddxf(x,c)(cid:1)=0foreveryc∈R.Given =−W−1(−e−u−1)−1−ln(−W−1(−e−u−1))
that x takes positive values only, we can say that for x → 0,
=u.
the value of f(x,c) is maximised or minimised depending
on whether f(x,c) is a decreasing or an increasing function, Substituting x with −W (−e−u−1)−1 and g(x) with u in
−1
respectively. (6) produces (7).
If f(x,c) is a decreasing function, then d f(x,c) < 0.
dx Remark. In summary, if we define
Taking into account that the denominator of the derivative is √
positive for x > 0, the numerator should be less than zero, F(u,c)=−1− 2u−cu
which implies that
we demonstrated that F(u,c) generates lower bounds on
x2 (cid:18) 1 (cid:19)2 W−1(−e−u−1) for c≥1 and upper bounds for c≤2/3. The
ln(1+x) < x− .
2 1+x(1−c) tightest lower and upper bounds are obtained for c=1 and
c=2/3,respectively,whenu∈(0,∞).However,iftheinterval
The above inequality holds for c = 2/3 as per (4) but is of u has a finite upper endpoint, that is, u ∈ (0,u ), we can
0
also valid for every c < 2/3. Thus, considering that the experimentally determine a value of 2/3<c<1 that generates
maximum value of f(x,2/3) is just below zero for x → 0 a lower bound which is tighter than F(u,1). For example,
and that f(x,2/3) is a decreasing function, we conclude that if u ∈ (0,1), the Lambert function W (−e−u−1) can be
−1
f(x,2/3)<0 for every x>0. bounded as follows
Following the same line of reasoning and invoking (5), we
can show that for c≥1, f(x,c) is an increasing function that F(u,3/4)<W−1(−e−u−1)<F(u,2/3) (8)
3
as shown in Fig. 2. Note that F(u,3/4) and W−1(−e−u−1) B. Requirements for beneficial cooperation
intersect for a value of u (not shown in the graph) that falls
If we divide both parts of (9) by −e, we obtain
outside the specified range (0,1).
(cid:18)−1− θ(cid:48) (cid:19)e−1−0.θ5(cid:48)γ <−e−1−γθ. (10)
III. APPLICATIONTOUSERCOOPERATION 0.5γ
In this Section, we consider a simple case of decode-and- Recalling that W(z) is the solution to W(z)eW(z)=z, where
forward cooperation [9] to illustrate the application of (7) and z = −e−1−γθ in this case, we can reduce (10) to
(8) to the outage probability analysis of the system.
θ(cid:48) (cid:16) (cid:17)
−1− >W −e−1−γθ . (11)
0.5γ
A. System model and problem formulation
Let two users equipped with identical wireless transceivers We observe that the input to the W function takes values
transmit to the same access point over orthogonal channels between −1/e and −1/e2, while the value of the output is
whicharesubjecttofrequency-flatquasi-staticRayleighfading expectedtobelessthan-1.WecanthusinferthattheLambert
and additive white Gaussian noise. Channel state information W functionin(11)correspondstotheW−1branch.From(11),
is available to the access point, which can use coherent we conclude that the SNR threshold θ(cid:48) needs to be bounded
detection to recover the received signals. as follows
In the first scenario, the two users transmit their own data γ (cid:16) (cid:16) (cid:17) (cid:17)
directly to the access point in a single step. We denote as θ ≤θ(cid:48) <−2 W−1 −e−1−γθ +1 , (12)
θ the signal-to-noise ratio (SNR) threshold that characterises
the modulation and coding (M&C) scheme employed by in order to ensure that condition (9) is met and cooperation
each transceiver [10]. If γ is the average SNR of the uplink offers a lower outage probability than non-cooperation.
channels, we assume that a suitable M&C scheme has been Even though (12) is accurate, it does not provide sufficient
employedsothatθ <γ.Theoutageprobabilityofdirectnon- insight into the dependence between θ(cid:48), θ and γ. A more
cooperative transmission is given by [11] conservative but more tractable upper bound on θ(cid:48) can be
obtained if we invoke the right-hand side inequality in (7),
Pnc =1−e−γθ. which gives
(cid:114)
In the second scenario, each user dedicates the first of a θ ≤θ(cid:48) ≤ γθ + θ. (13)
two-stepprocesstotransmititsowndatadirectlytotheaccess 2 3
point while listening to the transmission of its partner over a
The above expression implies that θ (left-hand side) is always
perfect inter-user channel. In the second step, each user uses (cid:112)
less than or equal to γθ/2 + θ/3 (right-hand side) or,
its own uplink channel to transmit the successfully recovered
equivalently,θ ≤(9/8)γ.Thisistruegiventhatweworkunder
data of its partner to the access point. The SNR threshold in
the assumption that θ <γ from the beginning of the analysis.
this cooperative scenario is taken to be θ(cid:48) ≥ θ. The reason
Wehaveestablishedthatifthevalueofθ(cid:48)isbetweenthetwo
for our choice is that the two cooperating users, in an effort
endpoints in (13), cooperation will definitely provide gains in
to match the data rate of non-cooperative transmission, might
reliability compared to non-cooperation. Taking into account
use a higher code rate or a higher modulation order, which
would make the adopted M&C scheme more susceptible to that θ/γ ∈ (0,1), we can use the tight lower bound on W−1
shown in (8) and combine it with (12) to state with certainty
channelerrorsandwouldincreasetherequiredSNRthreshold.
that if
Ifeachuserallocateshalfofitspowertotransmititsowndata
(cid:114)
directly to the destination and the remaining half to transmit γθ 3θ
θ(cid:48) ≥ + (14)
the data of its partner, the outage probability of cooperative 2 8
communication is [11]
cooperation should be avoided.
Pc =1−(cid:18)1+ 0.θ5(cid:48)γ(cid:19)e−0.θ5(cid:48)γ beeRnegpiolontsteodfibnenFeifig.ci3al. cMooopreerastpieocnifiacsaldley,finFeidg.b3ya(1s2h)owhasvae
snapshot of θ(cid:48) as a function of γ for θ =5 dB, while Fig. 3b
provided that the access point uses maximal ratio combining.
depicts the dependence of θ(cid:48) on θ for γ =5 dB. As expected,
Cooperationwillbebeneficialforthetwousersonlyifdata
whenthequalityoftheuplinkchannel(γ)increasesrelativeto
transfertothedestinationismorereliablethannon-cooperative
the SNR threshold of non-cooperative transmission (θ), users
transmission. This implies that P <P or, equivalently,
c nc havetheflexibilitytochooseanSNRthresholdforcooperative
(cid:18)1+ θ(cid:48) (cid:19)e−0.θ5(cid:48)γ >e−γθ. (9) t(rFaings.m3ias)s.ioTnhi(sθ(cid:48)t)refnrodmisarnevinecrsreedaswinhgelynbθroaapdperroaracnhgesetohfevvaaluluees
0.5γ
of γ (Fig. 3b). We observe that the top border of the shaded
Our objective is to determine the range of values of θ(cid:48) as a regions is tightly enveloped by the bounds in (13) and (14),
function of θ and γ for which condition (9) is met. which have a much simpler representation than (12).
4
10 5
Regionofbeneficialcooperation
9.5
Upperboundonhighestθ′forcooperation 4
9 Lowerboundonhighestθ′forcooperation
8.5 3
8
2
B) B)
d7.5 d
( (
′θ ′θ 1
7
6.5 0
6 Regionofbeneficialcooperation
−1 Upperboundonhighestθ′forcooperation
5.5
Lowerboundonhighestθ′forcooperation
5 −2
5 6 7 8 9 10 11 12 13 14 15 −2 −1 0 1 2 3 4 5
γ(dB) θ(dB)
(a)θ(cid:48) asafunctionofγ forθ=5dB (b)θ(cid:48) asafunctionofθ forγ=5dB
Fig.3. Cooperationyieldsaloweroutageprobabilitythannon-cooperationforanyvalueofθ(cid:48) withintheshadedregions,whichsatisfy(12).Lower(blue
dashedline)andupper(reddash-dottedline)boundsonthetopborderoftheregioncorrespondtotheright-handsideof(13)and(14),respectively.
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To illustrate the application of the Lambert function and
the proposed bounds, we assumed that inter-user channels are
perfectanduplinkchannelsarestatisticallysimilar.Follow-on
workwillaimtoderiveconditionsforcooperationinnetworks
whereinter-userchannelscanbereciprocalorindependentand
uplink channels can be statistically similar or dissimilar.